AFCRL --72-0222______

PULSE PAIR ESTIMATION OF

DOPPLER SPECTRUM PARAMETERS

By

Fredrick C. BenhamHerbert L. Groginsky

Aaron S. SoltesGeorge Works

RAYTHEON COMPANYEquipment Development Labor.-,tories

Wayland, Massachusetts 01778

Contract No. F19628-71-C-0126

Project No. 6672 LiTask No. 66 04rWork Unit No. 66720401

" CPeriod Covered: 1 February 1971 through 31 January 1972

30 March 1972

Approved for public release; distribution unlimited

Contract Monitor: Graham M. Armstrong, Meteorology Laboratory

Prepared for

AIR FORCE CAMBRIDGE RESEARCH LABORATORIESAIR FORCE SYSTEMS COMMAND

UNITED STATES AIR FORCEBEDFORD, MASSACHUSETTS 01730

rNATIONAL TECHNICALINFORMATION SERVICE

j' A 22151

UnclassifiedS,- Securttr Clasuficetton

DOCUMENT CONTROL DATA. R A D(SecuP#ty CeleslIcation of title. body of abstract and indexng annotatilon muir be entered when the overall report In claselefled)

I. ORIGINATING ACTIVITY (Cotporale author) 2a. REPORT SECURITY CLASSIFICATION

Raytheon Company UnclassifiedEquipment Development Laboratory 2b. GROUP "Wayland, Massachusetts 01778 N/A

3. REPORT TITLE

PULSE PAIR ESTIMATION OF DOPPLER SPECTRUM PARAMETERS

4. DESCRIPTIVE NOTES (T'ype O report and inclusive dlefo)

Scientific Final I February - 31 January 1972 Approved 24 May 1972S AUTO:ORIS) (Firlt name, middle Initial, elt nafml)

Herbert L. Groginsky George A. WorksAaron S. Soltes Frederick C. Benham

6. REPORT DATE 70. TOTAL NO. OF PAGES 7b. NO. OF Pr$

30 March 1972 148 780. CONTRACT OR GRANT NO 90. ORIGINATOR'S REPORT NUIIOERMII

F19628-71-C-0126b. Project, Task, Work Unit Nos.

667 2-04-01 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

C. 66 2 04 0 b. OTHER4 REPORT NolsI (Any other nunmber. that may be assignedDoD Element 62101F thiso r .,.

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A-Approved for public release; distribution unlimited.

S1 SUPPLEMENTARY NOTES 12. SPONSORING MILI TARY ACTIVITY

TAir Force Cambridge ResearchTech, other I Laboratories (LY)

L. G. Hanscom Fieldl Bedford, Massachusetts 017303. ABStRACT

The results of an expanded study and investigation of the Pulse Pair technique forestimating the first and second moments (mean and variance) of doppler spectra forradar backscatter from atmospheric phenomena are presented. The theory isextended to include the effects of non-ideal conditions, such as noise, and experi-mentally verified by extensive performance tests using simulated weather signalswith controllable parameters. A proposed experimental model of a real-timedigital pulse pair processor is defined and compared with alternate processingtechniques. Based on the encouraging results of the study, recommendations aremade to carry the theory into practice; these include the construction of a real-timedigital pulse pair processor with flexible characteristics to gather and reduce datafor evaluation while operating with real radars, and the development of additionalrelated theory needed to guide the experimental effort.

DD NY51473 UnclassifiedSecurntV C.-ssIiicSton

UnCliseffieclsecurity L0immmif1icmtiOn________ _____ ___ ________

4.LINK A LINK 19 LINK CKF'V 111101-1-1-8

"Pulse Pair" Estimation Theory"Pulse Pair" ProcessorsSpectral AnalysisSimulating Weather Radar ReturnPulse Doppler RadarWeather Radar

I

UnlssfeSPI-y lSMCto

L

AFCRL- 72-0222

PULSE PAIR ESTIMATION OF

DOPPLER SPECTRUM PARAMETERS

By

Fredrick C. BenhamHerbert L. Groginsky

Aaron S. SoltesGeorge Works

RAYTHEON COMPANYEquipment Development Laboratories

Wayland, Massachusetts 01778

Contract No. F19628-71-C-0126

Project No. 6672Task No. 667204Work Unit No. 66720401

FINAL REPORT

Period Covered: I February 1971 through 31 January 1972

30 March 1972a

Approved for public release; distribution unlimited

Contract Monitor: Graham M. Armstrong, Meteorology Laboratory

Prepared for

AIR FORCE CAMBRIDGE RESEARCH LABORATORIESAIR FORCE SYSTEMS COMMAND

UNITED STATES AIR FORCEBEDFORD, MASSACHUSETTS 01730

.2 -

ABSTRACT

The results of an expanded study and investigation of the

Pulse Pair technique for estimating the first and second

moments (mean and variance) of doppler spectra for radar

backscatter from atmospheric phenomena are presented.

The theory is extended te include the effects of non-ideal

conditions, such as noise, and experimentally verified by

extensive performance tests using simulated weather

signals with controllable parameters. A proposed experi-

mental model of a real-time digital pulse pair processor

is defined and compared with alternate processing tech-

niques. Based on the encouraging results of the study,

recommendations are made to carry the theory into

practice; these include the construction of a real-time

digital pulse pair processor with flexible characteristics

to gather and reduce data for evaluation while operating

with real radars, and the development of additional

related theory needed to guide the experimental effort.

; ii

TABLE OF CONTENTS

Section No. Page

1.0 INTRODUCTION 1-11. 1 Scope 1-11.2 Summary 1-11. 3 Technical Background 1-4

2.0 THEORY OF PULSE PAIR ESTIMATION 2-12. 1 The Pulse Pair Estimators 2-12. 2 Estimation Accuracy 2-32. 3 Hardware Implications 2-82. 4 Waveform Flexibility 2-82. 5 Suirnmary 2-14

3.0 EXPERIMENTAL EVALUATION 3-13. 1 Porcupine Radar Weather Return 3-13. 1. 1 Estimation of Mean and Spread by Conventional

Spectral Analysis 3-13. 1.2 Estimates of Mean and Spread by Pulse Pair

Processing 3-43. 1. 3 Comparison of the Estimators 3-4 A3.2 Simulated Radar Return 3-83.2. 1 Purpose, Capabilities and Advantages 3-83.2.2 Generating Sequences of Simulated Doppler Radar

Target Return with Controllable Parameters 3-103.2.3 Performance of Pulse Pair Estimators 3-18 I3. 2.4 Performance of Spectral Analysis Estimator 3-283. 2.5 Comparison of the Width Estimators 3-34

4.0 HARDWARE REALIZATION 4-14. 1 General Description of the Pulse-Pair Processor 4-14.2 Technical Description of the Pulse-Pair Processor 4-24.3 System Design 4-34.3. 1 IF Amplifier - Phase Detector 4-34. 3.2 Digitizer 4-7A4. 3.3 Correlator 4-74.3.4 Digital Integrators 4-134.3.5 Function Generators 4-134.4 Performance 4-144.4.1 Comparison with tne CMF 4-144.4.2 Comparison with Analog Spectrum Analyzers 4- 164.4.3 FFT Processing 4-164. 5 Conclusions 4-17

5.0 CONCLUSIONS AND RECOMIMENDATIONS 5-1

References

iii

TABLE OF CONTENTS (Cont'd.)

APPENDIX TITLE

A Analysis of Accuracy of Spectral Parameter Estimatesby Pulse Pair Estimation

B Mathematical Derivations Relating to the Generationof Simulated Doppler Return

C Fortran Main Programs

C-1 DOPGENC-2 PPSTATC-3 SPEC

D Fortran SubroutinesD-I FORTD-2 PAIRD-3 UNPKD-4 CDCCD-5 BESID-6 STARD-7 GNRN

E Spectral Analysis Plots of Typical Sequences ofSimulated Doppler Radar Target Return

- i

t iv

I -

LIST OF ILLUSTRATIONS

Figure No. Title Pase1 Spectral Density Parameters 2-la

I a Standard Deviation of Pulse Pairs Center Frequency 2-4Estimate (f~ ) vs Spectral Width (wT) with per Pulse

S/N a Parameter, for Uniform Pulse SpacingA

2 Standard Deviation of Pulse Pairs Width Estimate (wT) 2-5

vs True Spectral Width (wT) with Per Pulse S/N a

Parameter, for Uniform Pulse Spacing

3 Accuracy of Pulse Pairs Estimate of Mean Velocity vs 2-6

Spectral Width, for Independent Pairs (from Rummler,

Ref. (4))

4 Fractional Accuracy of Pulse Pairs Estimate of Spectral 2-7

Width vs True Spectral Width for Independent Pairs

(from Rummlcr, Rcf. (4))

5 P'alse Pair Waveforms 2-9

6 Tradeoff Characteristic of Pulse Pair Mean Frequency 2-11

Measurement for Fixed Radar EnergyI

7 Tradeoff Characteristic of Pulse Pair Width Measurement 2-12

for Fixed Radar Energy '

8 Typical Data Spectrum for N 1024 3-2

9 Overall Block Diagram of the Software Simulation System 3-9

for Investigating and Comparing the Performance of the

Pulse Pair and Spectrum Analyzer Estimators of

Spectral Parameters

10 Block Diagram of Simulated Doppler Return Generator 3-13

11 Flowchart of DOPGEN 3-15

12 FFT Processed Simulated Doppler Spectrum of Weather 3-17

Data

v

rE

LIST OF ILLUSTRATIONS (Cont.)

Figure No. Title Page

13 Flowchart of PPSTAT 3-19

14 Theoretical Accuracy of Pulse Pair Estimators of Mean 3-21A

Frequency (fo) vs Number of Pulse Pairs Compared with

Simulation Results

15 Theoretical Accuracy of Pulse Pair Estimates of Mean 3-22

Frequency (fo) vs Spectral Width (w) compared with

Simulation Results

16 Theoretical Accuracy of Pulse Pair Estimates of Spectral 3-23(AWidth (') vs Number of Samples Compared with Simulation

Results

17 Theoretical Accuracy of Pulse Pair Estimates of Spectral 3-24 iA

Width ('v vs Actual Spectral Width (w) Compared with

Simulation Results

18 Theoretical Accuracy of Pulse Pair Estimates of Spectral 3-25 jWidth (w) vs S/N Compared with Simulation Results (for

f = 300 Hz; w = 78 Hz)o0

19 Theoretical Accuracy of Pulse Pair Estimates of Spectral 3-26

Width (w) vs S/N Compared with Simulation Results (for

f z 600 Hz; w = 156 Hz)0

20 Theoretical Accuracy of Pulse Pair Estimates of Spectral 3-27A

Width (w) vs S/N Compared with Simulation Results (for

(f = 900 Hz; w = 234 Hz)

21 Flowchart of SPEC 3-29

Levl or imlatonIncr ata(f = 00Hz; w =78 Hz)z Spectral Analysis Width Estimates (w) vs. Threshold 3-31

Level for Simulation Input Data (fo = 600 Hz; w 165 Hz)0

vi- --

LIST OF ILLUSTRATIONS (Cont.)

Figure No. Title PageA24 Spectral Analysis Width Estimates (w) vs. Threshold 3-33

Level for Simulation Input Data (fo = 900 14z; w 234 Hz)

25 Pulse Pair Processor Signal Flow Diagram 4-4

26 PPP Block Diagram 4-5

27 Porcupine IF Amplifier 4-7

28 FFT Complex Multiplier 4-9

vii

7=

LIST OF TABLES

Table No. Title Pae

1 Comparison of Pulse-Pair P with Spectral Analysis, 3-5A 0

f for T a 20:2:30t A

4 2 Comparison of Pulse-Pair w' with Spectral Analysis 3-6Awfor T = 20:2:30

0 3 List of Recorded Sequences of Simulated Doppler Return 3-16

Generated, with Parameters as shown.

(PRF = 3300 pps in all cases)

4 Compari3on of Pulse Pair and Spectrum Analysis; 3-35

Estimates of Spectral Width

5 Peak Signal-to-Noise Improvement of Spectrum 3-36

Analyzer

6 Comparison of Pulse Pair and Spectrum Analyzer 3-37

Derived Width Estimates (Spectrum Analyzer Threshold

Set 5 dB above Average Background Noise)

7 Proposed Design Goals for Pulse Pair Processor 4-15

viii

__ _ _ _ __ _ _ _ _ __ _ _ _ _

1.0 INTRODUCTION

1. 1Scp

This is the Final Report on Contract F19628-71-C-0126, which is

devoted to the study and investigation of the pulse pair measurement tech-

niqu for determination of the first and second moments of Doppler spectra

of radar backscatLer from atmospheric phenomena. This technique pro-

mises to greatly simplify and thereby reduce the costs of deriving quantita-

tive data from a pulse Doppler weather radar. Initial efforts in the applica-

tion of the pulse pair technique to weather radar signals were conducted

under a portion of a prior contract with the Weather Radar Branch of the

AFCRL Meteorology Laboratory (F19628-68-C-0345). The emphasis on the

present contract has been to extend the theory and to experimentally verify

the performance of the pulse pair estimating technique to include the effects

of non-ideal conditions, such as noise, that are encountered in an operational

environment with real equipment; and to prepare a design specification for a

real time pulse pair signal processor that could be utilized in weather radar

research.

In the interests of overall continuity and perspective, and to facilitateunderstanding, this report provides a unified, self-sufficient presentation ofthe theory and experimental work to date, including some of the early results

from the previous contract.

1.2 Summary

Major steps forward that were taken during the present contract include

(1) establishing a theoretical foundation for quantitatively predicting the per-

formance capabilities of the pulse pair estimators under various combinations

* of conditions, and optimizing the design parameters of the radar-processor I

system; and (2) experimentally verifying the theory by hundreds of quantitatively

controlled performance tests conducted with the aid of the simulation systemdeveloped for this purpose.

These studies have demonstrated 6nalytically, the feasibility of im-

plementing a real time signal processor, using the pulse pair computing

algorithm as its central core to obtain quantitative estimates of mean velocity

and velocity spread under a wide variety of conditions and for a wide variety

of radars. This report describes the initial concepts and features of a processor

enabling experimental confirmation of its predicted performance, in a form

L' 1-

suitable for performing meteorological research with Doppler radars as well.

With the instrument proposed here it will be possible to collect and reduce

data under a wide variety of conditions economically. This will rmake it

possible to establish a broad data base from which the operational utility of

a Doppler weather radar may be inferred. The profosed signal processor

will then prov-ide a benchmark for sizing and designing a signal processor

for operational use.

The key features available in a pulse pair signal processor -ire

I. quantitative (digital) estimates of spectral mean, spectral width

and target reflectivity in real time

Z. large numbers of contiguous range gates may be processed

simultaneously (up to 10Z4 cells in the proposed implementation)

3. selectable dwell time may be used to adjust the block length of the

signal processor to the radar scanning requirement and to the

meteorological conditions (integration of up to 1024 samples per

range cell to be provided in proposed signal processor)

4. range resolution may be obtained (resolution equivalent to

ft (. 5us) in the proposed implementation)

5. predictable performance based on self contained data (confidence

bounds on all estimates are computable based on signal processor

derived S/N data)

6. sliding window operation is feasible (estimates based on the

most recent N seconds of data)

7. the storage required consists essentially of three complex words

per range cell (last sample, summary crosscorrelation data,

summary power data) vhile the computing power required is to be

able to calculate 2 conplex multiplies and adds at the input data

rate

8. additional waveform flexibility (non uniform p. r. f. and pulse -idths--

limited frequency agIlity

9. sensitivity comparable to spectrum analyzers

10. independent AGC for each range bin to improve overall dynamic

range

1-2

IA

as well as the features common to all digital signal processors, namely

11. non critical adjustmentsr

7 12. long termdrift free,stable performance without calibration

13. flexibility in interfacing with a variety of radars.

The digital outputs of the processor can be displayed, recorded, or

rerioted for further displhiy or processing.

To fully utilize the processor capabilities it must operate in conjunction

with a coherent radar receiver that has a wide dynamic range and is able to

accept AGC control voltages from the processor at a range cell-to-range

cell rate. To insure compaHIillity, it is proposed to incorporate in the

*processor a self-contained receiver with the required characteristics that

will accept 1.F. and reference signals from the radar, gain control voltages

from the processor, and deliver the required in-phase and quadrature complex

video signals to the processor. Since the processor measures signal amplitudes

to form the AGC control voltage for each range cell, this information is avail-

able as a digital output. Together with the large useful receiver dynamic

range, as an added benefit the processor amplitude output in conjunction with

range information can also be utilized for other purposes, such as the cal-

culation of target reflectivity.

It should be noted that although there are other methods of estimating

spectrai mean. the pulse pair technique provides quantitative estimates of

this parameter in digital form, and is the only known nethod for estimating

spectral width in real tine.

* •The balance of this report is devoted primarily to the technical aspects

of the pulse pair program. The technical background ..s presented in the

succeeding purtion of this Introduction; the theory of Pulse Pair Estimation,

including original analyses performed on the present contract, appears in

Section 2; Section 3 deals with the methods and results of Experimental Eval-

uations of the pulse pair technique, including that performed with real radar

data on the prior contract and that utilizing simulated radar return on the

present contract; Section 4 describes the proposed Hardware Realization

of a Pulse Pair Processor capable of operating on real radar data in real

time; and Section 5 presents the conclusions that have been drawn and the

Recommendations that have oeen formulated as a result of the Study and

Investigation.

1-3

IL

Detailed Analyses, Computer Programs, and Spectral Plots are in-

cluded in the Appendices in support of the main text of this report.

1.3 Technical Background

A commonly occurring problem in radar measurements is to estimate

the spectral mean and variance of a unimodal Gaussian signal imbedded in

white Gaussian noice. The classical technique used is to form spectral

estimates from the time sequence, and then form estimates of the spectral

mean and variance by standard formulae. It is not difficult to show that these

techniques are not optimum and result in signal processor dependent bias

errors which affect the system sensitivity.

When both mean and variance are unknown, the optimum spectral para-

meter estimates can only be obtained implicitly from the solution of a trans-

cendental equation in which the form of the underlying spectrum is known.

To overcome these defects, various investigators have resorted to threshold

tests to reject the noisy frequencies, in order to apply classical mean and

variance estimators to the residue after thresholding.

This report describes a new technique, known as pulse pair analysis,

which bypasses the stage of spectrum analysis, and directly estimates spectral

mean and variance. Not only does this technique result in a less complex set

of calculations to be made by the processor, but it has a form which is based on

an optimum estimation theory and as such is relatively free of arbitrary choices.

Furthermore, the technique is robust with respect to underlying spectral

shape, and permits more flexibility in the choice of radar waveform.

Its major disadvantage with respect to classical spectral parameter esti-

mation is the fact that the technique does not indicate the presence of multi-

peaked spectra. When the full spectral analysis is available clutter regions

may be identified and possibly rejected, and separated non-standard spectral

shapes may be detected. Pulse pair analysis, being a spectral shape inde-

pendent estimation technique, requires a clutter rejection device preceding it

(where necessary), and yields only indirect evidence of peculiar spectrum

conditions (abnormally broad widths) for indicating the presence of multiple

lobed spectra.

As for sensitivity, simulation evidence, to be presented in a later sec-

tion, indicates that the sensitivity of the pulse pair estimation is comparable

to that of the shape independent spectrum processing techniques.

1-4

2. 0 THEORY OF PULSE PAIR ESTIMATION

2. 1 The Pulse Pair Estimators

It may be shown 1 that the optimum estimate of the spectral mean and

variance of a sequence of pairs of measurements which are uncorrelated is

given by

=

df "N (z-2)

where N

x N L r lk (2-3)

k=1

N

= {N Irk +Ir I](2-4)k=l

rik is the observed signal, S(f) is the underlying spectral density

function, and aN is the mean equivalent noise cross-section (see Figure 1).

It is possible to rewrite these equations in a more illuminating form by

defining

fs~fei~Rft dtR(t) = S(f) e Z (2-5)

We get

AirfT (A

arg & R (wT) = argX (-6)

I R(/T) = X (2-7)

Z- 1

wA

crNT

f f

Figure 1. Spectral Density Parameters'

2-la

Thus the latter equation alone determines w, and furthermore, if R(T) is real,A

the first equation is independent of w.

Essentially this result says that if the underlying spectral density function

of the signal is characterized by a spectrum whose shape is invariant to frequencyL

shift and broadening in the specified manner, then these two parameters may beC estimated from the pair wise measurement directly by estimating the complexi: correlation function of the data.

.4 It is important to note that for small T

A 2 (2A8)

R IwT) R(O)- w/ZwR (0) (2-,T) (2-8)

(No first order term is present because S(f) by definition has nean zero.) Also by defi-

nition R(O) and R" (0) are both unity (the first to have unit energy, the secondIA

to define the parameter w, i.e., unit variance for the underlying signal shape).

As a result, in this case, the estimator of spectral width parameter is given

by

2 x1 (2-9)(ZT

•(2,RT) Y

and the estimator of the spectral mean is given by

2 T" arg X.

These are the pulse pair spectral parameter estimators suggested by[ .: Rummnler.()3(4

The approximations made in arriving at these results indicate that noI- * difficulty in utilizing these estimators should arise beyond that expected from

sampling considerations. Thus frequencies in excess of will be arribiguously

folded by the mean estimator, and the variance estimator will be degraded when

the underlying spectral width begins to approach the sampling frequency.

I°:,I I~llms•

. . " -

It is also important to note that the explicit estimators do not depend

directly on any particular spectral shape. This is not true of estimators of

these same parameters based on spectral measurements.

2.2 Estimation Accuracy

Since (2-9) and (2-10) are explicit functions of the data, it is straightforward

to analyze the expected performance of the estimators. The analysis is com-

plicated by the nonlinearity of the estimator functions.Appendix A gives an approximate analysis of the performance of both the

mean and the width estimators. It is shown there that the estimator is unbiased

and that when the per pulse signal to noise ratio is useful, the variance of these

estimators Is given by

ZM Var (Zwf 0 T) = [E(+ Ze "2( + 2nwT 12-11

and

2M (2TwT) Var (ZTwT) N N ( Ze "( 2 1wT)) 2 ( 1 -(2wT) (2-12)

+ (ZiTwT) 3

where

M iumber of pairs in the estimate (r number of data points in the

sample)

N/S = per pulse noise to signal ratio

These forMulae were derived for Gaussian random variables with Gaussian

shaped spectra. More general formulae are available in Appendix A.

The general formulae were derived for a uniform train of samples andaccount for all correlations in the data.

(4)Figures la and 2 show plots of these functions. Comparable figures

derived for the case of independent pulse pair measurements, (i.e., from

a nonuniform pulse train, with successive pairs spaced so far apart that the

echoes are completely decorrelated) are shown in Figures 3 and 4. Comparison

of these figures, which show negligible differences at useful signal to noise

ratios, are indication again of the robustness of the technique.

2-3

101

Figure Ia. Standard Deviation of Pulse Pairs -Center Frequency Estimate (f)vs Spectral

0Width (wT) with per pulse S/N a Parameter,

*for Uniforrm Pulse Spacing

6

0

15 dB

10

(-b

Oo1

0 0. 1 0.2 0. 3wT*

LL- 2-4

100 Figure 2. Standard Deviation of Pulse Pairs

Width Estimate (wT) vs True Spectral Width

(wT) with Per Pulse SIN a Parameter, for

Uniform Pulse Spacing

lb

10

0. 1

00. 1 0.20.3w

2-5

10

1.0

-6 dB

-320dB

0 ~~ - 0.dBZ03 3

-5d

Figur 3.AcrcdfPlePisEsiaeo enVlctSpcta Witfr1dpnetPis fo umeRf 4

15d

v'i-M(S. D.

t 1000

100

0

.3 d4 150d

0 d

0.

0 0. 1 0.2 0.3

Spectral Width Parameter, w

Figure 4. Fractional Accuracy of Pulse Pairs Estimate of Spectral Widthvs. True Spectral Width fo~r Independent Pairs (from Ru-.mmier, Ref. (4))

2.3 Hardware Implications

The pulse pair technique calculation in essence requires 2 complex pair

multiplies and adds for each new data point in order to form X and Y as indicated

in (2-3) and (2-4). The conversion of these to mean and variance estimates is

an operation which need not be carried out at the input data rate. To bring this

I into sharp focus, a digital spectrum analyzer capable of processing up to 1024L

complex samples at a single range, would require 5 times the number of con-

parable arithmetic operations and up to 500 times the data storage of a corn-

parable pulse pair processor.

This efficiency in signal processing makes it possible to consider im-

plementing a real time signal processor which can obtain spectral mean and

variance estimates in real time with selectable dwell time per gate of up to a

1000 complex samples per dwell, with as many as 1000 range cells processed

simultaneously. Available technology currently permits these calculations

to be made at a 2 MHz data rate, thereby permitting range resolution 0. 5es

(i. e., 250 ft.) and even higher thruput rate processors are being developed

now.

In a later section we describe a candidate signal processor implementing

this technique to the above performance specifications. That section detailsthe flexibility and performance which may be expected of a relatively modest

signal processor.

2. 4 Waveform Flexibility

Tie restriction of the processor to consecutive pairs makes possible

a great deal of waveform flexibility which can be used to advantage to

overcome certain inherent radar limitations. Prime among these are the

*range-Doppler coupling which makes the unambiguous Doppler interval of the

radar vary inversely with its unambiguous range interval. For weatherradars, this amounts to a reduction of the Doppler analysis band of the radar

as its range is extended.

The pulse pair technique makes it possible to use the waveform shown in

Figure Sa to change the range-Doppler coupling relationship to

f < 1 (2-13)d Tz

2-8+-I

Ii

II

4.b

'.40

i-I4) .4-

4)4

4

3.4'-4 .44 '-4

~4 4

he 4)0 4)

4) -

4 p44)

4

4)4)

4 .4

~34

ULl 1~e.J

H*1IA~u~nba~j

A~.zau~

2-9

This makes it possible to increase the Doppler coverage band on relatively

small cells located at greater distances from the radar. Selection of T1

and T? permits the radar to operate with an unambiguous Doppler spread

reciprocally related to the dimension of an isolated cell. In weather radars

such a technique might permit the unambiguous Doppler measurement

of such conditions as severe convective cells and possibly local CAT con-

ditions at low angles and long ranges.

The second waveform possibility, shown in Figure 5b, is available with

both equally and unequally spaced pairs. The accuracy equations, (2-11) and

(2-12), reveal that a key perfcrmance parameter is the per pulse signal to

noise ratio. The equations indicate that there is an optimum S/N per pulse

which minimizes the measurement errors for a fixed radar energy. That is,

if S/N and M are related by the equation

= M (S/N ) (2-14)

(which indicates that a fixed energy is available to the measurement that

may be distributed in any way between the number of pairs processed and the

S/N used per pulse) then either equation (2-11) or (2-12 has a minimum at a

definite value of M.

In particular, it may be shown that for the measurement of mean

velocity, the optimum per pulse S/N is given by

(S/N)opt = 1T wT Cfr) 1/2 (2-15)

and at that point the measurement variance is given by

^(ZT) 2 [ - 2 (2n wT) 2 ( -wTVn)1/2l

Var f l0T) opt e e + 2

e (2-16)

Figures 6 and 7 indicate the sensitivity of this condition to nonoptimum

conditions for mean Doppler and width measurements respectively. It is

important to note that minimum error for both measurements does not occur

at the same value of signal to noise ratio. Indeed, the curves show that sub-

stantially more per pulse S/N is required to minimize width errors.

2-10

100

S MxS/N wT 2 'wT .1

where M = no. of radar pulses transmitted

SIN = per pulse signal to noise ratio

10 .05

wT 0. 1

N1 0 IInd 02

2-11

100

A

e= M x SIN .

where M uo. of radar pulesmwtransmitted w

LSIN= per pulse signal10 to noise ratio

.05 4

Id.

N

.. 0

0 SIN (dB; 10 20

Figure 7. Tradeoff Charactieristic of Pulse Pair WidthMeasurement for Fixed Radar Energy e~

Z--1

The tradeoff characteristics may be used to establish the benefits to

be obtained by the frequency modulation waveform of Figure 5b as follows.

Suppose that the radar and meteorological parameters are such that only normalized

widths of 0. 05 or greater are of interest. Then Figure 7 indicates that the data A

should be collected at the signal to noise ratio of 7.5 dB per pulse. Suppose

then that it were required to make that measurement to an accuracy of 109/6.

Then

2 2

w=(ZwT 2 0.01 (2- 17)w(2Tr wT) 2 i -

For wT 0. 05, Figure 7 indicates that A

2 2Ze (ZrwT) 1 2o T = 0.8 (2-18)

so that

" T0.4 (2-19)

(21T wT) 4 2OrA

w

w

J

S36 dB

Thus,at least M = (36 - 7.5) dB

= 930 pulses must be processed to achieve the requisite

accuracy.

2If the peak power in the pulse were such that 13. 5 dB per pulse were

available for the measurement, this theory suggests that transmission of

four frequency coded subpulses each with /. 5 dB per pulse signal to noise

ratio for a total transmission of 250 groups of four might result in bette

overall accuracy than would the same length of uncoded transmission. Tie

improvement in performance would be negligible at wT = .05 for the con-

ditions given, but at wT = 0. 1 the frequency modulation would improve the

accuracy by 40% = ( -I).

2-13

rhere is an important caveat which must be considered in this regard;namely, that additional receivers may be required to implement the frequency

modulation system for extended targets.

2. 5 Summary

The pulse pair technology offers new potential for Doppler weather radar

systems. It permits the economic implementation of a real time signal processor

which can produce prodigious data reduction, reducing the enormous amount

of data produced by such a radar to more manageable dimensions. It does sowithout a sacrifice in sensitivity in cases of interest.

It makes it possible to consider new radar modulation techniques per-mitting the radar to collect the data more efficiently. Thus a frequency and pulse

pair spacing agile radar,which has merit in both increasing the data rate

(number of pulse pairs available) and suppressing some of the range-Doppler

ambiguities,can be processed without penalty in a pulse pair analyzer.

The pulse pair technique produces estimates which have predictable

accuracy and are free of arbitrary parameter selection (such as the threshold

level in the spectrum analyzer technique). In a digital processor, the block

length of the analysis interval may be varied on command so that the processor

can easily be properly matched to the radar operating conditions (p. r. f. and

scan rate).

If data rates were not a consideration, it is clear from the structure of theprocessor, that it could be implemented as a sliding window processor, so that

running averages of the most recent Ni pulse pair data could be used to obtain

continuously updated mean and variance estimates.

In short, it appears that the pulse pair processing technique offers anefficient means of automating pulse Doppler weather radar processing with a

t technique which is accurate, sensitive and flexible. The equipment used inEl conjunction with clutter cancelling techniques should permit operation in a

significant background of ground clutter.

Its ability to signal the presence of unusual spectral shape is perhapsits ultimate limitation.

2-14/215

- ----- --

3.0 EXPERIMENTAL EVALUATION

This section describes several efforts to date to experimentally verify

the theoretically predicted accuracy capaoilities of the pulse pair estimators

of spectral mean and spread and to compare their performance with other

methods of estimation using identical input data. The initial experiment, which

was conducted on the previous contract, compared the results of pulse pair and

conventional spectral analysis processing of real Porcupine Radar weather

return . The limitations encountered with this approach to evaluation moti-

vated the more extensive and precise experimental evaluations accomplished on

the present contract using simulated radar return. Although all of the experi-

ments to date have involved non-real time pulse pair processing of the tape Irecorded radar output data, real time processing is both feasible and advan-

tageous; a proposed real time hardware processor is described in Section 4.

3. 1 Porcupine Radar Weather Return

A sequence of 10240 consecutive raw complex video samples (both in-phase

and quadrature) were tape recorded from the output of the Porcupine weather

radar during a light rain. The PRF was 3300 pps, the antenna elevation angle wasI 40* , and the data was obtained from the range cell with the maximurn S/N.

The analog data was then quantized for subsequent processing on a digital com-

puter by the "pulse pair" method, with conventional spectral analysis as a~control. The latter was required since the true spectral parameters of the

radar doppler return from the rain were unknown.

3. 1. 1 Estimates of Mean and Spread by Conventional Spectral Analysis

The recorded data was processed by conventional spectral analysis in

three different block lengths: one block of 8192 complex samples; two blocks

of 4096 samples; and ten blocks of 1024 samples. A typical plot of the amplitude

of the spectral components of one of the ten blocks of 1024 samples (representing

approximately 1 second of weather data) as analyzed by a digital computer pro-

gramrnmed to execute a Fast Fourier Transform is shown in Figure 8, "Typical

Dat; Spectrum for a Sequence of 1024 Samples. " The raS. -e9 appearance of the

spectrum is due partly to noise and partly to the distributeA nature of the weather

target.

3-1

IJ

IIP

-I.r.

I ai~ ' l I

I I

K I 1

-I I

IIFiueg ypclDt Setu orN=1I

I I

C) I!

I~I!The computer was programmed as follows to estimate the first moment

(f of the doppler spectrum and the second moment (w ) about

Letting z(n) = x(n) 4 jy(n) denote the nth sample in a block of length N,

the spectral amplitudes N

S(k) \ a(n) z(n)o NIlk = N-Ub, - .. - 3In=zO

are computed, where the a(n) are 60 dB Dolph Chebyshev weight coefficientsi

for a block of length N.

The unambiguous frequency interval is then shifted from 0 k S N-Ito - + 1 5 n I by the transformation

Pm-l1), 1 :5 m - N /Z

Zl) ( (3-2)

N

as shown in Figure (4).

The spectral mean and width could be calculated in accordance with

A PRF (m3f - n , where m Z(m) (I o N ' -Z(m)

and

PRF Mlm - m Z (3-4)

However, these classical algorithms for the mean and width of the distribution

make no provision for the effects of noise, although 44 is ev'dent from Figure 8

L that portions of the spectral plots are predominantly x._-se, which would tend to

obscure and degrade the estimates of the spectral parameters. Neither is there

a developed theory for the effects of noise on their accuracy for spectra derived

data. Nevertheless, despite the lack of guidance from theory, it is necessary

to employ some additional (though not necessarily optimum) technique together

* L with the spectral analysis estimators to improve their accuracy so that they

may serve as standards of comparison for the pulse pair estimators. A commonA

3-3 !A

method is to select threshold level referenced to the noise level or to the

peak of the sp, trum, and to use only those spectral components whose ampli-

tudes exceed the threshold in the computation of spectral mean and width as a

special set, which falls within the general class of spectral analysis estimators.Such a preliminary computational step, prior to the above calculation of themean and width, has been introduced into the spectral analysis estimator com-

puter program.

Rather than restrict the results of the spectral analysis estimates to aA A

single, possibly poor choice of threshold, the estimates for fo and w were

computed for each block of data over a range of assumed noise thresholds from

20 dB to 30 dB below the spectral peak, and are shown in Tables I and 2 respec-A

tively. It will be noted that the estimates of fo vary with threshold level within0I

each block over a range of from 5. 5% to 12. 9% of the nominal value, depending

upon the specific block.

The spectral analysis estimatee of W are even more sensitive to the

value of threshold. They vary with threshold level within each block over amuch larger range of from 51% to 68/6 depending upon the specific block. The -

potential problem of using such measurements as a standard of comparison for

evaluating the performance of the pulse pair estimator is noted here in passing,

and will be discussed further later on.

3. 1. 2 Estimates of Mean and Spread by Pulse Pair Processing

The identical Porcupine radar data processed by spectral analysis were

also processed by the pulse pair technique. The ten blocks of digitized se-

quences of 1024 samples were inputted to a computer programmed to imple-

ment the pulse pair estimators of equations (2. 9) and (2. 10). These results

are also shown in Table 1 and 2 respectively. The estimate of W includes a

correction for noise, the level of which was deduced from the experimental

data by assuming that the peaked portion of the spectrum was composed of

signal plus noise, while the flat skirts were noise alone.

3. 3. 3 Comparison of the Estimators

Numerical comparisons of the pulse pair and spectral analysis estimatesA

of f0 and % obtained from the Porcupine weather return data may be made from

the results in Tables I and 2.

3-4

%0 0 N

44~~~ In.4~%O O % r 4No

tn N -n --a 00'40

0 ~ orj r-0t- 0

OD V u m %o - N en C- t-

NN

M - " N V 'D " 00 N No' 0 "

C; '; .4 N *f * N * a * . .- N 0M 4 u -4 - N N ~ -

4 0

a4 0

Vo N r- t- *D N *

aO r N ' 00 M4 - -0H - --- N N-- -- -4

0

Lm O r- o0 0 0

cc 00 * * . . *

*0 _ _ _ _ _ _ _ 1 0

U.d

a4 00 N 3

NN N NN N S'a

3-5~

N LAN W MN'O O %- n o0 I

- ~fnN~ fn N

1: n m D t -

N v D 0 V. NN. co C% 0

Na c,. NO '~'~ 1 000 C'N% 0 nN N NL~'~00~N N N

N~~O -, o C

N4 N O' ON %D ~ -4 now N N N4

N--N N (7 -l N - -n

NU

N4 ON N- -

- - -4 - - - - - - - 0

.0 0

ii VN LA ~ I 'D -.c c, -Z O0 N O"~ 0-

NN~NNN- N N

AThese tables show the values of f and 40 as estimated block by block

0 0by pulse pairs and for each threshold level for the spectral analysis method.

The blocks of 1024 samples have been numbered sequentially to identify the

data from which the estimates were derived. The different estimates for a

given block of data appear on a horizontal line alongside that block.

As was noted above, the spectral analysis estimate of f varies with the0

threshold, T. It agrees beat with the pulse pair estimate of ? when T = 20 dB,

for which case the agreement is within better than 1% when the block length,A

N, is 4096 or greater. Since the spectral analysis estimates of fog which were

used as the control, do not vary too widely, it was generally concluded that

pulse pair estimate of f 0 was definitely satisfactory under the conditions

for which the data was taken.

On the other hand, the spectral analysis estimates of doppler spread,W, are quite sensitive to the value of T. Varying as they do over a range of

about 60%, they do not provide a satisfactory standard of comparison for

evaluating the performance of the pulse pair estimates of w. Without a know-

ledge of the true doppler spread of the weather radar return, it was impossible

to draw any conclusions as to the effectiveness of the pulse pair method for

estimating %0 with this particular data.

It appeared clear from the above results, that real radar weather

return of opportunity did not provide a promising source of data from which

the behavior of the pulse pair estimators could be quantitatively explored.

Not only is there no assurance of obtaining data with the range of spectral

and S/N parameters required for a valid exploration, but the accuracy with

which the true values of the parameters embedded in the data (which are

nended as a basis for comparison) can be established is in itself a function

& of the parameters. Since we cannot reasonably evaluate one unknown against

another unknown, the alternate approach of using simulated radar return with

prescribed parameters was indicated.

S3-7

3. 2 Simulated Radar Return

3.2. 1 Purpose, Capabilities and Advantages

In view of the limitations encountered in attempting to evaluate the pulse-

pair estimators by means of real Porcupine Radar weather return data, (see

3. 1 above) more extensive experirnentatl verification of their theoretically

predicted performance was undertaken on the present contract using simulated

radar weather return.

The use of simulation techniques to generate the input data for evaluating

the performance of the estimators provides many advantages which overcome the

principal limitations of the use of real radar data for evaluation purposes. Pri-

marily, it makes possible the generation of statistically significant numbers of

simulated complex radar samples with prescribed spectral distributions (mean

frequency and width)and S/N. This permits the generation of data tapes with

controllable parameters made to order and in sufficient quantities per case to

suit the evaluation needs. Since the true parameters are known in advance, a

direct and precise determination of estimator performance accuracy can be made

at all times. The ensemble of precise results can then be analyzed by the com-

puter to determine the statistical behavior of the estimators against the synthetic

distributed weather targets as a function of the controllable parameters, number

of samples entering into the estimate, etc., and the results compared with the

theory.

The particular cases chosen for the simulation tests were tailored to facilitate

comparison with the theoretical performance predictions derived in Section 2.

A PRF of 3300, the highest available for the Porcupine Radar was used for all

cases. A spectral mean of 300 Hz and width of 78 Hz were selected for the

initial cases. These values are equivalent to a weatheretarget velocity mean

and spread of approximately 8 meters per second and 2 meters per second

respectively, as observed with the Porcupine Radar. To vary the S/N para-

meter, two extreme values of S/N were chosen, including 0 dB and OOdB (no

noise present) and two cases closer to the poor S/N condition. To conveniently

vary the theoretically significant wT parameter, several multiples of the initial

mean and spread were selected to yield target velocity distributions with 16 meters

mean, 4 meters spread; and 24 meters mean, 6 meters spread. Each of these

cases was varied over the range of four S/N values established initially.

3-8

A

An overall block diagram of the software simulation system utilized for

investigating the pulse pair technique appears in Figure 9. The system

is capable of generating sequences of simulated radar doppler return signals

with controllable parameters; estimating the mean and width of the spectral dig-

tributions by both pulse pair and spectral analysis tecnniques; performing statis-

tical analyses of estimator accuracy; and plotting and/or printing out the results.

The simulation system is programmed in FORTRAN and was run on the

CDC 6600 Computer at AFCRL. To conveniently suit the capacity and capabilities

of the computer, the simulation sybtem was divided into separable steps, which

could be run at different times. To facilitate such division into steps, the Be-

quences of simulated radar doppler return which were to be used an input data to

3--

t

F.I

I

r

f 3- 8a

r

PUL.S.E PA7 PR(77SS17 !-.ATISTICADOPPLER RADAR RETURN GENERATOR ANA IY ItL" T . ....(DOPGEN)

A U!.1-'.- I.. . . . . .. r-; .... .Si ult on i tu

of Si.,netic t nnagic v,-

Doppler CONVE;NI [ t P£.'i 1'LM ANAYI. Y

Return . . LI

Spectral L

' i I i'[,',f. i Ii~ i" e,, "

Analysis for visual

. check of._ l simulated AMiP I

signals ,1. .Pen = ,.

Plot

Figure 9. Overall Block Diagraim ol th, 'o m*' S; o it for

Investigating and Comparing mr: !, c " : O' tlP, I-ds

Pair and Spectrum Anally.'er Plti., .-i or e i ara I a,-,rnet, roi

3-9

2N

the signal processors under investigation were recorded on magnetic tape for

later processing and reference. A library of such input data tapes was generated i

to provide desired numbers of statistically independent sequences of simulated

return, with desired combinations of spectral parameters and S/N. The signal

processing schemes, and modifications thereto, could thereby be statisticallyevaluated and compared on the basis of identical input data. Provision was also

made to visually verify the validity of the simulated data; designated sequences I

of returns from among those generated could be selected for spectral analysis

by conventional FFT and automatically plotted for visual check.

The forthcoming sub-sections deal in greater detail with the generation of4

the simulated doppler return; the comparative performance of the pulse pair

and spectrum analysis estimators against the simulated input data; and a com-

parison of pulse pair estimator performance as experimentally determined by

simulation with that predicted from theory. Excellent correlation between theory

and experiment are shown.

3. 2. 2 Generating Sequences of Simulated Doppler Radar Target Return

with Controllable Parameters

In the real Porcupine Radar case, a uniform PRF was used to generate

sequences of uniformly spaced coherent echoes reflected from distributed -

meteorological targets as observed in a fixed range gate. The uniformly spaced

echoes, in the form of digitized complex numbers derived from the phase detected

coherent video output,were then processed two at a time to yield the pulse pair

estimates of the target spectral parameters.

The object of the simulation program is to generate similar sequences of

uniformally spaced complex numbers equivalent to radar echoes reflected from a

distributed target with a given radial velocity distribution, as sampled at a given

uniform PRF and with a given output S/N. The problem is mathematically equiva-

t lent to that of numerically generating a sequence of sampled values of a stationary

Gaussian process having a specified correlation function or power spectrum. The

method employed is adapted from Levin(7)

3-10

I' _ ___ __ __ _ __ ____ _ _ _

A. rheory

The following parameters of the desired sequence must be controllable:

Term Symbol Units

MEAN DOPPLER f Hz0

SECOND MOMENT m= w= Hz2

SAMPLING RATE PRF Hz

S/N RATIO (per pulse) B dB

1. Spectral Distribution - An approximation to the desired spectral

distribution of the radar video data is achieved by playing Gaussian

noise of zero mean and unity variance through a filter consisting of

two cascaded, low-pass networks. The transfer function of the

filter is given by:

Za

HI(s) _ where a = Zirw and (3-5)(a + al 2 a = complex frequency (rad/sec)

From this e)pression it is-possible to derive the second moment and

half power point of the filter. The second moment is given by:- I

2 +j=

( )2 +jH-(s) H(s(s)do (3-6)

H(s) H(-s) da

The half power point (-3dB) occurs at

ga(42-1A . (3-7)

3-11

By L-transform theory, the recursion relation for the equivalent digital

filter is given by the impulse invariant transformation; namely, if

(s+ a)2

then, Z-_sa T e"T1 (3-8)

H(Z) =;where T ae[ ewT z-1 )2 PRF

The resulting recursion relation is then

a e-aT -aT -ZaTX n . , a- + 2 e Y n - e Y n - " ( 3 9

n n-(

Simulation of radar quadrature data is achieved as illustrated in 14

Figure 10. The sequences [xi. xzj are two independent Gaussian sep.ences

which are then played through the same filter, H(Z), specified by (3-8) and

(3-9). jZf nT~0Complex multiplication of the output sequences (Yi. y,) by e

shifts the mean frequency from zero to f.

Trans;ent effects of the filter are minimized by discarding a nurrberII

of initial terms. This number has been set at three times the fractional

V • bandwidth of the filter (with respect to the PRF) and it is given by:

[ INo. of discarded initial terms (3-10)I i I

2. Signal to Noise Ratio - When Gaussian noise is added to sequences

y, Y' 1 (See Figure 10 ), the per-pulse signal to noise ratio (S/N) is1 2 -

specified by:

3-1Z

U -0

144

0

0 1

u 84N.. 0

Z I4

3-13

aT a

S/N: a 4 T ' (3eaT + eaNaT -aT - B [dB] (3-11)a2 (E -T e aT

N

where a, is the variance of the added, zero mean, Gaussian noise. A givenN

S,.'N ratio is then achieved by specifying az as

a4T a T -z 4T z (3e a + ea T 10"Te )(3-12)

N aT - aT(z e

where B is the desired S/N ratio in dB.

Mathematical derivations relating to the above processes appear in

Appendix B.

B. Description of Computer Program

The program developed for generating the simulated doppler target return

on the AFCRL CDC 6600 is called DOPGEN. The flowchart for DOPGEN is shown

in Figure 11 , and the program statements coded in FORTRAN appear in

Appendix C 1.

Input to the program is in the form of data cards on which are specified:

(1) mean Doppler frequency;

(2) standard deviation of Doppler frequency;

(13) per-pulse signal-to-noise ratio;

(4) FFT plot control; and,

(5) number of blocks (N) of 1024 complex samples.

Upon reading the first data card, the program generates 1024 coriplex samples

which simulate Doppler return (a PRF of 3300 is used throughout). The

data block iE then written on magnetic tape along with twelve ancillary words

containing information about the data. This procedure continues until N inde-

pendent data blocks are written with the specified Doppler mean and standard

deviation.* A second data card can be used to change the mean and the width

of the Doppler spectrum or to change the S/N ratio. If no more data blocks

* Since each data b'ock occupies about 3. 5 ft. of magnetic tape, 685 data blocks willfit on a standard Z400 ft. long tape.

3-14

i r f

00

U)

r. 0

0 44

MM

00a4 a4

+ 0

.1I. - 3

are to be generated, as indicated by a blank data card, the program rewinds

the tape and reads each block to verify the readability of the tape. In addition,

for those data blocks on which the FFT plot control parameter (one of the

ancillary words) is set to one, the program performs an FFT on the data and

plots the transformed data on a Calcomp Plotter (available at AFCRL). This

is used as a check to verify the validity of the simulated data - a sample plot

is shown in Figure 12.

A library of 390 separate sequences of 1024 complex samples was generated

by the DOPGEN simulation program and recorded on magnetic tape for use au input

data. Each sequence of 1024 samples constitutes a data block. Three pairs of I. mean doppler and doppler standard deviation were run, each at four conditions

of S/N; a PRF of 3300 pps., representative of the Porcupine Radar. was used in

all cases. Thirty independent data blocks with specific parameters as shown in

Table 3 were generated for each combination of parameters. A sample

spectral analysis plot for each of the twelve combinations is prevented in

Appendix E. A

~fo w !:Mean Std. Des.

Block No. Doppler Doppler SIN Tape A(Hz) (Hz) (dB)

41-70 300 78 5 RPK07101-130 600 156 00 RPK08131-160 600 156 0 RPK08161-190 600 156 5 RPK08191-220 600 156 15 RPK08221-250 900 234 00 RPK08251-280 900 234 0 RPK08251-Z80 900 234 5 RPK08311-340 900 234 15 RPK0834 1-370 300 78 00 RPK09371-400 300 78 0 RPK09401-430 300 78 5 RPK09

.431-460 300 78 15 RPK09

Table 3. List of Recorded Sequences of Simulated Doppler ReturnGenerated, with Parameters as shown.(PRF 3300 pps in all cases)

3-16

km.

2,

P4a

4) >

_ _ __

A. -~ -- '~- -_lad_____ ____ _____ __ 0

O1' .- w-~-- LO4O N N O .~ - ~ . -4-A

4.4

Coll 00 o 124

_ _ _ _ _ _ _ 00.

3-1

3.2.3 Performance of Pulse Pair EstimatorsA. Computer Program

As shown in Figure 9 of the overall simulation system, the tape

recorded sequences of simulated doppler return are processed by the pulsepair algorithms to yield estimates of spectral mean and width as a iunction of

the number of pulse pairs entering into the estimate. The resulting estimatesare then compared with the known given spectral parameters of the input data

and the performance of the pulse pair estimators is statistically analyzed andsummarized to produce printouts of estimator accuracy versus number of pulse

pair samples for each simulated test condition.

The program for pulse pair processing and statistical evaluation is calledtPPSTAT. A flowchart of PPSTAT appears in Figure 13, and the coded

FORTRAN statements for this program may be found in Appendix C-2. As

shown in the flowchart, before reading the magnetic tape generated by DOPOEN,PPSTAT reads from a data card two data block numbers (the numbers are assignedby DOPGEN and are contained in the ancillary data associated with each block).Pulse Pair estimates are then made on consecutive data blocks starting with thefirst block rumber and continuing to the data block with the second block number.The estimates are made on samples of size 16, 32, 64, 128, 256, 512, and 1024.

After operating on the variable sample sizes from each data block, the program

performs a statistical summary of all the Pulse Pair estimates to yield estimator A

accuracy versus sample size.

B. Results~The hundreds of pulse pair estimates of spectral mnean and width produced,

analyzed, and printed out by the simulation system are presented in a mrannerintended to facilitate comparison with theoretically predicted performance capa-bilities. The close correlation shown between theory and experiment in the

t Figures that follow confirms the validity of the theory under the practical condi-tions of noise background, correlation between pulse pairs, etc. Analytic predic-tions of pulse pair estimator performance for particular sets sf conditions can,

therefore, be made with confidence.

The following comparisons between pulse pair experimental data and theory

have been calculated and plotted from the results of the simulation tests. Za

3-18

tL

AM

Start

IptN and K- Pulse PairRead estimates will be made anData - - data blocks N to M and the

results statistically summnarized.

Read Magnetic Generated byData Block g- -- Tp DOPCENNumber I

Pulesi Pair 1LineProces sing I---- Printer

a -- OW Data Flow

Save IW Program FlowPulse PairI

Results

No Does Yte Summary of

Figure 13. Flowchart of PPSTAT

3-i9

Figure Data Compared

14 Accuracy of Mean Frequency Estimates vs Number of Pulse Pairs

15 Accuracy of Mean Frequency Estimates vs Spectral Width

16 Accuracy of Spectral Width Estimates vs Number of Pulse Pairs

17 Accuracy of Spectral Width Estimates vs Actual Spectral Width

18 Accuracy of Spectral Width Estimates vs Signal-to-Noise Ratio(for fo--300 Hz;w = 78 Hz)

19 Accuracy of Spectral Width Estimates vs Signal-to-Noise ratiot (for f = 600 Hz; w = 156 Hz)2.0 Accuracy of Spectral Width Estimates vs Signal-to-Noise Ratio

(forf 0 a 900 Hz; w= Z34 Hz)

17.j

f

3-20

100

Theory A

I

10

a10

A 15

100 1000No. of Pulse Pairs

Figure 14. Theoretical Accurpcy of Pulse Pair Estimates ofMean Frequency ('fo) vs Number of Pulse PairsCompared with Similation Results

3-21

1-00

t44

10

SYMBOL /N dB -4

0

01.1500HFiue1.T ertclAcrcIfPlePi siaeof Man Fequncy f vsSpetralWidt (w

wit Si ultonR sul A3IZ

__1 Ig--

I -71000

0I. d

0100 -

44

4 1-Tb~ory

o 0 SIN rO0dB

0 0 S/N. - dD

10 100 ~o

Figure 16. Theoretical AL.:~~ Pi A! I'a 1;rn'L?of Spectral Width iv't vsNnh4ro ~jCompared with i iw

3-23

10001SIN9 - d

Thor

0

100-

10

15

N0

0V 100 w= Hz20A

Fiue17.TertclAcrc5fPlePi siae

o fO crlWdh()v cttlSeta it w

Co prdwt imlto eut

k -2

A

300

104 Samples

200 , { Pulse Pair

Simulation Results

1 a Upper Limit Based on100 i Pulse Pair Theory

100

:F Actual Width

o0 5 SII-0 dB15I I

s/N - dB

Figure 18. Theoretical Ac uracy of Pulse Pair Estimates of Spectral

Width (w) vs S/N Compared with Simulation Results (for

f - 300 Hz; w = 78 Hz)0

3-25

300

k 1024 SampleS

N

Actual Width

100 IPulse PairSimulation Results

* 0 -15-

-

0 510 S/N- dB

Figue 1. Th~rei~alAccracy of pulse pair EfstimTates of Spectral

Figuth 19. vs /N compared with Sim uationR s ts io

f 6 0 0 1z; w 156 H7)

3-26

300

I I aUppor Uimit Based on Pulse! ..Pair Theory

200 9 Actual Width

Pulse Pair Simulation ResultsN

1001024 Samples

0- " I0 5 10 15 00

S/N - dB

Figure ZO. Theoretigal Accuracy of Pulhe Pair Estimates of SpectralWidth (w) vs S/N Compared with Simulation Results (forf 0 900 Hz; w = Z34 Hz)

3-27

_ _ -..--- _-.,.

3. 2.4 Performance of Spectral Analysis Estimator

A. Computer Program

In order to provide a standard against which to compare the performance of

the pulse pair estimators under varying conditions of signal to noise, the simulated

doppler return generated by DOPGEN was also analyzed by the conventional spectral

analysis method. As in the case of real Porcupine radar data (Section 3. 1. 1 above), 4

the spectral parameter estimates were obtained in two steps. First, a spectral

analysis is performed on an entire data block to yield its spectrum; then the mean I

and spectral width parameters are calculated from all spectral components above a

given threshold level, for a range of threshold levels. The resulting estimates are

printed out as functions of threshold level. This spectral analysis program is

designated SPEC. As shown in the flowchart, Figure 21 , the Spectral Analysis

program searches the tape generated by DOPGEN and reads the data block whose

number is specified on the input data card. The 1024 complex samples are then

weighted using 60 dB Dolph-Chebyshev weighting coefficients and input into the FFT

sub-routine. If desired, the FFT output will be plotted. Next, the mean and

standard deviation are calculated based on all FFT output points above a specified

threshold level. The threshold level is automatically incremented in steps of 2 dB

starting at 2 dB below the maximum FFT output point and continuing to 40 dB below

the maximum point. The mean and standard deviation values are printed on the

line printer as a function of threshold level. The above procedure is repeated for

each input data card until a blank card is encountered. SPEC coded in FORTRAN

tappears in Appendix C3.V

B. Results

It was concluded in the discussion of results with Porcupine Radar input dataA

(Section 3. 1 above) that the spectral analysis estimator of fo was definitely sati-

* factory. Therefore, no further analysis and evaluation of the performance of this

estimator was done with the results of the simulation tests.

On the other hand, there was interest in how well the thresholded spectralA

analysis estimator of spectral width (w) would show up against the simulated input

data. Ip -

3-28

1

Read Data block number (N), and

Data --- FFT plot control

II

Re~d~dBII

OMPI~x[ eight FTYe enPo

C ompeFiguroie i.Fwcar ot ECbouin

Sam3-29

L1~~e Threshold _

Typical results are plotted in Figures 22, 23, and 24. Each figure

- -

depicts the width estimates produced by a single block of samples fordifferent cormbirhationo of input data parameters, f0 . w and SIN , as a

function of threshold level. As was indicated earlier, no claim is made

that the thresholde6 estimator of Eq.(3-4) is optimum. It is, however,t straightforward and readily implementable by computer. To preclude

possible bias effects from aliasing, the spectral width was calculated aboutthe true mean, f . Despite these precautions, the width estimates again areo0

* shown to be very sensitive to the value of the threshold. Only in the cases

of very high S/N do the estimates asymptotically approach the true width.

IL

IkI

19

. _i"i

'- 300 0 5

S"S/N -dB . I_ :FI

SIN dB

isi

N IOZ4 Samples

100

Actual Width

00

0 20 40 60

Threshold - dB

AFigure ZZ. Spectral Analysis Width Estimates (w) vs

Threshold Level for Simulated Input Data(fo = 300 Hz; w = 78 Hz)

3-31

I .. ... .. ... ----, --- _w r_._- ,.2.- - - . . - - -- _ _

3000jS0N- dB

15

It Actual Value

N

L

100

V

| lO24 Samples

* j

0 20 40 60Threshold - dB

Figure 23. Spectral Analysis Width Estimates (w ) vs ThresholdSLevel for Simulated Input Data (fo = 600 Hz; w :156 Hz).

3 03

i 3..3

. _ . . . . . . . . . . .I.. . . . .. . . . . . .

100i i I i i :i i .. "

33300 0 5 15

SIN dB

.1

00

Actual Width

100- 1024 Samples

-I t

0 20 40 60

Threshold - dB

Figure 24. Spectral Analysis Width Estimates (w) vs.Threshold Level for Simulated Input Data(fo = 900 Hz; w = 234 Hz)

3-33

3. 2. 5 Compariso of the Width Estimators

The accuracy achieved by the pulse pair width estimator in the simulation

tests is compared in Table 4 with the accuracy predicted from theory, and the

estimates yielded by the spectral analysis estimator. The results are shown

for all 12 cases of f0 , w and S/N. (It should be noted that the pulse pair data is

presented in the form of the standard deviation of the estimates from the trueI value, based on a statistical analysis of 20 estimates on as many independent

blocks of simulated data for each case shown; whereas the spectrum analysis

results are for a single block of samples for each case, as a function of threshold

level.)

"Table 4 and Figures 18-20 show that the pulse pair width estimates obtained

with the 1024 samples and S/N in excess of 5 dB per pulse produces width errors

of less than 3 dB for all cases tested. Translated into meteorological terms, for

the Porcupine radar, actual widths of . 2 rn/sec were estimated at . 3 m/sec

with the equivalent at a 1/3 sec look (at 3300 pulses/sec) at a signal-to-noise ratio

of 5 dB.

For a signal which is a Gaussian random variate with spectral density function

2w + 2

+ +a 2we

imbedded in white noise with spectral density coefficient No, the ratio of the peak

spectral value to the rma noise background level in the frequency domain is given byIo 2R,. . 2

N0 0

If this datum is sampled at the rate l/T, the per pulse signal-to-noise ratio is

given by

S1/N= Sc~f dfN0

2Ta

N0

3-34

[..... _.. .... ....

m~ Ln m q' LI 0 Ln m m - 0'

0- a ' N A N -n t.A N- 6

N nN m U) NtrLnn Nr

00~ L 0Ln S m m N a m-fA~~~L ONL 4 Ar

0 w____ ___0___ ___

o ^1 o 1. 00 0 aO 'o 0 '.

m U) 00 0 N 0 V L

-N m 0n 00 00 00 LA ,0 0'L

0- Ct' m %0 LA m LN n m )' D U 0 U -4

9 fn 00 N

N ~ mV fn CooLAs N (71 0

-4 < _____________

0%

o~~~ o* L "0 ' *f~ - N -

01 0 NA4 .A OD- s LA N wr.-

0 - -

to OLnLn ~ tLA Ln OtL U) 0

0 1.4

4.A 0

00 0

-4 ()

(Un

3-35

thus pte

tR,/S/N -2w wT

Thus the spectral peak output of the simulated signals exceeds the input signal-to-

noise ratio by approximately the values shown in Table 5.

Table 5Peak Signal-to-Noise Improvement of Spectrum Analyzer

f W Hz) D in dB

78 12.3

156 9.3

234 6.3

This calculation shows that the spectrum analyzer output reduction to mean

and variance may be expected to perform extremely poorly when the threshold is

set at approyrnately D + S/N (in dB) below the peak output. These points corres-

pond approximately to the break points shown on Figures 22 to 24. Assuming that

the spectrum analyzer threshold had been set 5 dB above the noise background level

(high enough to assure that the noise peaks would not disturb the measurement).Figures 22 to 24 indicate that the measured widths would be as shown in Table 6.

It should again be noted that the data shown on Figures 22 - 24 represents

only a single spectral sample and that the thresholds shown are measured with

respect to the sample peak value. Table 6 attempts to incorporate this factor by

using the breakpoints of the measurement, as the threshold level. These are

at approximately the threshold shown in Table 6.

In summary, the spectrum analyzer measurements appear to consistently

underestimate the width parameter while the pulse pair technique overestimates

it. The simulation indicates that the accuracies available for basically the same

data conditions are roughly comparable for the cases tested. Figures 22 - 24

indicate a marked sensitivity to threshold settings which is not directly evident in

the pulse pair measurements. It is important to note that the pulse pair measure-

ment of width uses an estimate of the noise background level also and thus has a

3-36J

ITatle 6

Comnparison of Pulse ftir and

Spectrum Analyzer Derived Wtdth Estttiutet4

(Spectrum Analyzer Threshold Set 5 .B Abovpe Average Backgrour-d Naise i

SIN Threshold Spectrumrr Pulse PairActual Width Avalvyis Fetimtrt Estimate

dB A

78 0 7.3 fl 180

5 1Z.3 44 Ilais ZZ. 3 6,4 75

156 0 4.s} 47 >09

5 9,3 9? 1t3

15 10. 95 !57

234 0 i. 37 2 63

5 6.3 70 240

i1

I'1 6 Gt

F4

corresponding parameter uncertainty in its process. In other words, 'loth

techniques require a measurement of the noise alone background to obtain

useful final results. -

Ij The conclusion to be drawn from this comparison is that the pulse pairtechnique shows no evident lack of sensitivity as compared with the spectrum

analyzer approach in situations characteristic of the meteorological applications.

3-3/31i

LIi .i

~3-38/3-39

!3

4.0 HARDWARE REALIZATION

This section describes a real time digital processor design to reduce the

pulse pair technique to practice. A brief theoretical review of the signal processing

algorithms to be implemented, is followed by discussions and block diagrams of the 7

processor and its key comporients. Design goals for the proposed pulse pair pro-

cessor are defined, and its performance compared with CMF, Analog Spectrum

Analyzer, and Fast Fourier Transform processors. Finally, the design tradeoffs

to be considered during the design phase are discussed.

4. 1 General Description of the Pulse-Pair Processor

This processor operates on the output of a radar, and is capable of pro-

ducing real-time independent estimates of the spectral mean and width of a

Doppler radar signal for each range cell viewed by the radar.

The processor accepts the radar IF signal and system trigger as inputs

and computes the mean frequency (mean velocity)-and spectral width (velocity

distribution) in each of 256, 512, 768 or 1024 range cells, selectable by a front

panel switch. Each range cell has a width of 0. 5, 1 or 2 microseconds selectable

by a front panel switch. The processor produces both analog outputs for viewing

on real time displays, and digital outputs which may be remoted or recordedA

for future computer processing. An internal digital automatic gain control sys-

tem independently measures the signal amplitude and adjusts the gain of the

processor for each range cell processed to maintain a 90 dB dynamnic range.

Although the pulse pair technique will work with either coherent or non-

coherent radars, a coherent receiver output is necessary in order to provide

* complex samples of the target return to the processor. In addition, to achieve

independent spectral estimates from range cell to range cell, the receiver gain

must have a wide dynamic range and be range cell-to-cell controllable. Accord-

ingly, the receiver requirements must be included as part of the processor con-

.F siderations; how much of an existing radar receiver can be used as is and how

much must be modified or incorporated as part of the processor depends upon&

the characteristics of the particular radar whose output is to be processed. The

receiver for the Pulse Pair Processor described below is interfaced to match the

Porcupine Radar.4

4-1

4. 2 Technical Description of the Pulse Pair Processor

The PPP performs spectral mean and width estimations according to equa-

tions (4-1) and (4-3) respectively, which are equivalent to earlier equations (2-9)

and (2-10). They are rewritten here in more convenient form for the discussion

of hardware.

A PRF - (4-1)

f *(z~ 1 ~

A PRF 1- I 1W 1(4-2)IT /_z2 2

where the z's are complex samples of signal plu_ noise, and N is the receiver0

noise power in the absence of signal. z* denotes the complex conjugate of z.

Equation (4- 1) estimates mean frequency by determining the phase diffei ences

in pairs of complex samples (zk+1, zk), averaging these differences to find

the mean phase difference, and scaling this mean phase difference by the PRF

to form a mean frequency estimate. The phase differences averaged are

weighted according to sample amplitude so that phase differences from strong

returns count more than differences from weak returns. The results in Section

3 showed that as the number of samples processed increases, the accuracy of

fo improves asymptotically.A

Equation (4-2) estimates spectral width w in terms of signal coherence,

including a correction for noise, N O,,0

a= 1 Zk+ 1 k (4-3)

which may be measured. The parameter a is well-defined for all sampled

signals including those with multirodal spectra. It may be seen from equation

(4-3) that in the absence of noise a = 1 for sinusoidal signals, and that a does not

depend on mean signal frequency, which may exceed the complex sampling fre-

quency (PRF). As is usual in sampled data systems, the signal bandwidth must

not exceed the PRF.

4-2

I

Figure 25 shows the signal flow of the PPP algorithm applied to the

sampled Doppler signal from one range cell. A delay and multiplier are used

to form the products zk+l z*, which are then averaged. The sample energies

2 A A

i are also averaged, and function generators are used to compute f and w.I zkl 0

There are a number of alternate ways of deriving the noise correction term

(N ) required for the calculation of the width estimate. One such method is de-02

picted in Figure 25. A measurement of receiver noise (n ) (which may be made,0

for example, at the output of the receiver at a range in excess of that in which any

signal return may be expected) is corrected for the receiver gain applicable to the_2

particular range cell, and introduced as the noise correction (N ) in the width esti-0

mate (w). The correction factor is obtained by calibrating the receiver. The mean

power output (p), which is made up of averaged measurements of signals plus noise,

is delivered for use as the range cell's control signal to set the receiver AGC at

its range. !I

Detailed design of the noise correction scheme has been deferred pending

a hardware tradeoff study of alternate schemes, to be performed as part of a

future hardware fabrication effort. The description of the system design which

follows does not include the noise correction feature.

4. 3 System Design

The PPP system, shown in Figure 26, comprises a fast AGC IF amplifier,

a two-point correlator, three (or four) digital integrators, a timing generator and

several function generators. The PPP accepts 30 MHz IF input and producesA A

digital and analog fo, w, range and integrated video outputs. The major sub-

systems of the PPP are scribed below.4.3. 1 IF Amplifier Phase Detector

Achievement of the processor 90 dB dynamic range requires independent

gain control in each range cell processed, which implies that the AGC voltage

must be a time varying signal with a bandwidth comparable to the IF bandwidth.

Such a system requires an IF amplifier of unusual design and closely integrated

IF amplifier and AGC loop designs. For this reason, the IF amplifier and phase

detectors are included in the PPP.

4-3

'a.4

CL 0

U. >

4-44

L -A --- -o

8 [ 8

COMPLEX / DIGITAL INTEGRATOR !5 HIFTMULTIPLY

A/DREGISTER

AGC 1024 WORDS

IF IINPUT17 IF AMPLIFIERL DIGITAL INTEGRATOR 1

AND PHASE IDETECTOR

3 MH2REFERENCE _____________

INPUT'7---------------- 6

PROGRAMMED 8 IARRAY X2 !-T EGI EGI DIGITAL INTEGRATOR

IS

CLOCK CLOCK)LO M L LO L__-

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8

PROLOGGREAM D)/LOG IM()/LOG PLOGW 2

PROGRAMMED a . PO R ME,

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IN TEGRATION PERIOD L 1 . 6 N'-

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512-256-128_

88 1

COMPLEX DIGITAL INTEGRATOR

BMULTIPLY

8

REGISTER DIGITAL INTEGRATOR DAOTU

I VP o

I 8

REGISER DGITA INTGRATR -D/AGOTUT

2ILOG P/LO)G W S PRGAMDMEAN

COKI REGISTER C ETARRAY REGISTER 0/A- > WEIDTY9 9 8 TAN-E-X 7 10OUTPUT

5 VP, 500

REGISER 3DIGITAL

ARRAY___ REGSNGE0D/

Fiur 262 1P BlocTDigra

4-5/4-60

An IF amplifier-phase detector, Figure 27, was recently designed for the -

PORCUPINE radar which is acceptable for the PPP with slight modifications, The

amplifier achieved approximately 60 dB ACC control range, with AGC applied totwo out of a total of three identical gain stages. ACC response time was less than

200 ns. By adding AGC to the third gain stage, this amplifier is expected to

achieve a 90 dB AGC control range without degradation of response time.

t4. 3. 2 Dipitize"

The digitizer accepts in-phase and quadrature bipolar video and produces Iidigitized estimates of these inputs. Eight bit quantization appears more than

Aadequate to compute f to the accuracy of the algorithm. A single A/D converter,

such as the ILC Data Devices VADC-B, may be switched to quantize both input iichannels. This is preferable to the use of two simultaneous A/D converters

because the single converter is less expensive and because problems of balancing

the input channels are simplified by the use of a single A/D.

The suggested A/D converter can take an analog sample every 150 ns and is

compatible with a range resolution of 500 ns. A 150 ns delay line must be used in

one input channel to compensate for the A/D sampling delay. This A/D includes

a sample and hold circuit, operates from standard +5V and * 15V power supplies

and is contained on a single 8. 8" x 4. 5" circuit card.

4.3. 3 Correlator

The correlator accepts complex video samples zk from the digitizer and

generates products of the form Zk+l zj where the sample Zk+l is delayed by one

range sweep time from the sample zk. This delay is accomplished using MOS

dynamic shift registers organized as 16 bits x 512 words. Recently available

1024 bit registers, such as the Intel 1404A, allow this delay to be fabricated

quite compactly.

The complex multiplier which forms the Zk+l z products will be similar

to the multiplier developed for a proprietary Raytheon FFT pulse compression

system. This multiplier, shown in Figure 28, employs programmed arrays and

adders to perform a 24 x 24 bit complex multiply and add every 0. 5 .is. Approxi-

mately 280 integrated circuits (DIP'S) are required, hence, the multiplier could

be fabricated on four 72-DIP Augat panels. The PPP requires only a 16 x 16 bit

complex multiplier; one rircuit card could probably be saved by redesigning the

FFT multiplier for the less demanding PPP requirement.

4-7/4-8

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IF 12 12 12 68 68 JRESo IIT I'I 68 J12 1.5 S TO R. PAR.ADE LTCto STOR.

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68 ADDER LATCH ---- S&M2 TO 12N WI

68~~41 JR/4-129 e

ADDER LACH__STRE __S&

4.3. 4 Digital Integrators

A minimum of three digital integrators are required, two to compute theA Areal and imaginary parts of u and one to compute average power p. These three

integrators will be similar to an integrator recently designed for AFCRL, to

minimize design costs, but the number of sweeps integrated will only be variable

up to 1024 by factors of 2.

An integrated power measurement is required to operate the AGC. This

output may be derived either from the power channel (p) of the PPP or through a

separate integrator. The integrator which produces AGC output, whether separate

or power channel, must be of the exponential rather than square moving window

type. Theory shows that the gain and phase of the square window integrator,

Sgv(O) -" VI -. coo wT (4-4)

W- Tan-l cosowT - 1sin(T -T (4-5)

makes it unsuitable for use in stable feedback loops with loop gain greater than 1.A design study should determine the performance differences in PPP3 employing

only three exponential window integrators,and PPP's employing three square

window integrators to compute u and p and one exponential integrator for AGC.

4. 3. 5 Function Generators

Three function generators are required for the PPP: a squared magnitude

generator, a log generator and an output function generator. As shown in Figure 27,all function generators will consist of programmed arrays combined with registers,

gating circuits and adders appropriate to the function computed.

2 -ro -XProgrammed arrays will contain the fuictions x , In x, tan e , andS (I - e x ) 2 . Each programmed array will consist of two integrated circuits

such as the Monolithic Memories MM6300. The Mr,/'300 is organized as an 8-bit in,

4-bit out array which may be programmed to contain any desired bit pattern; twosuch circuits form an 8-bit in, 8-bit out digital function generator. Approximately

50 ns after an 8-bit input word is supplied to the two programmed arrays, an 8-bit

output word is available, allowing each pair of arrays to compute, and an output

register to store, up to 4 outputs during an 0.5 4s range cell.

These arrays may be ordered preprogrammed by the manufacturer or may be

programmed by the user. Some vendors (e.g., Motorola) sell programmers for

their arrays. Raytheon is currently constructing a programmer for the Monolithic

Memories array in connection with a high speed pulse compression FFT develop-

ment; this programmer is expected to be available for use in PPP fabrication.

4-13

. . . . .- " ". . .I . .. . . . . . .° . . . - - " - =

'~ 4.4 Performance

The detailed performance of the PPP will depend upon a number of design

decisions, such as the AGC loop characteristics and the exact form of the function

generators; operational considerations such as the signal characteristics and the

number of pulses integrated; and the basic accuracy of the pulse pair algorithm.

Analysis of these factors should be made in a design study but is beyond the scope

of this hardware feasibility report.

By assuming a set of design goals for PPP hardware, subject to change in

the design study, a few general comparisons may be made between the PPP and

other related signal processing hardware. The design goals in Table7 are there-

fore assumed for the purposes of comparison.

4.4. 1 Comparison with the CMF

The Porcupine Coherent Memory Filter is a spectrum analyzer, and does

* not produce estimates of spectral mean and width directly. Modification cf the

CMF to accommodate different pulse repetition rates and pulse widths is expensive

and time consuming. By contrast, the PPP directly produces both spectral mean

and width estimates in digital form in real time and operates over a variety ofPRRs and pulse widths. Its quantitative estimates can also be recorded or

remoted for further proceseing, plotting or display.

Recently, the CMF has been used to generate displays from which an Qb-

server can estimate mean frequency. The PPP can generate such displays with

much finer time and frequency resolution by addition of a low cost voltage-to-delay

converter. In addition, the PPP meati and standard deviation outputs can be

directly displayed on a PPI scope to provide Plan Mean Indicator or Plan Width

Indicator displays. These displays could be contoured if desired by using existing

contour generator equipment (such as digital integrators).

The following compares the resolution capabilities of the CMF and PPP

CMF PPP Unit

Range Resolution 1860 150 Meters (m)

Range Elements 192 1024

Velocity Resolution 0.5 X fo/50 0.5 fo/ 1024 i/s

Dynamic Range 30 90 dB

Input Data Rate 1.75 32 M Bits/s

Size 1.0 0.3 19" rack x61

4-14

Table 7. Proposed Design Goals for Pulse Pair Processor

DESIGN GOAL

Signal Input: 30 MHz IF, -80 dBrr to +10 d~mInp%-t Impedance 50 ohms

Reference Input: 3 0 MH z, 1 v p-p into 100 ohms

Trigger Input: Z V - 90 v p pulse0. 1 is - 100 i.±s pulse width

Analog Outputs: Mean frequency, standard deviation mean powerand bipolar video)Output Impedance - 50 ohms

Digital Outputs: Mea: -requency, standard deviation mean power

and cell count bufferedTTL level outputs 10, 8, and 10 bits respectively

Trigger Output: Nominal 15 v p into 50 ohms - coincident with1 st range cell output

Range Cells Processed: 256, 512, 768 zr 1024 nominal selectable by frontpanel switch

PRF: 300 pj~s - 5000 ppsI

Pa~ige Cell Wvidth: 0. 5 Ls, 1 ts, or 2 Ls - selectable by front panel switch

Samples Processcd. 16, 32, 64, 1.8, 256, 512, or 1024, selectable by

front panel switch

Internal Qudntization: Digital signals shall be represented by a minimumnof 8 bits at maximum input level

Dynamnic Range: Normal and quadrature bipolar video shall berepresente'd internally by a Tninimum of 7 bitcovc-r an IF input rangpe of 100 .±v rms - 1 v rms

with 1024 samnple.; processedj

AG C: The procz-ssor shall includc a dynamic AGC system

to provide inJc-ponrlent gain Kontrol for each range

cell processed.

Input lowcr: 11 E- Ac C --10"',. 60 1-iz ±2 1liz

Mechanical Configuration: IMouritable in a 19"' rack cabinet

Ambient Temperature: 0 0 '- 4t0

C o n str iicti o n: Ac co.-(I iig to best commercial practice

4-15

F.!

4.4. 2 Comparison with Analog Spectrum Analyzers

Presently available analog spectrum analyzers such as the Federal Scientific

."Ubiquitous" may, like the CMF, be used to generate a display from which an

observer can estimate spectral means and widths indirectly. Unlike the CMF and

PPP, however, such analyzers are not presently capable of processing the entire

output of a weather radar in real time, and cannot generate any real-time displays.

The following compares the resolutien of the PPP and Ubiquitous analyzer.

Ubiquitous PPP Unit

Range Resolution 12001 150 Meters (m)

Range Elements 10 1024

Velocity Resolution 0.5 Xfo/500 0.5 ? f/10Z4 m/sDynamic Range 40 90 d/B

* Input Data Rate 0.602 32 M Bits/sSizeSize 0.3 0"3 19" racO. 6'

1 Analyzer, operating with.,ut exterr&Ai sampne .. n h,-Id system

2 Mailmurn continuous

4.4.3 Fast Fourier Tr..nsior. Pt .m:hir

FFT processing to compte spectra; -ccan andz %idth may be perfermed

either by computer or by ,peciai purpotie harlware. Ccumputer processing re-

quires that the weather r;:.dar signa be reccrded and an . yvzcd in non-real t ne,

but subject to this restrif:tion can jrovi-ie r.str.nmativ aTLy desired frequency reso-

lution. The PPP is comparr3 below, to a ;ec;za Yorpost. hardware F2YT recently

developed by Raytheon for pulse, oir,-i.sco,.

-F -T _PPP Unit

Range Resoluti," IS( ISO t/.etcrs (rr,)

Ran% .lemente 2(1 Z4 10214

Velocity Resolution 0.S -If ,C'4 C. '3 ,1024

Dynamic Range 9O0 d 1Input Data Rate I

Size .19" - ]" rack x W),

Numbers in ( ) m ndiate kt , v~au .- ,r a 1' 24 Cat,, nachmfe.

4- 1 '

4.5 Conclusions

* A hardware PPP capable of computing spectral mean and width for each of

1024 1/2 -4s radar range gates appears completely feasible at this time. The

processor can be fabricated primarily from available, designed components and

is expected to occupy no more than 24" of 19" rack space. No other signal pro-

cessing device now available can approach the dynamic range and data throughput

capabilities of the PPP, except possibly the FFT, which would cost approximately

10 times more than a PPP of comparable resolution.

FTable 7 gives a summary of the design goals for the proposed signal pro-

cessor.

In the design phase of the PPP, detailed design tradeoffs of the following

items must be accomplished.

1. The dynarlic AGC loop

2. The digital function generation

3. The radar calibration system

e. !;%e" noise monitoring system

4.17'4-16

5.0 CONCLUSIONS AND RECOMMENDATIONS

The preceding sections of this report show analytically and verify by data

from controlled simulation experiments that the pulse pair technique produces

estimates of spectral mean and width that have predictable accuracy over a wide

range of non-ideal operating conditions. Performance for unimodal spectra is

insensitive to spectral shape and the degree of correlation between successive

pulse pairs; the theory stands up under ccnditions of poor signal-to-noise ratio.

From a processor standpoint, it is shown that a flexible digital pulse pair

processor capable of handling a large number of range cells simultaneously is

feasible and that it could be arranged to easily match a variety of radars and

radar operating conditions, including PRFs, pulse widths and data rates. A

theoretical basis for optimizing the waveform parameters of a radar for pulse

pair processing is also presented.

A hardware implementation of a flexible digital pulse pair processor suitable

for exploring the utility of the technique in real time processing of weather radar

data is proposed. The design goals are defined, and its performance is compared

with other contemporary signal processors.

The experimental work together with the simulation studies demonstrate

that the pulse pair technique can yield good quality estimates of mean and variance

%s1itv raar data at signal-to-noise ratios obtainable with existent Doppler weather

raaars. !-_..ware tradeoff studies as well as the comparative performance

cbwaracteristics demonstrate that the pulse pair technique is not only feasible but

B ett-d on these conclusions, the construction of a flexible experimental

ra ol such t real-time pulse pair processor is recommended for the purpose

of cullectii.g a large data base on observable conditions. Experience gained while

&_.',ductinR reiearch with the experimental model will provide the basis for defining

te spvcifications for simpler models intended for operational use. A study should

aiso be mad,: to determine the best means to display, plot, and/cr record the re-

sultinri. data 'or further analysis, reduction and evaluation.

r 5-1

The fabrication phase for the experimental model should be preceded by a

design evaluation phase, during which the final parameters for the equipment can

be established from the results of tradeoff studies.

It is further recommended that some additional analysis be performed to

provide theoretical guidelines for comparing pulse pair anilysis to spectra data

: processors. In particular, although the theory of pulse pair estimation has now

been established, no theoretical foundation exists at present for predicting the

performance of width estimates derived from spectral analyzer data; such a

theoretical foundation should be developed. Analyses are also required to

properly relate and interface the pulse pair processor with particular radars,

ancillary devices, and operational usages as a research tool.

I6=

i -2

REFERENCES

1. Hofstetter, E. M. , "Simple Estimates of Wake Velocity Parameters, "Tech. Note 1970-11, 23 April 1970, M.I.T., Lincoln Laboratory,Lexington, Mass.

2. Rummler. W. D., "Introduction of a New Estimator for VelocityL Spectral Parameters," Tech. Memo MM-68-4121-5, 3 April 1968,

Bell Telephone Laboratories, Whippany, N.J.

3. Rummler, W. D., "Accuracy of Spectral Width Estimators UsingPulse Pair Waveforms, 1

1' Tech. Memo MM-68-4121-14, 29 October1968, Bell Telephone Laboratories, Whippany, N.J.

4. Rummier, W. D., "Two-Pulse Spectral Measurements," Tech. MemoMM-68-4121-15, 7 November 1968, Bell Telephone Laboratories,Whippany, N.J.

5. "Doppler Signal Processing and Instrumentation for Modified 'Porcupine'C-Band Pulse Doppler Radar, " Final Report, May 1971, on Raytheon,Wayland, Mass. Contract No. F19628-68-0345 with AFCRL (pp 1-39through 1-49).

6. Berger, T., "Analysis of Data from Pulse Pair Feasibility Experiment,"Tech. Memo TB-108, September 9, 1970, Raytheon Co., Wayland, Mass.

7. Levin, M. J., "Generation of a Sampled Gaussian Time Series Havinga Specified Correlation Function, " IRE Trans. on Information Theory,Vol. IT-6, pp. 545-548, December 1960.

?I

APPENDIX A

ANALYSIS OF ACCURACY OF SPECTRAL PARAMETER ESTIMATESBY PULSE PAIR ESTIMATION

H.L. Groginsky

ABSTRACT

This appendix gives a mathematical analysis of the performance of a

pulse pair estimator of spectral mean and variance. It aasumes that 4

complex data (I & Q) is collected at a uniform rate from a distributed

target whose underlying spectral density function is G(f) and that G(f) 21JA

is observed only after it is mixed with white additive noise.

I. IntroductionJ

If the signal spectrum is written as

G(f) = , (1)

with R(f) real and positive for all f,

R(-f) = R(f), (Z) A

J R(f) df = 1, (3)

ff R(f) df = 0, (4)

ff2 R(f) df = I, (5)

S = signal power,

and if the data r k is given by

kn (6)

A-I

then the pulse pair estimatora of f and W are given by0

..L tan1 (7)_x}

and =1- ] (8)

where M-1

X - r k r k+ (9)n=O

M-1Y = 1 r (10)

n=O

n k nk > <I nk' >

2. Spectral Mean Estimates

Let

g(') =fG(f) aeJ2 f& r df (12)fo

then

<s k akl3= g (jT) (13)

.Sk ak~j = 0 (14)

<n k n k+j > =N 6kj (15)

<n k n k+j> 3 0 (16)

where

ak = s(kT).

and is the Kronecker delta function.

A-Z

A-Z

A

Note thatA

s (T) R d ef,wgf 7

T 2J (f-fowTj2irf T 0

j Zwrrf T.- = Se p (wT) (17)

and that p (wT) is real.

Now M-1M-1

(X> =KM rk rk+ 1 = g(T)k= 0

j ZTrf T- Se 0 p (wT) (18)

Since S and p are real,

arg(X>= 2i-f T (19)0

which shows that the nean estimator is unbiased.

Now considerU

tan Re (20)

with 0 = ZrfoT (21)

A AIf ( is to be a useful estimator, 0 6 OTwhere OT is the true mean

• phase change per pair. Writing

= e + 66, (22)

T

Re {X} = Re {g(T)}+ E 1, (23)

IMr(X} =Im {g(T)} + ( 124)

A-3

we have A

tan T -a = .. -coo e Re (g(T)) + t

- ~jaj~jj +~ 2Z ii1 (25)ReIC'J t'~mg(T) -7-()) \ Re jg(T)J

from which it follows that

Re {g(T)) E I n{g(T)) (26

Ig(Tfl

mince con eT Refs(Tfl (27)T g(T)l

Now by defining a complex error

c C1 + je (28)

we may writew w (g(T) + g*(T (r -E (T) - (T)) (E-]

4j I g(T) 2

g *(T) e g (*liT)

zj jg(T)Z

Iml 1 E)

Igj

Im (29)g (T)

Now define a new complex estimator error

A-4

E4m.

th e n ..6€ + j 6e l-6 (T) -

Thus 2 .3.

<(z >+ <(6e)> <I-gr--I > (31)

and<(6)'- <(68)2> Re<gf- >

2 (3g (T) (3

from which it follows that• ; :Vat 60 -- ( : =R < g I)-- >_

<e> F( I>Re <1)

. -1 " - >• (34)gg(T) (

Now

X = glT) + E (35)

Therefore

I..< i .I > ( (36)

and< E z Var X2

< > - (27)g (T) g (T)

I Tote that Var X 2 is a complex quantity defined by

Var X < (X -g(T))2 >

a<X > " g2 (T) (38)1)and

g(T) (38 bi)

A-5

I ;

Now we can evaluate the requisite expresslons using the assumption ofGaussian statistics for the raw input data as follows:

M-1 M-1XI 2- k& k+ r >l

M-1 M-1- = 0<rkrk+1> <r r1+1 >

M kO 9=0I

+ < rk r rk+l ri+ >

-+ ( rk re+I>(2rk+l (39)

From (13)-(16) it follows that

term 1 : Ig(T)I 2

I g ((2k)T) k

term 2 2IS +NI z =

term 3 0.

Note S/N is the per sample signal to noise ratio. For notational con-

venience we will write

gn g (nT). (41)

Then

A-6

... ...

(x2~ g1+ M- M- 2 ii.

M- M-1+ 1

g1- 2+ ". [(S+gN)2S2 ] ~)(2

M-1 21

n=-(M-1)

where

f - - (43)

Similarly

M-1 M-1 , ,S rkrk+l rr 2 +I >MZ k=O 1=0

M-1 M-1

m k=l *=1

+ <rk r,> <rk+l r +1 >

+ <rk r> <r k+l r, >

(44)

A-7

rThen (13) - (16) yields

term 1 g,

term 2- 0

1+l.k g * I - k - I 2 k-1, k+l

terin3 = (S + N) g 2 2 = k-I

g (S+N) 2 k+l

and substituting these in (44) yields

M-1 M-12 0 *< X,> g, + -- 9 g lkg--

M k= 1:

N+ g2 + g*z ]I (M-1)M z 2

M-i: g, + @ 9 n+l gn-l M

+ (2 + -- zn " (45)

N 1

Now substituting (44) and (45) in (38 bis) yields

__S+N)2_ $ 1 (gz+g 2 ) ]A} Re(S+ )S 1-VrM - N (1- - ) g1 g!

M-1 * I+gn gn+l g n-In=-(M-I) 9 l Zi 1 '~)t(6

-(46)

A- 8

For large M and small wT (46) yields approximately

a 1 + N/S)l- N 2p (T)Var{6e)~ =- I(wT)Z S p (wT) I

+ (irwT) IT

I p(wT)I

2v S [ I- p(ZwT) + (2nwT)4J

2 i p(wT)j 2

(47)

Equation (47) is a general expression for predicting the variance of thepulse pair estimate of the mean Doppler frequency accounting for the correla-

tions in the data itself. It is virtuaUy independent of actual spectral shape.

3. Width Estimator

We have shown that the spectral width estimate is given by

w - [ - LN_ :_ B (48)(Z -rr T) z LJ

If the estimate is useful, it must have small errors and hence

2W + 2w (6w)= B T + 7B

TBSBT + - 2 ..

gs a (Z.T)

where

I lr + g + (gIrn) + ) (50)

and Y= S+N+17 (51)

A-9

Note that

g l r '1+ gl Im2 Re(g* E (52)

a o that

2 A _ g1(Z=T) W" (6w) -Re . (53)Sj

It follows that

(ZwT) 4 (-WS ) Var A = Var (64) + Var(.!)- 2 cov{(S4) (--) (54)

From (32) and (33) it follows that

Var r(6 - Re <I I> + < >

+~ ~ (g+1 I S+N) 2 " 2s 1 (gz + g-2

=m + N (1-- T4 2- I gil z 91

M-1+ ( 2) ( 2 + g n l1 g n- 1 f M (n )-(M-) I g 1l + 9 g-

n g2

(55)

Following the methods used earlier we have

M M2 2 2< 2> 2SN (56)

A-10

s2 [~ '~' 2 + (S+N ) 2 S 2 -

l{~I ~I ~fM+l ()+ (S+N) 2 - 2

Simila rly

C 0v(6 < 3/S) - R (Y-S-N) >

= Re r +1 r , S2 I

S9 [ 9g *~~ t (n) + N(,+g

1 gn gn-- fM(n + N g1 + g 1

(57)Now assembling all the pieces yields

S (ZwT) 4 Var (W)

1 (1t [(+N/S)2.l]+ .i

A-11

M-1 * *

+g2 + 2 + 1- fMn,

n=-(M-l)(58)

This is the requisite expression for the variance of the width parameter

estimator.

4. Example

To reduce the equations to useful form, it is of interest to evaluate

them for a Gauss shaped spectrum. In this case

-fZR(f) = e -( (59)

so that

p(t) = e (60)

- (f-fo)20

G(f) = S/W w2 (61)

jZir £ T

gn S e 0 P (nwT)

j 2nf T - (21r nwT)2

= Se e 2 (62)

N (g2 + g- N (63)

A-12

~4!j

: gn+ l n - 1 2 "

1 n (64)

* Z-e + g-.

1 g2 (65)

- (T2 (2n 2

22- 1

-(2TrTw)2 (m+ 1/2) (mn- 1/2)

e

1 (2TrTw) 2 - (Z mTw) 2

Se (66)which leads to

A1;]

2M Var {0} 1 + NIS. 1 -2N!S I P2 (l

M-1+ 1 phi2 1 - 1P 21 fM (n))

(67)

A - 1 3

2M lasT) ) Var W -12 (1 + N/S) 2 [ +2 IP1 o ]

N 8N 2

M-1" z (2rTw)Z+ : ionia 2 + 3 IP J 2- 4 1 P 11 e ( 8n=-(M-1) (68)

These expressions may be simplified as follows (for large M and2rTw <<1

(I + N/S) - 1 - 2 N/S I 2 (1 -) = N/S (2 + N S) - 2 NIS I p2 l

SN/S [(N/S + 2 -1 P 2 )] (69)

Pnl , 2)f M (n)

00

-00

00 2? E.-(2 TrnwTl2

(ZlrwT) 2 e-00

t

2irwT J e dt

- 00ZTfwT i- (70)

A-14

I Similarly

[(I+ N/S) 2 - ] [+ 2 Zp + INi - Pl 2

r ' [i -- ~NIS (NIS + 2) (1 + 21 P,[ I) + S ,Z " N11

- N/S N/ +l+-21pl 1 2) + 2+ 21 p2 4 1p12]

=N/S [N/S (1+ 2 1 p, 1 2 + 2 (1-I pl1 2)2] (71)

1 (2 wT)Z 2

p PI ?-[1+3 1 P2 4 1I2 en

2 [w + 3 2 32-4e T)

;3 -1 2 wT)3p

3 (wT 4w. (72)

As a result it is possible to reduce the variance expressions to

2 MVar{0) e ( + ( 2 e-)

+ ZTwT (73)

L A-15

N N ( ~ ~ (Z irw T ) . Z w T ;2M (2TvwT) 2 Var (2iwT) 1 +2 [a(~ z ) + z (i. (2wT))

+ (2irwT)3 4 (74)

Figures I and 2 are essentially plots of (73) and (74) and enable the

prediction of system performance for various choices of parameters.

A-16

r1- l l

U0

-4

4'r4

I -.-- tz:~27 T .

' 44- Wi ~ - . I d *i. Z

A 17

Ica

CI

* ~~~ ~ A 18i z - - 4

APPENDIX B

MATHEMATICAL DERIVATIONS RELATING TO THEGENERATION OF SIMULATED DOPPLER RETURN

a. Second Moment:

m2 -w

I C H(s) H(-s) d

-ja

I aI

-7 H(s) H(-) das

Zrj -j Zr -j .(s + a)' s r

do(.=+a)

a .where a = 2w4

Similarly,

1 \ H(s) H(-s) (-a ) d s a3

~j 4.BI

B3-1

therefore,

MI

b, Half Power Points (w3):

a. I= IHOW) .I - 4

"w 3 a [ - '*

c. Recursion Relation:

Z' a' T e -aTY(Z) - H(Z) : -ZaT -I

•(1-2e z • ZZ Te. aT -~ aT - -ZaT Z-z

Y(Z) a' Te X(Z)Z I + 2" Y(Z) - e Y(Z) Z

which yieldsa T -eaT -Za.T

Yn = a A T e"-a x n- " 2 e - T Y n- - e .- :n-Xn-I - e n-2

d. Signal to Noise:

Signal Power = 2 H(Z) H(I/Z) -

2Trj '/

The factor 2 is due to the power from both the real and imaginary channels.

The contour of integration is the unit circle. The Z-transform is used

because a digital signal is the desired output from the simulator; therefore,

signal power should be found in terms of the Z-transform. To continue,

12 I 4 T' Z dZ

Z~rj (Z- aT)z (Z-eaz

Since there is only one (second order) pole inside the unit circle at Z e- aT

the value of S is given by:

B-2

i-4

_ w* d a 4 T'ZZifi dZ L aT J

(Z-e ..aT

Resulting in

•aT aT(3e + e

Za 4 T'aT - aT(e -e )

Noise power. N. is given by:I N -- 2 a z

[ The factor 2 results frorm. noise power in the real and imaginary channels.

Therefore the S/N ratio is given by:

4 aT aTS/N= a T' ( 3- e

[ 5N -aT(

T N (e e 1IN

I.

t

•B 3/B-4

I.

"F

APPENDIX C

I FORTRAN MAIN PROGRAMS

C-1 DOPGENC-2 PPSTAT

C-3 SPEC

C-

APPENDIX C-1

DOPOEN Program Coding

C -2

0- U-L

1- 0

ze -

V)~

-l. 0- a.. 0-ceI D _> DU-u-.co. C ) rj _j 0 -

6M.- - he I. - *

Dc 0- CL-0- - .~

U. 0z 0 0

o0 - .4 T C

WrI C- -- < - 0

-U) 0. tfs cr -W w 70a :ye D tw

t3- c- 00 D

> 0~ N- -~ iC < I

LL 1 0l f~ (1 11Zi/ C - C- it 0i C- L- u ')o77Z

Q: u u4 ID 7 A ) LL U

x - -J 90.

bov O-L)Ut I-) u - 0 L)

Cz~ S - - Z.0 00-3

x 4

0-L .eLL

'4 &L Mnu-- -

8 0 z~ 1. - i4 LN 0 w -

CL I-. If 4AX 1b C L 0 eI.-J I- Z *

0Uw4x - a x- W LU N1%* 0 L C - z Co -L lo -C

0 0- * %k- i 0 .1J (.1 0 M -Z NN

- 4 0 CL "c i 0%.I ., .-I it CC isZ 0 0 0 f-* 0 *C)LL 0 CDIt-0 0 . L-1 -0 L) 4. c

it< y-6u- - 4- n-- I N 0- & Se , P- ~l -mO

0. If It 9.I - -IfI L):)-i i ZjZ

N m 0 LWA. 0 * 1O' 0 *W'z7 L- a W - Z 0W LL Co -C

ILA z~-s a - z u t 00 --z 3- x~-.j~

C-4

r - - ~Lit

IU*1.

LU.

zz

z LLLU 0

ww

0. 0 U.

Z 0 cm U0 I.J0

0 : 0t 0 LL U.

C Ic 0 '

U' .j 0

I- (j -4 o.Z Z n a zc )0C- LU 03 9 *j co0

- 3 Z X 0 + 1 rq 41a: zf +- it -- * ZN X). VJj to 4 CC U LL 0 Za: 00 - CI-- j U

0. *- - _z Z Z *( - ,D C, 0' N.*

U LU Ii It <~J 0 oi a.~~ UILD).j J Z -Q 0of 1 -- - OIt cc z Y-311 Z 0 .i w1 LU 0 IL zwUw w

4 4 wS~ X- -d LU f . l U- - - .1 x- J,- 10- 1.-i- 0)- 0 - Z - Z If It 0 11- .. 0 - Z Q0 . 0 :- 0 0- w ix -a3- LL D. 4if~ cc 00 Z cl: cc w U.

NJ 4 0f-4 0 Q 3a f.44 -9 0 w(~N x --c WC30 0

LO 0 0- 0 0 -

-d - .4- N r- U'%U

C-5

cc rI.16

re

iii- 0

- ILU..

U. a -

1. -4U-6

0 4

-; x - U C; -

U. Z *4<U 11- a- 0sO

o . -ju c - Q

P- - UJ-t- - b- t0 .- t- - - - -6I. f Z - - f 1 f - Z p 1 4 0 a - >- If Iof 2P 1cc U t c 0 O a 0 .IL NJ- e0 0 0 b Le 0 a d

x - 04 U :z L. - .h D a >-(3 c

0 a~ 0% 0 0 - ~ 0 CI0y WY m .0- Nm *I z0 1-N ~ ~ ~ ~ ~ ~ ~ u u -S -0 4 II -Ha C

U. w.OW & N ~ w Zc-6w

PQ PGEMI.5

I-j

00

0

ON u

xr w x0 a 00JL- L.)

CC -a . W

00

C- al D. LL X

>. 4U in 4A .d C0 1 0- - -D<* Z - J~ D4Fkm.3

b.i 2 - ;0 0 N -cc0 0 -X - 1- .4 Z 2: 0 T ~ -4

al * ii -i XI 0i cc co 0 = M

X- C).- - 0--o wj 0c4 0 LL ) -i 4 &- 00 0

~0 N Q0 00000000

N- $d 0 0000000

-Y y N ('.j N M~ ~.-~ L) 0 - 00 00

C-7

APPENDIX C-2

PPSTAT Programi Coding

C-8J

U.1A~ %AF0

ce .

N 0 U

'00U.

0. U. - o

i'2 4. .e coZ o A ol o- 4_>6

z. Z %A

t- 00I -z 0 ,c

0 co0 -. -

LA CL -i>u

Qa Wi U.1 C 0

No wr ... I -: u

-4 CL~ -. ac 9Z- 0 a

I D/ CL s Z;+ l .- 0 w;

* U 4-9 0 0 C, L -u 1f D0 Vx- l-w (9 U; Z w1' Z -7- o .1

IX LZ -.1 ~ Z - N ^7 * iL ft .a. m -j a . U *- 1- z z zi - z O.4 -b.I- 1 4aY- zCxx Z 0 _ s 1 f fl 4 - a o

0 0 -coo a / QW-noZXz zw - x-

- U 0N- Ul LJ -I.- A~ 4 w> Ui'J~ -.

0L) IdZ i N

C. O" N *-Z I C- -

I4

FPSTATz

C-j

* 5

N LA IN

4' -u w-a "

- 4 0 r-10-~ + it 6 L 3- .-

4 Y.N )e beI -+ l- *; 0 z 4-j +Y -. 0~ "U '4 x*

- 4 -' 0 4i.._j- 0+ 6W 4.I 1.1. 60 -

+- 0- -U z T- beu IU%.O 0 L40 g - 4A4% .CJ -V.f +

to0-n 0 1 V) 0- to z- LA if ~ &O w 0 f it L n 1 hX..~ ofJ .- 4 umJ3 " 2 OLL *)U 11 it- IC #Ii =)"1'C4I r V

4xx% bc 0' 12 -, .1. 0 .0 zz n~ x be - xx- * - ) x w

WU)JU X .t. 0m ZA0 0 ) -'' 3- .L w f CL uD > x .U.1 N

C-10

1= 014r4I

U.13

"4 1CL IrU. o u

Z

uIL

4AJ 14N .

0 0 g --) 1-- 7 z 0 z +* + .1 N >~ > *cc

w w0 00 00-0 0 4 *mZwfi 1 l000 . A - .-

0- 1-(41 t1 e A AV )LtL 41 11If1%z Z .N '-NAV) -4-.4AL ww0 -- 4NMCCjM' N- cw " ym? 2d 1: - W ' w- t or 0,o C D0

0 . Of 0- LL- wwU UU JU JL. J I JL JU L JLI CAKA004 )u ) -̂ 61 >>>>6 A' 1'- 9 - ~ -------------------

in 0C.,~'C O ao O ~ .~--

0 -4-~ C-11

PPSTAT-4

x. w

x 0

ey NNZ 0

z z04 x 4

*> W

N um o e -l

N~O > Uf A on1;U -4 X %t r-t

-A M Ub - . IJ'

4J le . N I*--0 LAz a zO KN NIN4 49 IK 0 ui x1 -4 x z 9uJ -U . jZ =. - w Ol 4 aU

4 - ; :!04x% :-

a% -d F%

b. ^1- 7 1 NxXINX aZ- x~

w e w&b w 000 .~a oc K o

it 0 c o 00 00 CC)

l-W 0 00 ZNo -. * 0W 00-

ty o N4 UN w 00'Jh~.e 0 0j

a ~ aC-1I

0 ~ 0 - a~-4OaC ~ f~0 z~A

APPENDIX C

E SPEC Program Coding

I.

G-1

i.4

C-1-

PROGRAM SPEC TRACE CDC 6600 FTN V3*0-P)2704 OPflaO 03/2

PROGRAM S"EC(INPUT#OUTPurTAPE5=INPUrTAPEG=ouTPUrTAPESTAPE9)

COMMENr--SYSTE'M SU**LrE0 SUBROUTINES REQUIREU--OQPLOTqENDlPLTCOM~Mt--UjS qt;2.j tFfl StIRROUITNrS QE"~n-nrgSfFR

c THE EflLLOWIlNfl COfNT~ttL CARfl ARE mP~flFf WHEN 12 0S ARE~ LOF fl51C FTN(A)r ATTArHP-:MN,PxIJ7I flTq,MO=Ij,C SETCOREe

- LaADPF'd.C REQUESTttAPE3).

r RPOIIFqTTA&AruTIacwIalnTmfi Vc REnUEST,TAPES,Hl. (RPKO$/NORING)

C UNLOA,i&PE3).

c U N. OAO,.T A'Egs

REAL MEAN

DIMENSION Z(t02'.),A(i024) ,X(1024),Y(1536)T flMFNCZTl n-Sf51S '-aa

IF=-14= flJ4 . .:I:IO _ _ _ _

N=1024 ~ fA _ __ _ __ _ _ __ _ _ __ _ _

.. FieLvi) O i~ A -_ __ __ __ __ _

1.-. O..

-- -. _ __ _ __ _ __ _ _-I

REWNDN 8__ __ _ __ __ _ __ _ __ __ _ __ _

RE21 TFA T00- OK,; Vyfop 100 11", t[,r 1 !oil i Iu L IMT2 ,IW 9IPLOT9191

---------------- L ia41 -

L R I. EI I 1 pti +

12'39Z(oo=9124

PROIRAM SPEC TRACE flOC 6600 FTN V390-P270, OPTz0 03/2

CALL FORT (Z, 4, SqIFSIFERR)

30 Y-(I) =REAL Q (1)e42AIMAGZfI~ ______ __

r ~'YWAX=Y(t)D0 40 11ip10?4

k- 40 IF(YMAXsLTaYfI))YNAXSY(I)

_____________F(TPLOTa0,O).Ifl2L__ TO_______75____

00 50 1:1,10?450 -Y(I)=Y(I)/ymkx

FFF=-i. p_ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _

00 60 I=10102.- ~FF~F=FFF4..

XII) :FF760 YQI10.*AL0O;1OY(T))

CALL fQL0IT(L.A8,XPYqN1 16 1 1.12H'fl [fiG (MAGI. 2,fl,fliKPLOT1l

00 70 Jz1,102470 Y(J)=i lP*(Y(J)/i8.)*YMAX75 CONTINUT-

IF(IAVG*EQo)S0 TO 200

ui0 76 II'41,BL.076, AIIG=AVG+YtI)

AVG=AVGI100.

00 77 I:1,10?L.________ 77 (... A I) Y (r) -AVrv______________________

200 IF(IqIAS.E-,IGo ro 230

00 210 1=1,1024

Ki1i02L461-I210 Y(JiY((I)_________________

00 220 1=1,13-IFTS JP=TSHFT+1-T

L * K2=1024.1sHFr,1-1_________ 220 Y(J?):Y(C)__________

230 04ONTINUE

(3 Ifn T= , f

Y MTNV=YN~iX/( (1 ff #T4R* l. I~ II

U _ _ _ _ _ _ _-._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _C _15

hLI

:I'ROGRAN SPEC TRACE COC 600 FYN V3*0-P2704 OPT0 013/2

SIzO.

FNzLINt-t

00 80 JzLIMITjLI14IT2

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IF(KPL0ToEQ.1) CALL ENDPLT

REWINO 9C~AII FVT

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2000 FORMAT(t4l)

C-16

APPENDIX D

FORTRAN SUBROUTINES

D-i FORT

D-2 PAIR,.4

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APPENDIX D- I

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APPENDIX E

SPECTRAL ANALYSIS PLOTS OF TYPICAL SEQUENCES OF

SIMULATED DOPPLER RADAR TARGET RETURN

The radar PRF is 3300 PPS in all cases. Mean, width, and single pulse

S/N parameters are as listed below:

Mean 1 Width (a) Single-Pulse S/NCase Hz Hz d]B__

1 300 78 02 300 78 53 300 78 154 300 78 m5 600 156 06 600 156 57 600 156 158 600 1569 900 234 0

10 900 234 511 900 234 151Z 900 234

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