OPTIMUM TWO-DIMENSIONAL TRANSMIT-RECEIVER DESIGN

Hui Sheng, Kaizhi Wang, Xingzhao Liu

Department of Electronic Engineering, Shanghai Jiao Tong University

ABSTRACT

In this paper, a theory of two-dimensional transmit-receiver

design is presented. Only deterministic targets and random

clutters and noises are considered in this model, and we reach

two-dimensional optimization based on maximization signal-

to-interference-and-noise ratio(SINR). New result focuses on

extending the previous one-dimensional waveform design to

two-dimensional one, in order to satisfy some practical appli-

cation like synthetic aperture radar(SAR). Both the theoretic

derivation and simulation result proves that 2D radar imaging

can gain high SINR with proper transmit-receiver design.

Index Terms— 2D, transmit-receiver design, SINR,SAR.

1. INTRODUCTION

Shortly after the concept of waveform design came to pub-

lic, researchers realized its importance to improve the radar

system performance. Since its good application potential,

scholars pay increasing attention on ”transmit-centric” close-

loop radar system design rather than ”receiver-centric” adap-

tive signal and knowledge based algorithms. These transmit

waveforms can be determined by prior knowledge gained by

previous received echoes.

Since now, the critical criterion of one-dimensional radar

performance lies on signal-to-noise ratio(SNR) and mutual

information(MI).Back to 1967, Dunbridge,B had already paid

his attention to signal design theory concerning the problem

of determining transmitter signal waveforms for maximizing

the probability of correct reception and mentioned SNR as a

metric for better system performance in [1]. Then, in [2], [3],

SNR appeared as the essential evidence to prove the superi-

ority of the designed system or algorithm in solving the prob-

lem of clutter suppression and spectral estimation. Real clas-

sical issue considering approaching the optimum waveform

for maximum SNR and MI under the interference from both

clutter and noise was written by Bell in [4], [5]. When the

problem tended to matching a known target response in the

signal-dependent interference, Pillai talked about it in [6]. In

[7], [8], the method presented in [6] was applied into target-

recognition application. There are also some breakthroughs

in signal design for clutter rejection and waveform selection

from the perspective of target classification in [9], [10].

Bell and Goodman attributed a lot to one-dimensional

waveform design based on the measure of both SNR and MI

in [5],[11]. Firstly,They provide us with two different scenes

including situation of noise only and that of clutter plus noise

one. What’s more, based on mutual information(MI) and

signal-to-noise(SNR), four different transmit-received pairs

are developed and tested. Finally, with mathematic deriva-

tion, Goodman’s paper shows the connection between SNR-

based results and MI-based ones.By now, one-dimensional

waveform design for radar detection and recognition has been

well exploited.

But then, the idea of extending the one-dimensional radar

waveform design to a two-dimensional processing for syn-

thetic aperture radar (SAR) is considered and investigated.

Along with clutter and noise, deterministic target is presented

in analysis in order to reach the optimum-designed waveform

for established targets. Through a transfer function repre-

senting the characteristics and feathers of known targets, a

designed transmit waveform and the related matched filter

make echoes resonate at a particular azimuth-domain time

and range-domain time, thus the receiving data maximizes

at a particular position which helps a lot in target detection.

Therefore, we regard signal-to-noise ratio (SNR) as a measure

to judge the system performance and believe that maximum

SNR leads to the optimal state of radar imaging system.Our

results mainly focus on 2D SNR-based transmit-receiver de-

sign for deterministic target in signal-dependent interference.

This paper is organized as followed: Section II describes

the transmit channel’s model used for a deterministic tar-

get in signal-dependent interference and its two-dimensional

transmit-receiver design procedure for maximum SINR out-

put. Section III proposes the specific theoretic derivation to

approach an optimal result and offers mathematic expressions

of designed transmit waveform, corresponding matched filter

and value of final SINR. Section IV includes many simulation

results to support the former theory.

2. SYSTEM MODEL

Like the most mature technologies used in 2D radar image,

we segment the transmit-receiver design into two processes:

the range-domain waveform design and the azimuth-domain

waveform design.

From Fig.1, we obtain the main structure of two-dimensional

imaging model. Here, h(t, τ) stands for the transfer func-

4610978-1-4673-1159-5/12/$31.00 ©2012 IEEE IGARSS 2012

Fig. 1. Block Diagram of Two-dimensional System

tion of extended target. xr(t, τ) and rr(t, τ) represents the

range-domain transmit-receiver pair, in addition, xa(t, τ)and ra(t, τ) represents the azimuth-domain transmit-receiver

pair. Just like the case in most papers, n(t, τ) is regarded as

a complex-valued, zero-mean channel noise process. And,

we set c(t, τ) to be a complex-valued, zero-mean Gaussian

random process representing an interference component. In

fact, two-dimensional transmit-receiver designs lead to two

transmit signals in both range domain and azimuth domain,

which obviously goes against physical truth. However, using

mathematic optimization, we can replace the two transmit

signals in dotted line box with one in future work.

3. PROBLEM FORMULATION

Unlike the model only under the interference of Gaussian

noise, the signal-dependent influence from clutter should be

taken into consideration. As shown in Fig.1, two-dimensional

waveform design means deviation in both dimensions. Just

like some classical methods like Range-Doppler algorithm,

we create a range-domain transmit-receiver pair at first and

then a range-domain one.

3.1. Range-domain Deviation

As seen in Fig.1, the signal yr(t, τ) defines the output af-

ter the range-domain receive filter. And its signal compo-

nent ys,r(t, τ) and signal-dependent interference component

yn,r(t, τ) are defined by

ys,r(t, τ) = rr(t, τ) ∗τxr(t, τ) ∗

τh(t, τ) (1)

and

yn,r(t, τ) = rr(t, τ) ∗τxr(t, τ) ∗

τc(t, τ) + n(t, τ) (2)

According to the definition, we obtain the output signal-

to-interference-plus-noise ratio(SINR) at t0,τ0:

(SINR)t0,τ0 =|ys,r(t0, τ0)|2

E[|yn,r(t0, τ0)|2] (3)

And the waveform energy constraint in range domain can be

written as: ∫w

|Xr(t0, fτ )|2dfτ ≤ Eazimuth (4)

which means that each transmit pulse has an energy con-

straint. Now, we express SINR into range frequency domain,

and it can be written as:

(SINR)t0,τ0 =| ∫∞−∞Rr(t0, fτ )H(t0, fτ )Xr(t0, fτ )e

j2πfττdτ |2∫∞−∞ |Rr(t0, fτ )|2L(t0, fτ )dfτ

(5)

in which L(t0, fτ ) = |Xr(t0, fτ )|2Scc(t0, fτ ) + Snn(t0, fτ )To reach the optimal result, we put Schwarz’s inequality

into practice. Therefore, the maximum SINR would be:

(SINR)t0,τ0 =

∫ ∞

−∞

|H(t0, fτ )Xr(t0, fτ )|2L(t0, fτ )

dfτ (6)

And its relevant matched filter is

Rr(t, fτ ) =[kH(t, fτ )Xr(t, fτ )e

j2πfττ0 ]∗

Scc(t, fτ )|Xr(t, fτ )|2 + Snn(t, fτ )(7)

We recognize that the transmit signal has an energy con-

straint expressed as (4), and the integration kernel of (6) is

concave. As a result, based on Lagrangian multiplier tech-

nique, the optimal range-domain input signal arrives at:

|Xr(t, fτ )|2 = max[0,−Snn(t, fτ )

Scc(t, fτ )+A

√Snn(t, fτ )|H(t, fτ )|2

S2nn(t, fτ )

]

(8)

in which A is a constant received by putting (8) into (4).

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3.2. Azimuth-domain Deviation

The former section dictates the process to approach the range-

domain optimization. However, it brings this system with new

transmit waveform and receive filter. Thus, before extend-

ing the range-domain waveform design to an azimuth-domain

one, we should redefine the parameters in derivation.

Ha(t, fr) = Rr(t, fr)H(t, fr)Xr(t, fr) (9)

Ca(t, fτ ) = Rr(t, fτ )C(t, fτ )Xr(t, fτ ) (10)

Na(t, fτ ) = Rr(t, fτ )N(t, fτ ) (11)

Also, here comes an azimuth-domain energy constraint and it

means that the total energy of particular range-domain time is

limited during a finite azimuth-domain transmit period.∫W

|Xa(ft, τ0)|2dft ≤ Erange (12)

Similarly, if and only if the matched filter has the form of

Ra(ft, τ0) =[kHa(ft, τ0)Xa(ft, τ0)e

j2πftt0 ]∗

|Xa(ft, τ0)|2Sacc(ft, τ0) + Sa

nn(ft, τ0)

(13)

the SINR achieves its maximization:

(SNR)t0,τ0 =

∫ ∞

−∞

|Ha(ft, τ0)Xa(ft, τ0)|2Sa

nn(ft, τ0)dft (14)

To achieve receive signal, we use the similar theory in

previous section to gain the optimum waveform, and its ex-

pression comes as:

|Xa(ft, τ0)|2 = max[0,

√|Ha(ft,τ0)|2Sa

nn(ft,τ0)

Sacc

(ft,τ0)

× (Aa −√

Sann

(ft,τ0)

|Ha(ft,τ0)|2 )](15)

the same as that of range domain, Aa is achieved by replacing

(15) into (12).

4. RESULT

In this section, we apply the transmit-receiver design into

scene simulation and investigate the performance of SNR-

based waveform design under the impact from signal-dependent

interference. For a deterministic target, it is easy to calculate

the transfer function and power spectrum. Therefore, with the

help of clutter PSD and noise PSD, we can decide value of

SINR and shape of transmit waveform. However, these three

spectra are not the only parameters that matters to designed

transmit signal. For a given target power spectrum, clutter

PSD and noise PSD, transmit waveforms and corresponding

SINR yet differ with changes in energy constraints. In order

to express this system’s performance, we generate two zero-

mean complex-valued random matrixes using fixed variation

to simulate two-dimensional clutter and noise and model

Fig. 2. Receiving Signal After Range-domain Design

Fig. 3. SINR After Range-domain Design

eight-point rectangle target’s power spectrum and transfer

function.

As expected, Fig.2 shows the transmit-receiver design

in range-domain yields a similar echo compared with that

of range compression in Range-Doppler algorithm. Each

recorded pulse arrives at a peak value through corresponding

matched filter. At the same time, the radar system experiences

a high SINR unless the energy constraint is too small to get it.

Common sense tells us that the greater transmit energy is, the

higher SINR will be, and data simulation never go against the

law. Just like the result in Fig.3, average SINR for each row

in receiving signal increases tremendously with the boom-

ing of energy constraints. The SINR after one-dimensional

transmit-receiver design performs slightly worse than that of

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two-dimensional transmit-receiver designs. Final SINR af-

ter two-dimensional optimization process presented in Fig.4

provides both high value in SINR and a quick increment with

energy constraints.

Fig. 4. Final SINR After Two-dimensional Designs

5. SUMMARY AND CONCLUSIONS

A comprehensive theory of two-dimensional transmit-receiver

design matched to deterministic targets under signal-dependent

interference has been analyzed. A detailed deviation about

this process has been presented and challenged by a real

scene simulation. Based on eight-point target’s transfer, clut-

ter PSD and noise PSD, we calculate optimal waveform and

its matched filter to maximize SINR under this scene. We

evaluate receiving echoes by their related SINR, and achieve

acceptable good consequences.

Though the above example shows a good application po-

tential of this model, lots of works yet needs be done. In

future experiment, multiple metrics can be put into practice

like mutual information between target and receiving signal

and target recognition possibility for a finite group of deter-

ministic targets.

6. REFERENCES

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[3] Vachon, P.W., West,J.C, “Spectral estimation techniques

for multilook SAR images of ocean waves,” IEEETransaction on Geoscience and Remote sensing

[4] Mark R. Bell, “Information theory and radar: Mutual in-

formation and the design and analysis of radar waveform

and systems,” Ph.D. dissertation, California Institute ofTechnology, Sept, 1998.

[5] Mark R. Bell, “Information theory and radar wave-

form design,” IEEE Transactions on Information The-ory, Sept, 1993.

[6] Pillai, S., Oh, H., Youla, D., and Guerci, J., “Opti-

mum transmit-receiver design in the presence of signal-

dependent interference and channel noise,” IEEE Trans-actions on Information Theory, Mar, 2000.

[7] Guerci, J. R. and Pillai, S. U., “Theory and application

of optimum transmit-receive radar,” Proceedings of theIEEE 2000 International Radar Conference, May, 2000.

[8] Garren, D. A., Osborn, M. K., Odom, A. C., Gold-

stein, J. S., and Guerci, J. R., “Enhanced target detection

and identification via optimised radar transmission pulse

shape,” IEE Proceedings-Radar, Sonar and Navigation,

June, 2001.

[9] DeLong, J. D. F. and Hofstetter, E. M., “Enhanced

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[10] Sowelam, S. M. and Tewfik, A. H., “Waveform selection

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[11] Ric A. Romero, Junhyeong Bae, Nathan A. Goodman,

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