communication systems, 5e

Communication Systems, 5e

Chapter 3: Signal Transmission and Filtering

A. Bruce Carlson

Paul B. Crilly© 2010 The McGraw-Hill Companies

Chapter 3: Signal Transmission and Filtering

• Response of LTI systems

• Signal distortion

• Transmission Loss and decibels

• Filters and filtering

• Quadrature filters and Hilbert transform

• Correlation and spectral density

© 2010 The McGraw-Hill Companies

Power Gain

• Gain is defined as the power out divided by the power in

3

2

in

out 0HP

Pg

• Cascaded gain is the gain of the receiver chain

in

out10dB P

Plog10g

321 GGGG

1in

3out

3in

3out

2in

2out

1in

1out

P

P

P

P

P

P

P

PG

dB3dB2dB1dB GGGG

Noise Figure (App. A p. 828)

• The Noise Figure Described the expected output noise power reflected to the input of the device. Thus, it defines the additional “equivalent input” noise contribution. (p. 840)

4

0

device

0

device0

N

N1

N

NNF

• Cascaded noise figure is effected by the gain

21

3

1

21 GG

1F

G

1FFF

Receiver Stages Gain/NF Example

• T/R Switch

• RF Filter and Amplification

• Zero-IF Quadrature Downconversion

• Anti-Aliasing Filters

• ADC

5

LNAFilter

ADC

ADC

Quad.Hybrid

LPF10 MHz

LPF10 MHz

InPhase

QuadPhase

T/R Ctrl LO

1 3a 4 5a 6

RF Input RF Filter and LNAQuadrature

DownconversionDigitization with

Anti-Aliasing Filter

AMP

3b

Attn

Attn

5b

RFT/RSW

Xmt

2

• Isolate Transmit and Receive (TDD comm)

• Reduce Interference BW and Limit NF

• Gain Adjustment

• Downconvert

• ADC Input Signal Conditioning

Receiver Cascaded Analysis

6

LNAFilter

ADC

ADC

Quad.Hybrid

LPF10 MHz

LPF10 MHz

InPhase

QuadPhase

T/R Ctrl LO

1 3a 4 5a 6

RF Input RF Filter and LNAQuadrature

DownconversionDigitization with

Anti-Aliasing Filter

AMP

3b

Attn

Attn

5b

RFT/RSW

Xmt

2

Stage 1 2 3a 3b 4 5a 5b

Gain (dB) -0.90 -0.50 20.00 20.00 14.00 -6.00 -3.00Noise FIgure (dB) 0.90 0.50 2.90 5.30 35.00 6.00 3.00

Noise Figure (linear) 1.23 1.12 1.95 3.39 3162.28 3.98 2.00

Total Gain (dB) -0.90 -1.40 18.60 38.60 52.60 46.60 43.60Total Gain (linear) 0.81 0.72 72.44 7.2E+03 1.8E+05 4.6E+04 2.3E+04

Total NF (linear) 1.230 1.380 2.692 2.725 3.161 3.161 3.161Total NF (dB) 0.90 1.40 4.30 4.35 5.00 5.00 5.00

Receiver Sensitivity (Advanced)

• Receiver Noise Floor– kTB (p. 413)

– Cascaded NF

• Min. Detectable Signal– Detection Threshold

• ADC Considerations– Max. Signal

– ADC SNR

• Dynamic Range– Instantaneous

– Spur Free Dynamic Range

7

-174 dBm/Hz

-104.00 dBm

-33.60 dBm

-103.60 dBm

10 MHz - 70.00dB

Receiver Gain43.6 dB

AD

C S

NR

70 dB

-99.00 dBm

NF 5.00 dB

+10.0 dBm

Receiver NoiseFloor

-91.00 dBm

DT 8 dB

Min. DetectableSignal (MDS)

Maximum InputSignal Power

Thermal NoiseFloor

Maximum SignalPower to ADC

Received SignalPower

Dynamic Range

kT Noise Floor

57.4 dB

ECE 6640 8

RF Propagation & Range Equations

• The power density in a sphere from a “point source” antenna (surface area of a sphere)

• Receiving power collected by an antenna (using the effective area of the receiving antenna so that p(d) can be collected)

sphereofarea

P

r

Prp tt

__4 2

𝑃 = 𝑝 𝑟 ⋅ 𝐴 =𝑃 ⋅ 𝐴

4𝜋 ⋅ 𝑟densityfluxpowerincident

extractedpowertotalAer

• Effective Antenna Area

𝐴 =𝐺 ⋅ 𝜆

4𝜋𝑃 = 𝑝 𝑟 ⋅ 𝐴 =

𝑃 ⋅ 𝐺 ⋅ 𝜆

4𝜋 ⋅ 𝑟

9

Free-Space Path Signal Loss

• As an RF signal propagates, there is path loss.

tPtG rG

f

R

rP

22

c

Rf4R4L

• As shown above

22

2

2

44 fR

cGGP

R

GGP

L

GGPP rt

trt

trt

tr

fc

Note

10

1st Order RF Range Estimate

• Friis Transmission Formula– Direct, line-of-sight range-power equation

– No real-world effects taken into account

where: rP is the received (or transmitted)

tG is the effective transmitter (or receiver) antenna gain

R is the distance between the transmitter and receiver, and is the wavelength f is the frequency

11

System Range

• Maximum Range (Pr is the receiver sensitivity - MDS)

dBmPt

dBmGt

dBmGr

dBmPr

m0 mR1

tPtG rG

f

R

rP

dBPdBGRfdBc

dBdBGdBP rrtt

2

42

Note: as f increases range decreases

12

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

6 GHz4 GHz

dB13.1997e8.9048e3

6e369e64

c

Rf4L 2

22

u

dB60.1957e18.6038e3

6e369e44

c

Rf4L 2

22

d

Satellite relay system Ex. 3.3-1 (1 of 2)

Path Losses

13


dBW1.89dB20dB1.199dB55dBW35P rcv_sat

dBWdBdBdBdBWPout 6.110516.1951618

22

rtt

rttr

R4

GGP

L

GGPP

dB1.107dBW1.89dBW18gamp

Satellite relay system (2 of 2)

Pt=35 dBW

Pt=18 dBW

Error in 4th ed.

Power Received

Satellite Gain

RF Interference/Jamming

• What happens if interference is stronger than the signal of interest?

• Jamming …

– Cellular telephone: TX 824-849 and RX 869-894 MHz

– Pico-cell transmitter: power +10 dBm, Gt=+3 dB, Rt=100 ft.

– Jammer: 1 mW→0 dBm, Gj=0 dB

14

Jam cell phone if less than 22.3 ft. away ….

Example Commercial JammerManufacturer Specifications

• Affected Frequency Ranges:– CDMA/GSM: 850 to 960MHz

– DCS/PCS:1805 to 1990MHz

– 3G:2,110 to 2,170MHz

– 4G LTE:725-770MHz

– 4G Wimax:2345-2400MHz or 2620-2690MHz

– WiFi:2400-2500MHz

• Total output power: 3W

• Jamming range: up to 20m, the jamming radius still depends on the strength of signal in given area

• Power supply: 50 to 60Hz, 100 to 240V AC

• With AC adapter (AC100-240V-DC12V), 4000mA/H battery

• Dimension:126 x 76 x 35mm not including antenna(roughly 5” x 3” x 1.5”)

• Full set weight:0.65kg15

These are not legal in the USby FCC Regulations.

~$256 from China

Filters and filtering

• Ideal filters

• Bandlimiting and timelimiting

• Real filters

• Pulse response and risetime


17

The Ideal Filter

• To receive a signal without distortion, only changes in the magnitude and/or a pure time delay are allowed. 0ttxKty

02exp tffXKfY

• The transfer function is

0tf2expKfH

• A constant gain with a linear phase KfH 0tf2f

18


(b) Bandpass Filter

Ideal filters

(a) Lowpass Filter

19

Ideal LPF Filter

• For no distortion, the ideal filter should have the following properties:

fjexpfHfH

u

u

fffor,0

fffor,KfH

u

u0

fffor,arbitrary

fffor,tf2f

• The impulse response for an ideal LPF is

u

u

u

u

f

f

0

f

f

0

dfttf2jexpKth

dftf2jexptf2jexpKth

20

Ideal Filter (2)

0uu

0

0u

0

0u

0

0u

f

f0

0

f

f

0

ttf2sincKf2th

tt2

ttf2sinK2th

tt2j

ttf2jexpK

tt2j

ttf2jexpKth

tt2j

ttf2jexpKth

dfttf2jexpKth

u

u

u

u

• Continuing

• The sinc function– A non-causal filter

Notes and figures are based on or taken from materials in the course textbook: Bernard Sklar, Digital Communications, Fundamentals and Applications,

Prentice Hall PTR, Second Edition, 2001.

21


(a) Transfer function (b) Impulse response

For a causal approximation, eliminate negative time from h(t).

t

dttB2sincKB2th

Ideal lowpass filter

22

Band-limiting and Time-limiting

• Band-limiting and Time-limiting are mutually exclusive!!– Easy to show with rect <==>sinc transform pair

• The engineering solution– Negligibly small can be ignored

– Values less than a defined value are ignored

– The non-ideal design is used and, if it isn’t good enough, a smaller threshold to ignore value is set (repeating until the desired result achieved)

Filter types

• Low pass: rejects high frequencies

• High pass: rejects low frequencies

• Band pass: rejects frequencies above and below some limits

• Notch: rejects one frequency

• Band reject: rejects frequencies between two limits


24

Real Filters: Terminology

• Passband– Frequencies where signal is

meant to pass

• Stopband– Frequencies where some defined

level of attenuation is desired

• Transition-band– The transitions frequencies

between the passband and the stopband

• Filter Shape Factor– The ratio of the stopband

bandwidth to the passband bandwidth

PB

SB

BW

BWSF

PBBW

SBBW

25


Typical amplitude ratio of a real bandpass filter

Figure 3.4-3

Real Bandpass Filter

The -3 dB or half-power bandwidth is shown

26

Bandwidths that are Used



Note: Sinc freq. domain is appropriate for digital symbols

27

Bandwidth Definitions

(a) Half-power bandwidth. This is the interval between frequencies at which Gx(f ) has dropped to half-power, or 3 dB below the peak value.

(b) Equivalent rectangular or noise equivalent bandwidth. The noise equivalent bandwidth was originally conceived to permit rapid computation of output noise power from an amplifier with a wideband noise input; the concept can similarly be applied to a signal bandwidth. The noise equivalent bandwidth WN of a signal is defined by the relationship WN = Px/Gx(fc), where Px is the total signal power over all frequencies and Gx(fc) is the value of Gx(f ) at the band center (assumed to be the maximum value over all frequencies).

(c) Null-to-null bandwidth. The most popular measure of bandwidth for digital communications is the width of the main spectral lobe, where most of the signal power is contained. This criterion lacks complete generality since some modulation formats lack well-defined lobes.



28

Bandwidth Definitions (2)

(d) Fractional power containment bandwidth. This bandwidth criterion has been adopted by the Federal Communications Commission (FCC Rules and Regulations Section 2.202) and states that the occupied bandwidth is the band that leaves exactly 0.5% of the signal power above the upper band limit and exactly 0.5% of the signal power below the lower band limit. Thus 99% of the signal power is inside the occupied band.

(e) Bounded power spectral density. A popular method of specifying bandwidth is to state that everywhere outside the specified band, Gx(f ) must have fallen at least to a certain stated level below that found at the band center. Typical attenuation levels might be 35 or 50 dB.

(f) Absolute bandwidth. This is the interval between frequencies, outside of which the spectrum is zero. This is a useful abstraction. However, for all realizable waveforms, the absolute bandwidth is infinite.



29

Selecting RF/IF Filter Types Based on Shape Factors

Vectron International, General technical information, http://www.vectron.com/products/saw/pdf_mqf/TECHINFO.pdf

Ban

dwid

th (

kHz)

Center Freq. (MHz)

Filter Design Notes

• Butterworth Filter Definition– Poles on the unit circle

– Frequency Scaling

• Active Audio Filter Implementations– One Pole Op Amp design

– Sallen-Key LPF Active Filter• 2-pole filter implementation per Op Amp (non-inverting)

– Multiple Feedback (MFB) Circuit Lowpass Filter• Alternate 2-pole design (inverting)

– Cascading stages for higher order filters• Texas Instruments, Active Low-Pass Filter Design, Application Report,

SLOA049B

• Passive LC filter– T and Pi Filters

– Buy it from Coilcraft30

31

Butterworth Low Pass Filter

• Maximally Flat, Smooth Roll-off, Constant 3dB point for all orders

n2

0ww1

1jwHjwH

n2

0

n

n2

0

n2

n2

0

2

ws11

1

wsj1

1

wjs1

1sH

M.E. Van Valkenburg, Analog Filter Design, Oxford Univ. Press, 1982. SBN: 0-19-510734-9

10-1

100

101

102

103

-120

-100

-80

-60

-40

-20

0

Butterworth Filter Family

Frequency (normalized)

Att

enua

tion

(dB

)

1st order

2nd order

3rd order4th order

5th order

32

Butterworth Filter PSD

10-1

100

101

102

103

-120

-100

-80

-60

-40

-20

0

Butterworth Filter Family


Att

enua

tion

(dB

)

1st order

2nd order

3rd order4th order

5th order

33

Butterworth Filter PSD (2)

10-1

100

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1Butterworth Filter Family


Att

enua

tion

(dB

)

1st order

2nd order

3rd order4th order

5th order

34

Matlab Script: ButterPlot.m%% Butterworth filter plots%

freqrange = logspace(-1,3,1024)';wrange=2*pi*freqrange;

[B1,A1]=butter(1,2*pi,'s');[H1] = freqs(B1,A1,wrange);





Hmatrix=[H1 H2 H3 H4 H5];

figure(1)semilogx(freqrange,dB(psdg(Hmatrix)));gridtitle('Butterworth Filter Family');xlabel('Frequency (normalized)');ylabel('Attenuation (dB)');legend('1st order','2nd order','3rd order','4th order','5th order','Location','SouthWest');axis([10^-1 10^3 -120 3]);

figure(2)semilogx(freqrange,dB(psdg(Hmatrix)));gridtitle('Butterworth Filter Family');xlabel('Frequency (normalized)');ylabel('Attenuation (dB)');legend('1st order','2nd order','3rd order','4th order','5th order','Location','SouthWest');axis([10^-1 3 -9 1]);

35

Chebyshev Type IFilter PSD (Cheby1Plot.m)

10-1

100

101

102

103

-120

-100

-80

-60

-40

-20

0

Chebyshev Type I Filter Family


Att

enua

tion

(dB

)

1st order

2nd order

3rd order4th order

5th order

36

Chebyshev Type IFilter PSD (2)

10-1

100

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1Chebyshev Type I Filter Family


Att

enua

tion

(dB

)

1st order

2nd order

3rd order4th order

5th order

37

Available MATLAB Filters(Signal Proc. TB)

• Analog or Digital– Butterworth

– Chebyshev Type I

– Chebyshev Type II

– Elliptic or Cauer

• Special – Linear Phase– Bessel

• Digital Windows/Filters– barthannwin

– bartlett

– blackman

– blackmanharris

– bohmanwin

– chebwin

– flattopwin

– gausswin

– hamming

– hann

– kaiser

– nuttallwin

– parzenwin

– rectwin

– triang

– tukeywin

38

Analog Lowpass Filter Design

• Butterworth – Monotonic Decreasing

Magnitude– All poles

• Chebyshev (Cheby Type 1) – Passband Ripple– All poles

• Inverse Chebyshev (Cheby Type2) – Stopband Ripple

• Elliptical or Cauer Filter – Passband Ripple– Stopband Ripple

• Bessel Filter– Linear Phase Maximized

101

102

103

104

105

106

107

-160

-140

-120

-100

-80

-60

-40

-20

0

20Filter Comparison: Magnitude

Butter

Bessel

Cheby1Cheby2

Ellip

Spec

Butterworth Order PredicationFilter Order = 4 3dB BW = 1778.28 Hz

Bessel Order PredicationFilter Order = 4 3dB BW = 1778.28 Hz

Chebyshev Type I Order PredicationFilter Order = 3 3dB BW = 1000 Hz

Chebyshev Type II Order PredicationFilter Order = 3 3dB BW = 8972.85 Hz

Elliptical or Cauer Order PredicationFilter Order = 3 3dB BW = 1000 Hz

39

Matlab Filter Generation (1)

• Passband

• Stopband

• Passband Ripple (dB)

• Stopband Ripple (dB)

• fpass=1000;

• fstop=10000;

• AlphaPass=0.5;

• AlphaStop=60;

• w#### = 2 x pi x f####

[Nbutter, Wnbutter] = buttord(wpass, wstop, AlphaPass, AlphaStop,'s');

[Ncheby1, Wncheby1] = cheb1ord(wpass, wstop, AlphaPass, AlphaStop,'s');

[Ncheby2, Wncheby2] = cheb2ord(wpass, wstop, AlphaPass, AlphaStop,'s');

[Nellip, Wnellip] = ellipord(wpass, wstop, AlphaPass, AlphaStop,'s');

Filter Order and other design parameters

40


Filter Transfer Function Generation

[numbutter,denbutter] = butter(Nbutter,Wnbutter,'low','s')

[numbesself,denbesself] = besself(Nbutter,Wnbutter)

[numcheby1,dencheby1] = cheby1(Ncheby1,AlphaPass, Wncheby1,'low','s')

[numcheby2,dencheby2] = cheby2(Ncheby2,AlphaStop, Wncheby2,'low','s')

[numellip,denellip] = ellip(Nellip,AlphaPass,AlphaStop, Wnellip,'low','s');

Spectral Response from Transfer Function[Specbutter]=freqs(numbutter,denbutter,wspace);

[Specbesself]=freqs(numbesself,denbesself,wspace);

[Speccheby1]=freqs(numcheby1,dencheby1,wspace);

[Speccheby2]=freqs(numcheby2,dencheby2,wspace);

[Specellip]=freqs(numellip,denellip,wspace);

41


figure(11)

semilogx((fspace),dB(psdg([Specbutter Specbesself Speccheby1 Speccheby2 Specellip])), ...

specfreq1,specmag1,'k-.',specfreq2,specmag2,'k-.',specfreq3,specmag3,'k-.');

title('Filter Comparison: Magnitude')

legend('Butter','Bessel','Cheby1','Cheby2','Ellip','Spec')

101

102

103

104

105

106

107

-150

-100

-50

0

Filter Comparison: Magnitude

Butter

Bessel

Cheby1Cheby2

Ellip

Spec

42

Matlab Code

• AnalogFilterCompare.m

• Additional Resources– Dr. Bazuin’s Filter Notes – on web site

– Dr. Bazuin’s Draft Filter Manual – for ECE 4810 folks (or pre 4810) see me

43

Pulse Response and Risetime

• Low Pass Filters cause sharp signal edges to be smoothed.

• The amount of smoothing is based on the bandwidth of the filter– More smoothing smaller bandwidth

• Fourier relationship:– a narrow rect function in time results in a broad (wide

bandwidth) sinc function in frequency

– a wide rect function in time results in a narrow (small bandwidth) sinc function in frequency

44

Filter Step Response

• 1 Hz and 10 Hz 4th order Butterworth LPF Filters

• The step response can be used to help define the bandwidth required for pulse signals.

10-1

100

101

102

103

104

-120

-100

-80

-60

-40

-20

0

Butterworth Filters


Att

enua

tion

(dB

)

1 Hz

10 Hz

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (sec)

Am

plitu

de

1 Hz

10 Hz

45

Filter Bandwidth for Pulses

• Pulse of length T

• Null-to-null BW of

• Single Sided BW desired

• B/2 may be acceptable in some cases– See textbook discussion

TfcsinTT

trect

T

2nulltonull

T

1B

-3 -2 -1 0 1 2 3

0

0.5

1

1.5

2

46

Pulse Filtering

• Four one-sided BW filters

• 0.1 sec pulse responses

0 10 20 30 40 50 60 70 80 90 100-160

-140

-120

-100

-80

-60

-40

-20

0

20Butterworth Filters

Frequency (fs = 100 Hz)

Att

enua

tion

(dB

)

2.5 Hz

5.0 Hz

10. Hz

20. Hz

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2Butterworth Filters

Time (fs=100Hz)

Am

plitu

de (

dB)

Test Signal

2.5 Hz

5.0 Hz10. Hz

20. Hz

PulseTest1.m

(digital filters)

47Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Pulse response of an ideal LPF

Figure 3.4-10

Text Comparison Chart(2.5, 5.0 and 20 Hz Plots)

48


See PulseTest2.m or PulseTest3.m

(digital filters)

Pulse resolution of an ideal LPF. B = 1/2t

49

Hilbert Transform

• It is a useful mathematical tool to describe the complex envelope of a real-valued carrier modulated signal in communication theory.

• The precise definition is as follows:

http://en.wikipedia.org/wiki/Hilbert_transform

dt

x1

t

1txtx̂

t

1thQ

f0j

0f,0

0f,j

fsgnjfHQ

1fHfHfH *QQ

2

Q

50


(a) Convolution; (b) Result

Hilbert transform of a rectangular pulse

51

Hilbert Transform of Cos

• This is useful in generating a complex signal from a real input signal as follows …

tf2cosAtx 0

00

Q00

ffff2

Aj

fHffff2

AfX̂

tf2sinAtx̂ 0

52

Real to Complex Conversion

t

1

t

tx

tx

tx̂

tx̂jtxty

53

Hilbert Transform Real to Complex Conversion

• Original Real

• Hilbert Transform Complex

fXtx

fXfjjfXtxjtxthtc sgnˆ

fXffXtxjtxthtc sgnˆ

0,0

0,2ˆ

ffor

fforfXtxjtxthtc

The Hilbert Transform can be used to create a single sided spectrum! The complex representation of a real signal.

54

Quadratic Filters

• We may want to process real signals using complex filtering or translated into the complex domain.

• Quadrature Signal Processing involves creating an “In-Phase” and “Qudrature-Phase” signal representation. – Usually this is done by “quadrature mixing” which

creates two outputs from a real data stream by mixing one by a cosine wave and the over by a sine wave.

phasequadraturejphasein

tf2sinjtf2costxtf2jexptx

55

Correlation and Spectral Density

• Using Probability and the 1st and 2nd moments– Assuming an ergodic, WSS process we use the time

average

• Properties:

• Schwarz’s Inequality

0tvtvtvP2

v

tzatzatzatza

tzttz

tztz

22112211

0

**

2

wv twtvPP

56

Autocorrelation and Power

• Autocorrelation Function

• Properties ttt tvtvtvtvR vv

tt

t

vvvv

vvvv

vvv

RR

R0R

P0R

57

Crosscorrelation

• Crosscorrelation Function

• Properties ttt twtvtwtvR vw

tt

t

wvvw

2

vwwwvv

RR

R0R0R

58

Application

• Correlation of phasors

2T

2T

21T

21 dttwwjexpT

1limtwjexptwjexp

2

Twwcsinlimtwjexptwjexp 21

T21

else,0

ww,1twjexptwjexp 21

21

59

Power Spectral Density

• The Fourier Transform of the Autocorrelation

• Remember ECE 3800!

60


Interpretation of spectral density functions

communication systems, 5e

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