Communication Systems, 5e - Homepages at WMUbazuinb/ECE4600/Ch03_3.pdf · Communication Systems, 5e...

Communication Systems, 5e

Chapter 3: Signal Transmission and Filtering

A. Bruce CarlsonPaul 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


3

Free-Space Loss

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

tPtG rG

f

RrP

22

cRf4R4L

• As shown above

22

2

2

44 fRcGGP

RGGP

LGGPP rt

trt

trt

tr

fc

Note

4

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

22

rtt2

2rt

tr Rf4cGGP

R4GGPP

RfdBcdBdBGdBGdBPdBP rttr

2

42

5

System Range

• Maximum Range (Pr is the receiver sensitivity)

dBmPt

dBmGt

dBmGr

dBmPr

m0 mR1

tPtG rG

f

RrP

rtr

t GGPP

f4cR

dBPdBGRfdBcdBdBGdBP rrtt

2

42

6

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

6 GHz4 GHz

dB13.1997e8.9048e3

6e369e64c

Rf4L 222

u

dB60.1957e18.6038e3

6e369e44c

Rf4L 222

d

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

Path Losses

7


dBW1.89dB20dB1.199dB55dBW35P rcv_sat

dBWdBdBdBdBWPout 6.110516.1951618

22

rtt

rttr R4

GGPL

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

8

22

2

2

44 t

rtt

j

rjj Rf

cGGPRfcGG

P

jtt

jjt R

GPGP

R

3.22210110100 2

3 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 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:0.65kg

9

These are not legal in the USFCC Regulations.

~$256 from China

Filters and filtering

• Ideal filters• Bandlimiting and timelimiting• Real filters• Pulse response and risetime


11

The Ideal Filter

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

02exp tffXKfY

• The transfer function is 0tf2expKfH

• A constant gain with a linear phase KfH 0tf2f

12


(b) Bandpass Filter

Ideal filters

(a) Lowpass Filter

13

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

f0

f

f0

dfttf2jexpKth

dftf2jexptf2jexpKth

14

Ideal Filter (2)

0uu

0

0u

0

0u

0

0u

f

f0

0

f

f0

ttf2sincKf2thtt2

ttf2sinK2th

tt2jttf2jexpK

tt2jttf2jexpKth

tt2jttf2jexpKth

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.

15


(a) Transfer function (b) Impulse response

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

t

dttB2sincKB2th

Ideal lowpass filter

16

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


18

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

BWBWSF

PBBW

SBBW

19


Typical amplitude ratio of a real bandpass filterFigure 3.4-3

Real Bandpass Filter

The -3 dB or half-power bandwidth is shown

20

Bandwidths that are Used



Note: Sinc freq. domain is appropriate for digital symbols

21

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.



22

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.



23

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 (k

Hz)

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 Coilcraft

24

25

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

1w

sj1

1wj

s1

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)

Atte

nuat

ion

(dB

)

1st order2nd order3rd order4th order5th order

26

Butterworth Filter PSD

10-1 100 101 102 103-120

-100

-80

-60

-40

-20

0

Butterworth Filter Family


Atte

nuat

ion

(dB

)


27

Butterworth Filter PSD (2)

10-1 100-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1Butterworth Filter Family


Atte

nuat

ion

(dB

)


28

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]);

29

Chebyshev Type IFilter PSD (Cheby1Plot.m)

10-1 100 101 102 103-120

-100

-80

-60

-40

-20

0

Chebyshev Type I Filter Family


Atte

nuat

ion

(dB

)


30

Chebyshev Type IFilter PSD (2)

10-1 100-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1Chebyshev Type I Filter Family


Atte

nuat

ion

(dB

)


31

Available MATLAB Filters(Signal Proc. TB)

• Analog or Digital– Butterworth– Chebyshev Type I– Chebyshev Type II– Elliptic or Cauer– Bessel

• Digital– barthannwin – bartlett – blackman – blackmanharris – bohmanwin – chebwin – flattopwin – gausswin – hamming – hann – kaiser – nuttallwin – parzenwin – rectwin – triang – tukeywin

32

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

ButterBesselCheby1Cheby2EllipSpec

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

33

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

34

Matlab Filter Generation (2)

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);

35

Matlab Filter Generation (3)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

ButterBesselCheby1Cheby2EllipSpec

36

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

37

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

38

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


Atte

nuat

ion

(dB

)

1 Hz10 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)

Ampl

itude

1 Hz10 Hz

39

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

TfcsinTTtrect

T2nulltonull

T1B

-3 -2 -1 0 1 2 3

0

0.5

1

1.5

2

40

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)

Atte

nuat

ion

(dB

)

2.5 Hz5.0 Hz10. Hz20. 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 (d

B)

Test Signal2.5 Hz5.0 Hz10. Hz20. Hz

PulseTest1.m

(digital filters)

41Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Pulse response of an ideal LPFFigure 3.4-10

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

42


See PulseTest2.m or PulseTest3.m

(digital filters)

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

43

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

dtx1

t1txtx̂

t

1thQ

f0j0f,00f,j

fsgnjfHQ

1fHfHfH *QQ

2Q

44


(a) Convolution; (b) Result

Hilbert transform of a rectangular pulse

45

Hilbert Transform of Cos

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

tf2cosAtx 0

00

Q00

ffff2Aj

fHffff2AfX̂

tf2sinAtx̂ 0

46

Real to Complex Conversion

t1

t

tx

tx

tx̂

tx̂jtxty

47

Hilbert Transform Real to Complex Conversion

• Original Real

• Hilbert Transform Complex fXtx

fXfjjfXtxjtxthtc sgnˆ

fXffXtxjtxthtc sgnˆ

0,00,2

ˆfforfforfX

txjtxthtc

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

48

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

49

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

0tvtvtvP 2v

tzatzatzatza

tzttz

tztz

22112211

0

**

2

wv twtvPP

50

Autocorrelation and Power

• Autocorrelation Function

• Properties tvtvtvtvR vv

vvvv

vvvv

vvv

RR

R0RP0R

51

Crosscorrelation

• Crosscorrelation Function

• Properties twtvtwtvR vw

wvvw

2vwwwvv

RR

R0R0R

52

Application

• Correlation of phasors

2T

2T

21T21 dttwwjexpT1limtwjexptwjexp

2Twwcsinlimtwjexptwjexp 21

T21

else,0

ww,1twjexptwjexp 21

21

53

Power Spectral Density

• The Fourier Transform of the Autocorrelation

• Remember ECE 3800!

54


Interpretation of spectral density functions

Communication Systems, 5e - Homepages at WMUbazuinb/ECE4600/Ch03_3.pdf · Communication Systems, 5e...

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