Communication Systems, 5ebazuinb/ECE4600/Ch03_1.pdf · • Method1: Apply an impulse and measure the time response. – “Impulse generator” and a time samples • Method 2: Apply

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 (ECE 3710)• Signal distortion• Transmission Loss and decibels• Filters and filtering• Quadrature filters and Hilbert transform• Correlation and spectral density


Linear Time Invariant (LTI) Systems

• Principle of Superposition– Summed input create transformed summed outputs

• Time-invariant– invariant result if shifted in time

• Homogeneity– Constant multiplication of input results in identical constant

multiplication of the output

• Transfer Functions – input to output relationship• Expected for “lumped-parameter” elements (R, L, C)

– At frequencies where component sizes and frequency wavelengths are in the same order, it doesn’t work anymore. (Distributed-parameter systems)

3

Transfer functions and frequency response

• System transfer function– Impulse response– Form convolution to generate output

• For a sinusoidal input:The output is a magnitude and phase shifted sinusoid


dttfjththfH 2exp

tftx 02cos 002cos0

fHtfAty fH

00fHA fH 000 arg fHfHfH

Example: Time Response of a First Order System

• RC Filter with 1Hz and 10Hz cutoff frequency

y(t) v(t)

RC1sRC

1

sC1R

sC1

sHsYsV

0t,RCtexp

RC1th

5

Step Responses: 1 Hz and 10 Hz RC• Matlab Code Fig3_1_4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Step Response

Time (sec)

Ampl

itude

1 Hz10 Hz

6

10-1 100 101 102 103 104-120

-100

-80

-60

-40

-20

0Butterworth Filters

Frequency (normalized)

Atte

nuat

ion

(dB

)

1 Hz10 Hz

10-1 100 101 102 103 104-100

-80

-60

-40

-20

0

Butterworth Filters

Phase (degrees)

Atte

nuat

ion

(dB

)

1 Hz10 Hz

Impulse and Square Wave Responses: 1 Hz and 10 Hz RC

• Matlab Code Fig3_1_4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

Impulse Response

Time (sec)

Ampl

itude

1 Hz10 Hz

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.5

1

Linear Simulation Results

Time (sec)

Ampl

itude

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.5

1

Linear Simulation Results

Time (sec)

Ampl

itude

7

Transfer Function Methods

• For continuous-time systems, the Laplace transform is used to define the filter transfer function

– Frequency Response

– or

8

11

11

011

1

ssssss

sXsYsH n

nn

n

nn

mm

1222

222

11

1

011

1

fjfjfjfjfjfj

fXfYfH n

nn

n

nn

mm

fXj

fYj

efXefY

fXfYfH arg

arg

Determining Transfer Functions

• Method1: Apply an impulse and measure the time response.– “Impulse generator” and a time samples

• Method 2: Apply known frequencies at known amplitudes and phases and measure the frequency response.– Sine wave generator and a dual trace oscilloscope– A one instrument equivalent is a Network Analyzer

coscos fffHfY

thdthty

9

Block Diagram Methods

• Extensively used to describe how signals are connected in and flow through a system– A flow chart similar to a software flow diagram– Allows engineers to handle complexity and check that

things are connected right.

• Multiple levels of block diagrams– Show the system hierarchy– Every Block can be broken into another lower-level

block diagram until you get to a single transfer function, equation, decision or operations.

10

Block Diagram Analysis

• Block diagrams consist of unidirectional, operational blocks that represent the transfer function of the variables of interest

sRGsRGsY

sRGsRGsY

2221212

2121111

Notes and figures are based on or taken from materials in the ECE 371 course textbook: R.C. Dorf and R.H. Bishop, Modern Control Systems, 9th ed., Prentice Hall, 2001. ISBN 0-13-030660-6.

11

Block Diagram Transformations

Notes and figures are based on or taken from materials in the ECE 3710 course textbook: R.C. Dorf and R.H. Bishop, Modern Control Systems, 9th ed., Prentice Hall, 2001. ISBN 0-13-030660-6.

12

Block diagram types


(a) Parallel

(b) Cascade

(c) Feedback

Note on Zero Order Hold BD• Typical ADCs use a “sample and hold” prior to

the ADC• Sampling is typically an integration of the signal

for a fixed sampling period• Hold is to insure the ADC has a stable signal for a

defined period of time

14

Frequency Selective Filtering

• Filters are intended to remove noise and interference– Maximize the amount that is removed

• Filters will modified the spectrum of the signal of interest in some way– Minimize any signal distortion that may be caused by the filter.

• The above restrictions can usually not both be performed!– Compromise between the two!!

See Figure 3.1-7 on p. 101.

See Matlab Example 15

Example: Figure 3.1-73 Filters Power Spectrum- 10, 100 & 1000 Hz, 4 pole

Signal Power Spectrum- 100 Hz, 1 pole roll-off

3 OutputPower Spectrum 16

100 101 102 103 104-45

-40

-35

-30

-25

-20

-15

-10

-5

0Signal

Freq

Am

plitu

de (d

B)

100 101 102 103 104-45

-40

-35

-30

-25

-20

-15

-10

-5

0Filters

Freq

Am

plitu

de (d

B)

FH1FH2FH3

100 101 102 103 104-45

-40

-35

-30

-25

-20

-15

-10

-5

0Output

Freq

Am

plitu

de (d

B)

O1O2O3

LTI System Energy

dffXfHdffYE22

y

dffXdffHdffXfHE

dffXfXfHfHE

dffXfHfXfHE

dffXfHfXfHE

y

y

y

y

2222

**

**

*

17

Product of the Signal and Filter Power Spectrum(See the previous example)

Spectral Modifications

• Matched Filter (ECE 3800)

• “Wiener Filter” (ECE 3800)

2XXYY wHwSwS

sXsHsY

utsKh

sSsSsFsF NNXXCC

PlaneHalfLeftC

XX

C sFsS

sFsH

1

18

Signal Filtering in the Real World (1)

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, 2001.19

Signal Filtering in the Real World (2)

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, 2001.20

Filter 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

21

Signal Distortion

• The RF channel (and any filtering) may distort the signal between what you think you transmitted and what you receive.

1. Amplitude Distortion

2. Delay Distortion

3. Nonlinear Distortion• Not an LTI operation

KfH

mtf2fHfHarg d

22

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

Test signal x(t) = cos 0t - 1/3 cos 30t + 1/5 cos 50t

Test Signal Example

23


(a) low frequency attenuated by ½ (by a HPF)

(b) high frequency attenuated by ½ (by a LPF)

Test signal with filtered, linear amplitude distortion

24

Square Wave Filtering

• MATLAB Fig3_2_4.m

25

10-1 100 101 102 103 104-120

-100

-80

-60

-40

-20

0

Butterworth Filters

Frequency (normalized)

Atte

nuat

ion

(dB

)

1 kHz LPF10 Hz HPF

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-2

-1

0

1

21000 Hz Low Pass Filter

Time (sec)

Ampl

itude

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-2

-1

0

1

210 Hz High Pass Filter

Time (sec)

Ampl

itude

Phase Delay and Delay Distortion

• Constant time delay results in linear phase delay and is desirable for distortionless transmissions.– See the time delay property of the Fourier Transform

• A constant phase shift is not linear in phase and results in significant signal distortion.– See the following constant phase delay of 90 deg.– Use an “equalizer” to fix the phase problems

26

Test signal with constant phase distortion


Test signal x(t) = cos(0t -90)- 1/3 cos(30t -90)+ 1/5 cos(50t-90)

shift all inputs by = -90

Not a square wave anymore

27

Equalization to over come linear distortion


Consider a channel with linear distortion

2To overcome linear distortion equalization

( ) ( ) 0( )

overall system function is ( ) ( ) ( )

dj ft

eqC

eq C

KeH f H fH f

H f H f H f

Time Delay Distortion

• The effective time delay due to phases

• The equalized time delay can be approximated

• For distortionless transmission,

ftfffH dCC 2arg

gCEQCg tfjfHfHfHtth 2exp

gCEQC tfffHfH 2arg

gdEQC tftffHfH 2arg

gd tft 0 0arg fHfH EQC

29

Phase or Delay Distortion

• If the phase shift is not linear, the phase delay time is

– For a pure time delay, the phase change is linear and the phase delay is a constant.

– Note: constant time delay is desirable, while constant phase delay is not

ffHftd

2

arg

30

Envelope or Group Delay

• The group delay is a constant for all frequencies, it is approximated as

• This is commonly envisioned to apply to the envelope of the signal, while the phase delay is applied to the carrier.

df

fdtg

21

31

Time Delay – Signal Distortion

• For a test signal

• The filtered output would become

tf2sintxtf2costxtx c2c1

cdcg2

cdcg1

fttf2sinttxA

fttf2costtxAty

Expect the carrier delay to be greater than the signal envelope delay!

32

Equalization

• Apply a filter that corrects for amplitude and delay distortion in the receiver.

• Compensate so that the difference between the phase and group delay becomes zero

021

2

ftdf

fdf

ffttft compcompgd

33

Logarithms and Decibels• RF signal power and voltage magnitudes are typically

described in:– Decibel-Watts (dBW)– Decibel-milliwatts (dBm)– Decibel-voltage (dBv)

• The Ratio of RF power or voltage is typically describe in:– Decibels (dB)

• The relative antenna power (transmitting or receiving) for a particular direction as compared to an “isotropic antenna” is describe in:– Decibles-isotropic (dBi)

34

• The “bel” is defined as the log base 10 of power ratios

• The magnitude was too small, therefore the Decibel was defined

Alexander Graham Bell

in

out

PPbels log

in

out

PPdBdecibels log10

35

Derived Power/Voltage Measures• Decibels for Power, Voltage, and Current

– Power

– Voltage

– For matched impedances where Rout=Rin

in

out

PPdBdecibels log10

in

out

RV

RV

dBdecibels

RVP

in

out

2

2

2

log10

in

out

VVdBdecibels log20

36

Derived Power/Current Measures• Decibels for Power, Voltage, and Current

– Power

– Current

– For matched impedances where Rout=Rin

in

out

PPdBdecibels log10

in

out

RIRI

dBdecibels

RIP

in

out

2

2

2

log10

in

out

IIdBdecibels log20

37

Power Measurement• Watts (P in Watts)

• Milliwatts (P in Watts)

• Milliwatts (Pm in milliwatts)

PdBWwattsdecibels log10

310)(log10 mWPdBmmilliwattsdecibels

mWPdBmmilliwattsdecibels log10

38

Common Decibel Values

Power Ratio (in

outP

P ) Decibels (dB)

100 20 dB

10 10 dB

5 6.99 dB

4 6.02 dB

2 3.01 dB

1 0 dB

0.5 -3.01 dB

0.25 -6.02 dB

0.1 -10 dB

0.01 -20 dB

39

Common Decibel Power ValuesPower (Watts) Decibels-Watts (dBW) Decibels-MilliWatts (dBm)

1 megaWatt 60 dBW 90 dBm

1 kilowatt 30 dBW 60 dBm

10 watts 10 dBW 40 dBm

1 watt 0 dBW 30 dBm

0.1 watt -10 dBW 20 dBm

ohmsvolt 501 2 -16.99 dBW +13 dBm

1 milliwatt -30 dBW 0 dBm

1 microwatt -60 dBW -30 dBm

1 nanowatt -90 dBW -60 dBm

1 picowatt -120 dBW -90 dBm

40

Suggested Matlab Functions

valuevaluedB log10)(

valuevaluedBv log20)(

1010)(value

valueidB

2010)(value

valueidBv

41

Logarithm/Decibel Reminders baba logloglog

baba logloglog

bdBadBbadB

bdBadBbadB

cbacb

a logloglog1log

cdBbdBadBcb

adB

1

42

Documents

Communication Systems, 5ebazuinb/ECE4600/Ch03_1.pdf · • Method1: Apply an impulse and measure the time response. – “Impulse generator” and a time samples • Method 2: Apply