Communication Systems 5eCommunication Systems, 5e

Chapter 3: Signal Transmission and Filtering

A. Bruce CarlsonPaul B. Crilly

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Chapter 3: Signal Transmission and FilteringFiltering

• Response of LTI systems• Signal distortion• Transmission Loss and decibels• Filters and filtering• Quadrature filters and Hilbert transform• Correlation and spectral density

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Pulse Response and Risetimep

• Low Pass Filters cause sharp signal edges to be h dsmoothed.

• The amount of smoothing is based on the b d idth f th filtbandwidth of the filter– More smoothing smaller bandwidth

• Fourier relationship:t f ti i ti lt i b d ( id– 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

3

(bandwidth) sinc function in frequency

Filter Step Responsep p

• 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.

-20

0

Butterworth Filters

1.2

1.4 Step Response

-60

-40

ttenu

atio

n (d

B)

0.6

0.8

1

Ampl

itude

120

-100

-80

At

1 Hz10 Hz

0

0.2

0.4

1 Hz10 Hz

4

10-1 100 101 102 103 104-120

Frequency (normalized)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

Time (sec)

Filter Bandwidth for Pulses

• Pulse of length T⎞⎛

2

( )TfcsinTTtrect ⋅⋅⇔⎟⎠⎞

⎜⎝⎛

1

1.5

• Null-to-null BW of

T2nulltonull ⇔−−

0

0.5

• Single Sided BW desiredT

T1B =

-3 -2 -1 0 1 2 3

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

T

5

Pulse Filteringg

F id d BW fil0

20Butterworth Filters

2.5 Hz5 0 Hz • Four one-sided BW filters

• 0.1 sec pulse responses -60

-40

-20

on (d

B)

5.0 Hz10. Hz20. Hz

-120

-100

-80

Atte

nuat

io

1

1.2Butterworth Filters

Test Signal2.5 Hz

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

-140

Frequency (fs = 100 Hz)

0.6

0.8

1

ude

(dB

)

5.0 Hz10. Hz20. Hz

0

0.2

0.4

Am

plitu

PulseTest1.m

60 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.2

Time (fs=100Hz)

PulseTest1.m

(digital filters)

Text Comparison Chart(2 5 5 0 and 20 Hz Plots)

Pulse response of an ideal LPFFi 3 4 10

(2.5, 5.0 and 20 Hz Plots)

See PulseTest2 m or PulseTest3 mFigure 3.4-10 See PulseTest2.m or PulseTest3.m

(digital filters)

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

See Fig03_4_11.m

8

(Butterworth filters)

The inherent time delay has been removed from the output

Pulse Resolution: Matlab

• Using a 1st order and 6th order Butterworth Filter– The text uses an “ideal filter”– Filters have group and phase delay!

11st Order Linear Sim

e

0 0.5 1 1.5 2 2.5 30

0.5

Time (sec)

Ampl

itude

0.5

16th Order Linear Sim

Ampl

itude

90 0.5 1 1.5 2 2.5 3

0

Time (sec)

A

Hilbert Transform• It is a useful mathematical tool to describe the

complex envelope of a real-valued carriercomplex envelope of a real valued carrier modulated signal – (i.e. make a complex signal from a real one)

• The precise definition is as follows:

( ) ( ) ( )∫∞ λx11ˆ ( ) ( ) ( )∫∞−

λ⋅λ−λ

⋅π

=⋅π

∗≡ dtx1

t1txtx̂

( ) 1thQ ≡ ( ) ( ) ⎪⎨

⎧=>−

=⋅−≡ 0f00f,j

fsgnjfH( )tQ ⋅π ( ) ( )

⎪⎩

⎨>==≡

f0j0f,0fsgnjfHQ

( ) ( ) ( ) 1*2=⋅= fHfHfH QQQ

10http://en.wikipedia.org/wiki/Hilbert_transform

( ) ( ) ( )fff QQQ

Quadrature filter

• Allpass network that shifts the phase of positive frequencies by 90° and negative frequencies by +90 °frequencies by -90° and negative frequencies by +90 °

{ 1 0( ) sgn ( )0Qj fH f j f h tj f t

− >= − = ⇔ =+ <

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{ 0Q j f t+ < π

Hilbert Transform Real to Complex ConversionComplex Conversion

• Original Real( ) ( )

• Hilbert Transform Complex( ) ( )fXtx ⇔

( ) ( ) ( )[ ] ( ) ( ) ( )fXfjjfXtxjtxthtc ⋅⋅⋅−⇔⋅+= sgnˆ

( ) ( ) ( )[ ] ( ) ( ) ( )fXffXtxjtxthtc +⇔+ sgnˆ( ) ( ) ( )[ ] ( ) ( ) ( )fXffXtxjtxthtc ⋅+⇔⋅+= sgn

( ) ( ) ( )[ ] ( )⎨⎧ >⋅

⇔⋅+=0,2

ˆfffforfX

txjtxthtc( ) ( ) ( )[ ]⎩⎨ < 0,0 ffor

j

The Hilbert Transform can be used to create a

12single sided spectrum! The complex representation of a real signal.

Hilbert transform propertiesp p

ˆ1. ( ) and ( ) have same amplitude spectrumx t x t

2. Energy and power in a signal and its Hilbert tranform are equal

ˆ3. ( ) and ( ) are orthogonal

x t x t

∞

⇒

∫ ˆ ( ) ( ) 0 (energy)x t x t dt−∞

=∫

li 1 ˆm ( ) ( ) 0 (power)2

T

TT

x t x t dtT→∞

−

=∫

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Hilbert transform of a rectangular pulse

(a) Convolution; (b) Result in time domain

( ) ( ) ( )txjtxty ˆ⋅+=

14

Hilbert Transform of Cosine

( ) ( )tf2cosAtx 0 ⋅⋅π⋅=

( ) ( ) ( )[ ] ( )

( ) ( )[ ]

Q00

A

fHffff2AfX̂ ⋅+δ+−δ⋅=

( ) ( )[ ]00 ffff2Aj +δ+−δ−⋅⋅=

( ) ( )tf2sinAtx̂ 0 ⋅⋅π⋅=

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

15

Real to Complex Conversionp

( )tδ ( )tx

( )tx ( ) ( ) ( )tx̂jtxty ⋅+=

t1⋅π

( )tx̂

16

Real to Complex Mixingp g

• Analog Devices AD8347: 0.8 GHz to 2.7 GHz Di C i Q d D d lDirect Conversion Quadrature Demodulator– RF input and LO input

Q adrat re O tp t– Quadrature Output

17

Quadratic Filters

• We may want to process real signals using l fil i l d i h lcomplex filtering or translated into the complex

domain. Q d t Si l P i i l ti• Quadrature Signal Processing involves creating an “In-Phase” and “Quadrature-Phase” signal representationrepresentation. – 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.

( ) ( ) ( ) ( ) ( )tf2sinjtf2costxtf2jexptx ⋅⋅π⋅+⋅⋅π⋅⇒⋅⋅π⋅⋅

18

phasequadraturejphasein −⋅+−⇒

Correlation and spectral densityp y

• Correlation of power signals• Correlation of energy signals• Spectral density functions

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Correlation and Spectral Densityp y

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

average( ) ( ) ( ) 0tvtvtvP 2 ≥⋅=≡ ∗

• Properties:

( ) ( ) ( ) 0tvtvtvPv ≥=≡

( ) ( ) **( ) ( )( ) ( )

( ) ( ) ( ) ( )tzatzatzatza

tzttz

tztz

22112211

0

**

⋅+⋅=⋅+⋅

=−

=

• Schwarz’s Inequality

( ) ( ) ( ) ( )tzatzatzatza 22112211 ++

20( ) ( )

2

wv twtvPP ∗⋅≥⋅

Autocorrelation and Power

• Autocorrelation Function

• Properties( ) ( ) ( ) ( ) ( )∗∗ ⋅τ+=τ−⋅≡τ tvtvtvtvR vv

( )( ) ( )

( ) ( )∗τ=τ−

τ≥

=

vvvv

vvv

RR

R0RP0R

( ) ( )τ=τ− vvvv RR

21

Sum and difference signalsg

[ ]( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )t t t R R R R R± ± [ ]( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

If ( ) and ( ) are uncorrelated ( ) ( ) 0

z v w vw wv

vw wv

z t v t w t R R R R R

v t w t R R

= ± ⇒ τ = τ + τ ± τ + τ

∀τ ⇒ τ = τ =( ) ( ) ( ) ( )

( ) ( ) ( )

vw wv

z v wR R R⇒ τ = τ + τ

with 0 z v wP P Pτ = ⇒ = +

Useful when computing signal plus noise.

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Crosscorrelation

• Crosscorrelation Function

• Properties( ) ( ) ( ) ( ) ( )∗∗ ⋅τ+=τ−⋅≡τ twtvtwtvR vw

( ) ( ) ( )( ) ( )∗τ=τ

τ≥⋅

wvvw

2vwwwvv

RR

R0R0R

23

Energy Spectral Densitygy p y

• The Fourier Transform of the Autocorrelation( ) ( )[ ] ( ) ( )∫

∞

∞−

⋅⋅⋅⋅−⋅=ℑ= ττπττ dfjRRfG vvv 2exp

( ) ( )[ ] ( ) ( )∫∞

ℑ dfffGfGR 21

• When Noise is a random variable:

( ) ( )[ ] ( ) ( )∫∞−

− ⋅⋅⋅⋅⋅=ℑ= dffjfGfGR vvv τπτ 2exp1

– Describe filtered output

• Compare deterministic and statistical signals– Deterministic have no information content; therefore,

only signals with varying, random messages are actually of interest!

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actually of interest!

System analysis in τ domainy y

( )h t( )( )x

x tR τ

( )( )ττ

yRy

( ) ( ) ( ) ( ) ( )y t x t h t h x t d∞

−∞

= ∗ = λ − λ λ∫

( )y

( ) ( ) ( ) ( ) ( )yx x xR h R h R d∞

−∞

τ = τ ∗ τ = λ τ − λ λ∫

* *

and with output autocorrelation is:

( ) ( ) ( ) ( ) ( )R h R h R d∞

μ = −λ

τ = −τ ∗ τ = −μ τ −μ μ∫

*

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

y yx yx

y x

R h R h R d

R h h R

−∞

τ τ ∗ τ μ τ μ μ

⇒ τ = −τ ∗ τ ∗ τ

∫

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System Analysis in f domainy y

( )h t( )( )x

x tR τ

( )fGx ( )fGy( )( )ττ

yRy( )fH

( ) ( )[ ]τxx RfG ℑ=

( ) ( ) ( )fGfHfG 2

( )y

( ) ( )[ ]τyy RfG ℑ=

( ) ( ) ( )fGfHfG xy ⋅=

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Interpretation of spectral densityInterpretation of spectral density functions

27