Lecture13

Slides by Prof. Brian L. Evans and Dr. Serene Banerjee

Dept. of Electrical and Computer Engineering

The University of Texas at Austin

EE445S Real-Time Digital Signal Processing Lab Spring 2011

Lecture 13

Matched Filtering and DigitalPulse Amplitude Modulation (PAM)

13 - 2

Outline

• Transmitting one bit at a time

• Matched filtering

• PAM system

• Intersymbol interference

• Communication performanceBit error probability for binary signals

Symbol error probability for M-ary (multilevel) signals

• Eye diagram

13 - 3

Transmitting One Bit

• Transmission on communication channels is analog• One way to transmit digital information is called

2-level digital pulse amplitude modulation (PAM)

b t

)(1 tx

A

‘1’ bit

Additive NoiseChannel

input output

x(t) y(t)

b

)(0 tx

-A

‘0’ bit

t

receive ‘0’ bit

receive‘1’ bit

)(0 ty

b

-A

b t

)(1 ty

AHow does the

receiver decide which bit was sent?

13 - 4

Transmitting One Bit

• Two-level digital pulse amplitude modulation over channel that has memory but does not add noise

h t

)(th

1

b t

)(1 tx

A

‘1’ bit

b

)(0 tx

-A

‘0’ bit

Model channel as LTI system with impulse response

h(t)

CommunicationChannel

input output

x(t) y(t)t

)(0 ty

-A Th

receive ‘0’ bit

th+bh

Assume that Th < Tb

t

)(1 ty receive‘1’ bit

h+bh

A Th

13 - 5

Transmitting Two Bits (Interference)

• Transmitting two bits (pulses) back-to-back will cause overlap (interference) at the receiver

• Sample y(t) at Tb, 2 Tb, …, andthreshold with threshold of zero

• How do we prevent intersymbolinterference (ISI) at the receiver?

h t

)(th

1

Assume that Th < Tb

tb

)(tx

A

‘1’ bit ‘0’ bit

b

* =)(ty

-A Th

tb

‘1’ bit ‘0’ bit

h+b

Intersymbol interference

13 - 6

Preventing ISI at Receiver

• Option #1: wait Th seconds between pulses in transmitter (called guard period or guard interval)

Disadvantages?

• Option #2: use channel equalizer in receiverFIR filter designed via training sequences sent by transmitterDesign goal: cascade of channel memory and channel

equalizer should give all-pass frequency response

h t

)(th

1

Assume that Th < Tb

* =

tb

)(tx

A

‘1’ bit ‘0’ bit

h+b

t

)(ty

-A Th

b

‘1’ bit ‘0’ bit

h+b

h

13 - 7

k

bk Tktgats ) ( )(

Digital 2-level PAM System

• Transmitted signal

• Requires synchronization of clocks between transmitter and receiver

Transmitter Channel Receiver

bi

Clock Tb

PAM g(t) h(t) c(t)1

0

ak{-A,A} s(t) x(t) y(t) y(ti)

AWGNw(t)

Decision

Maker

Threshold

Sample at

t=iTb

bits

Clock Tb

pulse shaper

matched filter

1

00 ln4 p

p

AT

N

bopt

13 - 8

Matched Filter

• Detection of pulse in presence of additive noiseReceiver knows what pulse shape it is looking for

Channel memory ignored (assumed compensated by other means, e.g. channel equalizer in receiver)

Additive white Gaussian noise (AWGN) with zero mean and variance N0 /2

g(t)

Pulse signal

w(t)

x(t) h(t) y(t)

t = T

y(T)

Matched filter

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

0 tntgthtwthtgty

T is the symbol period

13 - 9

power average

power ousinstantane

)}({

|)(|

SNR pulsepeak is where,max

2

20

tnE

Tg

Matched Filter Derivation

• Design of matched filterMaximize signal power i.e. power of at t = T

Minimize noise i.e. power of

• Combine design criteria

g(t)

Pulse signal

w(t)

x(t) h(t) y(t)

t = T

y(T)

Matched filter

)(*)()( thtwtn )(*)()(0 thtgtg

T is the symbol period

13 - 10

Power Spectra

• Deterministic signal x(t)w/ Fourier transform X(f)Power spectrum is square of

absolute value of magnitude response (phase is ignored)

Multiplication in Fourier domain is convolution in time domain

Conjugation in Fourier domain is reversal & conjugation in time

• Autocorrelation of x(t)

Maximum value (when it exists) is at Rx(0)

Rx() is even symmetric,i.e. Rx() = Rx(-) )( )()()( *2

fXfXfXfPx

)(*)( )( )( ** xxFfXfX

)(*)()( * xxRx

t

1x(t)

0 Ts

Rx()

-Ts Ts

Ts

13 - 11

Power Spectra

• Two-sided random signal n(t)Fourier transform may not exist, but power spectrum exists

For zero-mean Gaussian random process n(t) with variance 2

• Estimate noise powerspectrum in Matlab

22* )( )( )( )( )( fPtntnER nn

)( )( nn RFfP

N = 16384; % finite no. of samplesgaussianNoise = randn(N,1);plot( abs(fft(gaussianNoise)) .^ 2 );

approximate noise floor

dttntntntnERn )( )( )( )( )( **

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

nndttntntntnERn

13 - 122 2 2

0 | )( )(| |)(|

dfefGfHTg Tfj

Matched Filter Derivation

• Noise

• Signal

dffHN

dffStnE N202 |)(|

2 )(} )( {

f

20N

Noise power spectrum SW(f)

)()( )(0 fGfHfG

dfefGfHtg tfj )( )( )( 2 0

20 |)(|2

)( )()( fHN

fSfSfS HWN

g(t)

Pulse signal w(t)

x(t) h(t) y(t)

t = T

y(T)

Matched filter

)(*)()(0 thtgtg

)(*)()( thtwtn AWGN Matched

filter

T is the symbol period

13 - 13

dffH

N

dfefGfH Tfj

20

2 2

|)(|2

| )( )(|

Matched Filter Derivation• Find h(t) that maximizes pulse peak SNR

• Schwartz’s inequality

For vectors:

For functions:

upper bound reached iff

|||| ||||cos |||| |||| | | *

ba

bababa

TT

Rkxkx )( )( 21

-

2

2

-

2

1

2

*2

-

1 )( )( )( )( dxxdxxdxxx

a

b

13 - 14)( )( Hence,

inequality s' Schwartzby )( )(

whenoccurs which , |)(| 2

|)(| 2

|)(| 2

| )( )(

|)(| |)(| | )( )(

)()( and )()(Let

*

2 *

2

0max

2

020

2 2

222 2

2 *21

tTgkth

kefGkfH

dffGN

dffGN

dffHN

dfefGfH|

dffGdffHdfefGfH|

efGffHf

opt

Tfjopt

-

Tfj

-

Tfj

Tfj

Matched Filter Derivation

T is the symbol period

13 - 15

Matched Filter

• Impulse response is hopt(t) = k g*(T - t)

Symbol period T, transmitter pulse shape g(t) and gain k

Scaled, conjugated, time-reversed, and shifted version of g(t)

Duration and shape determined by pulse shape g(t)

• Maximizes peak pulse SNR

Does not depend on pulse shape g(t)

Proportional to signal energy (energy per bit) Eb

Inversely proportional to power spectral density of noise

SNR2

|)(| 2

|)(| 2

0

2

0

2

0max

N

Edttg

NdffG

Nb

13 - 16

t=nT T

Matched Filter for Rectangular Pulse

• Matched filter for causal rectangular pulse shapeImpulse response is causal rectangular pulse of same duration

• Convolve input with rectangular pulse of duration T sec and sample result at T sec is same asFirst, integrate for T sec

Second, sample at symbol period T sec

Third, reset integration for next time period

• Integrate and dump circuit

Sample and dump

h(t) = ___

13 - 17

k

bk Tktgats ) ( )(

Digital 2-level PAM System

• Transmitted signal

• Requires synchronization of clocks between transmitter and receiver

Transmitter Channel Receiver

bi

Clock Tb

PAM g(t) h(t) c(t)1

0

ak{-A,A} s(t) x(t) y(t) y(ti)

AWGNw(t)

Decision

Maker

Threshold

Sample at

t=iTb

bits

Clock Tb

pulse shaper

matched filter

1

00 ln4 p

p

AT

N

bopt

13 - 18

)( )( )( )(

)(*)()( where)()()(

,i

ikkbkbiii

kbk

tnTkipaiTtpaty

tctwtntnkTtpaty

k

bk Tktats ) ()(

Digital 2-level PAM System

• Why is g(t) a pulse and not an impulse?Otherwise, s(t) would require infinite bandwidth

Since we cannot send an signal of infinite bandwidth, we limit its bandwidth by using a pulse shaping filter

• Neglecting noise, would like y(t) = g(t) * h(t) * c(t) to be a pulse, i.e. y(t) = p(t) , to eliminate ISI

actual value(note that ti = i Tb)

intersymbolinterference (ISI)

noise

p(t) is centered at origin

13 - 19

) 2

(rect 2

1 )(

||,0

, 2

1

)(

W

f

WfP

Wf

WfWWfP

Eliminating ISI in PAM• One choice for P(f) is a

rectangular pulseW is the bandwidth of the

systemInverse Fourier transform

of a rectangular pulse isis a sinc function

• This is called the Ideal Nyquist Channel• It is not realizable because pulse shape is not

causal and is infinite in duration

) 2(sinc)( tWtp

13 - 20

WffW

fWfffW

Wf

W

ffW

fP

2 || 20

2 || 22

)|(|sin1

4

1

|| 0 2

1

)(

1

111

1

Eliminating ISI in PAM

• Another choice for P(f) is a raised cosine spectrum

• Roll-off factor gives bandwidth in excessof bandwidth W for ideal Nyquist channel

• Raised cosine pulsehas zero ISI whensampled correctly

• Let g(t) and c(t) be square root raised cosine pulses

W

f11

222 161

2cos

sinc )(

tW

tW

T

ttp

s

ideal Nyquist channel impulse response

dampening adjusted by rolloff factor

13 - 21

Bit Error Probability for 2-PAM

• Tb is bit period (bit rate is fb = 1/Tb)

v(t) is AWGN with zero mean and variance 2

• Lowpass filtering a Gaussian random process produces another Gaussian random processMean scaled by H(0)

Variance scaled by twice lowpass filter’s bandwidth

• Matched filter’s bandwidth is ½ fb

h(t)s(t)

Sample att = nTb

Matched filterv(t)

r(t) r(t) rn k

bk Tktgats ) ( )(

)()()( tvtstr

r(t) = h(t) * r(t)

13 - 22

Bit Error Probability for 2-PAM

• Binary waveform (rectangular pulse shape) has amplitude A over nth bit period nTb < t < (n+1)Tb

• Matched filtering by integrate & dumpSet gain of matched filter to be 1/Tb

Integrate received signal over symbol period, scale and sample

n

Tn

nTb

Tn

nTbn

vA

dttvT

A

dttrT

r

b

b

b

b

)(1

)(1

)1(

)1(

0-

Anr

)( nr rPn

AProbability density function (PDF)

See slide 13-16

13 - 23

Av

PAvPvAPAnTsP nnnb )( )0())(|error(

0 /A

/nv

Bit Error Probability for 2-PAM

• Probability of error given thattransmitted pulse has amplitude –A

• Random variable is Gaussian withzero mean andvariance of one

AQdve

AvPAnTsP

v

A

n 2

1))(|error( 2

2

nv

Q function on next slide

PDF for N(0, 1)

Tb = 1

13 - 24

Q Function

• Q function

• Complementary error function erfc

• Relationship

x

y dyexQ 2/2

21

)(

x

t dtexerfc22

)(

22

1)(

xerfcxQ

Erfc[x] in Mathematica

erfc(x) in Matlab

13 - 25

2

2

SNR where,

2

1

2

1

))(|error()())(|error()( error)(

A

Qσ

AQ

σ

AQ

σ

AQ

AnTsPAPAnTsPAPP bb

Bit Error Probability for 2-PAM• Probability of error given that

transmitted pulse has amplitude A

• Assume that 0 and 1 are equally likely bits

• Probability of errordecreases exponentially with SNR

)/())(|error( AQAnTsP b

2

1)( )(

22

e

Qx

exerfc

x

, positive largefor x

See slide 8-17

Tb = 1

13 - 26

PAM Symbol Error Probability

• Average transmitted signal power

GT() is square root of theraised cosine spectrum

Normalization by Tsym willbe removed in lecture 15 slides

• M-level PAM symbol amplitudes

• With each symbol equally likely

sym

nT

sym

nSignal T

aEdG

T

aEP

}{|)(|

2

1}{ 22

2

sym

M

isym

M

Mi

isym

Signal T

dMid

MTl

MTP

3)1( )12(

21

11 22

2

1

22

12

2

2, ,0 , ,1

2 ),12(

MMiidli ......

2-PAM

d

-d

4-PAM

Constellation points with receiver

decision boundaries

d

d

3 d

3 d

13 - 27

symNoise T

Nd

NP

sym

sym2

2

2

1 0

2/

2/

0

)()( symRnsym nTvanTx

PAM Symbol Error Probability

• Noise power and SNR

• Assume ideal channel,i.e. one without ISI

• Consider M-2 inner levels in constellationError if and only if

where

Probablity of error is

• Consider two outer levels in constellation

dnTv symR |)(|

d

QdnTvP symR 2)|)((|

2/02 N

d

QdnTvP symR ))((

two-sided power spectral density of AWGN

channel noise after matched filtering and sampling

0

22

3

)1(2 SNR

N

dM

P

P

Noise

Signal

13 - 28

Alternate Derivation of Noise Power

dnTwgnTv r

)()()(

22 )()()( dnTwgEnTvE r

})()()()({ 212211

ddnTwgnTwgE TT

212121 )}()({)()( ddnTwnTwEgg TT

TdGdg

sym

sym

rr

22/

2/

2222 )(2

1)(

Noise power

T = TsymFiltered noise

2 (1–2)

Optional

13 - 29

d

QM

MdQ

M

dQ

M

MPe

)1(2

2 2

2

PAM Symbol Error Probability

• Assuming that each symbol is equally likely, symbol error probability for M-level PAM

• Symbol error probability in terms of SNR

13

SNR since SNR1

3

12 2

2

22

1

2

Md

P

P

MQ

M

MP

Noise

Signale

M-2 interior points 2 exterior points

13 - 30

Visualizing ISI

• Eye diagram is empirical measure of signal quality

• Intersymbol interference (ISI):

• Raised cosine filter has zeroISI when correctly sampled

• See slides 13-31 and 13-32

knk

k

symsymknsymsymk g

kTnTgaagTkTnganx

)0(

)( )0() ( )(

nkk

sym

nkk

symsym

g

kTgdM

g

kTnTgdMD

,, )0(

)( )1(

)0(

)( )1(

13 - 31

Eye Diagram for 2-PAM• Useful for PAM transmitter and receiver analysis

and troubleshooting

• The more open the eye, the better the reception

M=2

t - Tsym

Sampling instant

Interval over which it can be sampled

Slope indicates sensitivity to timing error

Distortion overzero crossing

Margin over noise

t + Tsymt

13 - 32

Eye Diagram for 4-PAM

3d

d

-d

-3d

Due to startup transients.

Fix is to discard first few symbols equal to number of symbol periods in pulse shape.