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 2014

Lecture 14

Matched Filtering and DigitalPulse Amplitude Modulation (PAM)

14 - 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

14 - 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

How does the receiver decide

which bit was sent?

receive‘1’ bit

b t

)(1 ty

A

receive ‘0’ bit

)(0 ty

b

-A

t

14 - 4

Transmitting One Bit

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

h t

)(tc

1

b t

)(1 tx

A

‘1’ bit

b

)(0 tx

-A

‘0’ bit

Model channel as LTI system with impulse response

c(t)

LTIChannel

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

14 - 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

)(tc

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

14 - 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

)(tc

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

14 - 7

k

bk Tktgats ) ( )(

Digital 2-level PAM System

• Transmitted signal

• Requires synchronization of clocksbetween transmitter and receiver

Transmitter Channel Receiver

bi

Clock Tb

PAM g(t) c(t) h(t)10

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

boptN(0, N0/2)

p0 is the probability bit ‘0’

sent

bits

14 - 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

14 - 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

14 - 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

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

)( )( )( )( 2* tntnERn

)( )( nn RFfP

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

approximate noise floor

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

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

nndttntntntnERn

0 when 0 )( )( )( * tntnERn2)( fPn

14 - 11

14 - 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

14 - 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

14 - 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

14 - 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

14 - 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) = ___

14 - 17

k

bk Tktgats ) ( )(

Digital 2-level PAM System

• Transmitted signal

• Requires synchronization of clocksbetween transmitter and receiver

Transmitter Channel Receiver

bi

Clock Tb

PAM g(t) c(t) h(t)10

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

boptN(0, N0/2)

p0 is the probability bit ‘0’

sent

bits

14 - 18

)( )( )( )(

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

,i

ikkbkbiii

kbk

tnTkipaiTtpaty

thtwtntnkTtpaty

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

We limit its bandwidth by using a pulse shaping filter

• Neglecting noise, would like y(t) = g(t) * c(t) * h(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

14 - 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

14 - 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 h(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

14 - 21

Bit Error Probability for 2-PAM

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

w(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 filterw(t)

r(t) r(t) rn k

bk Tktgats ) ( )(

)()()( twtstr

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

Bit Error Probability for 2-PAM

• Noise power at matched filter output

14 - 22

dnTwhnTv

)()()(

22 )()()( dnTwhEnTvE

})()()()({ 212211

ddnTwgnTwgE TT

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

TdHdh

sym

sym

22/

2/

2222 )(2

1)(

Noise power

T = TsymFiltered noise

2 (1–2)

14 - 23

Bit Error Probability for 2-PAM

• Symbol amplitudes of +A and -A• Rectangular pulse shape with amplitude 1

• Bit duration (Tb) of 1 second

• Matched filtering with gain of one (see slide 14-15)Integrate received signal over nth bit period and sample

n

n

n

n

n

n

vA

dttwA

dttrr

)(

)(

1

1

0-

Anr

)( nr rPn

AProbability density function (PDF)

14 - 24

Av

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

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)

0 /A

/ nvforPDF

Tb = 1

14 - 25

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

14 - 26

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 error exponentiallydecreases with SNR (see slide 8-16)

)/())(|error( AQAnTsP b

2

1)(

2

e

Q

positive largefor

Tb = 1

14 - 27

PAM Symbol Error Probability

• Set symbol time (Tsym) to 1 second

• Average transmitted signal power

GT() square root raised cosine spectrum

• M-level PAM symbol amplitudes

• With each symbol equally likely

}{|)(| 2

1 }{ 222

nTnSignal aEdGaEP

3

)1( )12(2

1 2

22

1

22

12

2 dMid

Ml

MP

M

i

M

Mi

iSignal

2, ,0 , ,1

2 ),12(

MMiidli ......

2-PAM

d

-d

4-PAM

Constellation points with receiver

decision boundaries

d

d

3 d

3 d

14 - 28

2

2

2

1 0

2/

2/

0 Nd

NP

sym

sym

Noise

nnn var

PAM Symbol Error Probability

• Noise power and SNR

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

• Consider M-2 inner levels in constellationError only if

where

Probability of error is

• Consider two outer levels in constellation

dvn ||

d

QdvP n 2)|(|

2/02 N

d

QdvP n )(

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

14 - 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

14 - 30

Visualizing ISI

• Eye diagram is empirical measure of signal quality

• Intersymbol interference (ISI):

Raised cosine filter has zeroISI when correctly sampled

knk

k

symsymknsymsymk g

kTnTgaagTkTnganx

)0(

)( )0() ( )(

nkk

sym

nkk

symsym

g

kTgdM

g

kTnTgdMD

,, )0(

)( )1(

)0(

)( )1(

14 - 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

14 - 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.