Baseband ReceiverReceiver Design:

DemodulationMatched FilterCorrelator Receiver

DetectionMax. Likelihood Detector

Probability of Error

Transmit and Receive Formatting

Sources of Error in received Signal Major sources of errors:

Thermal noise (AWGN) disturbs the signal in an additive fashion (Additive)

has flat spectral density for all frequencies of interest (White)

is modeled by Gaussian random process (Gaussian Noise)

Inter-Symbol Interference (ISI) Due to the filtering effect of transmitter, channel and

receiver, symbols are “smeared”.

FREQUENCYDOWN

CONVERSIONTRANSMITTEDWAVEFORM

AWGN

RECEIVEDWAVEFORM RECEIVING

FILTEREQUALIZINGFILTER THRESHOLD

COMPARISON

FORBANDPASSSIGNALS

COMPENSATIONFOR CHANNELINDUCED ISI

DEMODULATE & SAMPLE SAMPLEat t = T

DETECT

MESSAGESYMBOL

ORCHANNELSYMBOL

ESSENTIAL

OPTIONAL

Demodulation/Detection of digital signals

Receiver Structure

Receiver Structure contd The digital receiver performs two basic

functions: Demodulation Detection

Why demodulate a baseband signal??? Channel and the transmitter’s filter causes ISI which

“smears” the transmitted pulses Required to recover a waveform to be sampled at t =

nT. Detection

decision-making process of selecting possible digital symbol

Important Observation Detection process for bandpass signals is

similar to that of baseband signals. WHY??? Received signal for bandpass signals is

converted to baseband before detecting Bandpass signals are heterodyned to baseband

signals Heterodyning refers to the process of frequency

conversion or mixing that yields a spectral shift in frequency.

For linear system mathematics for detection remains same even with the shift in frequency

Steps in designing the receiver Find optimum solution for receiver design with

the following goals: 1. Maximize SNR2. Minimize ISI

Steps in design: Model the received signal Find separate solutions for each of the goals.

Detection of Binary Signal in Gaussian Noise

The recovery of signal at the receiver consist of two parts Filter

Reduces the received signal to a single variable z(T) z(T) is called the test statistics

Detector (or decision circuit) Compares the z(T) to some threshold level 0 , i.e.,

where H1 and H0 are the two possible binary hypothesis0)(0

1

H

H

Tz

Receiver Functionality The recovery of signal at the receiver consist of two

parts:1. Waveform-to-sample transformation

Demodulator followed by a samplerAt the end of each symbol duration T, pre-detection point

yields a sample z(T), called test statistic

Where ai(T) is the desired signal component, and no(T) is the noise component

2. Detection of symbolAssume that input noise is a Gaussian random process and

receiving filter is linear

2,1)()()( 0 itntaTz i

2

0

0

00 2

1exp2

1)(nnp

Finding optimized filter for AWGN

channelAssuming Channel with

response equal to impulse function

Detection of Binary Signal in Gaussian Noise For any binary channel, the transmitted signal over a

symbol interval (0,T) is:

The received signal r(t) degraded by noise n(t) and possibly degraded by the impulse response of the channel hc(t), is

Where n(t) is assumed to be zero mean AWGN process For ideal distortionless channel where hc(t) is an

impulse function and convolution with hc(t) produces no degradation, r(t) can be represented as:

10)(00)(

)(1

0

binaryaforTttsbinaryaforTtts

tsi

2,1)()(*)()( itnthtstr ci

Ttitntstr i 02,1)()()(

Design the receiver filter to maximize the SNR

Model the received signal

Simplify the model: Received signal in AWGN

)(thc)(tsi

)(tn

)(tr

)(tn

)(tr)(tsiIdeal channels

)()( tthc

AWGN

AWGN

)()()()( tnthtstr ci

)()()( tntstr i

Find Filter Transfer Function H0(f)

Objective: To maximizes (S/N)T and find h(t) Expressing signal ai(t) at filter output in terms of filter transfer function H(f)

where H(f) is the filter transfer funtion and S(f) is the Fourier transform of input signal s(t)If the two sided PSD of i/p noise is N0/2 Output noise power can be expressed as:

Expressing (S/N)T :

dfefSfHta ftji

2)()()(

dffHN 202

0 |)(|2

dffHN

dfefSfH

NS

fTj

T 20

22

|)(|2

)()(

For H(f) = Hopt (f) to maximize (S/N)T use Schwarz’s Inequality:

Equality holds if f1(x) = k f*2(x) where k is arbitrary constant and * indicates complex conjugate

Associate H(f) with f1(x) and S(f) ej2 fT with f2(x) to get:

Substitute yields to:

dffSdffHdfefSfH fTj222

2 )()()()(

dxxfdxxfdxxfxf2

2

2

1

2

21 )()()()(

dffSNN

S

T

2

0

)(2

Or and energy E of the input signal s(t):

Thus (S/N)T depends on input signal energyand power spectral density of noise andNOT on the particular shape of the waveform

Equality for holds for optimum filter transfer function H0(f) such that: (3.55)

For real valued s(t):

0

2maxNE

NS

T

dffSE2

)(

0

2maxNE

NS

T

fTjefkSfHfH 20 )(*)()(

fTjefkSth 21 )(*)(

whereelse

TttTkSth

00)(

)(

The impulse response of a filter producing maximum output signal-to-noise ratio is the mirror image of message signal s(t), delayed by symbol time duration T.

The filter designed is called a MATCHED FILTER

Defined as:a linear filter designed to provide the maximum signal-to-noise power ratio at its output for a given transmitted symbol waveform

whereelse

TttTkSth

00)(

)(

Matched Filter Output of a rectangular Pulse

Replacing Matched filter with Integrator

A filter that is matched to the waveform s(t), has an impulse response

h(t) is a delayed version of the mirror image (rotated on the t = 0 axis) of the original signal waveform

whereelse

TttTkSth

00)(

)(

Signal Waveform Mirror image of signal waveform

Impulse response of matched filter

Correlation realization of Matched filter

Correlator Receiver This is a causal system

a system is causal if before an excitation is applied at time t = T, the response is zero for - < t < T

The signal waveform at the output of the matched filter is

Substituting h(t) to yield:

When t=T

So the product integration of rxd signal with replica of transmitted waveform s(t) over one symbol interval is called Correlation

dthrthtrtzt

)()()(*)()(0

dtTsr

dtTsrtzt

t

0

0

)(

)()()(

dsrtz T )()()( 0

Correlator versus Matched Filter The functions of the correlator and matched filter The mathematical operation of Correlator is correlation, where a

signal is correlated with its replica Whereas the operation of Matched filter is Convolution, where

signal is convolved with filter impulse response But the o/p of both is same at t=T so the functions of correlator

and matched filter is same.

Matched Filter

Correlator

Implementation of matched filter receiver

Mz

z1

z)(tr

)(1 Tz)(*

1 tTs

)(* tTsM )(TzM

z

Bank of M matched filters

Matched filter output:Observation

vector

)()( tTstrz ii Mi ,...,1

),...,,())(),...,(),(( 2121 MM zzzTzTzTz z

Implementation of correlator receiver

dttstrz i

T

i )()(0

T

0

)(1 ts

T

0

)(ts M

Mz

z1

z)(tr

)(1 Tz

)(TzM

z

Bank of M correlators

Correlators output:Observation

vector

),...,,())(),...,(),(( 2121 MM zzzTzTzTz z

Mi ,...,1

Example of implementation of matched filter receivers

2

1

z

zz

)(tr

)(1 Tz

)(2 Tz

z

Bank of 2 matched filters

T t

)(1 ts

T t

)(2 tsT

T0

0

TA

TA

TA

TA

0

0

DetectionMax. Likelihood Detector

Probability of Error

Detection Matched filter reduces the received signal to a single variable

z(T), after which the detection of symbol is carried out The concept of maximum likelihood detector is based on

Statistical Decision Theory It allows us to

formulate the decision rule that operates on the data optimize the detection criterion

0)(0

1

H

H

Tz

P[s0], P[s1] a priori probabilities These probabilities are known before transmission

P[z] probability of the received sample

p(z|s0), p(z|s1) conditional pdf of received signal z, conditioned on the class si

P[s0|z], P[s1|z] a posteriori probabilities After examining the sample, we make a refinement of our

previous knowledge P[s1|s0], P[s0|s1]

wrong decision (error) P[s1|s1], P[s0|s0]

correct decision

Probabilities Review

Maximum Likelihood Ratio test and Maximum a posteriori (MAP) criterion:If

else

Problem is that a posteriori probabilities are not known. Solution: Use Bay’s theorem:

010 )|()|( Hzspzsp

)()|()()|( 00110011

0

1

0

1

)()()|(

)()()|(

sPszpsPszpH

H

H

H

zPsPszp

zPsPszp

How to Choose the threshold?

101 )|()|( Hzspzsp

)()()|(

)|( zpispiszpzisp

This means that if received signal is positive, s1 (t) was sent, else s0 (t) was sent

1

Likelihood of So and S1

MAP criterion:

)()()(

)|()|(

)(1

0

0

1

0

1

LRTtestratiolikelihoodsPsP

szpszp

zLH

H

When the two signals, s0(t) and s1(t), are equally likely, i.e., P(s0) = P(s1) = 0.5, then the decision rule becomes

This is known as maximum likelihood ratio test because we are selecting

the hypothesis that corresponds to the signal with the maximum likelihood.

In terms of the Bayes criterion, it implies that the cost of both

types of error is the same

testratiolikelihoodH

H

szpszp

zL max1)|()|(

)(0

1

0

1

Substituting the pdfs

2

0

0

000 2

1exp2

1)|(:azszpH

2

0

1

011 2

1exp2

1)|(:azszpH

1

21exp

21

21exp

21

1)|()|()(

0

1

202

00

212

0

0

1

0

1

H

H

az

az

H

H

szpszpzL o

02

)()()}(ln{

0

1

20

20

21

20

01

H

H

aaaazzL

20

010120

20

21

0

1

20

01

2))((

2)()(

aaaaaa

H

H

aaz

12

)()(exp 20

20

21

20

01

aaaaz

Hence:

Taking the log, both sides will give

Hence

where z is the minimum error criterion and 0 is optimum threshold For antipodal signal, s1(t) = - s0 (t) a1 = - a0

)(2))((

0120

010120

0

1

aaaaaa

H

H

z

001

0

1

2)(

aa

H

H

z

0

0

1

H

H

z

Probability of Error

Error will occur if s1 is sent s0 is received

s0 is sent s1 is received

The total probability of error is sum of the errors

dzszpseP

sePsHP

0 )|()|(

)|()|(

11

110

dzszpseP

sePsHP

0

)|()|(

)|()|(

00

001

)()|()()|(

)()|()()|(),(

001110

0011

2

1

sPsHPsPsHP

sPsePsPsePsePPi

iB

If signals are equally probable

Hence, the probability of bit error PB, is the probability that an incorrect hypothesis is made

Numerically, PB is the area under the tail of either of the conditional distributions p(z|s1) or p(z|s0)

)|()|(21

)()|()()|(

0110

001110

sHPsHP

sPsHPsPsHPPB

dzaz

dzszpdzsHPPB

2

0

0

0

001

21exp

21

)|()|(

0

0 0

The above equation cannot be evaluated in closed form (Q-function)

Hence,

duu

dzdudzduthenazulet

dzazP

aa

B

2exp

21

1

21exp

21

2

2)(

000

0

2

0

0

0

0

01

0

0

01

2aaQPB

Co-error function Q(x) is called the complementary error

function or co-error function Is commonly used symbol for probability

Another approximation for Q(x) for x>3 is as follows:

Q(x) is presented in a tabular form

Co-error Table

To minimize PB, we need to maximize:

or Where (a1-a2) is the difference of desired signal components

at filter output at t=T, and square of this difference signal is the instantaneous power of the difference signal

i.e. Signal to Noise Ratio

00 22

21

2 NEQ

NEQSNRQP dd

B

20

201

aa

0

01

2aa

00

20

201 2

2NE

NEaa

NS dd

T

Imp. Observation