Performance of Digital Communications System CHAPTER 5 School of Computer and Communication Engineering, Amir Razif B. Jamil Abdullah EKT 431: Digital.

Performance Performance of Digital of Digital

CommunicatioCommunications Systemns System

CHAPTER CHAPTER 55

School of Computer and Communication School of Computer and Communication Engineering, Amir Razif B. Jamil AbdullahEngineering, Amir Razif B. Jamil Abdullah

EKT 431: Digital EKT 431: Digital Communications Communications

Chapter OverviewChapter OverviewError performance degradationError performance degradationDetection of signals in Gaussian Detection of signals in Gaussian noisenoiseMatched filter receiverMatched filter receiverOptimizing error performanceOptimizing error performanceError probability performance of Error probability performance of binary signalingbinary signaling

Introduction Introduction The received waveform are in the pulse-The received waveform are in the pulse-

shape form. shape form. And yetAnd yet the demodulator need the demodulator need to recover the pulse waveform.to recover the pulse waveform.

Reason: The arriving waveform are not in Reason: The arriving waveform are not in the ideal pulse shapes.the ideal pulse shapes.

Filtering caused ISI and signals appear to be Filtering caused ISI and signals appear to be “smeared” and not ready for sampling and “smeared” and not ready for sampling and detection.detection.

Demodulator goal ~ to recover baseband Demodulator goal ~ to recover baseband pulse with best SNR and free of ISI.pulse with best SNR and free of ISI.

Error performance Error performance Degradation Degradation

Page 118 textPage 118 text Detector ~ retrieve the bit stream from the Detector ~ retrieve the bit stream from the

received waveform as error free as possible.received waveform as error free as possible. Primary causes of error performance degradation;Primary causes of error performance degradation;

Effect of filtering Effect of filtering ~ at transmitted channel and ~ at transmitted channel and receiverreceiver

Non ideal transfer function Non ideal transfer function ~ caused ”smearing “ ~ caused ”smearing “ or ISI or ISI

Electrical noise & interference Electrical noise & interference ~ galaxy and ~ galaxy and atmospheric noise, switching transient, atmospheric noise, switching transient, intermodulation noise, signal from other intermodulation noise, signal from other source. * thermal noise cannot be elaminated.source. * thermal noise cannot be elaminated.

In digital communicationsIn digital communications Depends on EDepends on Ebb/N/Noo

EEbb/N/No o is a measure of normalized signal-to-noise is a measure of normalized signal-to-noise ratio (SNR)ratio (SNR) SNR ~ SNR ~ refers to average signal power to refers to average signal power to

average noise power ratio (S?N or SNR).average noise power ratio (S?N or SNR). In digital communication In digital communication EEbb/N/Noo a normalize a normalize

version of SNR. version of SNR. Where Where EEbb is the bit energy can be describe as is the bit energy can be describe as

signal power signal power SS times the bit time times the bit time TTbb

NN00 is noise power spectral density; noise is noise power spectral density; noise power divide by bandwidth power divide by bandwidth WW. .

Can be degrade in two waysCan be degrade in two ways

1.1. Through the decrease of the desired signal Through the decrease of the desired signal power.power.

2.2. Through the increase of noise power or Through the increase of noise power or interfering signal.interfering signal.


Lecture 8Lecture 8 662006-02-142006-02-14

•Example: Probability of symbol error Example: Probability of symbol error for M-PSKfor M-PSK~ ~ One of the performance in digital One of the performance in digital communication system is the plot of bit error communication system is the plot of bit error probabilityprobability P Pbb versusversus E Ebb/N/No.o.

bP

dB / 0NEb


Linear system – the mathematics of detection is Linear system – the mathematics of detection is unaffected by a shift in frequency.unaffected by a shift in frequency.

Equivalent theorem Equivalent theorem

Performing bandpass linear signal Performing bandpass linear signal processing, followed by heterodyning the processing, followed by heterodyning the signal to basebandsignal to baseband

yields the same result asyields the same result as

heterodyning the bandpass signal to heterodyning the bandpass signal to baseband, followed by baseband linear signal baseband, followed by baseband linear signal processing.processing.


Heterodyning ~Heterodyning ~ a frequency conversion or a frequency conversion or mixing process that yields a spectral shift in mixing process that yields a spectral shift in the signal. the signal.

The performance of most digital communication The performance of most digital communication systems will often be described and analyzed as if systems will often be described and analyzed as if the transmission channel is a the transmission channel is a BASEBAND CHANNELBASEBAND CHANNEL..


Figure 5.1: Two basic steps in demodulation & Figure 5.1: Two basic steps in demodulation & detection of digital signals.detection of digital signals.


Detection of signals in Detection of signals in Gaussian noiseGaussian noise

Pg 119 textPg 119 text Maximum likelihood receiver structureMaximum likelihood receiver structure

The decision making criterion in step2 The decision making criterion in step2 Figure 5.1Figure 5.1 was described by was described by equation 3.7equation 3.7. A popular criterion . A popular criterion for choosing the threshold level for choosing the threshold level γγ for the binary for the binary decision which is is based on minimizing the decision which is is based on minimizing the probability of error.probability of error.

The computation for minimum error value of The computation for minimum error value of γγ = =

γγ0 0 starts with forming an inequality expression starts with forming an inequality expression between the ratio of conditional probability between the ratio of conditional probability density functions and the signal a priori density functions and the signal a priori probabilities.probabilities.

The threshold The threshold γγ0 0 is theis the optimum threshold for optimum threshold for minimizing the probability of making an incorrect minimizing the probability of making an incorrect decision - decision - minimum error criterion.minimum error criterion.

A detector that minimizes the error probabilityA detector that minimizes the error probability - - maximum likelihood detector.maximum likelihood detector.

Note : Further reading – page 120, 121 & 122 Note : Further reading – page 120, 121 & 122 textbook.textbook.


Matched FilterMatched Filter Matched filter ~ Matched filter ~ a linear filter designed to a linear filter designed to

provide provide maximum signal-to-noise power ratiomaximum signal-to-noise power ratio at its output for a given transmitted symbol at its output for a given transmitted symbol waveform.waveform.

DefinitionDefinition A filter which immediately precedes circuit A filter which immediately precedes circuit

in a digital communications receiver is said in a digital communications receiver is said to be matched to a particular symbol pulse, to be matched to a particular symbol pulse, if it maximizes the output SNR at the if it maximizes the output SNR at the sampling instant when that pulse is present sampling instant when that pulse is present at the filter input.at the filter input.

The ratio of the instantaneous signal power to The ratio of the instantaneous signal power to average noise power,(S/N)average noise power,(S/N)TT

where where

aaii ~ ~ is signal componentis signal component

σσ²²00 ~ ~ is variance of the output noiseis variance of the output noise

Matched FilterMatched Filter

The maximum output (S/N)The maximum output (S/N)TT depends on the depends on the input input signal energysignal energy and the and the power spectral densitypower spectral density of of noise, noise, not on the particular shapenot on the particular shape of the waveform of the waveform that is used. that is used.


Correlation realization of the matched filterCorrelation realization of the matched filter Impulse response of the filterImpulse response of the filter


Correlator and matched filterCorrelator and matched filter The impulse response of filter is a delay version of the The impulse response of filter is a delay version of the mirror mirror

imageimage (rotate on the t=0 axis) of the signal waveform. (rotate on the t=0 axis) of the signal waveform.

Figure 5.2: Correlator and matched filter (a) Matched Figure 5.2: Correlator and matched filter (a) Matched filter characteristics (b) Comparison of matched filter filter characteristics (b) Comparison of matched filter

outputs.outputs.


Comparison of Comparison of convolution & correlationconvolution & correlation Matched FilterMatched Filter

The mathematical operation of MF is The mathematical operation of MF is Convolution Convolution

– – a signal is convolved with the impulse a signal is convolved with the impulse response of a filter.response of a filter.

The output of MF approximately sine wave that The output of MF approximately sine wave that is amplitude modulated by linear ramp during is amplitude modulated by linear ramp during the same time interval.the same time interval.

CorrelatorCorrelatorThe mathematical operation of correlator is The mathematical operation of correlator is

correlation – a signal is correlated with a replica correlation – a signal is correlated with a replica itself.itself.

The output is approximately a linear ramp The output is approximately a linear ramp during the interval 0 ≤ t ≤ Tduring the interval 0 ≤ t ≤ T


Matched Filter versus Matched Filter versus Conventional FiltersConventional Filters

Matched FilterMatched Filter Template that Template that matched tomatched to

the known shape of the the known shape of the signal being processed.signal being processed.

Maximizing the SNRMaximizing the SNR of a of a known signals in the known signals in the presence of AWGN.presence of AWGN.

Applied to Applied to known signalsknown signals with random parameters.with random parameters.

ModifyModify the the temporal temporal structure by gathering the structure by gathering the signal energy matched to its signal energy matched to its template & presenting the template & presenting the result as a peak amplitude.result as a peak amplitude.

Conventional FilterConventional Filter Screen outScreen out unwanted unwanted

spectral components.spectral components. Designed to provide Designed to provide

approximatelyapproximately uniform uniform gain, minimum gain, minimum attenuation.attenuation.

Applied to Applied to random random signals defined only by signals defined only by their bandwidth.their bandwidth.

Preserve the temporalPreserve the temporal or spectral structure of or spectral structure of the signal of interest.the signal of interest.

In generalIn general Conventional filters : Conventional filters :

~ isolate & extract a high fidelity estimate ~ isolate & extract a high fidelity estimate of the signal for presentation to the of the signal for presentation to the matched filtermatched filter

Matched filters : Matched filters :

~ gathers the signal energy and when its ~ gathers the signal energy and when its output is sampled, a voltage proportional to output is sampled, a voltage proportional to that energy is produced for subsequent that energy is produced for subsequent detection & post-detection processing.detection & post-detection processing.

Matched Filter versus Matched Filter versus Conventional FiltersConventional Filters

Optimizing error Optimizing error performanceperformance

Text Pg 127Text Pg 127 To optimize PTo optimize PB, B, in the context of AWGN channel & in the context of AWGN channel &

the Rx shown in figure below, need to select the the Rx shown in figure below, need to select the optimum optimum receiving filter in waveform to sample receiving filter in waveform to sample

transformation transformation (step 1) (step 1) And the optimum decision thresholdAnd the optimum decision threshold (step 2)(step 2)

For binary case the optimum decision threshold For binary case the optimum decision threshold given asgiven as --

Find a minimum required bandwidth for the baseband transmission of a four levelPAM pulse sequence having a data rate of R = 2400 bits/s if the system transfercharacteristic consists of a raised-cosine spectrum with 100% excess bandwidth(r = 1).

Solution 1-43:

M = 2k; since M = 4 levels, k = 2.

Symbol or pulse rate Rs = r/k = 2400/2 = 1200 symbols/s

Minimum bandwidth W = 1/2(1+r)Rs = 1/2(2)(1200) = 1200Hz

Figure 3.19a (text) ~ baseband received pulse in time domain

Figure 3.19b (text) ~ Fourier transform of h(t)*Note that bandwidth starts at zero frequency and extend to f=1/T twice the size of Nyquist theretical minimum bandwidth.

Example 5.1:Example 5.1: Bandwidth Requirement Bandwidth Requirement (a)(a)

The same 4-ary PAM sequence is modulated onto a carrier wave, so that thebaseband spectrum is shifted and centered at frequency f0. Find the minimumrequired DSB bandwidth for transmitting the modulated PAM sequence. Assumethat the system transfer characteristic is same as in part .

Solution1-43:

From above example (a)

Rs= 1200 symbols/s

WDSB=(1+r)Rs = 2(1200) =2400 Hz

Continue in class

Example 5.2:Example 5.2: Bandwidth Requirement Bandwidth Requirement (b)(b)

For minimizing For minimizing PPBB need to choose the matched filter need to choose the matched filter that maximizes the argument of that maximizes the argument of Q(xQ(x)) that maximizes that maximizes

wherewhere

(a(a11 –a –a22) ~ ) ~ is the difference of the desired signal is the difference of the desired signal components at the filter output at time components at the filter output at time t = Tt = T

~ the square of ~ the square of (a(a11 –a –a22)) is the instantaneous is the instantaneous power of the different signal.power of the different signal.

so, an output SNRso, an output SNR A matched filter is the one maximize the output of A matched filter is the one maximize the output of

the SNR.the SNR. 2E2Edd/N/N00 is the maximum possible output of SNR. is the maximum possible output of SNR.


Optimizing error Optimizing error performanceperformance Binary signal vectorsBinary signal vectors

AntipodalAntipodalr=-1; correspond to two signals are r=-1; correspond to two signals are

“anticorrelated”“anticorrelated”The angle between the signal vectors is The angle between the signal vectors is

180°180°Vectors are mirror imagesVectors are mirror images

OrthogonalOrthogonalAngle between the signal vectors is 90°Angle between the signal vectors is 90°Vectors are in Vectors are in “L shape”“L shape”

0

)1(

N

EQP b

B

Antipodal

Orthogonal


Binary signal vectorsBinary signal vectors

Error probability Error probability performance of binary performance of binary

signalingsignaling Unipolar signalingUnipolar signaling

Baseband orthogonal signalingBaseband orthogonal signaling

~ by definition, it Requires S~ by definition, it Requires S11(t) and S(t) and S22(t) to (t) to have “0” (zero) correlation over each have “0” (zero) correlation over each symbol time duration.symbol time duration.


signalingsignaling

Bit error performance at the output, Bit error performance at the output, PPBB

Average energy per bit, Average energy per bit, EEbb


signalingsignaling

Bipolar signalingBipolar signalingBaseband antipodal signalingBaseband antipodal signalingBinary signals that are mirror images of Binary signals that are mirror images of

one another, Sone another, S11(t) = - S(t) = - S22(t) (t)


signalingsignaling


signalingsignaling

Bit error performance at output, Bit error performance at output, PPBB

Average energy per bit, Average energy per bit, EEbb


signalingsignaling

Bit error performance of unipolar & bipolar Bit error performance of unipolar & bipolar signalingsignaling


signalingsignaling

Consider a binary communication system that received equally likely signals s1(t) and s2(t) plus AGWN. See Figure below. Assumed that the receiving filter is matched filter, and that the noise-power spectral density No is equal to 10-12 Watt/Hz Use the value of receive signal voltage and time shown in figure below to compute the bit error probability.

Solution1-30:

We can graphically determine the received energy per bit s1(t) and s2(t) from the plot below.

The waveform is antipodal, we can find the bit error probability as

From the table B.1 Pb=3*10-4

Example 5.3:Example 5.3: Matched Filter Detection of Matched Filter Detection of Antipodal SignalsAntipodal Signals

joule

sVsVsV

dttvEb

12

623623623

3

0

2

10*6

)10(*)10()10(*)10*2()10(*)10(

)(

)46.3(1210

10*1212

12

QQQPb

Performance of Digital Communications System CHAPTER 5 School of Computer and Communication Engineering, Amir Razif B. Jamil Abdullah EKT 431: Digital.

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