Chapter 5
Performance of Digital
Communications System
Chapter Overview
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Error performance degradation
Detection of signals in Gaussian noise
Matched filter receiver
Optimizing error performance
Error probability performance of binary
signaling
Error performance Degradation
 Primary causes
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Effect of filtering
Non ideal transfer function
Electrical noise & interference
 In digital communications
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Depends on Eb/No
Cont’d...

Eb/No is a measure of normalized signal-to-noise
ratio (SNR)
SNR refers to average signal power &
average noise power
 Can be degrade in two ways
1.Through the decrease of the desired
signal power.
2.Through the increase of noise power or
interfering signal.

Cont’d...
 Linear system – the mathematics of
detection is unaffected by a shift in
frequency.
 Equivalent theorem
Performing bandpass linear signal
processing, followed by heterodyning the
signal to baseband
yields the same result as
heterodyning the bandpass signal to baseband,
followed by baseband linear signal processing.
Cont’d...
 Heterodyning – a frequency conversion or
mixing process that yields a spectral shift in
the signal.
 The performance of most digital
communication systems will often be
described & analyzed as if the transmission
channel is a BASEBAND CHANNEL.
Cont’d...
Detection of signals in Gaussian
noise
 Maximum likelihood receiver structure

A popular criterion for choosing the threshold
level γ for the binary decision in Equation 3.7
page 110
is based on minimizing the probability of error.

The computation for minimum error value of γ =
γ0 starts with forming an inequality expression
between the ratio of conditional probability
density functions and the signal a priori
probabilities.
Cont’d...
 The threshold γ0 is the optimum threshold for
minimizing the probability of making an
incorrect decision - minimum error criterion.
 A detector that minimizes the error probability
- maximum likelihood detector.
Note : Further reading – page 120, 121 & 122 textbook.
Matched Filter
 Definition


A filter which immediately precedes circuit in a
digital communications receiver is said to be
matched to a particular symbol pulse, if it
maximizes the output SNR at the sampling
instant when that pulse is present at the filter
input.
A linear filter designed to provide the
maximum signal to noise power ratio at its
output for a given transmitted symbol
waveform.
Cont’d...
 The ratio of the instantaneous signal power to
average noise power,(S/N)T
where
ai is signal component
σ²0 is variance of the output noise
Cont’d...
 The maximum output (S/N)T depends on the
input signal energy and the power spectral
density of noise, not on the particular shape
of the waveform that is used.
Cont’d...
 Correlation realization of the matched filter

Impulse response of the filter
Cont’d...
 Correlator and matched filter
Cont’d...
 Comparison of convolution & correlation
 Matched Filter
 The mathematical operation of MF is Convolution
– a signal is convolved with the impulse response of
a filter.
 The output of MF approximately sine wave that is
amplitude modulated by linear ramp during the same
time interval.
 Correlator
 The mathematical operation of correlator is
correlation – a signal is correlated with a replica
itself.
 The output is approximately a linear ramp during the
interval 0 ≤ t ≤ T
Matched Filter versus Conventional
Filters
 Matched Filter
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Template that matched to the
known shape of the signal being
processed.
Maximizing the SNR of a known
signals in the presence of
AWGN.
Applied to kwon signals with
random parameters.
Modify the temporal structure by
gathering the signal energy
matched to its template &
presenting the result as a peak
amplitude.
 Conventional Filter
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Screen out unwanted
spectral components.
Designed to provide
approximately uniform
gain, minimum
attenuation.
Applied to random
signals defined only by
their bandwidth.
Preserve the temporal or
spectral structure of the
signal of interest.
Cont’d...
 In general
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Conventional filters : isolate & extract a high
fidelity estimate of the signal for presentation
to the matched filter
Matched filters : gathers the signal energy &
when its output is sampled, a voltage
proportional to that energy is produced for
subsequent detection & post-detection
processing.
Optimizing error performance
 To optimize PB, in the context of AWGN
channel & the Rx, need to select the
optimum
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
receiving filter in waveform to sample
transformation (step 1)
Decision threshold (step 2)
 For binary case
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Threshold result
Cont’d...

For minimizing PB need to choose the matched
filter that maximizes the argument of Q(x) that
maximizes
where
(a1 –a2) is the difference of the desired signal
components at the filter output at time t = T
so, an output SNR
Cont’d...
 Binary signal vectors
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Antipodal
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The angle between the signal vectors is 180°
Vectors are mirror images
Orthogonal
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Angle between the signal vectors is 90°
Vectors are in “L shape”
Cont’d...
Antipodal
Orthogonal
Error probability performance of binary
signaling
 Unipolar signaling


Baseband orthogonal signaling
Requires S1(t) & S2(t) have 0 correlation over
each symbol time duration.
Cont’d...
Cont’d...
 Bit error performance at the output, PB
 Average energy per bit, Eb
Cont’d...
 Bipolar signaling
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
Baseband antipodal signaling
Binary signals that are mirror images of one
another, S1(t) = - S2(t)
Cont’d...
Cont’d...
 Bit error performance at output, PB
 Average energy per bit, Eb
Cont’d...
 Bit error performance of unipolar & bipolar
signaling
END OF CHAPTER 5