CHAPTER 5
Performance of
Digital
Communications
System
School of Computer and Communication Engineering,
Amir Razif B. Jamil Abdullah
EKT 431: Digital Communications
Chapter Overview
Error performance degradation
Detection of signals in Gaussian noise
Matched filter receiver
Optimizing error performance
Error probability performance of binary
signaling
Introduction
The received waveform are in the pulse-shape form. And yet
the demodulator need to recover the pulse waveform.
 Reason: The arriving waveform are not in the ideal pulse
shapes.
 Filtering caused ISI and signals appear to be “smeared” and
not ready for sampling and detection.
 Demodulator goal ~ to recover baseband pulse with best
SNR and free of ISI.

Error performance Degradation


Page 118 text
Detector ~ retrieve the bit stream from the received waveform as
error free as possible.

Primary causes of error performance degradation;
 Effect of filtering ~ at transmitted channel and receiver
 Non ideal transfer function ~ caused ”smearing “ or ISI
 Electrical noise & interference ~ galaxy and atmospheric
noise, switching transient, intermodulation noise, signal
from other source. * thermal noise cannot be elaminated.

In digital communications
 Depends on Eb/No
Error performance Degradation

Eb/No is a measure of normalized signal-to-noise
ratio (SNR)





SNR ~ refers to average signal power to average noise
power ratio (S?N or SNR).
In digital communication Eb/No a normalize version of
SNR.
Where Eb is the bit energy can be describe as signal power
S times the bit time Tb
N0 is noise power spectral density; noise power divide by
bandwidth W.
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.
Error performance Degradation
•Example:
Probability of symbol error for M-PSK
~ One of the performance in digital communication system is
the plot of bit error probability Pb versus Eb/No.
Pb
2006-02-14
Eb / N0 Lecture
dB8
6
Error performance Degradation


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.
Error performance Degradation

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 and analyzed as if the transmission
channel is a BASEBAND CHANNEL.
Error performance Degradation
Figure 5.1: Two basic steps in demodulation & detection of digital
signals.
Detection of signals in
Gaussian
noise
 Pg 119 text

Maximum likelihood receiver structure
 The decision making criterion in step2 Figure 5.1 was
described by equation 3.7. A popular criterion for
choosing the threshold level γ for the binary decision
which is 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.
Error performance Degradation

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

Matched filter ~ a linear filter designed to provide maximum
signal-to-noise power ratio at its output for a given
transmitted symbol waveform.

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.
Matched Filter

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
Matched Filter

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.
Matched Filter

Correlation realization of the matched filter
 Impulse response of the filter
Matched Filter


Correlator and matched filter
The impulse response of filter is a delay version of the mirror image
(rotate on the t=0 axis) of the signal waveform.
Figure 5.2: Correlator and matched filter (a) Matched filter characteristics (b)
Comparison of matched filter outputs.
Matched Filter

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
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 known 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
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.

Matched Filter versus
Conventional Filters

In general
 Conventional filters :
~ isolate & extract a high fidelity estimate of the signal for
presentation to the matched filter

Matched filters :
~ gathers the signal energy and when its output is
sampled, a voltage proportional to that energy is produced
for subsequent detection & post-detection processing.
Optimizing error performance

Text Pg 127

To optimize PB, in the context of AWGN channel & the Rx
shown in figure below, need to select the optimum
 receiving filter in waveform to sample transformation
(step 1)


And the optimum decision threshold (step 2)
For binary case the optimum decision threshold given as
 -
Example 5.1: Bandwidth Requirement (a)
Find a minimum required bandwidth for the baseband transmission of a four level
PAM pulse sequence having a data rate of R = 2400 bits/s if the system transfer
characteristic 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 5.2: Bandwidth Requirement (b)
The same 4-ary PAM sequence is modulated onto a carrier wave, so that the
baseband spectrum is shifted and centered at frequency f0. Find the minimum
required DSB bandwidth for transmitting the modulated PAM sequence. Assume
that 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
Optimizing error performance

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
~ the square of (a1 –a2) is the instantaneous power of the
different signal.


so, an output SNR
A matched filter is the one maximize the output of the
SNR.
2Ed/N0 is the maximum possible output of SNR.
Optimizing error performance

Binary signal vectors
 Eb (1   ) 
PB  Q 

N0



Antipodal
 r=-1; correspond to two signals are “anticorrelated”
 The angle between the signal vectors is 180°
 Vectors are mirror images

Orthogonal
 Angle between the signal vectors is 90°
 Vectors are in “L shape”
Optimizing error performance
Binary signal vectors
Antipodal
Orthogonal
Error probability performance
of binary signaling

Unipolar signaling
 Baseband orthogonal signaling
~ by definition, it Requires S1(t) and S2(t) to have “0”
(zero) correlation over each symbol time duration.
Error probability performance of
binary signaling
Error probability performance of
binary signaling
Bit error performance at the output, PB
Average energy per bit, Eb
Error probability performance of
binary signaling

Bipolar signaling
 Baseband
antipodal signaling
 Binary signals that are mirror images of one
another, S1(t) = - S2(t)
Error probability performance of
binary signaling
Error probability performance of
binary signaling

Bit error performance at output, PB

Average energy per bit, Eb
Error probability performance of
binary signaling

Bit error performance of unipolar & bipolar signaling
Example 5.3: Matched Filter Detection of Antipodal
Signals
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 s 1(t) and s2(t) from the
plot below. E  3 v 2 (t )dt
b

0
 (10 3V ) 2 * (10 6 s )  (2 *10 3V ) 2 * (10 6 s )  (10 3V ) 2 * (10 6 s )
 6 *10 12 joule
The waveform is antipodal, we can find the bit error probability as
 12 *10 12
Pb  Q
12

10

From the table B.1 Pb=3*10-4

  Q 12  Q(3.46)

