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Chapter 5 (part 1)
Performance of Digital
Communications System
EKT 357
Digital Communication Engineering
Chapter 5 (part 1) Overview
Error performance degradation
Detection of signals in Gaussian noise
Matched filter
Error performance Degradation
Primary causes
Effect of filtering
Non ideal transfer function
Electrical noise & interference
Thermal noise
In digital communications
Depends on Eb/No
Error performance Degradation
Eb is bit energy and can be described as
signal power S times the bit time Tb. N0 is
noise power spectral density, and can be
described as noise power N divided by
bandwidth W.
Power spectral density (PSD) is the average
power, Px of a real- valued power signal x(t),
defined over the interval/ period.
Where the time average is taken over the
signal period T0 as follows:
Error performance Degradation
Since the bit time and bit rate Rb are
reciprocal, we can replace Tb with 1/Rb and
write
Therefore simplify the notation throughout the
book, by using R instead of Rb to represent
bits/s.
Eb/N0 is just a version of S/N normalized by
bandwidth and bit rate, as follows
Error performance Degradation
The important metrics of performance in
digital communication systems is a plot of the
bit-error probability PB versus Eb/N0.
For Eb/N0 ˃ x0, PB ˂ P0.
Error performance Degradation
Plot of the bit-error probability PB versus
Eb/N0.
Error performance Degradation
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.
Probability Density Function (pdf)
The pdf of the Gaussian random noise n0 can
be expressed as
Where σ0 is the noise variance. Thus, the
conditional pdf p(z|s1) and p(z|s2) can be
expressed as
and
Probability Density Function (pdf)
From the decision-making criterion,
The computation for the minimum error value will be
Consider a1 is the signal component of z(T) when
s1(t) is transmitted, and a2 is the signal component for
z(T) when s2 (t) is transmitted.
The threshold level
, will then represented by (a1 +
a2)/2, which is the optimum threshold for minimizing
the probability of making an incorrect decision.
This strategy is known as the minimum error criterion.
Probability Density Function (pdf)
We can therefore compute PB, by integrating
p(z|s1) between the limit -∞ and ɣ0 , or by
integrating p(z|s2) between the limits ɣ0 and ∞:
ɣ0 = (a1 + a2)/2 is the optimum threshold
Replacing the likelihood p(z|s2) with its
Gaussian equivalent and we have
Where σ0 is the variance of the noise out of
the correlator.
Probability Density Function (pdf)
Let u = (z-a2)/σ0. Then σ0 du = dz and
But x > 3,
approximation
for Q(x) is:
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.
Matched Filter
At time t= T, the sampler output z(T) consists
of a signal component ai and the variance of
the output noise (average noise power) is
denoted by
The ratio of the instantaneous signal power to
average noise power, (S/N)T, at time t= T, out
of the sampler is
Matched Filter
Now, we going to maximizes the (S/N)T by
finding expressing the signal ai(t) at the filter
output in terms of the filter transfer function
H(f) and Fourier transform of the input signal,
s(t) that is S(f). That is:
Then, we express the output noise power as :
Where N0/2 is the input noise.
Matched Filter
Thus, we obtain
Matched Filter (Summary)
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
Correlation realization of the
matched filter
Matched filter’s basic property:
The impulse response of the filter is a
delayed version of the mirror image
(rotated on the t=0 axis) of the signal
waveform.
Therefore, if the signal waveform is s(t), its
mirror image is s(-t), and the mirror image
delayed by T seconds is s(T-t)
Correlation realization of the
matched filter
The process of deriving the matched filter
does not help much in developing an
understanding of what the matched filter is all
about. There is an alternate way to look at
the matched filter process that is much more
intuitive. This method is called the correlation
receiver.
Correlation realization of the
matched filter
Correlator and 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
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 &
when its output is sampled, a voltage
proportional to that energy is produced for
subsequent detection & post-detection
processing.
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.