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

2. Chapter Overview Error performance degradation
Detection of signals in Gaussian noise
Matched filter receiver
Optimizing error performance
Error probability performance of binary signaling

3. 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.

4. 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

5. 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
Through the decrease of the desired signal power.
Through the increase of noise power or interfering signal.

6. 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.
Lecture 8

7. 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.

8. 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.

9. Error performance Degradation
Figure 5.1: Two basic steps in demodulation & detection of digital signals.

10. 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.

11. 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.

12. 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.

13. 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

14. 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.

15. Matched Filter Correlation realization of the matched filter
Impulse response of the filter

16. 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.

17. 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

18. Matched Filter versus Conventional Filters
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
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.

19. 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.

20. 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 -

21. 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.

22. 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

23. 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.

24. Optimizing error performance
Binary signal vectors
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”

25. Optimizing error performance
Binary signal vectors
Antipodal
Orthogonal

26. 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.

27. Error probability performance of binary signaling

28. Error probability performance of binary signaling
Bit error performance at the output, PB
Average energy per bit, Eb

29. Error probability performance of binary signaling
Bipolar signaling
Baseband antipodal signaling
Binary signals that are mirror images of one another, S1(t) = - S2(t)

30. Error probability performance of binary signaling

31. Error probability performance of binary signaling
Bit error performance at output, PB
Average energy per bit, Eb

32. Error probability performance of binary signaling
Bit error performance of unipolar & bipolar signaling

33. 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 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