1. Performance of Digital Communications System
Chapter 5
Performance of Digital Communications System

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. Error performance Degradation
Primary causes
Effect of filtering
Non ideal transfer function
Electrical noise & interference
In digital communications
Depends on Eb/No

4. Cont’d... SNR refers to average signal power & average noise power
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
Through the decrease of the desired signal power.
Through the increase of noise power or interfering signal.

5. 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.

6. 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.

7. Cont’d...

8. 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.

9. 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.

10. 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.

11. 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

12. 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.

13. Cont’d... Correlation realization of the matched filter
Impulse response of the filter

14. Cont’d...
Correlator and matched filter

15. 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

16. 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 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.

17. Cont’d...
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.

18. Optimizing error performance
To optimize PB, in the context of AWGN channel & the Rx, need to select the optimum
receiving filter in waveform to sample transformation (step 1)
Decision threshold (step 2)
For binary case
Threshold result

19. 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

20. Cont’d... Binary signal vectors Antipodal Orthogonal
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”

21. Cont’d...
Antipodal
Orthogonal

22. Error probability performance of binary signaling
Unipolar signaling
Baseband orthogonal signaling
Requires S1(t) & S2(t) have 0 correlation over each symbol time duration.

23. Cont’d...

24. Cont’d... Bit error performance at the output, PB
Average energy per bit, Eb

25. Cont’d... Bipolar signaling Baseband antipodal signaling
Binary signals that are mirror images of one another, S1(t) = - S2(t)

26. Cont’d...

27. Cont’d... Bit error performance at output, PB
Average energy per bit, Eb

28. Cont’d...
Bit error performance of unipolar & bipolar signaling

29. END OF CHAPTER 5