Performance of Digital Communications System Chapter 5 Performance of Digital Communications System

Chapter Overview 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 Effect of filtering Non ideal transfer function Electrical noise & interference In digital communications Depends on Eb/No

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.

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 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.

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.

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

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 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”

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 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