Having the understanding of some of the maths behind what matched filtering does when detecting a single symbol buried in noise, you will now implement the matched filter in Matlab for detecting the given symbol waveform shown in Fig in the presence of noise and test how the filter works

**Numircals matlab project**, Create the signal waveform S1 = s_{1}(t) shown in Fig 1(a) representing symbol 1. To simulate the effect of a noisy channel add White Gaussian noise of single sided power spectral density N =0.8 Watts/Hz to the signal using the following command:

S1 = S1 + randn(1,length(S1))*sqrt(power); where power = 0.4.

Plot this signal at the input of the receiver ie: the noisy waveform you have generated

Create the impulse response h_{opt}(t) of the optimum receiver filter to detect this noisy symbol produced in (k). ie: design the matched filter to detect the symbol. Plot the impulse response h_{opt}(t).

Filter the received noisy signal using the optimum receiver filter you designed. Plot output of y_{1}(t) of the optimum receiver filter. **Online matlab experts, **compare this Matlab simulated output with what you sketched for y_{1}(t) in part (d) above and comment on the comparison.

**Numerical projects in matlab**, Compare also the input to the matched filter (ie the noisy s1(t) ) plotted in (k) and the output of the matched filter y1(t) plotted in (m). Observe how the matched filter reduces the noise and peaks the signal so that the presence of the symbol is more evident. Note also how the matched filter maximise the signal to noise ratio at the end of the symbol period by peaking its output at this particular instant at which you can sample the signal to detect the bit.

**online matlab assignment helper **Determine the peak value of the output of the receiver from the Matlab simulated output in (m) and the time instant during which this peak occurs and compare with part (e) above which you obtained from your theoretical sketch. Comment on the comparison. What is the peak instantaneous output signal power from the Matlab simulation?

Now you will compute the peak instantaneous signal power-to-average noise power ratio of your Matlab implemented filter receiver output and compare this with the theoretical value of (So/No) you have found in (i).

First apply the signal waveform only (no noise) to the receiver input and determine the peak output signal power. Next apply the noise only (ie: white Gaussian noise of single sided power spectral density N =0.8) to the receiver and determine the output noise power. Hence determine the Matlab simulated peak instantaneous signal power-to-average noise power ratio (S_{o}/N_{o})_{.} Compare this value of (S_{o}/N_{o}) obtained from the Matlab simulations with the theoretical value of (S_{o}/N_{o}) found in (i) and comment on the result

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