# Diseño de filtro digital para detección se señal en ruido gaussiano

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DSP Project

I. Background:

Design a digital filter with the goal of detecting whether or not a signal is present in additive
uncorrelated Gaussian noise.

In this project, we would like to design a digital filter to decide whether x[n] = s[n] + w[n], or x[n] = w[n], the only two situations that can occur. x[n] is observed data vector, s[n] is the signal sequence to be detected, andw[n] is zero-mean, unit variance (σn2 = 1), statistically independent, additive Gaussian noise.

In part II, we will try some best ways to do this detection problem, but for part I use your best intuition and use FIR and IIR filter design technique of your choice.

Part 1:

In this part, use your best intuition at this point in the course to specify the desired characteristics of thefrequency response of your filter for this particular application.

Assume that s[n] is a square wave consisting of 10 periods. Each period also consists of 10 samples, the first five values are constant and the last five values are zeros. The length of your data, signal, and noise vectors, in this part, are therefore each 100. We will be considering at least three cases, Es/σn2 equal to 1, 4and 16, where the amplitude of the signal samples are such the Es = ∑99i=0 si2.

Part 2:

Design the optimum detector for the same square-wave sequence of 10 periods, s[n].
The design of the filter will be that of the inverse of the signal.

i. Plot the impulse response of your optimum detector (Filter):
ii. Plot the magnitude and phase response of your optimum detector (Filter):iii. Plot With signal, s[n], but no noise, w[n] = 0, compare and plot the input and output waveforms of your optimum detector (Filter) & Comment.
iv. Let the input be signal with additive noise, x[n] = s[n] + w[n], compare and plot the input and output waveforms of your optimum detector (Filter). Comment.
v. Let the input be noise only, x[n] = w[n], compare and plot the input andoutput waveforms of your optimum detector (Filter). Comment.

Part 3:

Repeat Problem 1 except this time; assume that s[n] is the square wave of 100 periods (instead of 10).

Part 4:

Compare and comment your results from Problems 2 and 3.

II. Problem 1

Derive an expression of the output sequence, y[n], in terms of the observed data sequence x[n] and signal sequence s[n].

x[n]= s[n] + w[n]

h[n] = s[N-1-n]

y[n] = x[n] * h[n] (where * represents Linear Convolution)

Solution:

We see that this particular system is a Linear Time Invariant System, which indicates that the output signal can be acquired from x[n] and h[n] through Linear Convolution. We see that linear convolution is:

y[n] = ∑ { x[k] × h[n-k] }
(Where -∞ ≤ k ≤ +∞; and × representsmultiplication)

Therefore,

y[n] = ∑ { x[k] × h[n-k] }
= ∑ { ( s[k] + w[k] ) × s[N-1-n-k] }
(Where -∞ ≤ k ≤ +∞; and × represents multiplication)

Problem 2 Solution:
noise = randn(1,100);
for i=0:9 % square-wave sequence of 10 periods
for j=1:5
S1(10*i+j)=1; % s(n)
h(10*i+j) =0;
end;
for j=6:10
S1(10*i+j)=0;
h(10*i+j) =1;
end;
end;
X1 = S1 + noise;figure(1);stem(h); % Plots impulse response of optimum detector
title('Impulse Response of Optimum Detector When N = 100 (AV)');
figure(2);freqz(h);
B = sgolay(2,5);
title('Magnitude and Phase Response of Optimum Detector When N = 100 (AV)');
figure(3);
subplot(2,1,1);stem(S1);
title('Input of Optimum Detector When Input = S(n) for N = 100');
subplot(2,1,2);stem(filter2(h,S1));title('Output of Optimum Detector When Input = S(n) for N = 100 (AV)');
figure(4);
subplot(2,1,1);stem(X1);
title('Input of Optimum Detector When Input = X(n) for N = 100');
subplot(2,1,2);stem(filter2(h,X1));
title('Output of Optimum Detector When Input = X(n) for N = 100 (AV)');
figure(5);
subplot(2,1,1);stem(noise);
title('Input of Optimum Detector When Input = noise for N = 100(AV)');...