Question

Write Matlab code (with detailed comments) to perform spectral analysis of 1KHz sine wave, noise and noise corrupted sine wave. Consider the sampling frequency Fs=10KHz.                                                                                                                                                   What should be the ideal frequency response of filter required to remove noise? Justify your answer

Answer 1

(a) MATLAB code is given below:

fsin = 1000;%% sine wave frequency
fs = 10000;%% sampling frequency
fbin = fsin/100;%% bin frequency of FFT
npoints = fs/fbin*15;% Total length of sequence
nfft = fs/fbin;% No of points of FFT
A = 1;% amplitude of sinewave
vsin = A*sin(2*pi*fsin/fs*(0:1:npoints-1));% sine wave defined
noise = wgn(1,length(vsin),-40);% noise witj -40dB power defined
nfft_total = 10*nfft;% total length of sequence used for calculating PSD
vsin_final = vsin(end-nfft_total+1:end);
noise_final = noise(end-nfft_total+1:end);
noise_plus_sine = noise+vsin;% noise + signal
noise_plus_sine_final = noise_plus_sine(end-nfft_total+1:end);

%%% Generating 10 different FFT's by dividing sequence into ten segments
%%% and then taking FFT of each segment
fft_vsin(nfft_total/nfft,nfft) = 0;
fft_noise(nfft_total/nfft,nfft) = 0;
fft_noise_plus_sine(nfft_total/nfft,nfft) = 0;
for i = 1:nfft_total/nfft
fft_vsin(i,:) = fft(vsin_final((i-1)*nfft+1:i*nfft));
fft_noise(i,:) = fft(noise_final((i-1)*nfft+1:i*nfft));
fft_noise_plus_sine(i,:) = fft(noise_plus_sine_final((i-1)*nfft+1:i*nfft));
end
freq = fs/nfft*(0:1:nfft-1);

%%%%
psd_vsin_avg(1,nfft) = 0;
psd_noise_avg(1,nfft) = 0;
psd_noise_plus_sine_avg(1,nfft) = 0;
for i = 1:nfft_total/nfft
psd_vsin_avg = psd_vsin_avg + abs(fft_vsin(i,:)).^(2);
psd_noise_avg = psd_noise_avg + abs(fft_noise(i,:)).^(2);
psd_noise_plus_sine_avg = psd_noise_plus_sine_avg + abs(fft_noise_plus_sine(i,:)).^(2);
end
psd_vsin_avg = psd_vsin_avg(1:nfft/2)/(2000*2000);% scaling factor to normalize PSD
psd_noise_avg = psd_noise_avg(1:nfft/2)/(2000*2000);% scaling factor to normalize PSD
psd_noise_plus_sine_avg = psd_noise_plus_sine_avg(1:nfft/2)/(2000*2000);% scaling factor to normalize PSD
frequency = freq(1:nfft/2);

PSD of signal, PSD of noise and PSD of signal + noise is plotted below:

(b) From plot, we can see that signal is single sinusoid but noise spread over whole frequency range. So ideal filter to filter out noise will be a bandpass filter centered at 1kHz and infinite Q such that its bandwidth = 0.