LPF experiment i. Run LPF_example.m to understand the behavior and commands. ii. Observe that the high frequency compone

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LPF experiment i. Run LPF_example.m to understand the behavior
and commands.
ii. Observe that the high frequency component of the signal is
canceled by the LPF
iii. In particular, understand the fir1(.) function, which
performs low pass filter design.
code:
%% Low pass filtering example
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear
clc
close all
% Parameter settings
t_duration = 30;
Fs = 80; % sampling frequency. No smaller than double of maximum
frequency of x.
t = 0:1/Fs:t_duration; % sample points in the observed
interval.
x = 2*cos(2*pi*0.7*t) + 7*sin(2*pi*5*t) + 5*cos(2*pi*23*t); %
sampled signal
%% Freq domain analysis of the signal
% fft(arg1,arg2): Fast Fourier Transform function
% - arg1: a subject which is transformed
% - arg2: # of frequency response sample points
xfft = fft(x);
L = length(xfft);
P2 = abs(xfft/L);
P1 = P2(1:floor(L/2)+1); % Only positive frequency components
including DC component.
P1(2:end-1) = 2*P1(2:end-1); % One-sided spectrum has doubled
amplitude that of double-sided spectrum.
f = Fs*(0:(L/2))/L; % Components of x is only located below
Fs/2.
plot(f,P1)
xlim([0 Fs/2])
xlabel('frequency [Hz]'); ylabel('amplitude'); title('Spectrum
of x');
%% Low-pass filtering
% butter(arg1,arg2): butterworth filter model
% - arg 1: order of the filter (The higher, the steeper the
slope.)
% - arg 2: cutoff frequency
% arg 2 is usually specified with target signal's sampling
frequency. (Fs/2 term)
% Recommended not to use too much high value of arg1.
fc = 10 ; % filter cutoff frequency (in Hz)
Omega_c = (fc/Fs)*2*pi; % Omega_c (rad/sample)
filter_n = 60; % filter order
cutoff_w = 2*fc/Fs; % (rad/sample /pi)
b = fir1(filter_n,cutoff_w); % coefficients of x in difference
equations
a = [1];
% Freqeuncy response of the filter given
figure
freqz(b,a); % shows H(e^(jOmega)) in log scale
fvtool(b,a); % similar to freqz(.)
% Filtering the signal
y = filter(b,a,x);
%% Filtered signal: Frequency domain analysis
clear L P1 P2;
yfft = fft(y);
L = length(yfft);
P2 = abs(yfft/L);
P1 = P2(1:L/2+1);
P1(2:end-1) = 2*P1(2:end-1);
figure
f = Fs*(0:(L/2))/L;
plot(f,P1);
xlim([0 Fs/2]);
xlabel('frequency [Hz]'); ylabel('amplitude'); title('Spectrum
of y');
grid on;