# I am trying to use the rectangular pulse function, but I am not sure how to use it correctly to solve a Fourier problem.

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Connor Mondock on 14 Apr 2022
Commented: Star Strider on 16 Apr 2022
Ts = 1/100;
t = -4:Ts:4-Ts;
% Now looking to create the rectangular pulse
fx = rectangularPulse(t,2);
After this runs, it says invalid number of arguments. I also tried typing in 'rectpuls' because that's intially what I thought the function was, but that didn't work either.

Star Strider on 14 Apr 2022
It needs to have the beginning and end defined, and an arbitrary argument for the independent variable (here ‘t’) —
syms omega t
p = rectangularPulse(-4,4,t)
p = P = fourier(p,t,omega)
P = P = simplify(P, 500)
P = figure
fplot(real(P), [-10 10])
grid .
Star Strider on 16 Apr 2022
As always, my pleasure!

Paul on 15 Apr 2022
Edited: Paul on 15 Apr 2022
Hi Connor,
This question brings up some interesting aspects of Fourier analysis, and how transforms of the discrete-time samples relate to the Fourier transform of the underlying continuous-time signal.
First, define the continous-time signal and its continuous time Fourier transform (CTFT)
syms t omega_c
s_c(t) = rectangularPulse(-4,4,t);
S_wc(omega_c) = simplify(fourier(s_c(t),t,omega_c))
S_wc(omega_c) = The indpendent variable, omega_c, has units of rad/sec (using default Fourier settings in sympref).
Plot the CTFT over the interval from -10 to 10 rad/sec
figure;
subplot(211);fplot(real(S_wc),[-10 10]); ylabel('Real')
subplot(212);fplot(imag(S_wc),[-10 10]); ylabel('Imag')
legend('CTFT') Now we want to generate samples of s_c(t) for discrete-time analysis. Define an 801-point, symmetric index vector
n = -400:400;
and the associated time vector with sampling period Ts
Ts = 1/100;
t = n*Ts;
The first and last points of t fall exactly on the leading and trailing edge of s_c(t) and we have to decide what the samples should be on those edges. s_c(t) is an even signal, so it makes sense to choose equal samples on the edges. Let's choose the edge samples to be 1/2 the amplitude of the pulse.
s_n = 0*t + 1;
s_n(1) = 1/2;
s_n(end) = 1/2;
With the samples we can compute the discrete-time Fourier transform (DTFT). First we define the continous-time frequency vector over the frequency interval of interest (note the 10 rad/sec is well below the Nyquist frequency of pi/Ts = 314 rad/sec)
The discrete-time frequency is then
wn = wc*Ts;
wn has units of rad/sample. We can compute the DTFT directly using its definition. To make it easier, define a simple function that maps our mathematical index, n, to the Matlab array index
sfunc = @(n) s_n(n + 401);
dtft1 = 0*wn;
for nval = n
dtft1 = dtft1 + sfunc(nval)*exp(-1j*wn*nval);
end
Compare to the CTFT by scaling the DTFT by Ts; scale wn by 1/Ts to convert it to rad/sec
figure
subplot(211);
hold on
fplot(real(S_wc),[-10 10]); ylabel('Real')
plot(wn/Ts,real(Ts*dtft1))
xlim([-10 10])
subplot(212);
hold on
fplot(imag(S_wc),[-10 10]); ylabel('Imag')
plot(wn/Ts,imag(Ts*dtft1))
xlim([-10 10])
legend('CTFT','DTFT1') The DTFT at DC is
Ts*dtft1(wn == 0)
ans = 8
which is exaclty the value of the CTFT at wc = 0, and is a consequence of the selected sample values at the pulse edges.
The DTFT for s_n can also be computed using the function freqz(). freqz() assumes that the first sample in the input signal corresponds to n = 0, In this problem the first sample corresponds to n = -400, so we apply the shift theorem to the output of freqz()
dtft2 = freqz(s_n,1,wn).*exp(1j*400*wn);
figure
subplot(211);
hold on
fplot(real(S_wc),[-10 10]); ylabel('Real')
plot(wn/Ts,real(Ts*dtft1))
plot(wn/Ts,real(Ts*dtft2))
xlim([-10 10])
subplot(212);
hold on
fplot(imag(S_wc),[-10 10]); ylabel('Imag')
plot(wn/Ts,imag(Ts*dtft1))
plot(wn/Ts,imag(Ts*dtft2))
xlim([-10 10])
legend('CTFT','DTFT1','DTFT2') Next, we can use the Discrete Fourier Transform (DFT) as implemented by the fft() function. The DFT assumes the underlying signal is periodic with period being the lenght of the signal, which our signal isn't. So we'll need to zero-pad to fake out the DFT, so to speak, by making the period look big. Similarly to freqz(), fft() assumes that the first point in the input corresponds to n = 0. Together these issues can be addressed in a couple of ways. One approach used here is to zero pad the signal on both sides, and then use ifftshift before taking fft()
dft = fft(ifftshift([zeros(1,200) s_n zeros(1,200)]));
The discete-time discrete frequency vector corresponding to the DFT samples is
Note that wdft is one sample short of going all the way around the unit circle, i.e,. the last point is is not 2*pi.
We want our plot to cover the negative frequencies so use fftshift()
dft = fftshift(dft);
and correspondingly modify the frequency vector (there are probably better ways to do this, but I think this way makes it clear what's happening)
wdft = [wdft(wdft > pi-eps(pi)) - 2*pi, wdft(wdft <= pi-eps(pi))];
Because numel(dft) is odd, the first and last points of wdft are
wdft([1 end])
ans = 1×2
-3.1390 3.1390
which are symmetric around 0 and less than pi in absolute value. If nume(dft) were even, then wdft(1) == pi, and wdft(end) would be pi - dwdft, where dwdft is the spaicing between samples of wdft.
The dft should be samples of the DTFT
figure
subplot(211);
hold on
fplot(real(S_wc),[-10 10]); ylabel('Real')
plot(wn/Ts,real(Ts*dtft1))
plot(wn/Ts,real(Ts*dtft2))
stem(wdft/Ts,real(Ts*dft))
xlim([-10 10])
subplot(212);
hold on
fplot(imag(S_wc),[-10 10]); ylabel('Imag')
plot(wn/Ts,imag(Ts*dtft1))
plot(wn/Ts,imag(Ts*dtft2))
stem(wdft/Ts,imag(Ts*dft))
xlim([-10 10])
legend('CTFT','DTFT1','DTFT2','DFT') Finally, the question asked about the function rectpuls(). Generate samples using rectpuls()
s_nrect = rectpuls(t,8);
The samples on the edges are
s_nrect([1 end])
ans = 1×2
1 0
So rectpuls() takes the full amplitude at the leading edge and zero at the trailing edge. s_nrect is not symmetric around the origin. Consequently, its DTFT (and DFT) will pick up a small imaginary part and distort the real part relative to the CTFT, though the value at DC is maintained
dtftrect = freqz(s_nrect,1,wn).*exp(1j*400*wn);
Ts*dtftrect(wn == 0)
ans = 8
figure
subplot(211);
hold on
fplot(real(S_wc),[-10 10]); ylabel('Real')
plot(wn/Ts,real(Ts*dtft1))
plot(wn/Ts,real(Ts*dtft2))
stem(wdft/Ts,real(Ts*dft))
plot(wn/Ts,real(Ts*dtftrect))
xlim([-10 10])
subplot(212);
hold on
fplot(imag(S_wc),[-10 10]); ylabel('Imag')
plot(wn/Ts,imag(Ts*dtft1))
plot(wn/Ts,imag(Ts*dtft2))
stem(wdft/Ts,imag(Ts*dft))
plot(wn/Ts,imag(Ts*dtftrect))
xlim([-10 10])
legend('CTFT','DTFT1','DTFT2','DFT','DTFTRect') Paul on 15 Apr 2022
I think you meant to post this comment in StarStrider's answer.
I can see a few problems, assuming that you're trying to approximate the CTFT.
s and t must both be row vectors with the same number of elements.
The independent variable of integration in the CTFT is t, not w.
The frequency variable, w, has units of rad/sec. Because the plot is against w, the axis label should just be w.
clear
Ts = 1/100;
t = -4:Ts:4;
s = ones(1,801); % just cover the nonzero part of the pulse
% FTs = fft(s)/numel(s);
w = linspace(-pi*3, pi*3, numel(s));
f = exp(-1j*w(:)*t) .* s;
F = trapz(t,f,2); % ingerate wrt t, result is a function of w
% Fourier Transform
figure
plot(w, real(F), w, imag(F))
grid
ylabel('Amplitude')
legend('Real(F)','Imag(F)', 'Location','best') Result is an essentially identical plot to the others in this answer.

R2020a

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