How to make a nonperiodic signal periodic?

Question

Matthew 2023-9-18

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https://ww2.mathworks.cn/matlabcentral/answers/2022322-how-to-make-a-nonperiodic-signal-periodic

评论： William Rose 2023-9-21

I have the below code. If you run it, you'll see a graph that runs from t = 0 to t = 7. What I'd like to do is make this signal continuous, as in it runs forever from t → -∞, +∞. Of course I looked it up and I see many users use stem or syms functions to make signals but they never have signals with this many piece wise components.

Thanks so much!

clc
close all
%
t=linspace(-1,8,2356);
    v_t=-2*t.*fun_unit_step_gen(t,0);
     v_t= v_t+2*(t-1).*fun_unit_step_gen(t,1);
      v_t= v_t+3*fun_unit_step_gen(t,2);
       v_t= v_t+(t-2).*fun_unit_step_gen(t,2);
        v_t= v_t-5*(t-3).*fun_unit_step_gen(t,3);
         v_t= v_t+3*(t-4).*fun_unit_step_gen(t,4);
          v_t= v_t+5*(t-5).*fun_unit_step_gen(t,5);
           v_t= v_t-4*(t-6).*fun_unit_step_gen(t,6);
            v_t= v_t-fun_unit_step_gen(t,7);
%
plot(t,v_t,'r','LineWidth',2);
axis([-1 9 -4 4])
grid
title('Function f1(t)')
hold on
px=[0,.001];py=[-5,5];
plot(px,py,'-.k','LineWidth',1)
py=[0,.001];px=[0,9];
plot(px,py,'-.k','LineWidth',1)
hold off

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William Rose 2023-9-19

在 MATLAB Online 中打开

@Matthew,

Since you did not provide the function fun_unit_step_gen(), I made a guess. Is my guess correct?

t=linspace(-1,8,2356);

v_t=-2*t.*(t>=0);

v_t= v_t+2*(t-1).*(t>=1);

v_t= v_t+3*(t>=2);

v_t= v_t+(t-2).*(t>=2);

v_t= v_t-5*(t-3).*(t>=3);

v_t= v_t+3*(t-4).*(t>=4);

v_t= v_t+5*(t-5).*(t>=5);

v_t= v_t-4*(t-6).*(t>=6);

v_t= v_t-(t>=7);

%

plot(t,v_t,'r','LineWidth',2);

axis([-1 9 -4 4])

grid

title('Function f1(t)')

hold on

px=[0,.001];py=[-5,5];

plot(px,py,'-.k','LineWidth',1)

py=[0,.001];px=[0,9];

plot(px,py,'-.k','LineWidth',1)

hold off

The funciton as plotted will be discontinuous at t=0, 7, 14,..., if you repeat it at those intervals. Is that OK?

William Rose 2023-9-19

在 MATLAB Online 中打开

@Matthew,

If you want dot-dash lines along the axes, you can do it more simply with xline() and yline():

plot(-1:8,-3+6*rand(1,10),'-r');

xline(0,'-.k'); yline(0,'-.k'); grid on

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Answer 1

Fabio Freschi 2023-9-19

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2022322-how-to-make-a-nonperiodic-signal-periodic#answer_1313457

编辑：Fabio Freschi 2023-9-19

在 MATLAB Online 中打开

I had the same idea of @William Rose, here is a possible implementation

clear variables, close all

% anonymous function

v_t = @(t)-2*t.*(t>=0)+...

2*(t-1).*(t>=1)+...

3*(t>=2)+...

(t-2).*(t>=2)+...

-5*(t-3).*(t>=3)+...

3*(t-4).*(t>=4)+...

5*(t-5).*(t>=5)+...

-4*(t-6).*(t>=6)+...

-(t>=7);

% anonymous function with period T using mod

T = 8;

v_t_per = @(t,T)v_t(mod(t,T));

% time axis

t=linspace(-8,16,2356*3);

% plot

figure, hold on, grid on

plot(t,v_t(t),'r','LineWidth',2);

plot(t,v_t_per(t,T),'b:','LineWidth',2);

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William Rose 2023-9-20

@Matthew,

Since you are asking, it is worth being clear about the different forms of the Fourier transform (FT). The discrete Fourier transform (DFT) is the FT of a sampled signal. (The FFT is an algorithm for computing the DFT efficiently, and has become synonymous with the DFT. You don't have to compute the DFT with the FFT algorithm, but you might as well.) The DFT is defined for signals of finite length - call it length N (samples). The DFT assumes that x(n) repeats with period N. To put it another way, when you compute the DFT for a signal of length N, you get the coefficients for an infinite length signal with period N.

The DFT has a finite number of coefficients, and they correspond to discrete frequencies. Those discrete frequencies are the only frequencies you need to reconstruct a periodic signal with length N. The spacing between those discrete frequencies is

cycles per sample.

You say "I'll fast fourier transform both the nonperiodic signal and the periodic signal to observe if there are differences in the spectrum. I figured there would be because we were taught that nonperiodic signals have continuous spectrums, whereas periodic signals have discrete spectrums."

You were taught correctly: a non-periodic signal has a continuous spectrum. A non-periodic signal is a periodic one, in which the period N goes to infinity. As I said above, the frequency spacing of the DFT is 1/N, so the frequency spacing

as

. And

corresponds to a continuous spectrum.

How do you plan to "fast fourier transform the nonperiodic signal"? You can't. You cannot have a vector of infinite length in Matlab. You can get closer to an infinite length signal by adding a lot of zeros on either side of your signal, the computing the DFT (with the fft() function) of the signal with th added zeros.

If your non-periodic signal is a signal y(n) defined from n=0 to N-1, and 0 at all times before and after, then the way to think about it is:

y(n)=x(n)*b(n),

where x(n) is periodic with period N, and b(n) is the boxcar function, which is 1 from n=0 to N-1, and 0 elsewhere, and where * stands for standard multiplication at each time point. Multiplication in the time domain is equivalent to convolution in the frequency domain, so the FT of y(t) is given by

Y(f)=conv(X(f),B(f))

B(f) for a boxcar is well known, and you can get X(f) via the DFT (i.e. by using fft()).

Signal y(n) and b(b) are zero for an infinite time and non-zero for a finite time, so their FTs become vanishingly small.

Paul 2023-9-20

编辑：Paul 2023-9-21

@William Rose

There is also the Discrete Time Fourier Transform (DTFT) that can be applied to either finite duration or infinite duration signals (periodic or not), and the DTFT is a function of continuous frequency with a "continuous spectrum" in the sense that you're interpreting that phrase, i.e., delta-F -> 0, which is different than what I assumed the OP meant by that phrase. Further clarification needed from the OP as to what the goal of the problem actually is.

@Matthew

If you want to analyze continuous-time signals that can be expressed in closed form, as is the case with v_t, then fourier is a good place to start. If you want to analyze discrete-time signals, then fft and/or freqz are useful for finite duration signals, and fft is useful for periodic signals. As always one has to pick the right tool for the job and interpret the outputs correctly based on the problem statement.

William Rose 2023-9-21

@Paul, thank you.

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Answer 2

Image Analyst 2023-9-19

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How about making it for one chunk, and then using repmat to make copies of it? You can't go from t → -∞, +∞ but you can go for some finite number of elements (indexes).

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Answer 3

William Rose 2023-9-19

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2022322-how-to-make-a-nonperiodic-signal-periodic#answer_1313387

@Matthew,

If your goal is for the signal to repeat so that x(7...14)=x(0...7), and x(-7...0)=x(0...7), then I recommend using modulo division by 7 of the time argument.

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