Help Making a Piecewise Function Periodic

So I am trying to plot a piecewise function where z(t) = {t , 0 <= t < 1

e^(-5(t-1)) , 1 <= t < 2.5

0 , else

When I run it, it just gives the one period even though mod(t,2.5) should be keeping everything OK. How do I fix this? (I want to make it periodic)

(Oh yeah, my interval is -2 <= t <= 12)

t1 = -2 : 0.01 : 12; 
syms t z(t); % defining a symbolic variable
z(t) = (piecewise((mod(t,2.5)>=0)&(mod(t,2.5)<1),t, (mod(t,2.5)>=1)&(mod(t,2.5)<2.5), exp(-5.*(t-1)), 0));
z1 = z(t1);
figure
hold on
plot(t1, z1);

Answer 1

[Paul]

HI Connor,

Something like this?

syms t real
z(t) = piecewise(0 <= t <= 1,t, 1 < t <= 2.5, exp(-5*(t-1)),0);
fplot(z(t),[-2 12])

z1(t) = z(mod(t,2.5));
fplot(z1(t),[-2 12])

tval = -2:.01:12;
plot(tval,z1(tval))

[Walter Roberson]

Note by the way that piecewise is the only function that permits using 0 <= t <= 1 kind of syntax at the user level (at least that I can think of.)

(There are some symbolic functions that can return those structures.)

[Paul]

Also, bear in mind that that even though z1(t) is periodic, it is not the periodic extension of z(t). Rather z1(t) is the periodic extension of the product of z(t) and a rectangular window from t = 0 to t = 2.5.

[Anoop Kiran]

Hi Paul,

I tried your code, unfortunately it doesn't result in the periodic behavior at every 2.5 interval like the second and third plots that you have.

[Paul]

You'll need to show the exact code you ran and the results obtained before going any further.

Answer 2

[Alexander]

In the most recent R2024b Update 5 (24.2.0.2871072) version modular division by real number doesn't work as expected to plot periodic functions extentions. I tried the following approach and it worked. In short, do low level in special function.

clear;clf;clc;

% Define the period and the range for x
L = pi;      % Half-Period of the periodic function
t = -3*L:L/100:3*L;

% Define the given function
fl1 = @(t) (pi/2.*(t >= -pi & t <-pi/2));
fl2 = @(t) (-t.*(t >= -pi/2 & t < 0));
fr1 = @(t) (pi/2.*(t >= pi/2 & t <pi));
fr2 = @(t) (t.*(t >= 0 & t < pi/2));

f = @(t) (fl1(t)+fl2(t)+fr1(t)+fr2(t));
g = @(t) (-L + ((-L+t)/(2*L)-floor((-L+t)/(2*L)))*(2*L));
t_intervals = g(t);

%figure;
plot(t,f(t_intervals))
hold on;

title('f(t)', 'Interpreter','latex');
xticks([-2*pi -pi 0 pi 2*pi])
xticklabels({'-2\pi','-\pi','0','\pi','2\pi'})
ylim([-.1 pi/2*1.1])
yticks([0 pi/4 pi/2])
yticklabels({'0','\frac{\pi}{4}','\frac{\pi}{2}'})
xlabel('t'); %ylabel('f(t)');
set(gca,'TickLabelInterpreter', 'latex');
set(gca,'fontsize',12)
grid on;

hold off

Which should produce

[Walter Roberson]

I tested R2024b, and the code posted by @Paul works fine, including modular division by a real number.

[Steven Lord]

What does "doesn't work as expected" mean in this context?

• Do you receive warning and/or error messages? If so the full and exact text of those messages (all the text displayed in orange and/or red in the Command Window) may be useful in determining what's going on and how to avoid the warning and/or error.
• Does it do something different than what you expected? If so, what did it do and what did you expect it to do?
• Did MATLAB crash? If so please send the crash log file (with a description of what you were running or doing in MATLAB when the crash occured) to Technical Support so we can investigate.

[Alexander]

I understand the problem. By default Matlab assumes the variable is a complex number. So, it is the modular division for complex numbers what is different.

Need to define argument to the periodic function as a real variable explicitly then it works.

syms t real;

[Alexander]

Now I tried to define explicitly function argument as a real number, but plotting is still incorrect. My function is even and periodic, so it is symmetric about y-axis. When doing modular division by something goes wrong on the negative interval.

clear;clf;clc;
syms t real;

% Define the period and the range for x
L = pi;      % Half-Period of the periodic function
tval = -3*L:L/100:3*L;

% Define the given function
fl1 = @(t) (pi/2.*(t >= -pi & t <-pi/2));
fl2 = @(t) (-t.*(t >= -pi/2 & t < 0));
fr1 = @(t) (pi/2.*(t >= pi/2 & t <pi));
fr2 = @(t) (t.*(t >= 0 & t < pi/2));

f = @(t) (fl1(t)+fl2(t)+fr1(t)+fr2(t));
%g = @(t) (-L + ((-L+t)/(2*L)-floor((-L+t)/(2*L)))*(2*L));
%t_intervals = g(t);
%plot(t,f(t_intervals))

plot(tval,f(mod(tval,2*L)));
hold on;

title('f(t)', 'Interpreter','latex');
xticks([-2*pi -pi 0 pi 2*pi])
xticklabels({'-2\pi','-\pi','0','\pi','2\pi'})
ylim([-.1 pi/2*1.1])
yticks([0 pi/4 pi/2])
yticklabels({'0','\frac{\pi}{4}','\frac{\pi}{2}'})
xlabel('t'); %ylabel('f(t)');
set(gca,'TickLabelInterpreter', 'latex');
set(gca,'fontsize',12)
grid on;

hold off

Which produces incorrect plot:

[Walter Roberson]

L = pi;      % Half-Period of the periodic function
tval = -3*L:L/100:3*L;
plot(tval,mod(tval,2*L));

●

Notice that this mod result is nowhere negative.

plot(tval,rem(tval,2*L));

rem() does produce negatives, but only for the negative input range.

It is an error to think that mod() will produce negative outputs when the divisor is positive.

Compare to

plot(tval,mod(tval+L,2*L)-L);

[Alexander]

Confirmed. I also cleaned my code, which produces correct plot for even function by modulo division. No need to declare a function argument as real number.

clear;clf;clc;

% Define the period and the range for x
L = pi;      % Half-Period of the periodic function
t = -3*L:L/100:3*L;

% Define the given function
fl1 = @(t) (pi/2.*(t >= -pi & t <-pi/2));
fl2 = @(t) (-t.*(t >= -pi/2 & t < 0));
fr1 = @(t) (pi/2.*(t >= pi/2 & t <pi));
fr2 = @(t) (t.*(t >= 0 & t < pi/2));

f = @(t) (fl1(t)+fl2(t)+fr1(t)+fr2(t));

plot(t,f(mod(t+L,2*L)-L));
hold on;

title('f(t)', 'Interpreter','latex');
xticks([-2*pi -pi 0 pi 2*pi])
xticklabels({'-2\pi','-\pi','0','\pi','2\pi'})
ylim([-.1 pi/2*1.1])
yticks([0 pi/4 pi/2])
yticklabels({'0','\frac{\pi}{4}','\frac{\pi}{2}'})
xlabel('t'); %ylabel('f(t)');
set(gca,'TickLabelInterpreter', 'latex');
set(gca,'fontsize',12)
grid on;

hold off;

The resulting plot is symmetric about y-axis: