While working on **Fourier Series** or some other **Mathematical Problem**, you might sometime have to work with **Periodic Functions**.

Periodic Functions are those that give the same value after a particular period.

So we will use this definition to define a periodic function in **SCILAB**.

Let’s say that there is a function * f(x)* which is periodic with a period of

*and is already defined in the interval*

**2*T****[-T,T]**.

Then the function should have the same value at:** f(x), f(x+2*T), f(x+4*T), ….**

i.e.* f(x)=f(x+2*T)=f(x+4*T)=*……. since period=

**2*T**.

But I said that the function is defined only in the interval **[-T,T]**. So how is the computer supposed to calculate it’s value at **x>T**?

That’s easy. Since the value of the function at * f(x+2*T)* is simply

*, therefore we can generalize that whenever*

**f(x)***: then,*

**x>T**

**f(x)=f(x-2*T).****Note**: We have to keep taking

*back by*

**x***until it lies within*

**2*T i.e****(x-2*T)****[-T,T]**where the function is well-defined.

Similarly what about the value of function at * x* less than

*cause the function is not defined for values less than*

**(-T)****(-T)**?

Again, this time we use:

*. Note: We keep translating*

**f(x)=f(x+2*T)***forward by*

**x***until it lies within*

**2*T i.e****(x+2*T)****[-T,T]**where the function is well-defined.

Using the above two arguments we can create a function which will make any given function defined within** [-T,T]** and with a period **2*T** a periodic function.

**Here is the code for that:**

//Periodic Function function a=periodicf(T,f,x) if (x>=-T)&(x<=T) then a=f(x); elseif x>T then x_new=x-2*T; a=periodicf(T,f,x_new); elseif x<(-T) then x_new=x+2*T; a=periodicf(T,f,x_new); end endfunction

**Demo**:

deff('a=f(x)','a=x'); -->periodicf(2,f,0) ans = 0. -->periodicf(2,f,1) ans = 1. -->periodicf(2,f,2) ans = 2. -->periodicf(2,f,3) ans = - 1. -->periodicf(2,f,4) ans = 0.

In the above example I have created a function f(x)=x.

Then I called the function ‘periodicf’ with T=2(meaning period is 4) and passed the function ‘f’ as the second argument and then the third argument is the value of x at which I want the value of ‘f’.

The above code assumes the function to be defined within [-T,T] therefore the function’s starting point is (-T).

However, if you want to create a function that is defined in a different manner then you will have to define ‘f’ correspondingly.

Following are some examples of different kinds of periodic functions along-with their plots:

**Saw-tooth Wave(1):**

** Code:**

//Saw-Tooth Wave(all Positive) funcprot(0); function a=sawtooth(T,x) deff('a=sawtooth_temp(x)','if x>0 then; a=x; else; a=x+T; end;'); a=periodicf(T,sawtooth_temp,x); endfunction //Plotting a saw-tooth wave x=[-40:0.1:40]; for i=1:801 y(i)=sawtooth(5,x(i)); end plot(x,y);

**Output:**

**Saw-Tooth(2) Wave:**

**Code**:

//Sawtooth wave(2) funcprot(0); function a=sawtooth2(T,x) deff('a=sawtooth_temp(x)','a=x'); a=periodicf(T,sawtooth_temp,x); endfunction //Plotting a saw-tooth wave x=[-40:0.1:40]; for i=1:801 y(i)=sawtooth2(10,x(i)); end plot(x,y);

**Output:**

**Square Wave(or Rectangle Wave) :**

**Code:**

//Square-Wave or Rectangle Wave funcprot(0); function a=sqrwave(T,A,x) deff('a=sqwavetmp(x)','if x>0 then; a=A; else; a=0; end') a=periodicf(T,sqwavetmp,x); endfunction //Plotting a square wave x=[-40:0.1:40]; for i=1:801 y(i)=sqrwave(20,20,x(i)); end plot(x,y);

**Output:**

**Triangular Wave**:

**Code:**

//Triangle Wave: funcprot(0); function a=trianglewave(T,x) deff('a=trnglewavetmp(x)','if x>0 then; a=x; else; a=-x; end') a=periodicf(T,trnglewavetmp,x); endfunction //Plotting a Triangle wave x=[-40:0.1:40]; for i=1:801 y(i)=trianglewave(10,x(i)); end plot(x,y);

**Output:**

I did know the Youtube channel but ain’t Bragitoff. Thanks a lot, you spared me a new night spent to figure out how to get a saw-tooth.