Simple FFT and Filtering Tutorial with Matlab

Introduction

Let us understand FFT. It is Fast Fourier Transform, an algorithm to calculate DFT or discrete fourier transform in fast and efficient way. The first question that arises seeing the title is what the hell a tutorial on FFT doing in the new article section of code project in the year 2012 when the algorithm is about 50 years old. Great Question. By the way, you can refer following links for understanding the same in a complicated way.

http://en.wikipedia.org/wiki/Discrete_Fourier_transform

http://en.wikipedia.org/wiki/Fast_Fourier_transform

Background

One of the reasons, I thought I must write this tutorial is because I have seen many engineers finding it difficult to understand what exactly is a frequency domain, what exactly is filtering. How can we filter a signal in simple way. What is this symmetrical features of FFT and more importantly understanding the core facts about FFT. So if you are a programming Pro and a killer programmer having written your own DSP logics and algorithms, then this article may not give you any new or interesting things. But if you have just picked signal processing, you might find it quite helpful (Hopefully).

Using the Code

First Let us construct a simple sinusoidal signal of 50Hz with amplitude=5. I am considering a sampling frequency of 100 times the message frequency(it should it least be twice as par Nequist rate), which means I will collect 1000 samples from the actual analog sinusoidal signal. So if the frequency is 50, that means you will get one sample at 1/50 or .02 seconds. If you want 10 samples, you have to gather the samples for .2 seconds. You will sample the signal at every 1/5000 seconds.

f=50;
A=5;
Fs=f*100;
Ts=1/Fs;
t=0:Ts:10/f;
x=A*sin(2*pi*f*t);
plot(x),grid on

So the figure shows 10 cycles of sine wave of 50 Hz. You can change the plot to stem type to see the samples.

So lets take FFT.

F=fft(x); 

plot(real(F)),grid on  

What you see? The highest value is located at 11th position, which is nothing but 10th sample as matlab starts from 1.

10*5?=50Hz

It is so simple. But wait why do we get the imaginary part also? There is absolutely nothing called Imaginary values, they reflect past values.

Your signal is Sinusoidal, so you get negative cycle also right. Check this.

plot(real(imag(F))),grid on  

what you see? :)

Yes the same thing with one negative amplitude peak also. I hope you get the idea.

Now let us understand how you can have some filtering with FFT. What I will do is mix couple of more signal with different frequency with same amplitude and same number of samples with your signal x.

x1=A*sin(2*pi*(f+50)*t);
x2=A*sin(2*pi*(f+250)*t);
x=x+x1+x2; 

F2=zeros(length(F),1);
F2(10:11)=F(10:11);
xr=ifft(F2);
plot(real(xr)),grid on

 clear all;
clc;
close all
f=50;
 A=5;
Fs=f*100;
Ts=1/Fs;
t=0:Ts:10/f;
x=A*sin(2*pi*f*t);
x1=A*sin(2*pi*(f+50)*t);
x2=A*sin(2*pi*(f+250)*t);
x=x+x1+x2;
plot(x)
F=fft(x);
figure
N=Fs/length(F);
baxis=(1:N:N*(length(x)/2-1));
plot(baxis,real(F(1:length(F)/2))) 

Points of Interest

You can learn Matlab fundamentals from this source <here>

To know the details about any Matlab command, you can simply click on that command in the editor and press F1. Matlab help file explains the usage and other details about the commands like fft,sin and so on.

Behind all that complicated mathematics, there is a simple logic. I love that simple logic really.