Fast Fourier Transform of Time Domain Signals
The objective of this experiment is to take FFT of the time domain signal and plot the spectra and study the spectra of the signal.
Matlab Functions
Apart from already known functions that we have used in the previous experiments, list of few more MATLAB in built functions that may be used for the Experiment are listed below ,
1. fft() % to compute the fft of the sequence
2.linspace() % generate 'N' equally spaced points between two numbers
3. abs() % To take absolute value of the complex number [ Remember output sequence of FFT is complex valued]
4.nextpow2() % Ex: nextpow2(200) returns 8
5. fftshift() % Optional function
Let's Code
Example Code 1
Let us see how much FFT is faster than DFT in terms of computational time. Execute the following code in the editor window.
t=0:0.01:10;
x_n=sin(2*pi*t);
dft_mat=dftmtx(length(t));
disp('Using DFT')
tic;
X_k=dft_mat*x_n';
toc;
disp('Using FFT')
tic;
X_k=fft(x_n);
toc;
Let's consider a simple sequence x(n) = [1 1 1 1 1 1 ] that we have used in the class room, Computing the FFT for the sequence in the MATLAB is very simple.
Do the following In the command window,
>> x = [1 1 1 1 1 1] ;
>> fft(x,8)
ans =
Columns 1 through 5
6.0000 + 0.0000i -0.7071 - 1.7071i 1.0000 - 1.0000i 0.7071 + 0.2929i 0.0000 + 0.0000i
Columns 6 through 8
0.7071 - 0.2929i 1.0000 + 1.0000i -0.7071 + 1.7071i
Observation:
Number 8 in the second argument of fft() signals the function to compute 8-point FFT even though the input sequence is of length 6.
The output sequence of the fft() is complex valued
Appending two more zeros to compute 8 point fft() is automatic
Example Code 2:
Copy Paste the following code in the Editor Window ,save the file and run it.
Answer the Following:
What is your observation on the output plots?
What would happen if x(n) = [1 1 1 1] ?
Are there any symmetry exists in the plots? If so why it Exists?
Applying FFT for Basic Signals
This is the final task of the experiment. Applying fft to Basic signals and plotting their spectrum.
Example:
Generate 10 Hz sine wave and plot the spectra by computing fft for the sine wave. Here is the sample code to generate the sine wave.
Instructions to Plot the Correct Spectra:
Make use of 2^nextpow2() function to find N
Take N-point fft for the input 'x'
Find absolute value for the output sequence and plot the spectra.
Do Necessary adjustment to reproduce the output exactly as shown below
Refer the Documentation of fft to know more about using it.
Output to Reproduce
If you look at the spectra, there are two peaks at +10 Hz and -10 Hz as the input is 10 Hz Sine . Producing one side spectrum is also alright [i.e one peak at +10 Hz] . If the frequency varies, the position of the peak also moves to the respective frequency.
FFT for the rest of the basic Signals
Produce the fft plot for all the basic signals. The FFT plot for the step and ramp may look like the following. Refer to the external links to know the theory behind the plots
