**Fourier Transforms**

[IMP.: Along with the plots I also want to see the programs you used]

1. Take the sinusoidal function x = 0.5*sin(1+2*pi*0.01*t) where
t=[1:1:1024]

Obtain the power spectral density (PSD = |FFT|^{2}) of
the above function and plot the original function as well as the PSD vs
frequency and PSD vs period.

2. Take the function which is combination of two sinusoidal functions
having incommensurate frequencies

x = 0.5*sin(1+2*pi*0.01*t) + 0.5*sin(1+2*pi*sqrt(2)*0.01*t) where t=[1:1:2048]

Obtain the PSD - plot the original function as well as PSD vs
frequency and PSD vs period.

3. Now look at the effect of random noise. Add a random component =
random_strength*rand(1,2048) to the above function.

(The function rand generates uniformly distributed random numbers between
[0,1] and the parameter random_strength gives the relative magnitude of
the randomness compared to the periodic function)

Plot the function and the PSD vs frequency plots for random_strength
= 0.1, 0.5 and 1.

Note how the background intensity in the PSD plot rises with increasing
randomness.

4. Now instead of a sinusoidal function, consider a series of delta-functions
with the same frequency as the sinusoidal function in the 1st problem.
You can generate this by using the following function

>> x1=sign(x-0.99);

where x is defined in Problem 1.

But the values of this new function lie between -1 and 1. So we rescale
it to lie between 0 and 1.

>> x= 0.5*(1+x1);

Plot the original function as well as the PSD vs frequency plot

5. Now we go over to two dimensions. Design a 2-D function of diagonal
planes. One way you can do this is

>> a = zeros(100);

>> for i=1:100, for j=1:100, if mod(abs(i-j),5)==0, a(i,j)=1;end;end;end;

>> imagesc(a)

This will give you diagonal planes between (x=[1:100],y=[1:100])with
a spacing of 5 units between them.

Increase the spacing between planes to 10 and obtain the PSD.

Plot the contours of the original function as well as the PSD.

Given:

MATLAB is available in imsc5, imsc6, imsc7

MATLAB has a demo showing how to obtain the power spectral density

Let x be the vector whose fourier transform has to be obtained. MATLAB uses FFT - so it is most efficient if the number of elements in x is a power of 2, e.g, 128, 256, 512, 1024, etc.

>> Y=fft(x);

This computes the fast fourier transform of x. The first component of Y is simply the sum of the data, so that we do the following to get rid of it

>> Y(1)=[ ];

>> n=length(Y);

This gives the number of elements in Y - we need to take only half
of the total number, as the other half is simply a repetition of the first
half.

>> power = abs (Y(1:n/2)).^2;

or, equivalently.

>> power = abs(Y(1:n/2)).*abs(Y(1:n/2));

To obtain the frequency axis, note that we have to scale it by the Nyquist
frequency

When the data set is arranged according to a integer index 1,2,3,....

>> nyquist = 1/2;

>> frequency = (1:n/2)/(n/2)*nyquist;

>> plot(frequency, power)

To plot the data against the period instead of the frequency

>> period = 1./frequency;

>> plot(period, power)

Similar set of operations for two-dimensional fft using the command
fft2

See MATLAB help for further details