Exercises : Week 1 

Part 1 : Fourier Transforms and Matlab 

The purpose of these exercises is to gain some experience deriving Fourier 
Transforms analytically, and using Matlab 

*** The following are NOT formally assessed, but may be discussed in *** 
*** practicals and/or lectures                                       *** 

1. Derive the Fourier Transform pairs of the functions given in lectures 
   In Matlab it is possible to verify that kind of calculations using the 
   "symbolic math toolbox". With this toolbox it is possible to define symbolic 
   parameters and to calculate analytical expressions for the Fourier transformation. 
   General information about the functions in this toolbox can be found on the 
   following URL: 
   	"http://www-ccs.ucsd.edu/matlab/toolbox/symbolic/symbmath.html"
   By using the command "syms" a letter can be set as a symbolic parameter 
	e.g. >>syms x;
   Now the commands "fourier" or "ifourier" can be used to calculate the Fourier 
   or inverse fourier transformation of a function containing the parameter "x".
	e.g. >>fourier(sin(x)) will return an analytical expression for the Fourier 
	transformation of the sine function. 

2. Write Matlab definitions for the functions in the temporal domain, sample 
   these functions into a list, and use 'fft' from Matlab to get the Fourier 
   domain results. Plot these functions and explain any differences from that 
   expected in part 1. 
   Note : the results of the fft will be a list of _complex_ numbers. Try 
   plotting, abs(), real(), imag(), and ang(), and explain the results. 
   what happens if you "shift" the original data, starting the sampling at 
   a different point in time, but using the same period ? 

3. Similarly write Matlab definitions for the functions in the Fourier 
   domain and use the iftt function in Matlab to get the temporal domain 
   results. 
  


Part 2 : Discrete Fourier Transforms

The purpose of these exercises is to gain some experience plotting Fourier 
Transforms in Matlab 

*** The following are NOT formally assessed, but may be discussed in *** 
*** practicals and/or lectures                                       *** 

  
4. Consider a function func(t) that is periodic with period T.
To plot this function you need to create range and suitable sampling
If you have N samples then the interval between them is T/N.
	t = [0:T/N:T];
	f = func(t);
	plot(t,f);
Since the function is periodic, the value at 0 and T are the same, and
it turns out to cause problems in the discrete Fourier Transform (DFT)
if the repeated point is included. So we prefer to do this
	t = [0:T/N:T-T/N];
	f = func(t);
	plot(t,f);
If you want to see the function repeated explicitly, you can extend the
time range :
	tlong = [0:T/N:3*T-T/N];
	plot(tlong,func(tlong);

5. Now you can take the discrete Fourier Transform of the sampled function
	ff = fft(f);
This list is (generally) of complex numbers. We began with a list of
N numbers, f, and we got a new list of N complex numbers, ff. 
Note 1: Although complex numbers have two parts, (real and imaginary) we did 
not add any extra information. This is because ff appears in _conjugate pairs_.
If you examine it you will find that (for example) ff(2) is the complex 
conjugate of ff(N).
Note 2: There is nothing special about the choice N, but it turns out that
the fft algorithm is faster when the number is a power of 2 . Choose something
like N = 128 to begin with.

In order to plot the list ff, we need to know the frequency range. Recall that
the fundamental frequency is w0 = 2*pi/T. The range goes equally from 
negative frequencies to positive frequencies, and must have N numbers. 
We can get it like this
	w = [-(N-1)*pi/T:2*pi/T:N*pi/T];
We must plot the real and imaginary parts seperately. We can use colours
for this
	figure;
	plot(w,real(ff),'b');
	hold;
	plot(w,imag(ff,'r');
	hold;


Try doing this with functions such as Cos, Sin and others that you have
been shown.