Mathematical Methods, Algorithms and Implementations

S.Arridge 
Exercises 2 

Discrete Fourier Transforms, and Sampling

The purpose of these exercises is to gain some experience of the practical aspects of sampling 

*** This exercise is formally assessed and counts 10\% towards your result***

Consider the Band-limited function defined by the following MatLab M-File function :

   function res=BLF(t)
   res=cos(t)+4/5*sin(3*t)-1/6*cos(5*t)+2/7*cos(7*t);

   Determine the Nyquist frequency for this function and sample it
	  i)  above
   	  ii) below
   the Nyquist frequency

   write functions to reconstruct BLF at a higher sampling rate than either
   a) or b) using
	\begin{enumerate}
	  \item Time-domain interpolation with a Sinc function
	  \item Fourier domain interpolation by zero-filling
	\end{enumerate}
   compare the results to that obtained by directly sampling BLF at the
   higher sampling rate. Compare the timings in cases a) and b) [In
   MatLab use the "Tic Toc" function or some other timing function ].

   You may find that the reconstructed signal is scaled and time-shifted -
   explain why and find a correction either in the Fourier domain, or the
   temporal domain

Repeat the above using an unbounded function in the frequency domain such as a Gaussian. In this case the reconstruction can never avoid aliasing. Make estimates of errors as a function of sampling interval

Use Fourier zero-filling to implement interpolation for a 2D digital image and compare with any other method (nearest neighbour, linear 
interpolation)that you have been introduced to in the Machine Vision course.


Notes :

Your coursework should consist of program code, results and a short discussion (probably not more than 4 pages). 

Here is a MatLab routine to read in a bitmap image:

imread('filename.bmp','bmp')

Similarly other image formats can be read.

To interpolate an image, for example resize it, you can use the imresize function in the Image Processing Toolbox.