**DISCRETE FOURIER TRANSFORM (DFT)**

The discrete Fourier-transform of a signal is defined by:

X[k] = SUM(n=0,N-1) {x[n] exp(-j*2*pi*k*n/N)}

where j=sqrt(-1).

Try to determine the Discrete Fourier Transform (DFT) of the following signals:

- Dirac-delta at (discrete) time n=0,
- Shifted Dirac-delta at time n=1,
- Pulse function x[n] = 1 for n=0,1,
- Pulse function x[n] = 1 for n=0,..,3,
- Pulse function x[n] = 1 for n=0,...,7,
- sin(.) function,
- cos(.) function.

Explain Your answer. Try to determine the width of the Lobes in case of a pulse train signal (Note that the length of the DFT signal is equivalent to 2*pi!). To have a better understanding play with the following applet: