DFT, Calculating Transform for Simple Signals.

Cuthbert Nyack
It is possible to find closed expressions for the DFT of simple sequences, particularly those involving "discrete" steps and ramps. Sinusoids can also be summed but the expressions tend to be quite long.

Here a simple sequence x(n) given by the following expression is considered.
The DFT can be obtained from the definition:-
This expression is readily summed resulting in the following equation.

For comparison the DTFT of the same sequence is shown below. T here is the sampling time.

The coefficient in the complex form of the definition of the Fourier Series of a function with period T and equal to 1 for 0 £ t £ t is shown below.

The Fourier Transform of a pulse f(t) = 1 for 0 £ t £ t is also shown below for comparison.

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COPYRIGHT 2007 Cuthbert Nyack.