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We just covered ideal (and non-ideal) (time) sampling of CT signals . This enabled DT signal processing solutions for CTapplications ( ):
Much of the theoretical analysis of such systems relied on frequency domain representations. How do we carry out thesefrequency domain analysis on the computer? Recall the following relationships: where and are continuous frequency variables.
Consider the DTFT of a discrete-time (DT) signal
. Assume
is of finite duration
(
We want to work with on a computer. Why not just sample ?
The DTFT of the image in is written as follows:
where again we sampled at where . For example, we take . In the following section we will discuss in more detail how we should choose , the number of samples in the interval.
(This is precisely how we would plot in Matlab.)
Given (length of ), choose to obtain a dense sampling of the DTFT ( ):
Choose as small as possible (to minimize the amount of computation).
In general, we require in order to represent all information in Let's concentrate on : for and
Define
Represent in terms of a sum of complex sinusoids of amplitudes and frequencies
IDFT treats as though it were -periodic.
Proof that the IDFT inverts the DFT for
Given the following discrete-time signal ( ) with , we will compute the DFT using two different methods (the DFTFormula and Sample DTFT):
DFT consists of samples of DTFT, so , a -periodic DTFT signal, can be converted to , an -periodic DFT.
Also, recall,
Think of sampling the continuous function , as depicted in . will represent the sampling function applied to and is illustrated in as well. This will result in our discrete-time sequence, .
Why does equal ?
is -periodic, so it has the following Fourier Series :
So, in the time-domain we have ( ):
Combine signals in to get signals in .
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