DFT and the DTFT.
The expression for the DTFT is given below.
If the spectrum is sampled at angular frequencies
w = mws/N, the result is
shown below. Integrating this expression over w gives the DFT.
The DFT can be seen as a sampled version of the DTFT.
The effect is shown in the applet below.
The plots show that the DFT at any frequency is equal to the DTFT at that frequency.
Another way to show the connection between the 2 is to note
that the discreteness of the DFT arises from its assumed
periodicity. Therefore if periodicity is introduced into the
DTFT, then the result should be the DFT. This is illustrated
in the applet below by varying the parameter NP. When NP = 2, then the sequence for the DTFT is of length 2N - 1 with the
set between 0 and N - 1 being repeated between N and
2N - 1.
Fn = 2, 3, 4 and 5 are special
cases of Fn = 1 and show different number of periods.
As more repititions are added. The
height of the peaks increases with NP while their widths reduces
with NP with their areas remaining constant. In the limit of
NP going to infinity, each peak reduces to an impulse of magnitude equal to the DFT and centered at the DFT frequency.
The result is an impulsive form of the DFT. The DFT can be regarded as the DTFT with assumed periodicity in the time domain.
The relation between the DTFT and the DFT is therefore the same
as that between the Fourier Transform and the Fourier Series.
Another way of looking at the relation between the 2 is to increase
N, the total number of samples leaving n1(number of samples = 1)
constant. This has the effect of increasing the period of the
sequence and bringing the lines of the DFT closer together. As
N gets very large, the sequence used by the DTFT and the periodic
sequence assumed by the DFT begins to look alike and the DFT should look more like the DTFT.
Setting Fn = 6 shows this case and Fn = 7, 8, 9 and 10 are
special cases of Fn = 6.
In most of the applets in this section, the DTFT is shown along with
the DFT. There is nothing essential about this. It is done because
when the number of samples is small, it is easier to see the structure
of the DFT if it is accompanied by the DTFT.
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COPYRIGHT © 2007 Cuthbert Nyack.