DFT and the DTFT.

Cuthbert Nyack
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.