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## Discrete Time Fourier Transform (DTFT)

The Discrete Time Fourier Transform (DTFT) can be viewed as the limiting form of the DFT when its length is allowed to approach infinity: (3.2)

where denotes the continuous radian frequency variable,3.3 and is the signal amplitude at sample number .

The inverse DTFT is (3.3)

which can be derived in a manner analogous to the derivation of the inverse DFT .

Instead of operating on sampled signals of length (like the DFT), the DTFT operates on sampled signals defined over all integers .

Unlike the DFT, the DTFT frequencies form a continuum. That is, the DTFT is a function of continuous frequency , while the DFT is a function of discrete frequency , . The DFT frequencies , , are given by the angles of points uniformly distributed along the unit circle in the complex plane. Thus, as , a continuous frequency axis must result in the limit along the unit circle. The axis is still finite in length, however, because the time domain remains sampled.

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