Note 12. Discrete-Time Fourier Transform
The discrete-time Fourier transform (DTFT) is the appropriate Fourier technique to use in order to obtain a continuous-frequency spectrum for a signal that is a function of discrete time. The continuous-frequency spectrum obtained from the DTFT is periodic, with a period equal to T–1, where T is the discrete-time sampling interval. The DTFT finds widespread use within DSP, primarily because a digital filter’s unit sample response and frequency response comprise a DTFT pair.
The DTFT is defined by
12.1
and the corresponding inverse is given by
12.2
where ω is the continuous radian frequency and T is the discrete-time sampling interval. The z transform is defined in Note 44 as
12.3
If ejωT is substituted for z in this definition, the result is identical to Eq. (12.1). This result indicates that the DTFT is equal to the z transform of x[n] evaluated on the unit circle in the z-plane.
Math Box 12.1. Deriving the Discrete-Time Fourier Transform
The delta function model of ideal sampling can be used to derive the discrete-time Fourier transform (DTFT) from the continuous-time Fourier transform (CTFT). Begin with the delta function model of ideal sampling applied to the continuous-time function, x(t):
The CTFT for the sum in the equation above can be computed term by term. For any particular value of n, x(nT) is a constant, so the CTFT can be written as
Replacing x(nT) with the discrete-time sequence notation x[n] yields
which agrees with Eq. (12.1).
Table 12.1. Discrete-time Fourier transform pairs
Table 12.2. Properties of the DTFT
