Discrete Time Fourier Transform (DTFT)
Discrete Time Fourier Transform (DTFT)
The Discrete Time Fourier Transform (DTFT) is a type of Fourier analysis which allows the development of the spectral representation of a discrete signal using complex exponential signals as the underlying components. There are several ways to derive the concept of the DTFT, a common method being the assumption of a discrete signal having an infinitely long time period and using the Discrete Fourier Series (DFS) to formulate the DTFT. However, since the aim of the discussion is to provide an understanding of the concept of the DTFT while avoiding extraneous complex mathematical methods and also because the DFS has not been discussed yet. We take a simpler approach and provide a formulation derived from the z-transform. The bilateral z-transform of a signal is given by the equation
In order to form the DTFT from the z-transform, we let the variable z be a complex exponential i-e .
Where Ω is the frequency of the complex exponential. It is specified in radians/sec and ranges from -π to π where π refers to half the sampling frequency (fs/2).
is called the frequency response of x[n]. Like the z-transform, the DTFT of a discrete signal is also continuous. The output is a scaled sum of complex exponentials at some frequencies. The DTFT of a signal x[n] is a periodic signal with a period of 2π. The Inverse DTFT of a signal is given by the equation
Another point that needs to be understood is the graphical relation between the variable z andin the z plane. The range of the variable Ω is from -π to π which when plotted results in a unit circle as shown in the figure below
Therefore, the DTFT of a signal is equivalent to evaluating the z-transform over the unit circle. The frequency Ω varies along the unit circle as shown in the figure above. Since the DTFT is periodic, an increase in Ω greater than |Ω| =π will result in more revolutions around the circle.