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m12 - The Discrete-Time Fourier Transform

Module by: C. Sidney Burrus

Summary: The Discrete Time Fourier Transform is a generalization of the DFT to allow infinite duration signals. This results in a continuous frequency description rather than the discrete frequencies of the DFT.

The Discrete-Time Fourier Transform

In addition to finite length signals, there are many practical problems where we must be able to analyze and process essentially infinitely long sequences. For continuous-time signals, the Fourier series is used for finite length signals and the Fourier transform or integral is used for infinitely long signals. For discrete-time signals, we have the DFT for finite length signals and we now present the discrete-time Fourier transform (DTFT) for infinitely long signals or signals that are longer than we want to specify [1] . The DTFT can be developed as an extension of the DFT as N N goes to infinity or the DTFT can be independently defined and then the DFT shown to be a special case of it. We will do the latter. Some of these concepts are discussed further in Chapter (Reference).

Definition of the DTFT

The DTFT of a possibly infinitely long real (or complex) valued sequence f ( n ) f ( n ) is defined to be

F ( ω ) = ∑ − ∞ ∞ f ( n ) e − j ω n F ( ω ) = ∑ − ∞ ∞ f ( n ) e − j ω n (1)
and its inverse denoted IDTFT is given by
f ( n ) = 1 2 π ∫ − π π F ( ω ) e j ω n ⅆ ω . f ( n ) = 1 2 π ∫ − π π F ( ω ) e j ω n ⅆ ω . (2)
Verification by substitution is more difficult than for the DFT. Here convergence and the interchange of order of the sum and integral are serious questions and have been the topics of research over many years. Discussions of the Fourier transform and series for engineering applications can be found in [2] [3] . It is necessary to allow distributions or delta functions to be used to gain the full benefit of the Fourier transform.

Note that the definition of the DTFT and IDTFT are the same as the definition of the IFS and FS respectively. Since the DTFT is a continuous periodic function of ω ω , its Fourier series is a discrete set of values which turn out to be the original signal. This duality can be helpful in developing properties and gaining insight into various problems. The conditions on a function to determine if it can be expanded in a FS are exactly the conditions on a desired frequency response or spectrum that will determine if a signal exists to realize or approximate it.

Examples of DTFT

As was true for the DFT, insight and intuition is developed by understanding the properties and a few examples of the DTFT. Several examples are given below and more can be found in the literature [1] [2] [3] . Remember that while in the case of the DFT signals were defined on the region { 0 ≤ n ≤ ( N − 1 ) } { 0 ≤ n ≤ ( N − 1 ) } and values outside that region were periodic extensions, here the signals are defined over all integers and are not periodic unless explicitly stated. The spectrum is periodic with period 2 π 2 π .

  • D T F T { δ ( n ) } = 1 D T F T { δ ( n ) } = 1 for all frequencies.
  • D T F T { 1 } = 2 π δ ( ω ) D T F T { 1 } = 2 π δ ( ω )
  • D T F T { e j ω 0 n } = 2 π δ ( ω − ω 0 ) D T F T { e j ω 0 n } = 2 π δ ( ω − ω 0 )
  • D T F T { cos ( ω 0 n ) } = π [ δ ( ω − ω 0 ) + δ ( ω + ω 0 ) ] D T F T { cos ( ω 0 n ) } = π [ δ ( ω − ω 0 ) + δ ( ω + ω 0 ) ]
  • D T F T { ⊓ M ( n ) } = sin ( ω M k / 2 ) sin ( ω k / 2 ) D T F T { ⊓ M ( n ) } = sin ( ω M k / 2 ) sin ( ω k / 2 )

References

  1. A. V. Oppenheim and R. W. Schafer. (1989). Discrete-Time Signal Processing. Englewood Cliffs, NJ: Prentice-Hall.
  2. A. Papoulis. (1962). The Fourier Integral and Its Applications. McGraw-Hill.
  3. R. N. Bracewell. (1985). The Fourier Transform and Its Applications. (Third). New York: McGraw-Hill.

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