Skip to content Skip to navigation

Connexions

You are here: Home » Content » Discrete-Time Fourier Transform Pair

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

    • External bookmarks
  • E-mail the author

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual Connexions member, a community, or a respected organization.

This content is ...

In these lenses

  • richb's DSP

    This module is included inLens: richb's DSP resources
    By: Richard BaraniukAs a part of collection:"Signals and Systems"

    Comments:

    "My introduction to signal processing course at Rice University."

    Click the "richb's DSP" link to see all content selected in this lens.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

Discrete-Time Fourier Transform Pair

Module by: Don Johnson

Summary: Computing discrete-time frequencies by the use of Fourier transforms.

When we obtain the discrete-time signal via sampling an analog signal, the Nyquist frequency corresponds to the discrete-time frequency 12 1 2 . To show this, note that a sinusoid at the Nyquist frequency 12 T s 1 2 T s has a sampled waveform that equals

Sinusoid at Nyquist Frequency 1/2T

cos2π12 T s n T s =cosπn=-1n 2 1 2 T s n T s n 1 n (1)

The exponential in the DTFT at frequency 12 1 2 equals ⅇ-ⅈ2πn2=ⅇ-ⅈπn=-1n 2 n 2 n 1 n , meaning that the correspondence between analog and discrete-time frequency is established:

Analog, Discrete-Time Frequency Relationship

f D = f A T s f D f A T s (2)

where f D f D and f A f A represent discrete-time and analog frequency variables, respectively. The aliasing figure provides another way of deriving this result. As the duration of each pulse in the periodic sampling signal p T s t p T s t narrows, the amplitudes of the signal's spectral repetitions, which are governed by the Fourier series coefficients of p T s t p T s t , become increasingly equal. 1 Thus, the sampled signal's spectrum becomes periodic with period 1 T s 1 T s . Thus, the Nyquist frequency 12 T s 1 2 T s corresponds to the frequency 12 1 2 .

The inverse discrete-time Fourier transform is easily derived from the following relationship:

∫-1212ⅇ-ⅈ2πfmⅇ+ⅈπfndf=1ifm=n0ifm≠n 1 2 1 2 f 2 f m f n 1 m n 0 m n (3)

Therefore, we find that

∫-1212Sⅇⅈ2πfⅇ+ⅈ2πfndf=∫-1212∑msmⅇ-ⅈ2πfmⅇ+ⅈ2πfndf=∑msm∫-1212ⅇ-ⅈ2πfm-ndf=sn f 1 2 1 2 S 2 f 2 f n f 1 2 1 2 m m s m 2 f m 2 f n m m s m f 1 2 1 2 2 f m n s n (4)

The Fourier transform pairs in discrete-time are

Fourier Transform Pairs in Discrete Time

Sⅇⅈ2πf=∑nsnⅇ-ⅈ2πfn S 2 f n n s n 2 f n (5)

Fourier Transform Pairs in Discrete Time

sn=∫-1212Sⅇⅈ2πfⅇ+ⅈ2πfndf s n f 1 2 1 2 S 2 f 2 f n (6)

Footnotes

  1. Examination of the periodic pulse signal reveals that as Δ Δ decreases, the value of c 0 c 0 , the largest Fourier coefficient, decreases to zero: | c 0 |=AΔT c 0 A Δ T . Thus, to maintain a mathematically viable Sampling Theorem, the amplitude A A must increase as 1Δ 1 Δ , becoming infinitely large as the pulse duration decreases. Practical systems use a small value of Δ Δ , say 0.1 T s 0.1 T s and use amplifiers to rescale the signal.

Comments, questions, feedback, criticisms?

Send feedback