Skip to content Skip to navigation

Connexions

You are here: Home » Content » Uniform Sampling of a Continuous-Time Bandlimited Signal

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.

      What are tags? tag icon

      Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

    • External bookmarks
  • E-mail the author
  • Rate this module (How does the rating system work?)

    Rating system

    Ratings

    Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

    How to rate a module

    Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

    		(0 ratings)

Recently Viewed

This feature requires Javascript to be enabled.

Uniform Sampling of a Continuous-Time Bandlimited Signal

Module by: Phil Schniter

Summary: Covers methods involved in uniform sampling of a continuous-time bandlimited signal.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Uniform Sampling

Say that xn xn is sampled from x c t x c t with uniform sampling interval TT:

∀n,n∈ℤ:xn= x c nT n n xn x c n T (1)

Let us define the continuous-time "impulse train"

pt=∑n=-∞∞δt−nT p t n δ t nT (2)
where δt δ t denotes the Dirac delta function, defined by the properties:

unit area

∫-∞∞δtdt=1 t δ t 1 (3)

sifting property

∫-∞∞ftδt−τdt=fτ t ft δ t τ fτ (4)

Using Fourier series, it can be shown that

pt=1T∑k=-∞∞ⅇⅈ2πTkt p t 1 T k 2 T k t (5)

Figure 1: Signals used in sampling theorem.
Figure 1 (uni_sam.png)

Multiplying x c t x c t by the impulse train yields the "continuous-time sampled signal" x s t x s t which will help us to derive the sampling theorem. (See Figure 1)

x s t= x c t∑n=-∞∞δt−nT= x c t1T∑k=-∞∞ⅇⅈ2πTkt x s t x c t n δ t nT x c t 1 T k 2 T k t (6)

Taking the CTFT of xs t xs t,

Xs ⅈΩ=∫-∞∞ xc t1T∑k=-∞∞ⅇⅈ2πTktⅇ-ⅈΩtdt=1T∑k=-∞∞∫-∞∞ xc tⅇ-ⅈΩ−2πTktdt Xs Ω t xc t 1 T k 2 T k t Ω t 1 T k t xc t Ω 2 T k t (7)
Xs ⅈΩ=1T∑k=-∞∞ Xc ⅈΩ−2πTk Xs Ω 1 T k Xc Ω 2 T k (8)

Notice that Xs ⅈΩ Xs Ω is a summation of scaled and shifted copies of Xc ⅈΩ Xc Ω . When Xc ⅈΩ Xc Ω is bandlimited to πTrads T rad s the spectral copies do not overlap, and we say there is no "aliasing." Aliasing may result when Xc ⅈΩ Xc Ω is not bandlimited to πT T . (See Figure 2 and Figure 3) The frequency πTrads T rad s , i.e., 12THz 1 2 T Hz , is often called the Nyquist frequency.

Figure 2: Example of aliasing in Xs ⅈΩ Xs Ω
Figure 2 (aliasing.png)
Figure 3: Example of no aliasing in Xs ⅈΩ Xs Ω
Figure 3 (noaliasing.png)

We now investigate the relationship between the CTFT and the DTFT:

Xⅇⅈω=∑n=-∞∞xnⅇ-ⅈωn=∑n=-∞∞ xc nTⅇ-ⅈωn=∑n=-∞∞∫-∞∞ xc tδt−nTdtⅇ-ⅈωn=∑n=-∞∞∫-∞∞ xc tδt−nTⅇ-ⅈωndt X ω n x n ω n n xc n T ω n n t xc t δ t n T ω n n t xc t δ t n T ω n (9)
where xc tδt−nTⅇ-ⅈωn xc t δ t n T ω n is nonzero ⇔ n=tT n t T
Xⅇⅈω=∑n=-∞∞∫-∞∞ xc tδt−nTⅇ-ⅈωtTdt=∫-∞∞ xc t∑n=-∞∞ⅇ-ⅈωTtdt X ω n t xc t δ t n T ω t T t xc t n δ t n T ω T t (10)
where xc t∑n=-∞∞δt−nT= xs t xc t n δ t n T xs t
Xⅇⅈω= X s ⅈωT X ω X s ω T (11)
Plugging Equation 8 into Equation 11 yields:
Xⅇⅈω=1T∑k=-∞∞ Xc ⅈω−2πkT X ω 1 T k Xc ω 2 k T (12)

Notice that the DTFT of the sampled signal xn x n is a summation of scaled, stretched, and shifted copies of the CTFT of the continuous signal xc t xc t . As implied by Equation 11, the DTFT may show evidence of aliasing when xc t xc t is not bandlimited to the Nyquist frequency. (See Figure 4 and Figure 5)

Figure 4: Example of aliasing in Xⅇⅈω X ω .
Figure 4 (aliasing_scaling.png)
Figure 5: Example of no aliasing in Xⅇⅈω X ω .
Figure 5 (noaliasing_scaling.png)

Comments, questions, feedback, criticisms?

Send feedback