Connexions

You are here: Home » Content » Sampling CT Signals: A Frequency Domain Perspective
Content Actions
Lenses

What is 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 ...
Affiliated with (?)
This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • This module is included inLens: Rice University OpenCourseWare
    By: OpenCourseWare ConsortiumAs a part of collection:"Intro to Digital Signal Processing"

    Click the "Rice University OCW" link to see all content affiliated with them.

    Rice University OCW
Tags

(?)

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

Sampling CT Signals: A Frequency Domain Perspective

Module by: Robert Nowak

Summary: The module will provide an introduction to sampling a signal in the frequency domain and go through a basic example.

Understanding Sampling in the Frequency Domain

We want to relate x c t x c t directly to xn x n . Compute the CTFT of x s t=n=- x c nTδt-nT x s t n x c n T δ t n T
X s Ω=-n=- x c nTδt-nT-Ωtdt=n=- x c nT-δt-nT-Ωtdt=n=-xn-ΩnT=n=-xn-ωn=Xω X s Ω t n x c n T δ t n T Ω t n x c n T t δ t n T Ω t n x n Ω n T n x n ω n X ω (1)
where ωΩT ω Ω T and Xω X ω is the DTFT of xn x n .
Recall: X s Ω=1Tk=- X c Ω-k Ω s X s Ω 1 T k X c Ω k Ω s
Xω=1Tk=- X c Ω-k Ω s =1Tk=- X c ω-2πkT X ω 1 T k X c Ω k Ω s 1 T k X c ω 2 k T (2)
where this last part is 2π 2 -periodic.

Sampling

sec10_fig1.png
Figure 1
Example 1: Speech 
Speech is intelligible if bandlimited by a CT lowpass filter to the band ±4 kHz. We can sample speech as slowly as _____?
sec10_fig2.png
Figure 2
sec10_fig3.png
Figure 3: Note that there is no mention of TT or Ω s Ω s !

Relating x[n] to sampled x(t)

Recall the following equality: x s t=nxnTδt-nT x s t n n x n T δ t n T
sec10_fig4.png
Figure 4
Recall the CTFT relation:
xαt1αXΩα x α t 1 α X Ω α (3)
where αα is a scaling of time and 1α 1 α is a scaling in frequency.
X s ΩXΩT X s Ω X Ω T (4)

Comments, questions, feedback, criticisms?

Send feedback