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Sampling CT Signals: A Frequency Domain Perspective

Module by: Robert Nowak. E-mail the author

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

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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−nTⅇ-ⅈΩtdt=∑n=-∞∞ x c nT∫-∞∞δt−nTⅇ-ⅈΩtdt=∑n=-∞∞xnⅇ-ⅈΩnT=∑n=-∞∞xnⅇ-ⅈωn=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 ω≡ΩT ω Ω T and Xω X ω is the DTFT of xn x n .

Recall:

X s Ω=1T∑k=-∞∞ X c Ω−k Ω s X s Ω 1 T k X c Ω k Ω s
Xω=1T∑k=-∞∞ X c Ω−k Ω s =1T∑k=-∞∞ 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

Figure 1
Figure 1 (sec10_fig1.png)

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 _____?

Figure 2
Figure 2 (sec10_fig2.png)
Figure 3: Note that there is no mention of TT or Ω s Ω s !
Figure 3 (sec10_fig3.png)

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

Figure 4
Figure 4 (sec10_fig4.png)

Recall the CTFT relation:

xαt↔1αXΩα ↔ 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 Ω≡XΩT X s Ω X Ω T (4)

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