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Sampling CT Signals: A Frequency Domain Perspective

Module by: Robert Nowak. E-mail the author

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 x⁢n x n . Compute the CTFT of x s ⁢t=∑ n =−∞∞ x c ⁢n⁢T⁢δ⁢t−n⁢T x s t n x c n T δ t n T

X s ⁢Ω=∫−∞∞∑ n =−∞∞ x c ⁢n⁢T⁢δ⁢t−n⁢T⁢e(−i)⁢Ω⁢td t =∑ n =−∞∞ x c ⁢n⁢T⁢∫−∞∞δ⁢t−n⁢T⁢e(−i)⁢Ω⁢td t =∑ n =−∞∞x⁢n⁢e(−i)⁢Ω⁢n⁢T=∑ n =−∞∞x⁢n⁢e(−i)⁢ω⁢n=X⁢ω 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≡Ω⁢T ω Ω T and X⁢ω X ω is the DTFT of x⁢n 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
X⁢ω=1T⁢∑ k =−∞∞ X c ⁢Ω−k⁢ Ω s =1T⁢∑ k =−∞∞ X c ⁢ω−2⁢π⁢kT 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 2⁢π 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=∑ n nx⁢n⁢T⁢δ⁢t−n⁢T x s t n n x n T δ t n T

Figure 4
Figure 4 (sec10_fig4.png)

Recall the CTFT relation:

x⁢α⁢t↔1α⁢X⁢Ωα ↔ 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≡X⁢Ω⁢T X s Ω X Ω T
(4)

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