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  4. Sampling ct signals: a frequency

3.10 Sampling ct signals: a frequency domain perspective

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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 directly to x n . Compute the CTFT of x s t n ∞ ∞ x c n T δ t n T

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
where ω Ω T and X ω is the DTFT of x n .
X s Ω 1 T k ∞ ∞ X c Ω k Ω s
X ω 1 T k ∞ ∞ X c Ω k Ω s 1 T k ∞ ∞ X c ω 2 k T
where this last part is 2 -periodic.

Sampling

Speech

Speech is intelligible if bandlimited by a CT lowpass filter to the band±4 kHz. We can sample speech as slowly as _____?

Note that there is no mention of T or Ω s !

Relating x[n] to sampled x(t)

Recall the following equality: x s t n n x n T δ t n T

Recall the CTFT relation:

↔ x α t 1 α X Ω α
where α is a scaling of time and 1 α is a scaling in frequency.
X s Ω X Ω T

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Source:  OpenStax, Ece 454 and ece 554 supplemental reading. OpenStax CNX. Apr 02, 2012 Download for free at http://cnx.org/content/col11416/1.1
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