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2.5 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 t n T

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

Sampling

Speech

Speech is intelligible if bandlimited by a CT lowpass filter to the band4 kHz. We can sample speech as slowly as _____?

Note that there is no mention of T or s !
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Relating x[n] to sampled x(t)

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

Recall the CTFT relation:

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

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Source:  OpenStax, Fundamentals of signal processing. OpenStax CNX. Nov 26, 2012 Download for free at http://cnx.org/content/col10360/1.4
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