# 1.9 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 ${x}_{s}(t)=\sum$ x c n T t n T

${X}_{s}()=\int_{()} \,d t$ 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 $\equiv T$ and $X()$ is the DTFT of $x(n)$ .
${X}_{s}()=\frac{1}{T}\sum$ X c k s
$X()=\frac{1}{T}\sum$ X c k s 1 T k X c 2 k T
where this last part is $2\pi$ -periodic.

## Speech

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

## Relating x[n] to sampled x(t)

Recall the following equality: ${x}_{s}(t)=\sum x(nT)(t-nT)$

Recall the CTFT relation:

$(x(t), \frac{1}{}X(\frac{}{}))$
where  is a scaling of time and $\frac{1}{}$ is a scaling in frequency.
${X}_{s}()\equiv X(T)$

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