In this lesson, we will discuss *sampling* of continuous time signals.
Sampling a continuous time signal is used, for example, in A/D conversion, such as would be done in
digitizing music for storage on a CD, digitizing a movie for storage on a DVD, or taking a digital picture.

To start, we define the continuous time impulse train as:

As you just saw, *p*(*t*) is an infinite train of continuous
time impulse functions, spaced *T _{s}* seconds apart.

Now, *x*(*t*) is the continuous time signal we wish to sample.
We will * model * sampling as multiplying a signal by *p*(*t*).
Let *x _{s}*(

Let's show this graphically:

Now, we will derive the * Sampling Theorem *. To do this, we will
examine our signals in the frequency domain. To start, let *p*(*t*) have a Fourier Transform
*P*(*ω*), *x*(*t*) have a Fourier Transform *X*(*ω*), and
*x _{s}*(

Then, because *x _{s}*(

Now let's find the Fourier Transform of *p*(*t*). Because the infinite impulse train is
periodic, we will use the Fourier Transform of periodic signals:

where *C _{k}* are the Fourier Series coefficients of the periodic signal.

Let's find
the Fourier Series coefficients *C _{k}* for the periodic impulse train

by the sifting property. Therefore

Thus, an impulse train in time has a Fourier Transform that is a impulse train in frequency. The spacing between impulses in time is *T _{s}*, and the spacing between impulses in frequency is

Now, to finish our derivation of the Sampling Theorem, we will go back and determine *X*_{s}(*ω*).

We saw that:

Therefore,

or we get replicated, scaled versions
of *X*(*ω*), spaced every *ω _{0}*
apart in frequency:

As you can
see from the figure if *ω*_{0} - *ω*_{c}
< *ω*_{c},
we would get overlap of the
replicates of *X*(*ω*) in frequency.
This is known as "aliasing."
Therefore, to avoid aliasing, we require
*ω*_{0} - *ω*_{c} > *ω*_{c} or *ω*_{0} > 2*ω*_{c}.
If we avoid aliasing, we can recover *x*(*t*) from its samples.
(Usually, we choose a sampling rate a bit higher than twice the highest frequency since
filters are not ideal.)

We hear music up to 20 *kHz* and CD sampling rate is 44.1 *kHz*.
(Dogs would need a higher quality CD since they hear higher
frequencies than humans.)

We can recover *x*(*t*) from its sampled version *x _{s}*
(

__Example 1__ Given a signal *x*(*t*) with
Fourier Transform with cutoff frequency *ω*_{c} as shown:

you are given three different pulse trains with periods

, | , and | , |

draw the sampled spectrum in each case. Which case(s) experiences aliasing?

__Example 2__ The inverse Fourier Transform of the
signal in the previous example is

Draw the sampled signals using the sampling trains of the previous example

( | , | , and | ). |

Notice how aliasing looks in the time domain.