##
Sampling

The function *x(t)* is sampled at times *t = kT*. The sampled function *y(t) * can be represented
as the product of *x(t) * and a sampling function *p(t)*.
| 1 |

For impulse sampling,
| 2 |

Combining equations 1 and 2.
| 3 |

Since *p(t)* is a periodic function it can be represented by the Fourier series shown in equation 4.
| 4 |

where for impulse sampling, *a*_{k} = 1 / T.
| 5 |

Multiplying equation 5 by *x(t)*.
| 6 |

Multiplying a time function by results in a shift of * kw*_{s}
in the frequency domain. That is,
| 7 |

Applying equation 7 to equation 6 results in the following for the frequency spectrum of *y(t)* the sampled signal.
| 8 |

Consider a signal with the spectrum *X(jw)*, of the baseband *x(t)*.
The spectrum of the sampled function, *Y(jw)*, contains duplicates of the baseband signal
shifted to multiples of *kw*_{s} , as shown below.
**Aliasing**

To avoid the overlap distortion called aliasing, *w*_{b} must be less than
*-w*_{b} + w_{s}

That is,

*w*_{s} > 2 w_{b}

This result, that the sampling frequency must be at least twice the largest baseband frequency is called the **
Sampling Theorem**.