## Aliasing of Sampled Signals

This section quantifies aliasing in the general case. This result is then used in the proof of the sampling theorem in the next section.

It is well known that when a continuous-time signal contains energy at
a frequency higher than half the sampling rate , sampling
at samples per second causes that energy to *alias* to a
lower frequency. If we write the original frequency as
, then the new aliased frequency is
,
for
. This phenomenon is also called ``folding'',
since is a ``mirror image'' of about . As we will
see, however, this is not a complete description of aliasing, as it
only applies to real signals. For general (complex) signals, it is
better to regard the aliasing due to sampling as a summation over all
spectral ``blocks'' of width .

### Continuous-Time Aliasing Theorem

Let denote any continuous-time signal having a Fourier Transform (FT)

*aliasing terms*. They are said to

*alias*into the

*base band*. Note that the summation of a spectrum with aliasing components involves addition of complex numbers; therefore, aliasing components can be removed only if both their

*amplitude and phase*are known.

*Proof: *
Writing as an inverse FT gives

The inverse FT can be broken up into a sum of finite integrals, each of length , as follows:

Let us now sample this representation for at to obtain , and we have

since and are integers. Normalizing frequency as yields

**Next Section:**

Sampling Theorem

**Previous Section:**

Introduction to Sampling