We first start with the most basic sampling ideas based on various forms of Fourier transforms [14], [3], [19].

**The Spectrum of a Continuous-Time Signal and the Fourier Transform
**

Although in many cases digital signal processing views the signal as simple sequence of numbers, here we are going to pose the problem as originating with a function of continuous time. The fundamental tool is the classical Fourier transform defined by

and its inverse

where

If the signal is periodic with period

with the expansion having the form

The functions

For the non-periodic case in Equation 1 the spectrum is a function of continuous frequency and for the periodic case in Equation 3, the spectrum is a number sequence (a function of discrete frequency).

**The Spectrum of a Sampled Signal and the DTFT
**

The discrete-time Fourier transform (DTFT) as defined in terms samples of a continuous function is

and its inverse

can be derived by noting that

The spectrum of a discrete-time signal is defined as the DTFT of the
samples of a continuous-time signal given in Equation 5. Samples of the
signal are given by the inverse DTFT in Equation 6 but they can also be
obtained by directly sampling

which can be rewritten as an infinite sum of finite integrals in the form

where

where

This result is very important in determining the frequency domain effects of sampling. It shows what the sampling rate should be and it is the basis for deriving the sampling theorem.

**Samples of the Spectrum of a Sampled Signal and the DFT
**

Samples of the spectrum can be calculated from a finite number
of samples of the original continuous-time signal using the DFT. If we
let the length of the DFT be

and

then from Equation 66 and Equation 10 samples of the DTFT of

therefore,

if

**Samples of the DTFT of a Sequence
**

If the signal is discrete in origin and is not a sampled function of a
continuous variable, the DTFT is defined with

with an inverse

If we want to calculate

which after breaking the sum into an infinite sum of length-

if

This a combination of the results in Equation 10 and in Equation 16.

**Fourier Series Coefficients from the DFT
**

If the signal to be analyzed is periodic, the Fourier integral in Equation 1 does not converge to a function (it may to a distribution). This function is usually expanded in a Fourier series to define its spectrum or a frequency description. We will sample this function and show how to approximately calculate the Fourier series coefficients using the DFT of the samples.

Consider a periodic signal

with the coefficients given in Equation 3. Samples of this are

which is broken into a sum of sums as

But the inverse DFT is of the form

therefore,

and we have our result of the relation of the Fourier coefficients to the DFT of a sampled periodic signal. Once again aliasing is a result of sampling.

**Shannon's Sampling Theorem
**

Given a signal modeled as a real (sometimes complex) valued function of a real variable (usually time here), we define a bandlimited function as any function whose Fourier transform or spectrum is zero outside of some finite domain

for some

such that

This is more compactly written by defining the *sinc* function as

which gives the sampling formula Equation 53 from Least Squared Error Design of FIR Filters the form

The derivation of Equation 53 from Least Squared Error Design of FIR Filters or Equation 56 from Least Squared Error Design of FIR Filters can be done a number of ways. One of the quickest uses infinite sequences of delta functions and will be developed later in these notes. We will use a more direct method now to better see the assumptions and restrictions.

We first note that if

but

therefore,

which is the sampling theorem. An alternate derivation uses a rectangle function and its Fourier transform, the sinc function, together with convolution and multiplication. A still shorter derivation uses strings of delta function with convolutions and multiplications. This is discussed later in these notes.

There are several things to notice about this very important result.
First, note that although