1.3 Sampling, up--sampling, down--sampling, and multi--rate  (Page 2/5)

For the non-periodic case in [link] the spectrum is a function of continuous frequency and for the periodic case in [link] , 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 $F_d\left(\omega \right)$ is periodic with period $P=2\pi /T$ and, therefore, it can be expanded in a Fourier series with [link] resulting from calculating the series coefficients using [link] .

The spectrum of a discrete-time signal is defined as the DTFT of the samples of a continuous-time signal given in [link] . Samples of the signal are given by the inverse DTFT in [link] but they can also be obtained by directly sampling $f\left(t\right)$ in [link] giving

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

where $F_p\left(\omega \right)$ is a periodic function made up of shifted versions of $F\left(\omega \right)$ (aliased) defined in [link] Because [link] and [link] are equal for all $Tn$ and because the limits can be shifted by $\pi /T$ without changing the equality, the integrands are equal and we have

where $F_p\left(\omega \right)$ is a periodic function made up of shifted versions of $F\left(\omega \right)$ as in [link] . The spectrum of the samples of $f\left(t\right)$ is an aliased version of the spectrum of $f\left(t\right)$ itself. The closer together the samples are taken, the further apart the centers of the aliasedspectra are.

This result is very important in determining the frequency domain effects of sampling. It shows what the sampling rate should be and it is thebasis 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 welet the length of the DFT be $N$ and separation of the samples in the frequency domain be $\Delta$ and define the periodic functions

and

then from [link] and [link] samples of the DTFT of $f\left(Tn\right)$ are

therefore,

if $\Delta TN=2\pi$ . This formula gives a method for approximately calculating values of the Fourier transform of a function by taking theDFT (usually with the FFT) of samples of the function. This formula can easily be verified by forming the Riemann sum to approximate the integralsin [link] or [link] .

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 $T=1$ as

with an inverse

If we want to calculate $H\left(\omega \right)$ , we must sample it and that is written as

which after breaking the sum into an infinite sum of length- $N$ sums as was done in [link] becomes

if $\Delta =2\pi /N$ . This allows us to calculate samples of the DTFT by taking the DFT of samples of a periodized $h\left(n\right)$ .

This a combination of the results in [link] and in [link] .