Definition

Forward transform

where .

Inverse transform

Periodic Data

If is -periodic, then we have

Since . And here , so that

where we have the Dirac Comb

Thus, we get

And we can rearrange this into

which gives

where , and is the discrete Fourier series of . This is quite similar to the continuous case.

Note that the discrete Fourier Series is periodic with period , then we get the period of here is .

Properties

PropertySignalDTFT of the signal
Linear
Time shift
Frequency shift
Conjugate
Time inverse
Time expand
Convolution
Multiplication
Time difference
Sum
Frequency differentiate
Frequency real part
Frequency imaginary part

Note

The singular part of the sum property, , is aroused when . In this case does not converge.

Examples

Periodic

SignalDTFT
, see here

Non-periodic

SignalDTFT