# Time / Frequency Properties

## The DFT and IDFT

• The Discrete Fourier Transform (DFT) is given by:
 (1)

• The operation of the DFT can be reversed to recover a time-domain signal from its frequency representation. This is done with the Inverse Discrete Fourier Transform (IDFT).

• The IDFT is given by:
 (2)

## DFT Theorems

• The DFT specifies the relationship between a time-domain signal and its frequency-domain representation.

• Many general relationships between these two representations can be derived from Eq. (1).

• For example, if a time-domain signal x[n] is time-shifted by an amount (resulting in a signal ), its frequency-domain representation is simply multiplied by a phase shift term of :
 = = =

• This result is referred to as the Shift Theorem'', given by a corresponding transform pair'':
 (3)

• Note that the phase shift term has unity magnitude ( ) and so, a time-shift produces no change in the magnitude response of the corresponding frequency-domain representation.

## Convolution

• Convolution is the operation performed'' by digital filters. We can say that the filter convolves an input signal x[n] with its impulse response h[n] to produce an output y[n].

• Convolution is described mathematically as:
 (4)

• Convolution can be interpreted as a sample-by-sample multiplication and sum of the signal x[n] and a time ''flipped'' and delayed version of the signal h[n].

• We can derive the time/frequency transform pair for the convolution process as follows:
 = = X[k] H[k]

• We have just derived the very important result that convolution in the time domain corresponds to multiplication in the frequency domain:
 (5)

• For long signal lengths (N > 100 or so), it is much faster to transform signals with the FFT (and IFFT) and to perform the convolution as a frequency-domain multiplication.