## Frequency domain filter implementation

The dual nature of time and frequency domains means a
filter in the time domain can be equivalently implemented in the frequency
domain. Depending on the application, however, one domain is usually more
convenient to work in than the other.

A recursive IIR filter can be implemented in the frequency
domain by taking the product of the frequency domain equivalents of the input
sequence and the filter.

Here, *X*(ω) and *Y*(ω) are the Discrete Fourier
Transforms (DFT) of the input and the output sequences respectively, and
*IDFT* represents the Inverse Discrete Fourier Transform operation.
*H*_{a}(ω) and *H*_{b}(ω) are the DFTs of the filter
coefficients *a*_{i} and *b*_{j}, respectively, as
given by the following difference equation:

The DFT’s *H*_{a}(ω) and
*H*_{b}(ω) must be of the same length as *X*(ω) and
*Y*(ω). To accomplish this, the filter coefficients must be 0-padded
appropriately. Consequently, the frequency domain implementation is
computationally inefficient and will not be discussed further.