In the design of either analog or digital filters, we often approximate a specified magn...

In the design of either analog or digital filters, we often approximate a specified magnitude characteristic without particular regard to the phase. For example, standard design techniques for lowpass and bandpass filters are typically derived from a consideration of the magnitude characteristics only.

In many filtering problems one would ideally like the phase characteristics to be zero or linear. For causal filters, it is impossible to have zero phase . However, for many digital filtering applications, it is not necessary that the unit sample response fie of the filter be zero for n < 0 if the processing is not to be carried out in real time.

One technique commonly use in digital filtering when the data to be filtered are of finite duration and stored, for example, on a disc or magnetic tape is to process the data forward and then backward through the same filter.

Let h[n] be the unit sample response of a causal filter with an arbitrary phase characteristic. Assume that h[n] is real, and denote its Fourier transform by . Let x[n] be the data that we want to filter. The filtering operation is performed as follows: