In the design of either continuous-time or discrete-time filters, we often approximate a...

In the design of either continuous-time or discrete-time filters, we often approximate a specified magnitude characteristic without particular regard to the phase. For example, standard design techniques for low pass and band pass filters are derived from a consideration of the magnitude characteristics only.

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

One technique commonly used in discrete-time filtering when the data to be filtered are of finite duration and are stored, for example, in computer memory is to process the data forward and then backward through the same filter.

Let h[n] be the impulse response of a causal filter with an arbitrary phase characteristic. Assume that h[n] is real, and denote its Fourier transform by H(e^jω). Let x[n] be the data that we want to filter.

(a) Method A: The filtering operation is performed as shown in Figure P5.74-1.

1. Determine the overall impulse response h₁[n] that relates x[n] to s[n], and show that it has a zero-phase characteristic.

2. Determine |H₁(e^jω)|, and express it in terms of |H(e^jω)| and H(e^jω).

(b) Method B: As depicted in Figure P5.74-2, process x[n] through the filter h[n] to get g[n]. Also, process x[n] backward through h[n] to get r[n]. The output y[n] is then taken as the sum of g[n] and r[−n]. This composite set of operations can be represented by a filter with input x[n], output y[n], and impulse response h₂[n].

1. Show that the composite filter h₂[n] has a zero-phase characteristic.

2. Determine |H₂(e^jω)|, and express it in terms of |H(e^jω)| and ∠H(e^jω).

(c) Suppose that we are given a sequence of finite duration on which we would like to perform a bandpass zero-phase filtering operation. Furthermore, assume that we are given the bandpass filter h[n], with frequency response as specified in Figure P5.74-3, which has the magnitude characteristic that we desire, but has linear phase. To achieve zero phase, we could use either method A or B. Determine and sketch |H₁(e^jω)| and |H₂(e^jω)|. From these results, which method would you use to achieve the desired bandpass filtering operation? Explain why. More generally, if h[n] has the desired magnitude, but a nonlinear phase characteristic, which method is preferable to achieve a zero-phase characteristic?