It is often desirable to transform a prototype discrete-time lowpass filter to another k...

It is often desirable to transform a prototype discrete-time lowpass filter to another kind of discrete-time frequency-selective filter. In particular, the impulse invariance approach cannot be used to convert continuous-time highpass or bandstop filters to discrete-time highpass or bandstop filters. Consequently, the traditional approach has been to design a prototype lowpass discrete-time filter using either impulse invariance or the bilinear transformation and then to use an algebraic transformation to convert the discrete-time lowpass filter into the desired frequency-selective filter.

To see how this is done, assume that we are given a lowpass system function H_1p(Z) that we wish to transform to a new system function H(z), which has either lowpass, highpass, bandpass, or bandstop characteristics when it is evaluated on the unit circle. Note that we associate the complex variable Z with the prototype lowpass filter and the complex variable z with the transformed filter. Then, we define a mapping from the Z-plane to the z-plane

Instead of expressing Z as a function of z, we have assumed in Eq. (P7.64-1) that Z⁻¹ is expressed as a function of z^-1. Thus, according to Eq. (P7.64-2), in obtaining H(z) from H_1p(Z), we simply replace Z⁻¹ everywhere in Hlp(Z) by the function G(z^-1). This is a convenient representation, because H_1p(Z) is normally expressed as a rational function of Z⁻¹.

If H_1p(Z) is the rational system function of a causal and stable system, we naturally require that the transformed system function H(z) be a rational function of z −1 and that the system also be causal and stable. This places the following constraints on the transformation Z ⁻¹ = G(z ⁻¹):

1. G(z ^{− 1} ) must be a rational function of z⁻¹.

2. The inside of the unit circle of the Z-plane must map to the inside of the unit circle of the z-plane.

3. The unit circle of the Z-plane must map onto the unit circle of the z-plane.

In this problem, you will derive and characterize the algebraic transformations necessary to convert a discrete-time lowpass filter into another lowpass filter with a different cutoff frequency or to a discrete-time highpass filter.

(a) Let θ and ω be the frequency variables (angles) in the Z-plane and z-plane, respectively, i.e., on the respective unit circles Z = e^jθ and z = e^jω. Show that, for Condition 3 to hold, G(z ⁻¹) must be an all-pass system, i.e.,

(b) It is possible to show that the most general form of G(z⁻¹) that satisfies all of the preceding three conditions is

From our discussion of all-pass systems in Chapter 5, it should be clear that G(z⁻¹), as given in Eq. (P7.64-4), satisfies Eq. (P7.64-3), i.e., is an allpass system, and thus meets Condition 3. Eq. (P7.64-4) also clearly meets Condition 1. Demonstrate that

Condition 2 is satisfied if and only if |α_k| < 1.

(c) A simple 1st-order G(z⁻¹) can be used to map a prototype lowpass filter H_1p(Z) with cutoff θ_p to a new filter H(z) with cutoff ω_p. Demonstrate that

will produce the desired mapping for some value of α. Solve for α as a function of θ_p and ω_p. Problem 7.51 uses this approach to design lowpass filters with adjustable cutoff frequencies.

(d) Consider the case of a prototype lowpass filter with θ_p = π/2. For each of the following choices of α, specify the resulting cutoff frequency ω_p for the transformed filter:

(i) α = −0.2679.

(ii) α = 0.

(iii) α = 0.4142.

(e) It is also possible to find a 1^st-order all-pass system for G(z⁻¹) such that the prototype lowpass filter is transformed to a discrete-time highpass filter with cutoff ωp. Note that such a transformation must map Z⁻¹ = e^jθp → z⁻¹ = e^jωp and also map Z⁻¹= 1 → z⁻¹ = −1; i.e., θ = 0 maps to ω = π. Find G(z⁻¹) for this transformation, and also, find an expression for α in terms of θ_p and ω_p.

(f) Using the same prototype filter and values for α as in part (d), sketch the frequency responses for the highpass filters resulting from the transformation you specified in part (e).

Similar, but more complicated, transformations can be used to convert the prototype lowpass filter H1_p(Z) into bandpass and bandstop filters. Constantinides (1970) describes these transformations in more detail.