Let Hlp(Z) denote the system function for a discrete-time lowpass filter. The implementa...

Let Hlp(Z) denote the system function for a discrete-time lowpass filter. The implementations of such a system can be represented by linear signal flow graphs consisting of adders, gains, and unit delay elements as in Figure P7.51-1. We want to implement a low pass filter for which the cutoff frequency can be varied by changing a single parameter. The proposed strategy is to replace each unit delay element in a flow graph representing H_lp(Z) by the network shown in Figure P7.51-2, whereαis real and |α| < 1.

(a) Let H(z) denote the system function for the filter that results when the network of Figure P7.51-2 is substituted for each unit delay branch in the network that implements H_lp(Z). Show that H(z) and H_lp(Z) are related by a mapping of the Z-plane into the z-plane.

(b) If H(e^jω) and H_lp(e^jθ ) are the frequency responses of the two systems, determine the relationship between the frequency variables ω and θ. Sketch ω as a function of θ for α = 0.5, and −0.5, and show that H(ejω) is a lowpass filter. Also, if θ_p is the passband cutoff frequency for the original low pass filter H_lp(Z), obtain an equation for ω_p, the cutoff frequency of the new filter H(z), as a function of α and θ_p.

(c) Assume that the original low pass filter has the system function

Draw the flow graph of an implementation of H_lp(Z), and also draw the flow graph of the implementation of H(z) obtained by replacing the unit delay elements in the first flow graph by the network in Figure P7.51-2. Does the resulting network correspond to a computable difference equation?

(d) If H_lp(Z) corresponds to an FIR system implemented in direct form, would the flow graph manipulation lead to a computable difference equation? If the FIR system H_lp(Z) was a linear-phase system, would the resulting system H(z) also be a linearphase

system? If the FIR system has an impulse response of length M + 1 samples what would be the length of the impulse response of the transformed system?

(e) To avoid the difficulties that arose in part (c), it is suggested that the network of Figure P7.51-2 be cascaded with a unit delay element, as depicted in Figure P7.51-3. Repeat the analysis of part (a) when the network of Figure P7.51-3 is substituted for each unit delay element. Determine an equation that expresses θ as a function of ω, and show that if H_lp(e^jθ ) is a low pass filter, then H(e^jω) is not a low pass filter.