Page 127 - DSP Integrated Circuits

112 Chapters Digital Signal Processing

Figure P3.14a Figure P3.14b Direct form I

(c) Assume that the filter is realized in cascade form with first- and second-
order direct form I sections. Further, assume that a new filter is obtained
by changing the sign of the coefficients ai and b\ in each section. Sketch
the magnitude function of this new filter between (»T = 0 and ofT = 2n as
well as the impulse and step responses.
3.15 Prove Equation (3.18).
3.16 Give an example of an application in which the group delay properties are of
major interest? What are the reasons?
3.17 Show that a filter with the transfer function

is an allpass filter. The coefficient a is real and I a I < 1.
3.18 Determine the transfer function for the filter shown in Figure P3.18. Sketch
the pole-zero configuration in the z-plane.

Figure P3.18

3.19 Show that the DFT of a finite-length real sequence has conjugate symmetry.
3.20 In a car phone an array of eight microphones is used to cancel noise from the
speech signal. Each signal from the array is provided with an adaptive