Reading Assignment: Read pp.202-230,250-266, and 290-317 of DTSP.

Homework Assignment: Hand in Problems 7.2*, 7.3*, 7.4*, 7.5*, and 7.6*.

Problem 7.1

A causal linear time-invariant system has system function

(a) Determine the poles and zeros of this system function and plot them in the z-plane. What is the region of convergence for this system function?

(b) >From the pole-zero plot, sketch the log-magnitude of the frequency response of this system as accurately as is possible.

(c) If you know MATLAB, use conv( ) to obtain third-order polynomials for the numerator and denominator and then use freqz( ) to evaluate and plot the log-magnitude of the the frequency response. Compare to your answer in part (b).

Problem 7.2*

Problem 5.23 of Chapter 5 of Discrete-Time Signal Processing.

Problem 7.3*

Problem 5.24 of Chapter 5 of Discrete-Time Signal Processing.

Problem 7.4*

Consider a causal linear time-invariant system whose system function is

(a) Draw the signal flow graphs for implementations of the system in each of the following forms:

(i) Direct form I

(ii) Direct form II

(iii) Cascade form using first- and second-order direct form II sections

(iv) Parallel form using first- and second-order direct form II sections

(v) Transposed direct form II

(b) Write the difference equations for the network of part (a)(v) and use z-transforms to show that this network has the correct system function.

Problem 7.5*

Work Problem 6.12 in Chapter 6 of Discrete-Time Signal Processing.

Problem 7.6*

Work Problem 5.40 in Chapter 5 of Discrete-Time Signal Processing.

Let be the impulse response of an FIR system; i.e., for n<0 and n>M. Assume that is real. We can impose the condition of generalized linear phase on the frequency response of the system by imposing certain symmetry conditions on the impulse response. For types I (M even) and II (M odd) generalized linear phase systems, the impulse response satisfies

The frequency response of type I (M even) systems has the form

where and otherwise. Similarly, the frequency response of type II (M odd) systems has the form

where .

For types III (M even) and IV (M odd) systems, the impulse response satisfies

The frequency response of type III systems has the form

where , and for type IV (M odd) systems, the frequency response has the form

where .

(a) Verify the formula for for type II systems.

(b) Show that for the type I and II cases (i.e., ), if is a zero of , then , , and are also zeros of .

(c) Show that for the type III and IV cases (i.e., ), if is a zero of , then , , and are also zeros of .

(d) Show that it is always true that for the type II case (M odd).

(e) Show that it is always true that for both the type III (M even) and the type IV (M odd) cases, and that it is also always true that for the type III case.

Let be the impulse response of an FIR system; i.e., for n<0 and n>M. Assume that is real. We can impose the condition of generalized linear phase on the frequency response of the system by imposing certain symmetry conditions on the impulse response. For types I (M even) and II (M odd) generalized linear phase systems, the impulse response satisfies

and for types III (M even) and IV (M odd) systems, the impulse response satisfies

This problem investigates the implications of these constraints.

(a) Show that the frequency response of type I and II systems has the form

where is a real, even, and periodic function of . Determine expressions for and for M even and M odd.

(b) Show that the frequency response of type III and IV systems has the form

where is a real, odd, and periodic function of . Determine expressions for and for M even and M odd.

(c) Show that for the type I and II cases (i.e., ), if is a zero of , then , , and are also zeros of .

(d) Show that for the type III and IV cases (i.e., ), if is a zero of , then , , and are also zeros of .

(e) Show that it is always true that for the type II case (M odd).

(f) Show that it is always true that for both the type III (M even) and the type IV (M odd) cases, and that it is also always true that for the type III case.