PROBLEM SET 3. Note: This problem set is a little shorter than usual because we have not covered inverse z-transforms yet.

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PROBLEM SET 3 Issued: /3/9 Due: 2/6/9 Reading: During the past week we concluded our discussion DTFT properties and began our discussion of z-transforms, covering basic calculation of the z-transform

1 PROBLEM SET 3 Issued: /3/9 Due: 2/6/9 Reading: During the past week we concluded our discussion DTFT properties and began our discussion of z-transforms, covering basic calculation of the z-transform and the relationship between the nature of the time function and the ROC of its z-transform. During the coming week we will examine z-transform properties and inverses, and the relationships between the z-transform of an LSI system and its corresponding difference equation, along with the relationship between locations of poles and zeros of the z-transform of a system in the z-plane and the magnitude and phase of the DTFT of that system. Our discussion of DTFTs and their properties follows OSYP 2.7 to 2.9. Our discussion of z-transforms will in the end cover all topics in Chapter 3 of OSYP except for Sec. 3.6, plus selected topics in OSYP Sections It is also reviewed in a set of notes on z-transforms that is posted to the course website. Note: This problem set is a little shorter than usual because we have not covered inverse z-transforms yet. Some reminders: Remember that if the input to an LSI system is e j 0 n, the output will be yn e j 0 n H e j 0 While we will discuss z-transform properties in some detail, the properties below may be useful in working some problems on the problem set. Note that Table 3.2 on OSYP 32 provides a more complete list of properties. x n+ x 2 n X z + X 2 z where the ROC is the overlap of the ROCs of X z and xn N z N where the ROC is the same as the ROC of *hn Xz where the ROC is the overlap of the ROCs of Xz and X 2 z

2 8-49 Problem Set 3-2- Spring, 209 Problem 3.: x[n] n Consider the time function depicted above: Note that can be thought of as an even function that is shifted in time. The following questions can be answered using the DTFT properties. There is no need to solve them directly. (a) Determine the value of Xe j for 0 Determine the value of Xe j for 2 (c) Determine Xe j 5n + 2 2n + + 3n+ 3n 2 2n 3 + 5n 4 (d) Determine the value of (e) Determine the value of Xe j d Xe j 2 d (f) Determine and sketch the signal whose DTFT is the real part of Xe j Copyright 209, Richard M. Stern

3 8-49 Problem Set 3-3- Spring, 209 Problem 3.2 : Determine the z-transform of each of the following sequences. Sketch the pole-zero plot and indicate the region of convergence. Indicate whether or not the Fourier transform of the sequence exists. Use of the properties of the z-transforms will make many of the solutions much easier. (a) -- 3 n n un n u5 n (c) nu7 nun 5 (Be sure to check both the transform tables and properties here.) (d) 3 n 2 (e) -- 3 n 4 sin2n 4 3un 4 Problem 3.3: The causal (or right-sided ) sequence Xz has the z-transform Sketch the pole-zero diagram and specify the region of convergence for each of the functions below. z z -- z -- + j z -- j y n y 2 n xn 2 (c) y 3 n n e j0.4n 2 Problem 3.4: In Problem.4d and again in Problem 2.2 we considered the convolution of the following two functions: -- and 5 n 2 un 2 hn 4 n + u n+ As you know from the homework solutions, the result of this convolution is Copyright 209, Richard M. Stern

4 8-49 Problem Set 3-4- Spring, 209 yn n n n n (a) Obtain the Xz and, the z-transforms for and hn. Be sure to specify the regions of convergence. Obtain Yz, the z-transform of yn, and its region of convergence. (c) Show that Yz Xz with the appropriate region of convergence. Problem 3.5: Consider a linear shift-invariant system with transfer function Yz Xz z z -- z -- + j z -- j (a) Sketch the pole-zero diagram associated with this transfer function. List all the possible regions of convergence that could be associated with. For each possible ROC, indicate whether the corresponding unit sample response hn is right-sided, left-sided, both-sided, or finite in duration. (c) Suppose that you are told that the system is actually causal. Is it stable as well? Why or why not? (d) As we have discussed in class, we can also write the transfer function as Yz Xz By cross-multiplying the two expressions on the right and taking the inverse transform of the result of the cross-multiplication, obtain a difference equation that expresses the current output yn as a linear combination of previous outputs and current and previous inputs. Your difference equation should have real coefficients. (e) Note that, as we argued in class on Monday, the initial conditions for this difference equation must be zero in order for the system to be both linear and shift invariant. Specifically, if the system input begins at n 0 the initial conditions would be that all prevous values of yn that are needed to calculate the output y0. If the input to this system is applied at 3z z z -- + j z -- j n 0, what are the required initial conditions? Copyright 209, Richard M. Stern

5 8-49 Problem Set 3-5- Spring, 209 (e) As you know, the unit sample response of the system hn is the inverse z-transform of the transfer function. We will develop a set of general procedures in class on Monday that will enable us to compute inverse z-transforms when the transfer function is of the form of a ratio of polynomials in z or z as is the case in the present example (and most situations that you will encounter in this course). An alternate way of obtaining hn on a sample-by-sample basis is to calculate the system output yn when the input is n. This can be done using iteration, which is described in Sec. III of the z- transform notes and which will be reviewed briefly on Monday. Using the iteration techniques, obtain numerical values of hn for n equal to 0 through 2. MATLAB Problems This week there are no MATLAB problems (!) because the material we are covering at the moment does not lend itself readily to them. Nevertheless, the MATLAB functions zplane, roots, poly, residuez, and freqz are especially useful in dealing with some of the issues addressed in this week s problem set. Look over the help files for these MATLAB functions as they will be very useful in the weeks to come. Copyright 209, Richard M. Stern

SIGNALS AND SYSTEMS. Unit IV. Analysis of DT signals

SIGNALS AND SYSTEMS. Unit IV. Analysis of DT signals SIGNALS AND SYSTEMS Unit IV Analysis of DT signals Contents: 4.1 Discrete Time Fourier Transform 4.2 Discrete Fourier Transform 4.3 Z Transform 4.4 Properties of Z Transform 4.5 Relationship between Z