数字信号处理(英文版)课后习题答案2

(Partial) Solutions to Assignment 2 pp.73-76 1.16 In each of the following systems, let or be the input and or be the output. Determine whether each systems is (1) linear, (2) time invariant, (3) causal, (4) BIBO stable (g).  (i). ans: omitted ---------------------------------------------------- 1.17 A linear time invariant system has impulse response Determine the output sequence for each of the followign input signals: (b) (f)  (b) ans: The z-transform of is given by where ROC1: z-transform of is given by where ROC2: Therefore, the z-transform of the output is given by Perform inverse z to get (f) ans:  using the same method as in (b) (details omitted ) ---------------------------------------------------- 1.18. A linear time invariant system is defined by the difference equation b. Determine  the output of the system when the intpu is c. Determine  the output of the system when the input is ans: omitted ---------------------------------------------------- 1.19 The following expressions define linear time invariant systems. For each one determine the impulse respnose (a) (e) (a) ans: the impulse response is (e) ans: the impulse response is ---------------------------------------------------- 1.20 Each of the following expressions defines a linear time invariant system. For each one determine whether it is BIBO stable or not (g)  (k)  BIBO: Bounded input and bounded output (g) ans: omitted (k) ans: omitted ---------------------------------------------------- 1.21. Using the geometric series, for each of the following sequence determine the z-transform and its ROC (d) (g) (i) (d) ans: where ROC: (g) ans: The first part is equal to where ROC1 is The second part is equal to where ROC2 is Therefore combining both parts: where ROC={ROC1 and ROC2}: (i) ans: where ROC: whole complex domain ---------------------------------------------------- 1.22. You know what the and are. Using the properties only (do not reuse the definition of  the z-transform.) determine the z-transform of the following signals (c) (g) where ROC1: where ROC2: (c) ans: using z-transform property: We have: where ROC: (g) ans: details omitted. The final answer is Therefore combining both parts: where ROC={ROC1 and ROC2}: ---------------------------------------------------- 1.23 Using partial fraction expansion, determine the inverse z-transform of the following functions: (c) , (e) , (c) ans: (e) ans: procedures are the same as above. details omitted. ---------------------------------------------------- 1.24. For each of the followign linear difference equations, determine the impulse response, and indicate whether the system is BIBO stable or not (a) (c) (a) ans: Take z-transform on both sides where ROC: Because is finite Therefore, the system is BIBO stable (c) ans: omitted (the same as (a)) ---------------------------------------------------- 1.25. Although most of the time we assume causality, a linear difference equation can be interpreted in a number of ways. Consider the linear difference equation (a) Determine the transfer function and the impulse response. Is the system causal ? BIBO stable ? (a) omitted. ---------------------------------------------------- 1.26. 1.26 Consider the linear difference equation (a) Determine the transfer  function . Do you have enough information to determine the region of convergence ans: Don't have enough information to determine ROC. ---------------------------------------------------- 1.27. Given the system described by the linear difference equation Determine the output for each of the following input signals (a) (e) (a) ans: Take z-transform on both sides: ---------------------------------------------------- 1.28. Repeat Problem 1.27 when the system is given in terms of the impulse response Before you do anything, is the system stable ? Does the frequency response exist ? ans: omitted. ---------------------------------------------------- 1.29. Repeat Problem 1.27 when the system isgiven by the linear difference equation  Before you do anything, is the system stable ? Does the frequency response exist ? Ans: omitted. ----------------------------------------------------