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详细说明： 信号与系统 奥本海默 习题答案 英文 Exercise Ch. 1 1.1 Express each of the following complex numbers in Cartesian form (x+jy): (3) ejp/2 = cos(p/2)+j sin(p/2)=j (6)ejp/4 =[cos(p/4)+j sin(p/4)]=1+j 1.2 Express each of the following complex numbers in polar form (r ejq, with -p< q £ p): (6) (1-j)2 = 1-2j + j2 = 1 + 2 e-jp/2- 1 = 2 e-jp/2 (7) j(1-j) = j +1 = ejp/2 + 1 = ejp/4 ( ejp/4 + e -jp/4 ) = ejp/4 1.4 let x[n] be a signal with x[n] = 0 for n < -2, and n > 4. For each signal given below, determine the values of n for whi ch it is guaranteed to be zero. (d) x [-n+2] Solution: (d) Since –n +2 < -2 g n > 4; –n +2 > 4 g n < -2, so, x [-n+2] will be zero for n < -2 and n > 4. 1.11 Determine the fundamental period of the signal x[n]=1+e-e. Solution: The period of e is N1=7. The period of e is N2=5. So N = N1 N2=35 1.12 Consider the discrete—time signal Determine the values of the integers M and so that x[n] may be expressed as Solution: While So, we can find that easily: M = -1, = -3. 1.13 Consider the continuous-time signal . Calculate the value of for the signal Solution: Because 2) So 1.14 Consider a periodic signal With period T=2. The derivative of this signal is related to the “impulse train” With period T=2. It can be shown that Determine the values of A1, t1, A2, and t2. Solution: We get the figure from the function, Then, in the interval of , We get A1＝3，t1＝0，A2＝－3，t2＝1.) 1.19 For each of the following input-output relationships, determine whether the corresponding system is linear, time invariant or both. (b) (i) Consider two arbitrary inputs x1[n]and x2[n] x1[n] →y1[n] = xı²[n - 2] x2[n] → y2[n] = x2² [n - 2] Let x3[n] be a linear combination of x1[n] and x2[n] That is, x3[n] = ax1[n] + bx2[n] where a and b are arbitrary scalars. If x3[n] is the input to the given system, then the corresponding output y3[n] is Therefore, the system is not linear. (ii)Consider an arbitrary input x1[n]. Let is th e corresponding output. Consider a second input x2[n] obtained by shifting x1[n] in time: The output corresponding to this input is Also note that Therefore, This implies that the system is time-invariant. 1.31 In this problem we illustrate one of the most important consequences of the properties of linearity and time invariance. Specifically, once we know the response of a linear system or a linear time-invariant (LTI) system to a single input or the responses to several inputs, we can directly compute the responses to many other input signals. Much of the remainder of this book deals with a thorough exploitation of this fact in order to develop results and techniques for analyzing and synthesizing LTI systems. (a) Consider an LTI system whose response to the signal x1(t) in Figure P1.31 (a) is the signal y1(t) illustrated in Figure P1.31(b). Determine and sketch carefully the response of the system to the input x2(t) depicted in Figure 1.31(c). (b) Determine and sketch the response of the system considered in part (a) to the input x3(t) shown in Figure P1.31 (d). x2(t) x3(t) Solution: so, , so, ...展开收缩
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