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《信号与系统》第1章信号与系统1.0 引言1.1 连续时间和离散时间信号1.1.1 举例与数学表示1.1.2 信号能量与功率1.2 自变数的变换1.2.1 自变数变换举例1.2.2 周期信号1.2.3 偶信号与奇信号1.3 指数信号与正弦信号1.3.1 连续时间复指数信号与正弦信号1.3.2 离散时间复指数信号与正弦信号1.3.3 离散时间复指数序列的周期性质1.4 单位冲激与单位阶跃函数1.4.1 离散时间单位脉冲和单位阶跃序列1.4.2 连续时间单位阶跃和单位冲激函数1.5 连续时间和离散时间系统1.5.1 简单系统举例1.5.2 系统的互联1.6 基本系统性质1.6.1 记忆系统与无记忆系统1.6.2 可逆性与可逆系统1.6.3 因果性1.6.4 稳定性1.6.5 时不变性1.6.6 线性1.7 小结习题第2章线性时不变系统2.0 引言2.1 离散时间LTI系统：卷积和2.1.1 用脉冲表示离散时间信号2.1.2 离散时间LTI系统的单位脉冲响应及卷积和表示2.2 连续时间LTI系统：卷积积分2.2.1 用冲激表示连续时间信号2.2.2 连续时间LTI系统的单位冲激响应及卷积积分表示2.3 线性时不变系统的性质2.3.1 交换律性质2.3.2 分配律性质2.3.3 结合律性质2.3.4 有记忆和无记忆LTI系统2.3.5 LTL系统的可逆性2.3.6 LTI系统的因果性2.3.7 LTI系统的稳定性2.3.8 LTI系统的单位阶跃响应2.4 用微分和差分方程描述的因果LTI系统2.4.1 线性常系数微分方程2.4.2 线性常系数差分方程2.4.3 用微分和差分方程描述的一阶系统的方框图表示2.5 奇异函数2.5.1 作为理想化短脉冲的单位冲激2.5.2 通过卷积定义单位冲激2.5.3 单位冲激偶和其它的奇异函数2.6 小结习题第3章周期信号的傅里叶级数表示3.0 引言3.1 历史回顾3.2 LTI系统对复指数信号的响应3.3 连续时间周期信号的傅里叶级数表示3.3.1 成谐波关系的复指数信号的线性组合3.3.2 连续时间周期信号傅里叶级数表示的确定3.4 傅里叶级数的收敛3.5 连续时间傅里叶级数性质3.5.1 线性3.5.2 时移性质3.5.3 时间反转3.5.4 时域尺度变换3.5.5 相乘3.5.6 共轭及共轭对称性3.5.7 连续时间周期信号的帕斯瓦尔定理3.5.8 连续时间傅里叶级数性质列表3.5.9 举例3.6 离散时间周期信号的傅里叶级数表示3.6.1 成谐波关系的复指数信号的线性组合3.6.2 周期信号傅里叶级数表示的确定3.7 离散时间傅里叶级数性质3.7.1 相乘3.7.2 一阶差分3.7.3 离散时间周期信号的帕斯瓦尔定理3.7.4 举例3.8 傅里叶级数与LTI系统3.9 滤波3.9.1 频率成形滤波器3.9.2 频率选择性滤波器3.10 用微分方程描述的连续时间滤波器举例3.10.1 简单RC低通滤波器3.10.2 简单RC高通滤波器3.11 用差分方程描述的离散时间滤波器举例3.11.1 一阶递归离散时间滤波器3.11.2 非递归离散时间滤波器3.12 小结习题第4章连续时间傅里叶变换4.0 引言4.1 非周期信号的表示：连续时间傅里叶变换4.1.1 非周期信号傅里叶变换表示的导出4.1.2 傅里叶变换的收敛4.1.3 连续时间傅里叶变换举例4.2 周期信号的傅里叶变换4.3 连续时间傅里叶变换性质4.3.1 线性4.3.2 时移性质4.3.3 共轭及共轭对称性4.3.4 微分与积分4.3.5 时间与频率的尺度变换4.3.6 对偶性4.3.7 帕斯瓦尔定理4.4 卷积性质4.4.1 举例4.5 相乘性质4.5.1 具有可变中心频率的频率选择性滤波4.6 傅里叶变换性质和基本傅里叶变换对列表4.7 由线性常系数微分方程表征的系统4.8 小结习题第5章离散时间傅里叶变换5.0 引言5.1 非周期信号的表示：离散时间傅里叶变换5.1.1 离散时间傅里叶变换的导出5.1.2 离散时间傅里叶变换举例5.1.3 关于离散时间傅里叶变换的收敛问题5.2 周期信号的傅里叶变换5.3 离散时间傅里叶变换性质5.3.1 离散时间傅里叶变换的周期性5.3.2 线性5.3.3 时移与频移性质5.3.4 共轭与共轭对称性5.3.5 差分与累加5.3.6 时间反转5.3.7 时域扩展5.3.8 频域微分5.3.9 帕斯瓦尔定理5.4 卷积性质5.4.1 举例5.5 相乘性质5.6 傅里叶变换性质和基本傅里叶变换对列表5.7 对偶性5.7.1 离散时间傅里叶级数的对偶性5.7.2 离散时间傅里叶变换和连续时间傅里叶级数之间的对偶性5.8 由线性常系数差分方程表征的系统5.9 小结习题第6章信号与系统的时域和频域特性6.0 引言6.1 傅里叶变换的模和相位表示6.2 LTI系统频率响应的模和相位表示6.2.1 线性与非线性相位6.2.2 群时延6.2.3 对数模和波特图6.3 理想频率选择性滤波器的时域特性6.4 非理想滤波器的时域和频域特性讨论6.5 一阶与二阶连续时间系统6.5.1 一阶连续时间系统6.5.2 二阶连续时间系统6.5.3 有理型频率响应的波特图6.6 一阶与二阶离散时间系统6.6.1 一阶离散时间系统6.6.2 二阶离散时间系统6.7 系统的时域分析与频域分析举例6.7.1 汽车减震系统的分析6.7.2 离散时间非递归滤波器举例6.8 小结习题第7章采样7.0 引言7.1 用信号样本表示连续时间信号：采样定理7.1.1 冲激串采样7.1.2 零阶保持采样7.2 利用内插由样本重建信号7.3 欠采样的效果：混迭现象7.4 连续时间信号的离散时间处理7.4.1 数字微分器7.4.2 半采样间隔延时7.5 离散时间信号采样7.5.1 脉冲串采样7.5.2 离散时间抽取与内插7.6 小结习题第8章通信系统8.0 引言8.1 复指数与正弦幅度调制8.1.1 复指数载波的幅度调制8.1.2 正弦载波的幅度调制8.2 正弦AM的解调8.2.1 同步解调8.2.2 异步解调8.3 频分多路复用8.4 单边带正弦幅度调制8.5 用脉冲串作载波的幅度调制8.5.1 脉冲串载波调制8.5.2 时分多路复用8.6 脉冲幅度调制8.6.1 脉冲幅度已调信号8.6.2 在PAM系统中的码间干扰8.6.3 数字脉冲幅度和脉冲编码调制8.7 正弦频率调制8.7.1 窄带频率调制8.7.2 宽带频率调制8.7.3 周期方波调制信号8.8 离散时间调制8.8.1 离散时间正弦幅度调制8.8.2 离散时间调制转换8.9 小结习题第9章拉普拉斯变换9.0 引言9.1 拉普拉斯变换9.3 拉普拉斯反变换9.4 由零极点图对傅里叶变换进行几何求值9.4.1 一阶系统9.4.2 二阶系统9.4.3 全通系统9.5 拉普拉斯变换的性质9.5.1 线性9.5.2 时移性质9.5.3 Ｓ域平移9.5.4 时域尺度变换9.5.5 共轭9.5.6 卷积性质9.5.7 时域微分9.5.8 Ｓ域微分9.5.9 时域积分9.5.10 初值与终值定理9.5.11 性质列表9.6 常用拉普拉斯变换对9.7 用拉普拉斯变换分析和表征LTI系统9.7.1 因果性9.7.2 稳定性9.7.3 由线性常系数微分方程表征的LTI系统9.7.4 系统特性与系统函数的关系举例9.7.5 巴特沃兹滤波器9.8 系统函数的代数属性与方框图表示9.8.1 LTI系统互联的系统函数9.8.2 由微分方程和有理系统函数描述的因果LTI系统的方框图表示9.9单边拉普拉斯变换9.9.1 单边拉普拉斯变换举例9.9.3 利用单边拉普拉斯变换求解微分方程9.10 小结习题第10章Z变换10.0 引言10.1 Ｚ变换10.2 Ｚ变换的收敛域10.3 Ｚ反变换10.4 由零极点图对傅里叶变换进行几何求值10.4.1 一阶系统10.4.2 二阶系统10.5 Ｚ变换的性质10.5.1 线性10.5.2 时移性质10.5.3 Ｚ域尺度变换10.5.4 时间反转10.5.5 时间扩展10.5.6 共轭10.5.7 卷积性质10.5.8 Ｚ域微分10.5.9 初值定理10.5.10 性质小结10.6 几个常用Ｚ变换对10.7 利用Ｚ变换分析与表征LTI系统10.7.1 因果性10.7.2 稳定性10.7.3 由线性常系数差分方程表征的LTI系统10.7.4 系统特性与系统函数的关系举例10.8 系统函数的代数属性与方框图表示10.8.1 LTI系统互联的系统函数10.8.2 由差分方程和有理系统函数描述的因果LTI系统的方框图表示10.9 单边Ｚ变换10.9.1 单边Ｚ变换和单边Ｚ反变换举例10.9.2 单边Ｚ变换性质10.9.3 利用单边Ｚ变换求解差分方程10.10 小结习题第11章线性反馈系统11.0 引言11.1 线性反馈系统11.2 反馈的某些应用及结果11.2.1 逆系统设计11.2.2 非理想组件的补偿11.2.3 不稳定系统的稳定11.2.4 采样数据反馈系统11.2.5 跟踪系统11.2.6 反馈引起的不稳定11.3 线性反馈系统的根轨迹分析法11.3.1 一个例子11.3.2 死循环极点方程11.3.3 根轨迹的端点：K＝0和｜K｜＝＋∞时的死循环极点11.3.4 角判据11.3.5 根轨迹的性质11.4 奈奎斯特稳定性判据11.4.1 围线性质11.4.2 连续时间LTI反馈系统的奈奎斯特判据11.4.3 离散时间LTI反馈系统的奈奎斯特判据11.5 增益和相位裕度11.6 小结。
Chapter 9 Answers（a ）The given integral may be written as(5)0t j t e e dt σω∞-+⎰If σ<-5 ,then the function (5)te σ-+ grows towards ∞ with increasing t and the given integral does not converge .but if >-5,then the integral does converge (b) The given integral may be written as0(5)t j tee d σω-+-∞⎰t If σ>-5 ,then the function (5)te σ-+ grows towards ∞as t decreases towards -∞and the given integral doesnot converge .but if σ<-5,then the integral does converge (c) The given integral may be written as5(5)5t j t e e d σω-+-⎰t Clearly this integral has a finite value for all finite values of σ. (d) The given integral may be written as(5)t j t e e d σω∞-+-∞⎰tIfσ>-5 ,then the function (5)t e σ-+ grows towards ∞as t decreases towards -∞and the given integraldoes not converge If σ<-5, ,then function (5)te σ-+ grows towards ∞ with increasing t and the given integral does not converge If σ=5, then the integral stilldoes not have a finite value. therefore, the integral does not converge for any value of σ. (e) The given integral may be written as0(5)t j tee d σω-+-∞⎰t+ (5)0t j t e e d σω∞-+⎰t The first integral converges for σ<-5, the second internal converges if σ>-5,therefore, the given internal converges whenσ<5.(f) The given integral may be written as0(5)t j t e e d σω-+-∞⎰tIf σ>5 ,then the function (5)te σ--+grows towards ∞ as t decrease towards -∞ and the given integral does not converge .but if σ<5,then the integral does converge. (a)X(s)= 5(1)t dt eu t e dt ∞---∞⎰- =(5)0s tedt ∞-+⎰ =(5)5s es -++As shown in Example the ROC will be {}Re s >-5. (b) By using eg., we can easily show that g(t)=A 5te-u(-t-0t ) has the Laplace transformG(s)= 0(5)5s t Ae s ++The ROC is specified as {}Re s <-5 . Therefore ,A=1 and 0t =-1Using an analysis similar to that used in Example we known that given signal has a Laplace transform of the formX(s)115s s β+++The corresponding ROC is {}Re s >max(-5,Re{β}). Since we are given that the ROC isRe{s}>-3, we know that Re{β}=3 . there are no constraints on the imaginary part of β. We know form Table that111()sin(2)()()()Lt x t e t u t X s X s -=-←−→=-, Re{s}>-1 We also know form Table thatx(t)= 1()Lx t -←−→X(s)= 1()X s -The ROC of X(s) is such that if 0s was in the ROC of 1()X s , then -0s will be in the ROC of X(s). Putting the two above equations together ,we havex(t)= 1x (-t) =sin(2)()t e t u t --L ←−→X(s)= 1()X s -=-222(1)2s -+, {}Re s <1the denominator of the form 2s -2s+5. Therefore, the poles of X(s) are 1+2j and 1-2j.(a) the given Laplace transform may be written as ()X s =24(1)(3)s s s +++.Clearly ,X(s) has a zero at s=-2 .since in X(s) the order of the denominator polynomial exceeds the order of the numerator polynomial by 1 ,X(s) has a zero at ∞. Therefore ,X(s) has one zero in finite s-plane and one zero at infinity.(b) The given Laplance transform may be written asX(s)=1(1)(1)s s s +-+= 11s -Clearly ,X(s) has no zero in the finite s-plane .Since in X(s) the of the denominator polynomial exceedsthe order the numerator polynomial by 1,X(s) has a zero at ∞.therefore X(s) has no zero in the finite s-plane and one zero at infinity.(c) The given Laplace transform may be written as22(1)(1)()1(1)s s s X s s s s -++==-++Clearly ，X （s ）has a zero at s= in X(s) the order of the numerator polynomial exceeds the order of the denominator polynomial by 1,X(s) has zeros at ∞ .therefore , X(s) has one zero in the s-plane and no zero at infinity .(a) No. From property 3 in Section we know that for a finite-length signal .the ROC is the entire s-plane .therefore .there can be no poles in the finite s-plane for a finite length signal . Clearly in this problem this not the case.(b) Yes. Since the signal is absolutely integrable, The ROC must include, the j ω-axis . Furthermore ,X(s) has a pole at s=2 .therefore, one valid ROC for the signal would be Re{s}<2. From property 5 in section we know that this would correspond to a left-sided signal(C) No . Since the signal is absolutely integrable, The ROC must include , the j ω-axis . Furthermore ,X(s) has a pole at s=2. therefore ,we can never have an ROC of the form Re{s}> α. From property 5 in section we knew that x(t) can not be a right-side signal(d) Yes . Since the signal is absolutely integrable, The ROC must include , the j ω-axis . Furthermore ,X(s) has a pole at s=2 .therefore, one valid ROC for the signal could be α<Re{s}<2 such that α<0 .From property 6 in section ,we know that this would correspond to a two side signalWe may find different signal with the given Laplace transform by choosing different regions of 02s =- 13s =- 2132s j =- 3132s j =-Based on the locations of the locations of these poles , we my choose form the following regions of convergence: (i) Re{s}>- 12(ii)-2< Re{s}<- 12(iii)-3<Re{s}<-2 (iv)Re{s}<-3Therefore ,we may find four different signals the given Laplace transform. From Table ,we know thatG(t)= 2()()(2)L te x t G s X s ←−→=-. The ROC of G(s) is the ROC of X(s) shifted to the right by 2We are also given that X(s) has exactly 2 poles at s=-1 and s=-3. since G(s)=X(s-2), G(s)also has exactly two poles ,located at s=-1+2=1 and s=-3+2=-1 since we are given G(j ω) exists , we may infer that j ω-axis lies in the ROC of G(s). Given this fact and the locations of the poles ,we may conclude that g(t) is a two sidesequence .Obviously x(t)= 2te g(t) will also be two sidedUsing partial fraction expansion X(s)= 4243s s -++Taking the inverse Laplace transform, X(t)=443()2()t t e u t e u t ---The pole-zero plots for each of the three Laplace transforms is as shown in Figure(a) form Section we knew that the magnitude of the Fourier transform may be expressed aswe se that the right-hand side of the above expression is maximum for ω=0 and decreases as ω becomesincreasing more positive or more negative . Therefore 1()H j ω is approximately lowpass (b) From Section we know that the magnitude of the Fourier transform may be express as 1313(length of vector from to -+j )(Length of vector from to --j )ωωwe see that the right-hand side of the above expression is zero for ω= then increams withincreasing |ω| until |ω| reach 1/2. Then it starts decreasing as |ω| increase even further. Therefore | 2H (j )ω| is approximately bandpass.(c) From Section we know that the magnitude of the Fourier transform may be express as21313(length of vector from to -+j )(Length of vector from to --j ) ωωWe see that the right-hand side of the above expression is zero for ω=0. It then increases withincreasing |ω| until |ω| reaches 21. Then |ω| increases,| 3()H j ω| decreases towards a value of1(because all the vector lengths became almost identical and the ratio become 1) .Therefore |3()H j ω| is approximately highpass.X(s) has poles at s=13-213-2(s) has zeros at s=132and 132.From Section we know that |X(j ω)| is1313(Length of vector from toto 22221313(length of vector from to -+j )(Length of vector from to --j )ωωωωThe terms in the numerator and denominator of the right-band side of above expression cancel ourgiving us |X(j ω)|=1.(a) If X(s) has only one pole, then x(t) would be of the form ate - such a signal violates condition 2. Therefore , this statement is inconsistent with the given information.(b) If X(s) has only two poles, then x(t) would be of the form A 0sin()ate t ω- .Clearly such a signal could be made to satisfy all three conditions(Example:0ω=80π,α=19200). Therefore, this statement is consistent with the given information. (c) If X(s) has more than two poles (say 4 poles), then x(t) could be assumed to be of theform 00sin()sin()at btAe t Be t ωω--+. Clearly such a signal could still be made to satisfy all three conditions. Therefore, this statement is consistent with the given information. We have1}Re{,1)(->+=s s s X β.Also,(Length of vector form ω to -1)(Length of vector form ω to 1 I m-1 -3 Re -4 Re 1 Im Im Re1}Re{1),()()(<<--+=s s X s X s G αTherefore, ].11[)(2s s s s G -++-=ααβComparing with the given equation for G(s), ,1-=α .21=β. Since X(s) has 4 poles and no zero in the finite s-plane, we many assume that X(s) is of the form .))()()(()(d s c s b s a s As X ----=Since x(t) is real ,the poles of X(s) must occur in conjugate reciprocal pairs. Therefore, we mayassume that b=*a and d=*c . This result in .))()()(()(**c s c s a s a s As X ----=Since the signal x (t) is also even , the Laplace transform X(s) must also be even . This implies thatthe poles have to be symmetric about the j ω-axis. Therefore, we may assume that c=*a -. This results in .))()()(()(**a S a s a s a s As X ++--=We are given that the location of one of the poles is (1/2)4πj e . If we assume that this pole is a, we have 4444AX(s)=.1111(s-)(s-)(s+)(s+)2222j j j j e e e e ππππ-- This gives us22().11()()4422AX s s s s s =Also ,we are give that()(0)4x t dt X ∞-∞==⎰Substituting in the above expression for X(s), we have A=1/4. Therefore, 221/4().11()()4422X s s s s s =. Taking the Laplace transform of both sides of the two differential equations, we haves X(s)=1)(2+-s Y and s Y(s)=2X(s) . Solving for X(s) and Y(s), we obtain4)(2+=s s s X and Y(s)= 22s 4+.The region of convergence for both X(s) and Y(s) is Re{s}>0 because both are right-hand signals. . Taking the Laplace transform of both sides of the given differential equations ,we obtain ).(])1()1()[(223s X s s s s Y =+++++αααα therefore,.)1()1(1)()()(223αααα+++++==s s s s X s Y s H(a) Taking the Laplace transform of both sides of the given equation, we haveG(s) = s H(s)+ H(s). Substituting for H(s) from above,.1)1()1()1()(22223αααααα++=++++++=s s s s s s s GTherefore, G(s) has 2 poles.(b) we know that H(s) =.))(1(122αα+++s s s Therefore, H(s) has poles at and j ),2321(,1+--α ).2321(j --α If the system has to be stable,then the real part of the poles has to be less than zero. For this to be true, we require that ,02/<-α .,0>α.The overall system show in Figure may be treated as two feedback system of the form shown in figure connected in parallel. By carrying out an analysis similar to that described in Section we find the system function of the upper feedback system to be.82)/2(41/2)(1+=+=s s s s HSimilarly, the system function of the lower feedback system is .21)2/1(21/1)(2+=+=s s s HThe system function of the overall system is now.1610123)()()(221+++=+=s s s s H s H s HSince H(s)=Y(s)/X(s), we may write]123)[(]1610)[(2+=++s s X s s s Y . Taking the inverse Laplace transform, we obtaindtt dx t x t y dt t dy dt t y d )(3)(12)(16)(10)(2+=++. ( a) From problem , we know that differential equation relating the input and output of the RLC circuit is2()()()().d y t dy t y t x t dtdt++=Taking the Laplace transform of this (while nothing that the system is causal and stable), we obtain 2()[1]().Y s s s X s ++= Therefore ,2()1(),()1Y s H s X s s s ==++ 1{}.2e s ℜ>-(b) We note that H(s) has two poles at132s =--132s =-+From Section we know that the magnitude of the Fourier transform may be expressed as1313(Length of vector from to -+j )(Length of vector from to --j )ωω We see that the right hand side of the above expression Increases with increasing |ω| until |ω| reaches 12. Then it starts decreasing as |ω| increasing even further. It finally reaches 0 for |ω|=∞.Therefore 2|()|H j ω is approximately lowpass.(c) By repeating the analysis carried out in Problem and part (a) of this problem with R =310-Ω, wecan show that2()1(),()1Y s H s X s s s ==++ {}0.0005.e s ℜ>-(d) We have33(Vect.Len.from to -0.0005+j )(Vect.Len.from to -0.0005-j )ωωWe see that when |ω| is in he vicinity , the right-hand side of the above equation takes onextremely large value. On either side of this value of |ω| the value of |H (j ω)| rolls off rapidly. Therefore, H(s) may be considered to be approximately bandpass. . (a) The unilateral Laplace transform isX(s) = 20(1)t st e u t e dt -∞--+⎰= 20t st e e dt -∞--⎰=21+s {} 2.e s ℜ>-(b) The unilateral Laplace transform is2(3)0()[(1)()(1)]t st X s t t e u t e dt δδ-∞-+-=++++⎰2(3)0[()]t st t e e dt δ-∞-+-=+⎰612e s -=++ {} 2.e s ℜ>- (c) The unilateral Laplace transform is240()[()()]t t st X s e u t e u t e dt -∞---=⎰240[]t t st e e e dt -∞---=+⎰1124s s =+++ {} 2.e s ℜ>-. In Problem , we know that the input of the RL circuit are related by ).()()(t x t y dtt dy =+Applying the unilateral Laplace transform to this equation, we have ).()()0()(s x s y y s sy =+--(a) For the zero-state response, set (0)0y -=.Also we have u s x =)(L{)(2t u et-}=21+s .Therefore,y(s)(s+1)=.21+sComputing the partial fraction expansion of the right-hand side of the above equation and then taking its inverse unilateral Laplace transform, we have ).()()(2t u e t u e t y t t ---=(b) For the zero-state response, assume that x(t) = we are given that (0)1y -=,.11)(0)(1)(+=⇒=+-s s y s y s sy Taking the inverse unilateral Laplace transform, we have ()().t y t e u t -=Figure2()2()().t t y t e u t e u t --=- . The pole zero plots for all the subparts are shown in figure . (a) The Laplace transform of x(t) isX(s)= 230()t t st e e e dt ∞---+⎰= (2)(3)00[/(2)]|[/(3)]|s t s te s e s -+∞-+∞-++-+ =211252356s s s s s ++=++++(b) Using an approach similar to that show in part (a), we have41(),4L t e u t s -←−→+ {} 4.e s ℜ>-Also,551(),55L t j t e e u t s j -←−→+-and(){}551,555LT t j t e e u t e s s j --←−→ℜ>-++.From this we obtain()()()()55555215sin 52525LTt t j t t j t e t u t e e e e u t js ----⎡⎤=-←−→⎣⎦++ ,where {}5e s ℜ>- .Therefore,()()(){}245321570sin 5,51490100LTt t s s e u t e t u t e s s s s --+++←−→ℜ>-+++. -2 -3Im R a -2 R Im e Im R f Im Rg Im R h-2 2 4 -4 ImRd R b Im c R Im(c)The Laplace transform of ()x t is ()()023t t st X s e e e dt --∞=+⎰()()()()2300/2|/3|s t s t e s e s ----∞-∞⎡⎤⎡⎤=--+--⎣⎦⎣⎦ 211252356s s s s s -=+=---+.The region of convergence (ROC) is {}2e s ℜ<.(d)Using an approach along the lines of part (a),we obtain(){}21,22LT t e u t e s s -←−→ℜ>-+. Using an approach along the lines of part (c) ,we obtain(){}21,22LT t e u t e s s -←−→ℜ<-.From these we obtain()()222224t LT t t s e e u t e u t s --=+-←−→-, {}22e s -<ℜ<. Using the differentiation in the s-domain property , we obtain(){}22222228,2244t LT d s s te e s ds s s -+⎡⎤←−→-=--<ℜ<⎢⎥-⎣⎦-. (e)Using the differentiation in the s-domain property on eq.,we get()(){}2211,222LT t d te u t e s ds s s -⎡⎤←−→-=ℜ>-⎢⎥+⎣⎦+.Using the differentiation in the s-domain property on eq ,we get ()(){}2211,222LT t d te u t e s ds s s ⎡⎤--←−→=-ℜ<⎢⎥-⎣⎦-.Therefore,()()()(){}222224,2222t LT t t st e te u t te u t e s s s ---=--←−→-<ℜ<+-.(f)From the previous part ,we have ()()(){}2221,22LT t t t e u t te u t e s s -=--←−→-ℜ<-.(g)Note that the given signal may be written as ()()()1x t u t u t =-- .Note that (){}1,0LTu t e s s←−→ℜ>.Using the time shifting property ,we get(){}1,0s LT e u t e s s--←−→ℜ>.Therefore ,()1x t()()11,sLT e u t u t s----←−→ All s . Note that in this case ,since the signal is finite duration ,the ROC is the entire s-plane.(h)Consider the ()()()11x t t u t u t =--⎡⎤⎣⎦that the signal ()x t may beexpressed as ()()()112x t x t x t =+-+ . We have from the previous part()()11sLT e u t u t s----←−→, All s . Using the differentiation in s-domain property ,we have()()()12111s s s LT d e se e x t t u t u t ds ss ---⎡⎤--+=--←−→=⎡⎤⎢⎥⎣⎦⎣⎦, All s . Using the time-scaling property ,we obtain()121s s LT se e x t s --+-←−→, All s .Then ,using the shift property ,we have()21212s sLT s se e x t es ---+-+←−→ ,All s . Therefore ,()()()21122112s s s sLT s se e se e x t x t x t e s s----+--+=+-+←−→+, All s. (i) The Laplace transform of ()()()x t t u t δ=+ is (){}11/,0X s s e s =+ℜ>.(j) Note that ()()()()33t u t t u t δδ+=+.Therefore ,the Laplace transform is the same as the result of the previous part. (a)From Table ,we have()()()1sin 33x t t u t =.(b)From Table we know that()(){}2cos 3,09LT st u t e s s ←−→ℜ>+. Using the time scaling property ,we obtain()(){}2cos 3,09LT s t u t e s s -←−→-ℜ<+Therefore ,the inverse Laplace transform of ()X s is()()()cos 3x t t u t =--.(c)From Table we know that ()()(){}21cos 3,119LTt s e t u t e s s -←−→ℜ>-+.Using the time scaling property ,we obtain ()()(){}21cos 3,119LTt s e t u t e s s -+-←−→-ℜ<-++. Therefore ,the inverse Laplace transform of ()X s is()()()cos 3t x t e t u t -=--.(d)Using partial fraction expansion on ()X s ,we obtain ()2143X s s s =-++ .From the given ROC ,we know that ()x t must be a two-sided signal .Therefore ()()()432t t x t e u t e u t --=+-.(e)Using partial fraction expansion on ()X s ,we obtain()2132X s s s =-++. From the given ROC ,we know that ()x t must be a two-sided signal ,Therefore,()()()332ttx t e u t e u t --=+-.(f)We may rewrite ()X s as ()2311s X s s s =+-+()()2211/23/2s =-+()()()()222211/23/21/23/2s s =+-+-+Using Table ,we obtain()())())()/2/23cos 3/23sin3/2t t x t t e t u t e t u t δ--=+.(g)We may rewrite ()X s as ()()2311s X s s =-+.From Table ,we know that(){}21,0LT tu t e s s ←−→ℜ>.Using the shifting property ,we obtain()(){}21,11LT t e tu t e s s -←−→ℜ>-+.Using the differentiation property ,()()()(){}2,11LT t t t d s e tu t e u t te u t e s dt s ---⎡⎤=-←−→ℜ>-⎣⎦+. Therefore,()()()()33t t x t t e u t te u t δ--=--.four pole-zero plots shown may have the following possible ROCs:·Plot (a): {}2e s ℜ<- or {}22e s -<ℜ< or {}2e s ℜ>.·Plot (b): {}2e s ℜ<- or {}2e s ℜ>-. ·Plot (c): {}2e s ℜ< or {}2e s ℜ>. ·Plot (d): Entire s-plane.Also, suppose that the signal ()x t has a Laplace transform ()X s with ROC R . (1).We know from Table that()()33LT te x t X s -←−→+.The ROC 1R of this new Laplace transform is R shifted by 3 to the left .If ()3t x t e - is absolutely integrable, then 1R must include the jw -axis.·For plot (a), this is possible only if R was {}2e s ℜ> . ·For plot (b), this is possible only if R was {}2e s ℜ>-. ·For plot (c), this is possible only if R was {}2e s ℜ> . ·For plot (d),R is the entire s-plane. (2)We know from Table that(){}1,11LT t e u t e s s -←−→ℜ>-+.Also ,from Table we obtain()()(){}2,11LT t X s x t e u t R R e s s -⎡⎤*←−→=ℜ>-⎡⎤⎣⎦⎣⎦+ If ()()te u t x t -*is absolutely integrable, then 2R must include the jw -axis.·For plot (a), this is possible only if R was {}22e s -<ℜ<. ·For plot (b), this is possible only if R was {}2e s ℜ>-. ·For plot (c), this is possible only if R was {}2e s ℜ< . ·For plot (d),R is the entire s-plane.(3)If ()0x t = for 1t > ,then the signal is a left-sided signal or a finite-duration signal . ·For plot (a), this is possible only if R was {}2e s ℜ<-. ·For plot (b), this is possible only if R was {}2e s ℜ<-. ·For plot (c), this is possible only if R was {}2e s ℜ< . ·For plot (d),R is the entire s-plane.(4)If ()0x t =for 1t <-,then the signal is a right-sided signal or a finite-duration signal ·For plot (a), this is possible only if R was {}2e s ℜ>.·For plot (b), this is possible only if R was {}2e s ℜ>- . ·For plot (c), this is possible only if R was {}2e s ℜ>.·For plot (d),R is the entire s-plane..(a)The pole-zero diagram with the appropriate markings is shown Figure .(b)By inspecting the pole-zero diagram of part (a), it is clear that the pole-zero diagram shown in Figure will also result in the same ()X jw .This would correspond to the Laplace transform()112X s s =-, {}12e s ℜ<.(c)≮()X jw π=-≮()1X jw .(d)()2X s with the pole-zero diagram shown below in Figure would have the property that ≮()2X jw =≮()X jw .Here ,()211/2X s s -=-. (e) ()()21/X jw X jw =.(f)From the result of part (b),it is clear that ()1X s may be obtained by reflecting the poles and zeros in the right-half of the s-plane to the left-half of the s-plane .Therefore, ()11/22s X s s +=+.From part (d),it is clear that ()2X s may be obtained by reflecting the poles (zeros) in the right-half of the s-plane to the left-half and simultaneously changing them to zeros (poles).Therefore,()()()()2211/22s X s s s +=++plots are as shown in Figure . Table we have()()(){}2111,22LT t x t e u t X s e s s -=←−→=ℜ>-+and()()(){}3111,33LTt x t e u t X s e s s -=←−→=ℜ>-+.Using the time-shifting time-scaling properties from Table ,we obtain()(){}22112,22s LT s e x t e X s e s s ---←−→=ℜ>-+and()(){}33223,33s LT s e x t e X s e s s---+←−→-=ℜ>--.Therefore, using the convolution property we obtain ()()()()23122323s s LTe e y t x t x t Y s s s --⎡⎤⎡⎤=-*-+←−→=⎢⎥⎢⎥+-⎣⎦⎣⎦. clues 1 and 2,we know that ()X s is of the form()()()AX s s a s b =++. Furthermore , we are given that one of the poles of ()X s is 1j -+.Since ()x t is real, the poles of ()X s must occur in conjugate reciprocal pairs .Therefore, 1a j =-and 1b j =+and ()()()11AH s s j s j =+-++. From clue 5,we know that ()08X =.Therefore, we may deduce that 16A = and ()21622H s s s =++ .Let R denote the ROC of ()X s .From the pole locations we know that there are two possible choices of R .R may either be {}1e s ℜ<-or {}1e s ℜ>-.We will now useclue 4 to pick one .Note that()()()()22LTt y t e x t Y s X s =←−→=-.The ROC of ()Y s is R shifted by 2 to the right .Since it is given that ()y t is not absolutely integrable ,the ROC of ()Y s should not include the jw axis -.This is possible only ofR is {}1e s ℜ>-..(a) The possible ROCs are(i) {}2e s ℜ<-.(ii) {}21e s -<ℜ<-. (iii) {}11e s -<ℜ<.( iv) {}1e s ℜ>.(b)(i)Unstable and anticausal. (ii) Unstable and non causal. （iii ）Stable and non causal. (iv) Unstable and causal. .(a)Using Table ,we obtain(){}1,11X s e s s =ℜ>-+and(){}1, 2.2H s e s s =ℜ>-+(b) Since ()()()y t x t h t =*,we may use the convolution property to obtain()()()()()112Y s X s H s s s ==++.The ROC of ()Y s is {}1e s ℜ>-.(c) Performing partial fraction expansion on ()Y s ,we obtain . ()1112Y s s s =-++.Taking the inverse Laplace transform, we get()()()2t t y t e u t e u t --=-. (d)Explicit convolution of ()x t and ()h t gives us()()()y t h x t d τττ∞-∞=-⎰()()20t e e u t d ττττ∞---=-⎰t t e e d ττ--=⎰ for0t >()2.t t e e u t --⎡⎤=-⎣⎦ the input ()()x t u t =, the Laplace transform is (){}1,0.X s e s s=ℜ>The corresponding output ()()1t t y t e te u t --⎡⎤=--⎣⎦ has the Laplace transform()()(){}221111,0111Y s e s s s s s s =--=ℜ>+++. Therefore,()()()(){}21,0.1Y s H s e s X s s ==ℜ>+ Now ,the output ()()3123t t y t e e u t --⎡⎤=-+⎣⎦has the Laplace transform()()(){}12316,0.1313Y s e s s s s s s s =-+=ℜ>++++ Therefore , the Laplace transform of the corresponding input will be()()()()(){}1161,0.3Y s s X s e s H s s s +==ℜ>+ Taking the inverse Laplace transform of the partial fraction expansion of ()1,X s we obtain ()()()3124.t x t u t e u t -=+.(a).Taking the Laplace transform of both sides of the given differential equation and simplifying, we obtain ()()()212Y s H s X s s s ==--. The pole-zero plot for ()H s is as shown in figure .b).The partial fraction expansion of ()H s is()1/31/321H s s s =--+. (i).If the system is stable ,the ROC for ()H s has to be {}12e s -<ℜ< . Therefore ()()()21133t t h t e u t e u t -=---.(ii).If the system is causal, the ROC for ()H s has to be {}2e s ℜ> .Therefore()()()21133t t h t e u t e u t -=-.(iii)If the system is neither stable nor causal ,the ROC for ()H s has to be {}1e s ℜ<-.Therefore ,()()()21133t t h t e u t e u t -=--+-. If ()2t x t e =produces ()()21/6t y t e =,then ()()21/6H =. Also, by taking the Laplace transform of both sides ofthe given differential equation we get ()()()()442s b s H s s s s ++=++.Since ()21/6H = ,we may deduce that 1b = .Therefore()()()()()222424s H s s s s s s +==+++. ()()()t t t x t e e u t e u t --==+-,()()(){}112,111111X s e s s s s s -=-=-<ℜ<+-+-. We are also given that ()2122s H s s s +=++.Since the poles of ()H s are at 1j -±, and since ()h t is causal ,we may conclude that the ROC of()H s is {}1e s ℜ>-.Now()()()()()22221Y s H s X s s s s -==++-. The ROC of ()Y s will be the intersection of the ROCs of ()X s and ()H s .This is {}11e s -<ℜ<. We may obtain the following partial fraction expansion for ()Y s :()22/52/56/5122s Y s s s s +=-+-++. We may rewrite this as ()()()222/521411551111s Y s s s s ⎡⎤⎡⎤+=-++⎢⎥⎢⎥-++++⎢⎥⎢⎥⎣⎦⎣⎦.0 -1 2 ReImFigureNothing that the ROC of ()Y s is {}11e s -<ℜ<and using ,we obtain ()()()()224cos sin 555t t t y t e u t e tu t e tu t --=-++know that()()(){}111,0LTx t u t X s e s s=←−→=ℜ> Therefore,()1X s has a pole at0s =.Now ,the Laplace transform of the output()1y t of the system with()1x t as the input is()()()11Y s H s X s =Since in clue 2, ()1Y s is given to be absolutely integrable ,()H s must have a zero at 0s =whichcancels out the pole of ()1X s at 0s =.We also know that()()(){}2221,0LT x t tu t X s e s s=←−→=ℜ> Therefore , ()2x s has two poles at 0s =.Now ,the Laplace transform of the output ()2y t of the system with ()2x t as the input is()()()22Y s H s X s =Since in clue 3, ()2Y s is given to be not absolutely integrable ,()H s does not have two zeros at0s =.Therefore ,we conclude that ()H s has exactly one zero at 0s =. From clue 4 we know that the signal ()()()()2222d h t dh t p t h t dt dt=++is finite duration .Taking the Laplace transform of both sides of the above equation ,we get ()()()()222P s s H s sH s H s =++. Therefore,()()222P s H s s s =++.Since ()p t is of finite duration, we know that ()P s will have no poles in the finite s-plane .Therefore, ()H s is of the form()()1222Ni i A s z H s s s =-=++∏,where i z ,1,2,....,i N =represent the zeros of ()P s .Here ,A is some constant.From clue 5 we know that the denominator polynomial of ()H s has to have a degree which is exactly one greater than the degree of the numerator polynomial .Therefore, ()()1222A s s H s s s -=++.Since we already know that ()H s has a zero at 0s = ,we may rewrite this as ()222As H s s s =++ From clue 1 we know that ()1H 0.2 this ,we may easily show that 1A = .Therefore,()222s H s s s =++. Since the poles of ()H s are at 1j -± and since ()h t is causal and stable ,the ROC of ()H s is {}1e s ℜ>-..(a) We may redraw the given block diagram as shown in Figure . From the figure ,it is clear that()()1F s Y s s=. Therefore, ()()1/f t dy t dt =. Similarly, ()()/e t df t dt =.Therefore, ()()221/e t d y t dt =.From the block diagram it is clear that()()()()()()()21111266d y t dy t y t e t f t y t y t dtdt=--=--.。
第7章采样第8章通信系统第9章拉普拉斯变换第10章Z变换第11章线性反馈系统第7章采样7.2连续时间信号x（t）从一个截止频率为的理想低通滤波器的输出得到，如果对x（t）完成冲激串采样，那么下列采样周期中的哪一些可能保证x（t）在利用一个合适的低通滤波器后能从它的样本中得到恢复？7.3在采样定理中，采样频率必须要超过的那个频率称为奈奎斯特率。
试确定下列各信号的奈奎斯特率：7.4设x（t）是一个奈奎斯特率为ω0的信号，试确定下列各信号的奈奎斯特率：7.5设x（t）是一个奈奎斯特率为ω0的信号，同时设其中。
7.6在如图7-1所示系统中，有两个时间函数x1（t）和x2（t）相乘，其乘积W （t）由一冲激串采样，x1（t）带限于ω17.7信号x（t）用采样周期T经过一个零阶保持的处理产生一个信号x0（t），设x1（t）是在x（t）的样本上经过一阶保持处理的结果，即7.8有一实值且为奇函数的周期信号x（t），它的傅里叶级数表示为7.9考虑信号x（t）为7.10判断下面每一种说法是否正确。
7.11设是一连续时间信号，它的傅里叶变换具有如下特点：7.12有一离散时间信号其傅里叶变换具有如下性质：7.13参照如图7-7所示的滤波方法，假定所用的采样周期为T，输入xc（t）为带限，而有7.14假定在上题中有重做习题7.13。
7.15对进行脉冲串采样，得到若7.16关于及其傅里叶变换7.17考虑理想离散时间带阻滤波器，其单位脉冲响应为频率响应在条件下为7．18假设截止频率为π/2的一个理想离散时间低通滤波器的单位脉冲响应是用于内插的，以得到一个2倍的增采样序列，求对应于这个增采样单位脉冲响应的频率响应。
7.19考虑如图7-11所示的系统，输入为x[n]，输出为y[n]。
零值插入系统在每一序列x[n]值之间插入两个零值点，抽取系统定义为其中W[n]是抽取系统的输入序列。
若输入x[n]为试确定下列ω1值时的输出y[n]：7.20有两个离散时间系统S1和S2用于实现一个截止频率为π/4的理想低通滤波器。
第二部分课后习题第7章采样基本题7.1已知实值信号x（t），当采样频率时，x（t）能用它的样本值唯一确定。
问在什么ω值下保证为零？解：对于因其为实函数，故是偶函数。
由题意及采样定理知的最大角频率即当时，7.2连续时间信号x（t）从一个截止频率为的理想低通滤波器的输出得到，如果对x（t）完成冲激串采样，那么下列采样周期中的哪一些可能保证x（t）在利用一个合适的低通滤波器后能从它的样本中得到恢复？解：因为x（t）是某个截止频率的理想低通滤波器的输出信号，所以x（t）的最大频率就为＝1000π，由采样定理知，若对其进行冲激采样且欲由其采样m点恢复出x（t），需采样频率即采样时间问隔从而有（a）和（c）两种采样时间间隔均能保证x（t）由其采样点恢复，而（b）不能。
7.3在采样定理中，采样频率必须要超过的那个频率称为奈奎斯特率。
试确定下列各信号的奈奎斯特率：解：（a）x（t）的频谱函数为由此可见故奈奎斯特频率为（b）x（t）的频谱函数为由此可见故奈奎斯特频率为（c）x（t）的频谱函数为由此可见，当故奈奎斯特频率为7.4设x（t）是一个奈奎斯特率为ω0的信号，试确定下列各信号的奈奎斯特率：解：（a）因为的傅里叶变换为可见x（t）的最大频率也是的最大频率，故的奈奎斯特频率为0 。
（b）因为的傅里叶变换为可见x （t）的最大频率也是的最大频率．故的奈奎斯特频率仍为。
（c）因为的傅里叶变换蔓可见的最大频率是x（t）的2倍。
从而知x 2（t）的奈奎斯特频率为2（d）因为的傅里叶变换为，x（t）的最大频率为，故的最大频率为，从而可推知其奈奎斯特频率为7.5设x（t）是一个奈奎斯特率为ω0的信号，同时设其中。
当某一滤波器以Y（t）为输入，x（t）为输出时，试给出该滤波器频率响应的模和相位特性上的限制。
解：p（t）是一冲激串，间隔对x（t）用p（t-1）进行冲激采样。
先分别求出P（t）和P（t-1）的频谱函数：注意0ω是x（t）的奈奎斯特频率，这意味着x（t）的最大频率为02ω，当以p（t-1）对x（t）进行采样时，频谱无混叠发生。
第9章拉普拉斯变换习题9.1 对下列每个积分，给出保证积分收敛的实参数σ的值：解：（a）因可见，要使积分收敛，当t→∞时，需满足5＋σ＞0，即σ＞－5，此时积分为。
（b）此题中的被积函数与（a）相同，只是积分区间不同。
利用（a）中的积分过程可见，要使积分收敛，当t→∞时，需满足5＋σ＜0，即σ＜－5。
σ＜－5即为实数σ的取值范围。
（c）与（b）相似。
由于积分是在一个有限的区间[－5，5]上进行的，所以无论σ取何实值，积分均收敛，即实数盯的取值范围为－∞＜1＜∞。
（d）与（b）相似。
要使积分收敛，当t→∞时，应有5＋σ＞0，即t＞－5；且当t→－∞时，又应有5＋σ＜0，即σ＜－5。
综合以上分析可知，无论σ取何值，该积分都不收敛。
（e）此积分可化为以上第一个积分要收敛，需满足σ＞－5。
对于第二个积分，由于可见要使其收敛，需满足σ＜5。
综上所述，当－5＜σ＜5时，积分收敛。
（f）此积分可化为由（e）中分析可知，当实数σ＜5时，积分收敛。
9.2 考虑信号x（t）＝e-5t u（t－1）其拉普拉斯变换记为X（s），（a）利用式（9.3）求X（s），并给出它的收敛域。
（b）确定有限数A和t0，以使g（t）＝Ae-5t u（-t－t0）的拉普拉斯变换G（s）与X （s）有相同的代数式。
对应于G（s）的收敛域是什么？解：（a）由拉普拉斯变换的定义得（b）由拉普拉斯变换的定义得要使G（s）收敛，当t→∞时，需满足，即，此时对比G（s）与X（s）的代数表达式可发现，要使两者相同，应有A＝－1，t0＝－1,G （s）的ROC为9.3 考虑信号x（t）＝e-5t u（t）＋e-βt u（t）其拉普拉斯变换记为X（s）。
若X（s）的收敛域是Re{s}＞-3，应在β的实部和虚部上施加什么限制？解：利用常用信号的拉普拉斯变换对可直接写出这里的β可为复数，也可为实数。
若β为复数，那么只有它的实部对X（s）的ROC有影响。
X（s）的ROC为Re{S}大于－5和Re{－β}中的大者。
第九章9.6 解：(a) 若是有限持续期信号Roc 为整个s 平面，故存在极点不可能，故不可能为有限持续期。
(b) 可能是左边的。
(c) 不可能是右边的，若是右边信号，它并不是绝对可积的。
(d) x(t)可能为双边的。
9.8 解：因为te t x t g 2)()(=的傅氏变换，)(ωj G 收敛所以)(t x 绝对可积若)(t x 为左边或者右边信号，则)(t x 不绝对可积故)(t x 为双边信号9.10 解：(a) 低通(b) 带通(c) 高通9.14 解：dt e t x s X st ⎰∞∞--=)()(， 由)(t x 是偶函数可得)()()(t d e t x s X st --=⎰-∞∞ dt e t x t s ⎰∞∞----=)()(dt e t x t s ⎰∞∞---=)()( )(s X -=421πj e s =为极点，故421πj e s -=也为极点，由)(t x 是实信号可知其极点成对出现，故421πj e s -=与421πj e s --=也为极点。
)21)(21)(21)(21()(4444ππππj j j j e s e s e s e s Ms X --++--=由⎰∞∞-=4)(dt t x 得 4)0(=x所以，M ＝1/4 即，42}Re{42<<-s 9.21 解： (a) 3121)(+++=s s s X 2}Re{->s (b) 25)5(541)(2++++=s s s X 4}Re{->s (c) 3121)(----=s s s X 2}Re{<s (d) 22)2(1)2(1)(--+=s s s X 2}Re{2<<-s (e) 22)2(1)2(1)(-++-=s s s X 2}Re{2<<-s (f) 2)2(1)(-=s s X 2}Re{<s (g) )1(1)(s e ss X --= 0}Re{>s (h) 22)1()(se s X s -=- 0}Re{>s(i) ss X 11)(+= 0}Re{>s (j) ss X 131)(+= 0}Re{>s9.23 解：1． Roc 包括 Re{s}=32． Roc 包括 Re{s}=03． Roc 在最左边极点的左边4． Roc 在最右边极点的右边图1：1，2}Re{>s2，2}Re{2<<-s3，2}Re{-<s4，2}Re{>s图2： 1，2}Re{->s2，2}Re{->s3，2}Re{-<s4，2}Re{->s图3： 1，2}Re{>s2，2}Re{<s3，2}Re{<s4，2}Re{>s图4： 1，S 为整个平面2，S 为整个平面3，S 为整个平面4，S 为整个平面9.25 解：图略9.27 解：)(t x Θ为实信号，)(s X 有一个极点为j s +-=1)(s X ∴另一个极点为j s --=1)1)(1()(j s j s M s X ++-+=∴ 又Θ8)0(=X16=∴M 则，)1(8)1(8)(j s j j s j s X -+-++= 1}Re{->s 或者1}Re{-<s 之一使其成立又 )(2t x e tΘ不是绝对可积的 ∴ 对任一个s ，右移2，不一定在Roc 中因此，1}Re{-<s9.35 解： (a) )(1)(*)(s X st u t x L −→−Θ 那么方框图表示的方程为)(*)(*)(6)(*)()()(*)(*)()(*)(2)(t u t u t y t u t y t y t u t u t x t u t x t x --=++即 ⎰⎰⎰⎰⎰⎰∞-∞-∞-∞-∞-∞---=++t tt t t t dt d y d y t y dt d x d x t x ττττττττ)(6)()()()(2)( 对两边求导可得)(6)()()()()(2222t x dt t dx dtt x d t y dt t dy dt t y d --=++ (b) 126)(22++--=s s s s s H 121-==s s 是)(s H 的二重极点，由于系统是因果的所以 1}Re{->sRoc 包含虚轴，所以系统是稳定的。
Chap 66.1 Consider a continuous-time LTI system with frequency response()()|()|H j H j H j e ωωω=and real impulse response h(t). Suppose that we apply an input 00()cos()x t t ωφ=+ to this system .The resulting output can be shown to be of the form0()()y t Ax t t =-Where A is a nonnegative real number representing anamplitude-scaling factor and 0t is a time delay.(a)Express A in terms of |()|H j ω.(b)Express 0t in terms of0()H j ω Solution:(a) For 0()()y t Ax t t =-So 0()()jt Y j AX j eωωω-= 0()()()j t Y j H j Ae X j ωωωω-== So |()|A H j ω=(b) for 0()H j t ωω=- So 0()H j t ωω=-6.3 Consider the following frequency response for a causal and stable LTI system:1()1j H j j ωωω-=+ (a) Show that |()|H j A ω=,and determine the values of A. (b)Determine which of the following statements is true about ()τω,the group delay of the system.(Note()(())/d H j d τωωω=-,where ()H j ωis expressed in aform that does not contain any discontinuities.)1.()0 0for τωω=>2.()0 0for τωω>>3 ()0 0for τωω<>Solution:(a) for |()|1H j ω== So A=1(b) for )(2)()()1()1()(ωωωωωωarctg arctg arctg j j j H -=--=+∠--∠=∠ 212)()(ωωωωτ+=∠-=d j H d So ()0 0for τωω>>6.5 Consider a continuous-time ideal bandpass filter whose frequency response is⎩⎨⎧≤≤=elsewherej H c c,03||,1)(ωωωω (a) If h(t) is the impulse response of this filter, determine a functiong(t) such that)(sin )(t g t t t h c πω=(b) As c ω is increased, dose the impulse response of the filter get more concentrated or less concentrated about the origin?Solution(a) Method 1. Let1()()()()()()2h t x t g t H j X j G j ωωωπ=↔=* They are shown in the figures,where1,sin ()(){0,c c ctx t X j t ωωωωωωπ<=↔=> So we can get()2cos(2)()2[(2)(2)]c c c g t t G j ωωπδωωδωω=↔=-++Method 2. Using the inverse FT definition,it is obtained331(){}2c c c cj t j t h t e d e d ωωωωωωωωπ--=+⎰⎰ 11{sin 3sin }{sin }{2cos 2}c c c c t t t t t tωωωωππ=-= (b) more concentrated.Chap 77.1 A real-valued signal x(t) is know to be uniquely determined by its samples when the sampling frequency is10,000s ωπ=.For what values ofω is ()X j ω guaranteed to be zero? Solution:According to the sampling theorem 2s M w w > That is 110000500022M s w w ππ<== So if 5000M w w π>=,0)(=jw X7.2 A continuous-time signal x(t) is obtained at the output of an ideal lowpass filter with cutoff frequency 1,000c ωπ=.If impulse-train sampling is performed on x(t), which of the following sampling periods would guarantee that x(t) can be recovered from its sampled version using an appropriate lowpass filter?(a) 30.510T -=⨯(b) 3210T -=⨯(c) 410T -= Solution: π1000==c M w wFrom the sampling theorem,∴π20002=>M s w w ,that is 3102000222-==<πππM s w T ∴the conditions (a) and (c) are satisfied with the sampling theorem,(b) is not satisfied.7.3 The frequency which, under the sampling theorem, must be exceeded by the sampling frequency is called the Nyquist rate. Determine the Nyquist rate corresponding to each of the following signals:(a)()1cos(2,000)sin(4,000)x t t t ππ=++ (b)sin(4,000)()t x t tππ=(c) 2sin(4,000)()()t x t t ππ= Solution: (a) )4000sin()2000cos(1)(t t t x ππ++=max(0,2000,4000)4000M w πππ==∴ the Nyquist rate is 28000s M w w π>= (b) sin(4000)()t x t tππ= 4000M w π=∴ the Nyquist rate is 28000s M w w π>= (c) 2sin(4000)()t x t t ππ⎛⎫= ⎪⎝⎭ 2sin(4000)()t x tt ππ⎛⎫= ⎪⎝⎭221(1cos(8000))2t t ππ=- ∴8000M w π=∴the Nyquist rate is 216000s M w w π>=7.4 Let x(t) be a signal with Nyquist rate 0ω. Determine the Nyquist rate for each of the following signals:(a)()(1)x t x t +- (b)()dx t dt(c)2()x t(d)0()cos x t t ωSolution:(a) we let 1()()(1)y t x t x t =+-So 1()()()(1)()j j Y j X j e X j e X j ωωωωωω--=+=+ So the Nyquist rate of signal (a) is 0ω.(b) we let 2()()dx t y t dt= So 2()()Y j j X j ωωω=So the Nyquist rate of signal (b) is0ω. (c) we let 23()()y t x t = So 31()()*()2Y j X j X j ωωωπ= So the Nyquist rate of signal (c) is 20ω.(d) we let 40()()cos y t x t t ω=For 000cos [()()]FT t ωπδωωδωω→-++ So 4001()((()(())2Y j X j X j ωωωωω=-++ So the Nyquist rate of signal (d) is 03ω7.9 Consider the signal 2sin 50()()t x t tππ= Which we wish to sample with a sampling frequency of 150s ωπ= to obtain a signal g(t) with Fourier transform ()G j ω.Determine the maximum value of 0ω for which it is guaranteed that0()75() ||G j X j for ωωωω=≤Where ()X j ω is the Fourier transform of x(t).Solution: 2sin(50)()t x t t ππ⎛⎫= ⎪⎝⎭))100cos(1(2122t t ππ-= ∴100M w π=But π150=s wthe figure about before-sampling and after-sampling of )(jw H isWe can see that only when π500≤w , the before-sampling and after-sampling of )(jw H have the same figure.So if 0..)..(75)(w w for jw X jw G ≤=The maximum value of 0w is π50.Chap 99.2 Consider the signal 5()(1)t x t e u t -=- and denote its Laplace transform by X(s).(a)Using eq.(9.3),evaluate X(s) and specify its region of convergence. (b)Determine the values of the finite numbers A and 0t such that the Laplace transform G(s) of 50()()t g t Ae u t t -=-- has the same algebraic form as X(s).what is the region of convergencecorresponding to G(s)?Solution:(a). According to eq.(9.3), we will getdt e t x s X st -∞∞-⎰=)()(dt e t u e st t --∞∞--=⎰)1(5dt e t s )5(1+-∞⎰=)5()5()5()5()5(1)5(+=+--=+-=+-+-∞+-s e s e s e s s t s ROC:Re{s}>-5 (b). )()(05t t u Ae t g t --=-−→←LT 0)5(5)(t s e s A s G ++-=, Re{s}<-5 ∴ If )()(s X s G =then it ’s obviously that A=-1, 10-=t , Re{s}<-5.9.5 For each of the following algebraic expressions for the Laplace transform of a signal, determine the number of zeros located in the finite s-plane and the number of zeros located at infinity: (a)1113s s +++ (b) 211s s +- (c) 3211s s s -++ Solution :(a).1, 1)3)(1(423111+++=+++s s s s s ∴ it has a zero in the finite s-plane, that is 2-=sAnd because the order of the denominator exceeds the order of the numerator by 1∴ X(s) has 1 zero at infinity.(b). 0, 111)1)(1(1112-=-++=-+s s s s s s ∴ it has no zero in the finite s-plane.And because the order of the denominator exceeds the order of the numerator by 1∴ X(s) has 1 zero at infinity.(c). 1, 011)1)(1(112223-=++++-=++-s s s s s s s s s ∴ it has a zero in the finite s-plane, that is 1=sAnd because the order of the denominator equals to the order of the numerator∴ X(s) has no zero at infinity.9.7 How many signals have a Laplace transform that may be expressed as 2(1)(2)(3)(1)s s s s s -++++ in its region of convergence?Solution:There are 4 poles in the expression, but only 3 of them have different real part.∴ The s-plane will be divided into 4 strips which parallel to the jw-axis and have no cut-across.∴ There are 4 signals having the same Laplace transform expression.9.8 Let x(t) be a signal that has a rational Laplace transform with exactly two poles located at s=-1 and s=-3. If2()() ()t g t e x t and G j ω=[ the Fourier transform of g(t)]converges, determine whether x(t) is left sided, right sided, or two sided.Solution:)()(2t x e t g t =∴)2()(-=s X s G ROC: R(x)+Re{2}And x(t) have three possible ROC strips:),1(),1,3(),3,(+∞-----∞∴g(t) have three possible ROC strips: ),1(),1,1(),1,(+∞---∞ IF jw s s G jw G ==|)()(Then the ROC of )(s G is (-1,1)∴)(t x is two sides. 9.9 Given that1(),{}Re{}sat e u t Re s a s a -↔>-+ Determine the inverse Laplace transform of22(2)(),Re{}3712s X s s s s +=>-++ Solution: It is obtained from the partial-fractional expansion:22(2)2(2)42()712(4)(3)43s s X s s s s s s s ++-===+++++++,Re{}3s >-We can get the inverse Laplace transform from given formula and linear property.43()4()2()t t x t e u t e u t --=-9.10 Using geometric evaluation of the magnitude of the Fourier transform from the corresponding pole-zero plot ,determine, for each of the following Laplace transforms, whether the magnitude of the corresponding Fourier transform is approximately lowpass, highpass, or bandpass. (a): 1}Re{,.........)3)(1(1)(1->++=s s s s H (b): 221(),{}12s H s e s s s =ℜ>-++(c): 232(),{}121s H s e s s s =ℜ>-++ Solution:(a). 1}Re{,.........)3)(1(1)(1->++=s s s s H It ’s lowpass.(b).21}Re{,.........1)(22->++=s s s s s H It ’s bandpass.(c). 1}Re{., (1)2)(223->++=s s s s s H It ’s highpass.9.13 Let ()()()g t x t x t α=+- ,Where ()()t x t e u t β-=. Andthe Laplace transform of g(t) is 2(),1{}11s G s e s s =-<ℜ<-. Determine the values of the constantsαand βSolution: ()()()g t x t x t α=+-,and ()()t x t e u t β-=The Laplace transform : ()()()G s X s X s α=+- and()1X s s β=+,Re{}1s >- From the scale property of Laplace transform, ()1X s s β-=-+,Re{}1s < So 2(1)(1)()()()111s G s X s X s s s s βαββαβαα--+=+-=+=+-+-,1Re{}1s -<< From given 2()1s G s s =-,1Re{}1s -<< We can determine : 11,2αβ=-=。
