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1、1,1-1 Introduction,Chapter 1 Fundamental Concepts,1-2 Discrete-Time Signals,1-3 Discrete-time Systems,1-4 Basic Properties of Discrete-time Systems,Problems,2,Problems1.1 Sketch the discrete-time signal xn(0n10)obtained by sampling the continuous-time signal x(t)=e2t with sampling interval T=0.1.1.2
2、 Sketch the following discrete-time signals(a)xn=un2un1+un4(b)xn=(n+2)un+22un nun4(c)xn=n+1n+un+1 un21.3 Let xn be a signal with xn=0 for n 4.for each signal given below,determine the values of n for which it is guaranteed to be zero.(a)xn3(b)xn+4(c)xn(d)xn+2(e)xn2,Answer:(a)n7;(b)n0;(c)n2;(d)n4;(e)
3、n0,3,1.4 Determine if the following discrete-time signals are periodic.If so,find the fundamental period.(a)xn=(b)xn=ej7n(c)xn=ej(n/8)r=2q/1.5 Consider a discrete-time system with input xn and output yn.The input-output relationship for this system is yn=xn xn 2(a)Is the system memoryless?(b)Determi
4、ne the output of the system when the input is An,where A is any real or complex number.,Answer:N=2,Answer:N=14,Answer:not periodic,Answer:No,Answer:0,4,1.5 Consider a discrete-time system with input xn and output yn.The input-output relationship for this system is yn=xn xn 2(a)Is the system memoryle
5、ss?Solution:By the definition of a system with or without memory,the system is not memoryless.For example,y0=x0 x 2,the output yn at the given time n=0 is related to the value of the input xn at the time n=2.(b)Determine the output of the system when the input is An,where A is any real or complex nu
6、mber.Solution:yn=AnAn 2=0,5,1.6 Consider a discrete-time system with input xn and output yn related by where n0 is a finite positive integer.(a)Is the system linear?(b)Is the system time-invariant?1.7 For each of the following input-output relationships,determine whether the corresponding system is
7、linear,time invariant or both.(a)yn=xn22(b)yn=xn+1+xn1,Answer:Yes,Answer:Yes,Answer:Not linear;Time invariant,Answer:Linear;Time invariant,6,1.6(a)Solution:Consider two inputs x1n and x2n,y1n and y2n are the outputs of the system with the two inputs,respectively.Suppose that C1 and C2 are any comple
8、x constants and let the input xn=C1x1n+C2 x2n,the corresponding output of the system is:By the definition of a linear system,this system is linear.,7,1.6(b)Solution:Suppose that k0 is a time shift,y1n is the output of the system with the input x1n,i.e.,Let the input xn=x1nk0,the corresponding output
9、 of the system is:By the definition of a time-invariant system,this system is time-invariant.,8,1.7(b)Solution:(1)Consider two inputs x1n and x2n,y1n and y2n are the outputs of the system with the two inputs,respectively.Suppose that C1 and C2 are any complex constants and let the input xn=C1x1n+C2
10、x2n,the corresponding output of the system is:By the definition of a linear system,this system is linear.,9,1.7(b)Solution:(2)Suppose that k0 is a time shift,y1n is the output of the system with the input x1n,i.e.,Let the input xn=x1nk0,the corresponding output of the system is:By the definition of a time-invariant system,this system is time-invariant.,
