信号与系统奥本海姆英文版课后答案chapter1.doc

Chapter 1 Answers1.1 Converting from polar to Cartesian coordinates: 1.2 converting from Cartesian to polar coordinates:, , , , , 1.3. (a) =, =0, because (b) , .Therefore, =, = (c) =cos(t). Therefore, =,= (d) , . Therefore, = =0，because . (e) =, =1. therefore, =, =. (f) =. Therefore, =, =1.4. (a) The signal xn is shifted by 3 to the right. The shifted signal will be zero for n7. (b) The signal xn is shifted by 4 to the left. The shifted signal will be zero for n0. (c) The signal xn is flipped signal will be zero for n2. (d) The signal xn is flipped and the flipped signal is shifted by 2 to the right. The new Signal will be zero for n4. (e) The signal xn is flipped and the flipped and the flipped signal is shifted by 2 to the left. This new signal will be zero for n0.1.5. (a) x(1-t) is obtained by flipping x(t) and shifting the flipped signal by 1 to the right. Therefore, x (1-t) will be zero for t-2. (b) From (a), we know that x(1-t) is zero for t-2. Similarly, x(2-t) is zero for t-1,Therefore, x (1-t) +x(2-t) will be zero for t-2. (c) x(3t) is obtained by linearly compression x(t) by a factor of 3. Therefore, x(3t) will bezero for t1. (d) x(t/3) is obtained by linearly compression x(t) by a factor of 3. Therefore, x(3t) will bezero for t9.1.6 (a) x1(t) is not periodic because it is zero for t3. (b) Since x1(t) is an odd signal, is zero for all values of t. (c) Therefore, is zero when 3 and when . (d) Therefore, is zero only when .1.8. (a) (b) (c) (d) 1.9. (a) is a periodic complex exponential. (b) is a complex exponential multiplied by a decaying exponential. Therefore, is not periodic. （c） is a periodic signal. =. is a complex exponential with a fundamental period of . (d) is a periodic signal. The fundamental period is given by N=m() = By choosing m=3. We obtain the fundamental period to be 10.(e) is not periodic. is a complex exponential with =3/5. We cannot find any integer m such that m( ) is also an integer. Therefore, is not periodic.1.10.x(t)=2cos(10t1)-sin(4t-1)Period of first term in the RHS =.Period of first term in the RHS = .Therefore, the overall signal is periodic with a period which the least common multiple of the periods of the first and second terms. This is equal to .1.11. xn = 1+Period of first term in the RHS =1.Period of second term in the RHS =7 （when m=2）Period of second term in the RHS =5 (when m=1)Therefore, the overall signal xn is periodic with a period which is the least commonMultiple of the periods of the three terms inn xn.This is equal to 35.1.12. The signal xn is as shown in figure S1.12. xn can be obtained by flipping un and thenShifting the flipped signal by 3 to the right. Therefore, xn=u-n+3. This implies that M=-1 and no=-3.0-1-2-3123XnnFigure S 1.121.13 y(t)= =Therefore 1.14 The signal x(t) and its derivative g(t) are shown in Figure S1.14.10-1210-1t1-2g(t)2-3-3tFigure S 1.14x(t)Therefore )This implies that A=3, t=0, A=-3, and t=1.1.15 (a) The signal xn, which is the input to S, is the same as yn.Therefore , yn= xn-2+ xn-3 = yn-2+ yn-3 =2xn-2 +4xn-3 +( 2xn-3+ 4xn-4) =2xn-2+ 5xn-3 + 2xn-4The input-output relationship for S is yn=2xn-2+ 5x n-3 + 2x n-4(b) The input-output relationship does not change if the order in which Sand S are connected series reversed. . We can easily prove this assuming that S follows S. In this case , the signal xn, which is the input to S is the same as yn.Therefore yn =2xn+ 4xn-1= 2yn+4 yn-1 =2( xn-2+ xn-3 )+4(xn-3+ xn-4) =2 xn-2+5xn-3+ 2 xn-4The input-output relationship for S is once again yn=2xn-2+ 5x n-3 + 2x n-41.16 (a)The system is not memory less because yn depends on past values of xn.(b)The output of the system will be yn= =0(c)From the result of part (b), we may conclude that the system output is always zero for inputs of the form , k . Therefore , the system is not invertible .1.17 (a) The system is not causal because the output y(t) at some time may depend on future values of x(t). For instance , y(-)=x(0).(b) Consider two arbitrary inputs x(t)and x(t).x(t) y(t)= x(sin(t)x(t) y(t)= x(sin(t)Let x(t) be a linear combination of x(t) and x(t).That is , x(t)=a x(t)+b x(t)Where a and b are arbitrary scalars .If x(t) is the input to the given system ,then the corresponding output y(t) is y(t)= x( sin(t) =a x(sin(t)+ x(sin(t)=a y(t)+ by(t)Therefore , the system is linear.1.18.(a) Consider two arbitrary inputs xnand xn.xn yn =xn yn =Let xn be a linear combination of xn and xn. That is :xn= axn+b xnwhere a and b are arbitrary scalars. If xn is the input to the given system, then the corresponding output yn is yn= =a+b = ayn+b ynTherefore the system is linear.(b) Consider an arbitrary input xn.Let yn =be the corresponding output .Consider a second input xn obtained by shifting xn in time:xn= xn-nThe output corresponding to this input is yn= = = Also note that yn- n= .Therefore , yn= yn- nThis implies that the system is time-invariant.(c) If B, then yn(2 n+1)B.Therefore ,C(2 n+1)B.1.19 (a) (i) Consider two arbitrary inputs x(t) and x(t). x(t) y(t)= tx(t-1) x(t) y(t)= tx(t-1)Let x(t) be a linear combination of x(t) and x(t).That is x(t)=a x(t)+b x(t)where a and b are arbitrary scalars. If x(t) is the input to the given system, then the corresponding output y(t) is y(t)= tx (t-1)= t(ax(t-1)+b x(t-1)= ay(t)+b y(t)Therefore , the system is linear.(ii) Consider an arbitrary inputs x(t).Let y(t)= tx(t-1)be the corresponding output .Consider a second input x(t) obtained by shifting x(t) in time:x(t)= x(t-t)The output corresponding to this input is y(t)= tx(t-1)= tx(t- 1- t)Also note that y(t-t)= (t-t)x(t- 1- t) y(t)Therefore the system is not time-invariant.(b) (i) Consider two arbitrary inputs xnand xn. xn yn = xn-2 xn yn = xn-2.Let x(t) be a linear combination of xnand xn.That is xn= axn+b xnwhere a and b are arbitrary scalars. If xn is the input to the given system, then the corresponding output yn is yn = xn-2=(a xn-2 +b xn-2) =axn-2+bxn-2+2ab xn-2 xn-2 ayn+b ynTherefore the system is not linear.(ii) Consider an arbitrary input xn. Let yn = xn-2 be the corresponding output .Consider a second input xn obtained by shifting xn in time:xn= xn- nThe output corresponding to this input is yn = xn-2.= xn-2- nAlso note that yn- n= xn-2- nTherefore , yn= yn- nThis implies that the system is time-invariant.(c) (i) Consider two arbitrary inputs xnand xn.xn yn = xn+1- xn-1xn yn = xn+1 - xn -1Let xn be a linear combination of xn and xn. That is :xn= axn+b xnwhere a and b are arbitrary scalars. If xn is the input to the given system, then the corresponding output yn is yn= xn+1- xn-1=a xn+1+b xn +1-a xn-1-b xn -1 =a(xn+1- xn-1)+b(xn +1- xn -1) = ayn+b ynTherefore the system is linear.(ii) Consider an arbitrary input xn.Let yn= xn+1- xn-1be the corresponding output .Consider a second input xn obtained by shifting xn in time: xn= xn-nThe output corresponding to this input is yn= xn +1- xn -1= xn+1- n- xn-1- nAlso note that yn-n= xn+1- n- xn-1- nTherefore , yn= yn-nThis implies that the system is time-invariant.(d) (i) Consider two arbitrary inputs x(t) and x(t).x(t) y(t)= x(t) y(t)= Let x(t) be a linear combination of x(t) and x(t).That is x(t)=a x(t)+b x(t)where a and b are arbitrary scalars. If x(t) is the input to the given system, then the corresponding output y(t) is y(t)= = =a+b= ay(t)+b y(t)Therefore the system is linear.(ii) Consider an arbitrary inputs x(t).Lety(t)= =be the corresponding output .Consider a second input x(t) obtained by shifting x(t) in time:x(t)= x(t-t)The output corresponding to this input is y(t)= =Also note that y(t-t)= y(t)Therefore the system is not time-invariant.1.20 (a) Given x= y(t)=x= y(t)=Since the system liner ) =1/2(+)Therefore (t)=cos(2t)=cos(3t)(b) we know that (t)=cos(2(t-1/2)= (+)/2 Using the linearity property, we may once again write(t)=( +) =(+)= cos(3t-1)Therefore,(t)=cos(2(t-1/2) =cos(3t-1)1.21.The signals are sketched in figure S1.21.t-10-112213x(t-1)10-112tx(2t+1)t4-1321012x(2-t)0.50.5t3/2-3/2t102x(4-t/2)t1012618412Figure S1.211.22 The signals are sketched in figure S1.22a11/2-1/2-1n73210x3- n1.23 The even and odd parts are sketched in Figure S1.2311/2-1/2-1n73210xn-4(b)1-12n0x3n+1(d)-111/2-1/2n210x3n(c)2112n0xnun-3=xn(f)-4-1-211/2n20x3- n/2 +(-1)nxn/2(g)011n2(h)Figure S1.22-1/2x0(t)1/2-1-2021t-201/22tx0(t)(a)t-1/2x0(t)1/2-1-2021t1-1/2x0(t)1/2-1-2021(b)x0(t)-t/20tt3t/2-3t/20x0(t)(c)Figure S1.237n10-7xon10-1/2-77-1/2nxn(a)1/2n1/2-771n(n)31/2071/2-1nxon1(b)03/2-3/2-1/24n1/2xon3/251-5nxen(c)Figure S1.241.24 The even and odd parts are sketched in Figure S1.241.25 (a) periodic period=2/(4)= /2(b) periodic period=2/(4)= 2 (c) x(t)=1+cos(4t-2/3)/2. periodic period=2/(4)= /2 (d) x(t)=cos(4t)/2. periodic period=2/(4)= 1/2 (e) x(t)=sin(4t)u(t)-sin(4t)u(-t)/2. Not period.(f) Not period.1.26 (a) periodic, period=7.(b) Not period.(c) periodic, period=8.(d) xn=(1/2)cos(3n/4+cos(n/4). periodic, period=8.(e) periodic, period=16.1.27 (a) Linear, stable(b) Not period.(c) Linear(d) Linear, causal, stable(e) Time invariant, linear, causal, stable(f) Linear, stable(g) Time invariant, linear, causal1.28 (a) Linear, stable(b) Time invariant, linear, causal, stable(c)Memoryless, linear, causal(d) Linear, stable(e) Linear, stable(f) Memoryless, linear, causal, stable(g) Linear, stable1.29 (a) Consider two inputs to the system such thatand Now consider a third input n= n+n. The corresponding system outputWill be therefore, we may conclude that the system is additiveLet us now assume that inputs to the system such thatandNow consider a third input x3 n= x2 n+ x1 n. The corresponding system outputWill betherefore, we may conclude that the system is additive(b) (i) Consider two inputs to the system such that and Now consider a third input t= t+t. The corresponding system outputWill betherefore, we may conclude that the system is not additiveNow consider a third input x4 t= a x1 t. The corresponding system outputWill beTherefore, the system is homogeneous.(ii) This system is not additive. Consider the fowling example .Let n=2n+2+2n+1+2n and n= n+1+ 2n+1+ 3n. The corresponding outputs evaluated at n=0 areNow consider a third input x3 n= x2 n+ x1 n.= 3n+2+4n+1+5nThe corresponding outputs evaluated at n=0 is y30=15/4. Gnarly, y30 .This Therefore, the system is homogenous.1.30 (a) Invertible. Inverse system y(t)=x(t+4)(b)Non invertible. The signals x(t) and x1(t)=x(t)+2give the same output(c) n and 2n give the same outputd) Invertible. Inverse system; y(t)=dx(t)/dt(e) Invertible. Inverse system y(n)=x(n+1) for n0 and yn=xn for n1,but y(t)=1 for t1.1.41. (a) yn=2xn.Therefore, the system is time invariant. (b) yn=(2n-1)xn.This is not time-invariant because yn- N0(2n-1)2xn- N0. (c) yn=xn1+(-1)n+1+(-1)n-1=2xn.Therefore, the system is time invariant .1.42.(a) Consider two system S1 and S2 connected in series .Assume that if x1(t) and x2(t) are the inputs to S1.then y1(t) and y2(t) are the outputs.respectively .Also,assume that if y1(t) and y2(t) are the input to S2 ,then z1(t) and z2(t) are the outputs, respectively . Since S1 is linear ,we may write where a and b are constants. Since S2 is also linear ,we may write We may therefore conclude that Therefore ,the series combination of S1 and S2 is linear.Since S1 is time invariant, we may write and Therefore, Therefore, the series combination of S1 and S2 is time invariant.(b) False, Let y(t)=x(t)+1 and z(t)=y(t)-1.These corresponds to two nonlinear systems. If these systems are connected in series ,then z(t)=x(t) which is a linear system.(c) Let us name the output of system 1 as wn and the output of system 2 as zn .Then The overall system is linear and time-invariant.1.43. (a) We have Since S is time-invariant.Now if x (t) is periodic with period T. xt=x(t-T). Therefore, we may conclude that y(t)=y(t-T).This implies that y(t) is also periodic with T .A similar argument may be made in discrete time .(b)1.44 (a) Assumption : If x(t)=0 for tt0 ,then y(t)=0 for t t0.To prove That : The system is causal.Let us consider an arbitrary signal x1(t) .Let us