信号与系统第二版课后习题解答(6-7-9)奥本海姆.doc

Signal and SystemChap 66.1 Consider a continuous-time LTI system with frequency response and real impulse response h(t). Suppose that we apply an input to this system .The resulting output can be shown to be of the formWhere A is a nonnegative real number representing an amplitude-scaling factor and is a time delay.(a)Express A in terms of .(b)Express in terms of Solution:(a) For So So (b) for So 6.3 Consider the following frequency response for a causal and stable LTI system:(a) Show that ,and determine the values of A.(b)Determine which of the following statements is true about ,the group delay of the system.(Note ,where is expressed in a form that does not contain any discontinuities.)1.2.3 Solution:(a) for So A=1(b) for So 6.5 Consider a continuous-time ideal bandpass filter whose frequency response is (a) If h(t) is the impulse response of this filter, determine a function g(t) such that (b) As is increased, dose the impulse response of the filter get more concentrated or less concentrated about the origin?Solution(a) Method 1. Let They are shown in the figures,where So we can get Method 2. Using the inverse FT definition,it is obtained (b) more concentrated.12Chap 77.1 A real-valued signal x(t) is know to be uniquely determined by its samples when the sampling frequency is .For what values of is guaranteed to be zero?Solution:According to the sampling theorem That is So if ,7.2 A continuous-time signal x(t) is obtained at the output of an ideal lowpass filter with cutoff frequency .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) (b) (c) Solution:From the sampling theorem,that is 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)(b)(c) Solution:(a) the Nyquist rate is (b) the Nyquist rate is (c) the Nyquist rate is 7.4 Let x(t) be a signal with Nyquist rate. Determine the Nyquist rate for each of the following signals:(a)(b)(c)(d)Solution:(a) we let So So the Nyquist rate of signal (a) is .(b) we let So So the Nyquist rate of signal (b) is .(c) we let So So the Nyquist rate of signal (c) is 2.(d) we let For So So the Nyquist rate of signal (d) is 7.9 Consider the signal Which we wish to sample with a sampling frequency of to obtain a signal g(t) with Fourier transform .Determine the maximum value of for which it is guaranteed thatWhere is the Fourier transform of x(t). Solution:But the figure about before-sampling and after-sampling of isWe can see that only when , the before-sampling and after-sampling of have the same figure.So if The maximum value of is .Chap 99.2Consider the signal 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 such that the Laplace transform G(s) of has the same algebraic form as X(s).what is the region of convergence corresponding to G(s)?Solution: (a). According to eq.(9.3), we will get ROC:Res-5(b). , Res-5If then its obviously that A=-1, , Res-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) (b) (c) Solution: (a).1, 1it has a zero in the finite s-plane, that is And because the order of the denominator exceeds the order of the numerator by 1 X(s) has 1 zero at infinity. (b).0, 1it 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, 0it has a zero in the finite s-plane, that is And 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 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. If the Fourier transform of g(t) converges, determine whether x(t) is left sided, right sided, or two sided.Solution: ROC: R(x)+Re2 And x(t) have three possible ROC strips: g(t) have three possible ROC strips: IF Then the ROC of is (-1,1) is two sides.9.9 Given thatDetermine the inverse Laplace transform of Solution:It is obtained from the partial-fractional expansion:,We can get the inverse Laplace transform from given formula and linear property.9.10Using 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): (b): (c): Solution:(a