《信号与系统教学资料》信号与系统奥本海姆英文版课后答案chapter7.pdfVIP

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《信号与系统教学资料》信号与系统奥本海姆英文版课后答案chapter7.pdf

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Chapter 7 Answers 7.1 From the Nyquist sampling theorem , we know that only if X (j w)=0 for |w| ws/2 will be signal be recoverable from its samples. Therefore, X(jw)5000л. 7.2 From the Nyquist theorem ,we know that the sampling frequency in this case must be at least ws=2000 п.In other words ,the sampling period should be at most T=2п/ (ws)=1＊10-3.Clearly ,only (a) and (e) satisfy this condition. 7.3 (a) We can easily show that X(j w)=0 for |w| 4000п.Therefore, the Nyquist rate for this signal is wN=2(4000п)=8000п. (b)From the Tables 4.1 and 4.2 we know that X(j w) is a rectangular pulse for which X(j w)=0 for |w| 4000п.Therefore, the Nyquist rate for this signal is wN =2(4000п)=800п. (c) From the Tables 4.1 and 4.2 we know that X(j w) is the convolution of two rectangular pulses each of which is zero for |w| 4000п.Therefore ,X(j w)=0 for |w| 8000пand the Nyquist rate for this signal is wN=2(8000п)=16000п. 7.4 If the signal x(t) has a Nyquist rate of wo ,then its Fourier transform X (j w)=0 for |w| wo/2. (a) From chapter 4, y(t) = x (t) + x (t-1) ←?FT → Y (jw) = X (jw) + e-jwt X (jw). Clearly, we can only guarantee that Y (jw) =0 for |w| wo/2. Therefore, the Nyquist rate for y(t) is also wo. (b) From chapter 4, y(t) = dx(t) ←?FT→ Y (jw)= jw X(jw). dt Clearly, we can only guarantee that Y (jw) =0 for |w| wo/2. Therefore, the Nyquist rate for y(t) is also wo. (c) From chapter 4, y(t) =x2(t) ←?FT→ Y (jw)= (1/2п)[X(jw)*X(jw)] Clearly, we can only guarantee that Y (jw) =0 for |w| wo. Therefore, the Nyquist rate for y(t) is also 2wo. (d) From chapter 4, y(t)=x(t)cos (wot) ←?FT → Y (jw)= (1/2)X(j(w- wo)) +(1/2)X(j(w+ wo)). Clearly, we can guarantee that Y (jw) =0 for |w| wo+ wo/2. Therefore, the Nyquist rate for y(t) is 3wo. 7.5 Using Table 4.2, p(t) ←?FT → 2π T ∞ ∑δ (ω ? K 2π / T ) K =?∞ From Table 4.1 ∑p(t-1) ←?FT → 2π e-jw ∞ δ (ω ? k 2π )e ? jk 2π T . T k =?∞ T Since y(t)=x(t)p(t-1),we have Y (jw)= (1/2п)[X(jw)*FT{P(t-1)}] ∑=(1/T) ∞ X ( j(ω ? k 2π ))e ? jk 2π T