《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chapter6.pdfVIP

《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chapter6.pdf

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Chapter 6 Answers 6.6 (b) the impulse response h1[n] is as shown in figure s6.6,as was increase ,it is clear that the significant central lobe of h1[n] becomes more concentrated around the origin. consequently. h[n]=h1[n](-1)^n also becomes more concentrated about the origin. 6.7 the frequency response magnitude |H(jw)| is as shown in figure s6.7.the frequency response of the bandpass filter G(jw) will be given by G( jω) = FT{2h(t) cos(4000π t)} = H ( j(ω ? 4000π )) + H ( j(ω + 4000π )) This is as shown in figure s6.7 H(j -4000 -2000 -1000 π π1000 2000 π4000 G(jω ) -6000π -4000π -2000π 0 2000π 4000π 6000π Figure S6.7 (a) from the figure ,it is obvious that the passband edges are at 2000∏rad/sec and 6000∏rad/sec. this translates to 1000HZ and 3000Hz,respectively. (b) (b)from the figure ,it is obvious that the stopband edges are at 1600∏ rad/sec.this translates to 800Hz and 3200 Hz, respectively. 6.8 taking the Fourier transform of both sides of the first difference equation and simplifying, we obtain the frequency response H(e^jw)of the first filter. M ∑∑H (e jω ) = Y (e jω ) X (e jω ) bk e? jωk = k=0 N 1? ak e? jωk . k =1 Taking the Fourier transform of both sides of the second difference equation and simplifying ,we obtain the frequency response H1(e^jw) of the second filter. M ∑∑H (e jω ) = Y (e jω ) X (e jω ) (?1)k bk e? jωk = k=0 N 1? (?1)k ak e? jωk . k =1 This may also be written as M ∑∑H (e jω ) = Y (e jω ) X (e jω ) = b e? j(ω ?π )k k k =0 N 1 ? ak e? j(ω ?π )k = H (e j(ω ?π ) ). k =1 Therefore .the frequency response of the second filter is obtained bu shifting the frequency response of the first filter by ∏.although the first fitter has its passband between-wp and wp. Therefore, the second filter will have its passband between ∏-wp and ∏+wp. 6.9 taking the Fourier transform of the given differential equation and simplifying .we obtain the frequency of the LTI system to be H (e jω ) = Y (e jω ) X (e jω ) = 2 5 + jω Taking the inverse Fourier transform, we o

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