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2021年春季学期-信号与系统-第十四次作业参考答案-第八小题参考答案

系统 2021 信号 作业 春季 参考答案 学期 第八
2023-09-11 14:15:20 时间

本文是 2021年春季学期-信号与系统-第十四次作业参考答案 中各小题的参考答案。

 

§08 第八小题

8、 以下序列的长度为\nN.，求其离散傅里叶变换的闭合表达式。

（1） x [ n ] = sin ⁡ ( ω 0 n ) R N [ n ] x\left[ n \right] = \sin \left( {\omega _0 n} \right)R_N \left[ n \right] x[n]=sin(ω0​n)RN​[n]

（2） x [ n ] = a n ⋅ R N [ n ] x\left[ n \right] = a^n \cdot R_N \left[ n \right] x[n]=an⋅RN​[n]

（3） n 2 ⋅ R N [ n ] n^2 \cdot R_N \left[ n \right] n2⋅RN​[n]

▓ 求解

（1）
X ( k ) = ∑ n = 0 N − 1 sin ⁡ ( ω 0 n ) ⋅ e − j 2 π k n N X\left( k \right) = \sum\limits_{n = 0}^{N - 1} {\sin \left( {\omega _0 n} \right) \cdot e^{ - j{{2\pi kn} \over N}} } X(k)=n=0∑N−1​sin(ω0​n)⋅e−jN2πkn​

= ∑ n = 0 N − 1 1 2 j ( e j ω 0 n − e − j ω 0 n ) ⋅ e − j 2 π k N ⋅ n = \sum\limits_{n = 0}^{N - 1} {{1 \over {2j}}\left( {e^{j\omega _0 n} - e^{ - j\omega _0 n} } \right) \cdot e^{ - j{{2\pi k} \over N} \cdot n} } =n=0∑N−1​2j1​(ejω0​n−e−jω0​n)⋅e−jN2πk​⋅n

= 1 2 j [ 1 − e j ω 0 N 1 − e j ( ω 0 − 2 π k N ) − 1 − e − j ω 0 N 1 − e − j ( ω 0 + 2 π k N ) ] = {1 \over {2j}}\left[ {{{1 - e^{j\omega _0 N} } \over {1 - e^{j\left( {\omega _0 - {{2\pi k} \over N}} \right)} }} - {{1 - e^{ - j\omega _0 N} } \over {1 - e^{ - j\left( {\omega _0 + {{2\pi k} \over N}} \right)} }}} \right] =2j1​[1−ej(ω0​−N2πk​)1−ejω0​N​−1−e−j(ω0​+N2πk​)1−e−jω0​N​]

= 1 2 j ⋅ ( 1 − e j ω 0 N ) ⋅ ( 1 − e − j ( ω 0 + 2 π k N ) ) − ( 1 − e − j ω 0 N ) ⋅ ( 1 − e j ( ω 0 − 2 π k N ) ) ( 1 − e j ( ω 0 − 2 π k N ) ) ⋅ ( 1 − e − j ( ω 0 + 2 π k N ) ) = {1 \over {2j}} \cdot {{\left( {1 - e^{j\omega _0 N} } \right) \cdot \left( {1 - e^{ - j\left( {\omega _0 + {{2\pi k} \over N}} \right)} } \right) - \left( {1 - e^{ - j\omega _0 N} } \right) \cdot \left( {1 - e^{j\left( {\omega _0 {\rm{ - }}{{{\rm{2}}\pi k} \over N}} \right)} } \right)} \over {\left( {1 - e^{j\left( {\omega _0 - {{2\pi k} \over N}} \right)} } \right) \cdot \left( {1 - e^{ - j\left( {\omega _0 + {{2\pi k} \over N}} \right)} } \right)}} =2j1​⋅(1−ej(ω0​−N2πk​))⋅(1−e−j(ω0​+N2πk​))(1−ejω0​N)⋅(1−e−j(ω0​+N2πk​))−(1−e−jω0​N)⋅(1−ej(ω0​−N2πk​))​

= sin ⁡ ω 0 e − j 2 π k N − sin ⁡ ( ω 0 N ) + sin ⁡ ( ω 0 N − ω 0 ) e − j 2 π k N 1 − 2 cos ⁡ ω 0 e − j 2 π k N + e − j 4 π k N = {{\sin \omega _0 e^{ - j{{2\pi k} \over N}} - \sin \left( {\omega _0 N} \right) + \sin \left( {\omega _0 N - \omega _0 } \right)e^{ - j{{2\pi k} \over N}} } \over {1 - 2\cos \omega _0 e^{ - j{{2\pi k} \over N}} + e^{ - j{{4\pi k} \over N}} }} =1−2cosω0​e−jN2πk​+e−jN4πk​sinω0​e−jN2πk​−sin(ω0​N)+sin(ω0​N−ω0​)e−jN2πk​​

（2）
X ( k ) = ∑ n = 0 N − 1 a n e − j 2 π N k n X\left( k \right) = \sum\limits_{n = 0}^{N - 1} {a^n e^{ - j{{2\pi } \over N}kn} } X(k)=n=0∑N−1​ane−jN2π​kn = 1 − ( a e − j 2 π k N ) N 1 − a ⋅ e − j 2 π k N = 1 − a N 1 − a ⋅ e − j 2 π k N = {{1 - \left( {ae^{ - j{{2\pi k} \over N}} } \right)^N } \over {1 - a \cdot e^{ - j{{2\pi k} \over N}} }} = {{1 - a^N } \over {1 - a \cdot e^{ - j{{2\pi k} \over N}} }} =1−a⋅e−jN2πk​1−(ae−jN2πk​)N​=1−a⋅e−jN2πk​1−aN​

（3）
∑ n = 0 N − 1 n W n = W + 2 W 2 + ⋅ ⋅ ⋅ + ( N − 1 ) W N − 1 \sum\limits_{n = 0}^{N - 1} {nW^n } = W + 2W^2 + \cdot \cdot \cdot + \left( {N - 1} \right)W^{N - 1} n=0∑N−1​nWn=W+2W2+⋅⋅⋅+(N−1)WN−1 = W + W 2 + ⋅ ⋅ ⋅ + W N − 1 + = W + W^2 + \cdot \cdot \cdot + W^{N - 1} + =W+W2+⋅⋅⋅+WN−1+ W 2 + ⋅ ⋅ ⋅ + W N − 1 + W^2 + \cdot \cdot \cdot + W^{N - 1} + W2+⋅⋅⋅+WN−1+ W 3 + ⋅ ⋅ ⋅ + W N − 1 + W^3 + \cdot \cdot \cdot + W^{N - 1} + W3+⋅⋅⋅+WN−1+ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ W N − 1 W^{N - 1} WN−1

= W − W N 1 − W + W 2 − W N 1 − W + ⋅ ⋅ ⋅ + W N − 1 − W N 1 − W = {{W - W^N } \over {1 - W}} + {{W^2 - W^N } \over {1 - W}} + \cdot \cdot \cdot + {{W^{N - 1} - W^N } \over {1 - W}} =1−WW−WN​+1−WW2−WN​+⋅⋅⋅+1−WWN−1−WN​ = ∑ n = 0 N − 1 W n − N ⋅ W N 1 − W = W − W N 1 − W − ( N − 1 ) W N 1 − W = {{\sum\limits_{n = 0}^{N - 1} {W^n } - N \cdot W^N } \over {1 - W}} = {{{{W - W^N } \over {1 - W}} - \left( {N - 1} \right)W^N } \over {1 - W}} =1−Wn=0∑N−1​Wn−N⋅WN​=1−W1−WW−WN​−(N−1)WN​ = W − 1 1 − W − ( N − 1 ) 1 − W = − N 1 − W = {{{{W - 1} \over {1 - W}} - \left( {N - 1} \right)} \over {1 - W}} = {{ - N} \over {1 - W}} =1−W1−WW−1​−(N−1)​=1−W−N​

∑ n = 0 N − 1 n 2 W n = W + 4 W 2 + 9 W 3 + ⋅ ⋅ ⋅ + ( 2 N − 3 ) W N − 1 \sum\limits_{n = 0}^{N - 1} {n^2 W^n } = W + 4W^2 + 9W^3 + \cdot \cdot \cdot + \left( {2N - 3} \right)W^{N - 1} n=0∑N−1​n2Wn=W+4W2+9W3+⋅⋅⋅+(2N−3)WN−1 = W + W 2 + W 3 +            ⋅ ⋅ ⋅            + W N − 1 + = W + W^2 + W^3 + \,\,\,\,\,\,\,\,\,\, \cdot \cdot \cdot \,\,\,\,\,\,\,\,\,\, + W^{N - 1} + =W+W2+W3+⋅⋅⋅+WN−1+ 3 W 2 + 3 W 3 +          ⋅ ⋅ ⋅            + 3 W N − 1 + 3W^2 + 3W^3 + \;\;\;\; \cdot \cdot \cdot \,\,\,\,\,\,\,\,\,\, + 3W^{N - 1} + 3W2+3W3+⋅⋅⋅+3WN−1+ 5 W 3 +          ⋅ ⋅ ⋅           + 5 W N − 1 + 5W^3 + \,\,\,\,\,\,\; \cdot \cdot \cdot \,\,\,\,\,\,\,\,\, + 5W^{N - 1} + 5W3+⋅⋅⋅+5WN−1+ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅                              + \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + ⋅⋅⋅⋅⋅⋅⋅⋅+ ( 2 N − 3 ) W N − 1 \left( {2N - 3} \right)W^{N - 1} (2N−3)WN−1
$$$$ = W − W N 1 − W + 3 ( W 2 − W N ) 1 − W + 5 ( W 3 − W N ) 1 − W + ⋅ ⋅ ⋅ + ( 2 N − 3 ) ( W N − 1 − W N ) 1 − W = {{W - W^N } \over {1 - W}} + {{3\left( {W^2 - W^N } \right)} \over {1 - W}} + {{5\left( {W^3 - W^N } \right)} \over {1 - W}} + \cdot \cdot \cdot + {{\left( {2N - 3} \right)\left( {W^{N - 1} - W^N } \right)} \over {1 - W}} =1−WW−WN​+1−W3(W2−WN)​+1−W5(W3−WN)​+⋅⋅⋅+1−W(2N−3)(WN−1−WN)​ = 1 1 − W [ ∑ k = 1 N − 1 ( 2 k − 1 ) W k − ∑ k = 1 N − 1 ( 2 k − 1 ) W N ] = {1 \over {1 - W}}\left[ {\sum\limits_{k = 1}^{N - 1} {\left( {2k - 1} \right)W^k } - \sum\limits_{k = 1}^{N - 1} {\left( {2k - 1} \right)} W^N } \right] =1−W1​[k=1∑N−1​(2k−1)Wk−k=1∑N−1​(2k−1)WN] = 1 1 − W [ 2 ∑ n = 1 N − 1 n W n − ∑ n = 1 N − 1 W n − ∑ n = 1 N − 1 ( 2 k − 1 ) ] = {1 \over {1 - W}}\left[ {2\sum\limits_{n = 1}^{N - 1} {nW^n } - \sum\limits_{n = 1}^{N - 1} {W^n } - \sum\limits_{n = 1}^{N - 1} {\left( {2k - 1} \right)} } \right] =1−W1​[2n=1∑N−1​nWn−n=1∑N−1​Wn−n=1∑N−1​(2k−1)] = 1 1 − W [ 2 ⋅ − N 1 − W − W − W N 1 − W − ( N − 1 ) 2 ] = {1 \over {1 - W}}\left[ {2 \cdot {{ - N} \over {1 - W}} - {{W - W^N } \over {1 - W}} - \left( {N - 1} \right)^2 } \right] =1−W1​[2⋅1−W−N​−1−WW−WN​−(N−1)2] = − 2 N − ( W − 1 ) − ( N − 1 ) ( 1 − W ) ( 1 − W ) 2 = N ( N − 1 ) W − N 2 ( 1 − W ) 2 = {{ - 2N - \left( {W - 1} \right) - \left( {N - 1} \right)\left( {1 - W} \right)} \over {\left( {1 - W} \right)^2 }} = {{N\left( {N - 1} \right)W - N^2 } \over {\left( {1 - W} \right)^2 }} =(1−W)2−2N−(W−1)−(N−1)(1−W)​=(1−W)2N(N−1)W−N2​

 

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