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5.6. Given that x[n] has Fourier transform X(ejw), express the Fourier transforms of the following signals in terms of X(ejw). You may use the Fourier transform properties listed in Table 5.1. (a) x₁[n] = x[1 − n] + x[−1 – n] (b) x2[n] = x*[−n]+x[n] - 2 (c) x3[n] = (n − 1)² x[n] TABLE 5.1 Section PROPERTIES OF THE DISCRETE-TIME FOURIER TRANSFORM Property Aperiodic Signal x*[n] x[−n] Fourier Transform X(e) periodic with Y(ej) period 2π aX(ej)+bY(ej) e- juno X (ejw) X(ej(w-w₁)) X* (e¯jw) X(e jw) x[n] y[n] 5.3.2 Linearity ax[n] + by[n] 5.3.3 Time Shifting x[n - no] 5.3.3 Frequency Shifting ejwon x[n] 5.3.4 Conjugation 5.3.6 Time Reversal 5.3.7 Time Expansion X(k)[n] = { x[n/k], 0, if n = multiple of k if n X(ejkw) multiple of k 5.4 Convolution 5.5 Multiplication x[n] * y[n] x[n]y[n] X(ejw)Y(ejw) 1 5.3.5 Differencing in Time - x[n] x[n-1] n 5.3.5 Accumulation Σ x[k] k = -x ·X(ejw) 2πT 2TT (1 - e-ju)X(ejw) 1 e-jw +x ')de +πX(e³) Σ (w - 2πk) dX(ejw) k=-x 5.3.8 Differentiation in Frequency nx[n] ¡ dw 5.3.4 Conjugate Symmetry for Real Signals x[n] real 5.3.4 Symmetry for Real, Even Signals x[n] real an even 5.3.4 Symmetry for Real, Odd Signals x[n] real and odd 5.3.4 Even-odd Decomposition 5.3.9 of Real Signals Parseval's Relation for Aperiodic Signals +x x,[n] = x,[n] = &{x[n]} [x[n] real] Od{x[n]} [x[n] real] Σ|x[n] == 2π 12T 2 | | _ \ X ( e ³ ³ ³ d w 11=-0 X(eju ) = X*(e-in) (e¯jw) Re{X(e)} = Re{X(e¯jw)} Im{X(e)} = −Im{X(e¯jw)} |X(e)| = |X(e¯jw)| XX(ej) = -*X(e¯jw) X(ej) real and even X(e) purely imaginary and odd Re{X(ej")} jIm{X(e³w)}

5.6. Given that x[n] has Fourier transform X(ejw), express the Fourier transforms of the following signals in terms of X(ejw). You may use the Fourier transform properties listed in Table 5.1. (a) x₁[n] = x[1 − n] + x[−1 – n] (b) x2[n] = x*[−n]+x[n] - 2 (c) x3[n] = (n − 1)² x[n] TABLE 5.1 Section PROPERTIES OF THE DISCRETE-TIME FOURIER TRANSFORM Property Aperiodic Signal x*[n] x[−n] Fourier Transform X(e) periodic with Y(ej) period 2π aX(ej)+bY(ej) e- juno X (ejw) X(ej(w-w₁)) X* (e¯jw) X(e jw) x[n] y[n] 5.3.2 Linearity ax[n] + by[n] 5.3.3 Time Shifting x[n - no] 5.3.3 Frequency Shifting ejwon x[n] 5.3.4 Conjugation 5.3.6 Time Reversal 5.3.7 Time Expansion X(k)[n] = { x[n/k], 0, if n = multiple of k if n X(ejkw) multiple of k 5.4 Convolution 5.5 Multiplication x[n] * y[n] x[n]y[n] X(ejw)Y(ejw) 1 5.3.5 Differencing in Time - x[n] x[n-1] n 5.3.5 Accumulation Σ x[k] k = -x ·X(ejw) 2πT 2TT (1 - e-ju)X(ejw) 1 e-jw +x ')de +πX(e³) Σ (w - 2πk) dX(ejw) k=-x 5.3.8 Differentiation in Frequency nx[n] ¡ dw 5.3.4 Conjugate Symmetry for Real Signals x[n] real 5.3.4 Symmetry for Real, Even Signals x[n] real an even 5.3.4 Symmetry for Real, Odd Signals x[n] real and odd 5.3.4 Even-odd Decomposition 5.3.9 of Real Signals Parseval's Relation for Aperiodic Signals +x x,[n] = x,[n] = &{x[n]} [x[n] real] Od{x[n]} [x[n] real] Σ|x[n] == 2π 12T 2 | | _ \ X ( e ³ ³ ³ d w 11=-0 X(eju ) = X*(e-in) (e¯jw) Re{X(e)} = Re{X(e¯jw)} Im{X(e)} = −Im{X(e¯jw)} |X(e)| = |X(e¯jw)| XX(ej) = -*X(e¯jw) X(ej) real and even X(e) purely imaginary and odd Re{X(ej")} jIm{X(e³w)}

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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5.6. Given that x[n] has Fourier transform X(ejw), express the Fourier transforms of the following signals in terms of X(ejw). You may use the Fourier transform properties listed in Table 5.1. (a) x₁[n] = x[1 − n] + x[−1 – n] (b) x2[n] = x*[−n]+x[n] - 2 (c) x3[n] = (n − 1)² x[n] TABLE 5.1 Section PROPERTIES OF THE DISCRETE-TIME FOURIER TRANSFORM Property Aperiodic Signal x*[n] x[−n] Fourier Transform X(e) periodic with Y(ej) period 2π aX(ej)+bY(ej) e- juno X (ejw) X(ej(w-w₁)) X* (e¯jw) X(e jw) x[n] y[n] 5.3.2 Linearity ax[n] + by[n] 5.3.3 Time Shifting x[n - no] 5.3.3 Frequency Shifting ejwon x[n] 5.3.4 Conjugation 5.3.6 Time Reversal 5.3.7 Time Expansion X(k)[n] = { x[n/k], 0, if n = multiple of k if n X(ejkw) multiple of k 5.4 Convolution 5.5 Multiplication x[n] * y[n] x[n]y[n] X(ejw)Y(ejw) 1 5.3.5 Differencing in Time - x[n] x[n-1] n 5.3.5 Accumulation Σ x[k] k = -x ·X(ejw) 2πT 2TT (1 - e-ju)X(ejw) 1 e-jw +x ')de +πX(e³) Σ (w - 2πk) dX(ejw) k=-x 5.3.8 Differentiation in Frequency nx[n] ¡ dw 5.3.4 Conjugate Symmetry for Real Signals x[n] real 5.3.4 Symmetry for Real, Even Signals x[n] real an even 5.3.4 Symmetry for Real, Odd Signals x[n] real and odd 5.3.4 Even-odd Decomposition 5.3.9 of Real Signals Parseval's Relation for Aperiodic Signals +x x,[n] = x,[n] = &{x[n]} [x[n] real] Od{x[n]} [x[n] real] Σ|x[n] == 2π 12T 2 | | _ \ X ( e ³ ³ ³ d w 11=-0 X(eju ) = X*(e-in) (e¯jw) Re{X(e)} = Re{X(e¯jw)} Im{X(e)} = −Im{X(e¯jw)} |X(e)| = |X(e¯jw)| XX(ej) = -*X(e¯jw) X(ej) real and even X(e) purely imaginary and odd Re{X(ej")} jIm{X(e³w)}
Transcribed Image Text:5.6. Given that x[n] has Fourier transform X(ejw), express the Fourier transforms of the following signals in terms of X(ejw). You may use the Fourier transform properties listed in Table 5.1. (a) x₁[n] = x[1 − n] + x[−1 – n] (b) x2[n] = x*[−n]+x[n] - 2 (c) x3[n] = (n − 1)² x[n] TABLE 5.1 Section PROPERTIES OF THE DISCRETE-TIME FOURIER TRANSFORM Property Aperiodic Signal x*[n] x[−n] Fourier Transform X(e) periodic with Y(ej) period 2π aX(ej)+bY(ej) e- juno X (ejw) X(ej(w-w₁)) X* (e¯jw) X(e jw) x[n] y[n] 5.3.2 Linearity ax[n] + by[n] 5.3.3 Time Shifting x[n - no] 5.3.3 Frequency Shifting ejwon x[n] 5.3.4 Conjugation 5.3.6 Time Reversal 5.3.7 Time Expansion X(k)[n] = { x[n/k], 0, if n = multiple of k if n X(ejkw) multiple of k 5.4 Convolution 5.5 Multiplication x[n] * y[n] x[n]y[n] X(ejw)Y(ejw) 1 5.3.5 Differencing in Time - x[n] x[n-1] n 5.3.5 Accumulation Σ x[k] k = -x ·X(ejw) 2πT 2TT (1 - e-ju)X(ejw) 1 e-jw +x ')de +πX(e³) Σ (w - 2πk) dX(ejw) k=-x 5.3.8 Differentiation in Frequency nx[n] ¡ dw 5.3.4 Conjugate Symmetry for Real Signals x[n] real 5.3.4 Symmetry for Real, Even Signals x[n] real an even 5.3.4 Symmetry for Real, Odd Signals x[n] real and odd 5.3.4 Even-odd Decomposition 5.3.9 of Real Signals Parseval's Relation for Aperiodic Signals +x x,[n] = x,[n] = &{x[n]} [x[n] real] Od{x[n]} [x[n] real] Σ|x[n] == 2π 12T 2 | | _ \ X ( e ³ ³ ³ d w 11=-0 X(eju ) = X*(e-in) (e¯jw) Re{X(e)} = Re{X(e¯jw)} Im{X(e)} = −Im{X(e¯jw)} |X(e)| = |X(e¯jw)| XX(ej) = -*X(e¯jw) X(ej) real and even X(e) purely imaginary and odd Re{X(ej")} jIm{X(e³w)}
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