lec20 - 6.003: Signals and Systems Relations among Fourier...

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6.003: Signals and Systems Relations among Fourier Representations November 19, 2009

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Fourier Representations We’ve seen a variety of Fourier representations: CT Fourier series CT Fourier transform DT Fourier series One more today: DT Fourier transform ... and relations among all four representations.
DT Fourier transform Representing aperiodic DT signals as sums of complex exponentials. DT Fourier transform X ( e j )= Ø n = −∞ x [ n ] e j n (“analysis” equation) x [ n ]= 1 2 π Ú < 2 π> X ( e j ) e j n d (“synthesis” equation)

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Comparison to DT Fourier Series From periodic to aperiodic. DT Fourier Series a k = a k + N = 1 N Ø n = <N> x [ n ] e j 0 kn ; Ω 0 = 2 π N (“analysis” equation) x [ n ]= x [ n + N ] = Ø k = <N> a k e j 0 kn (“synthesis” equation) DT Fourier transform X ( e j )= Ø n = −∞ x [ n ] e j n (“analysis” equation) x [ n ]= 1 2 π Ú < 2 π> X ( e j ) e j n d (“synthesis” equation)
Comparison to DT Fourier Series Sum over an inﬁnite number of time samples instead of N . DT Fourier Series a k = a k + N = 1 N Ø n = <N> x [ n ] e j 0 kn ; Ω 0 = 2 π N (“analysis” equation) x [ n ]= x [ n + N ] = Ø k = <N> a k e j 0 kn (“synthesis” equation) DT Fourier transform X ( e j )= Ø n = −∞ x [ n ] e j n (“analysis” equation) x [ n ]= 1 2 π Ú < 2 π> X ( e j ) e j n d (“synthesis” equation)

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Comparison to DT Fourier Series Sum over an inﬁnite number of frequency components ( s ). DT Fourier Series a k = a k + N = 1 N Ø n = <N> x [ n ] e j 0 kn ; Ω 0 = 2 π N (“analysis” equation) x [ n ]= x [ n + N ] = Ø k = <N> a k e j 0 kn (“synthesis” equation) DT Fourier transform X ( e j )= Ø n = −∞ x [ n ] e j n (“analysis” equation) x [ n ]= 1 2 π Ú < 2 π> X ( e j ) e j n d (“synthesis” equation)
DT Fourier Series and Transform Example of series. Let x [ n ] represent the following periodic DT signal. n x [ n ] = x [ n + 8] 1 1 8 8 a k = 1 N Ø n = <N> x [ n ] e j 0 kn ; Ω 0 = 2 π N = 1 8 ± 1 + 1 2 e j 2 π 8 k + 1 2 e j 2 π 8 k ² = 1 + cos πk 4 8 k a k 1 1 8 8 1 4

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DT Fourier Series and Transform Example of transform. Let x [ n ] represent the aperiodic base of the previous signal. n x [ n ] 1 1 H ( e j ) = Ø n = −∞ x [ n ] e j n = ± 1 + 1 2 e j + 1 2 e j ² = 1 + cos Ω X ( e j ) 2 π 2 π 2
DT Fourier Series and Transform Similarities.

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