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MIT6_003S10_lec20

MIT6_003S10_lec20 - 6.003 Signals and Systems Relations...

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6.003: Signals and Systems Relations among Fourier Representations April 22, 2010

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Mid-term Examination #3 W ednesday, April 28, 7:30-9:30pm No recitations on the day of the exam. Coverage: Lectures 1–20 Recitations 1–20 Homeworks 1–11 Homework 11 will not collected or graded. Solutions will be posted. Closed book: 3 pages of notes ( 8 1 2 × 11 inches; front and back). Designed as 1-hour exam; two hours to complete. Review sessions during open oﬃce hours. .
Fourier Representations We’ve seen a variety of Fourier representations: CT Fourier series CT Fourier transform DT Fourier series DT Fourier transform Today: relations among the four Fourier representations.

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Four Fourier Representations We have discussed four closely related Fourier representations. DT Fourier Series DT Fourier transform 1 a k = a k + N = x [ n ] e kn X ( e j Ω ) = x [ n ] e j Ω n 2 j π N N n = <N> n = −∞ 2 π N 1 2 π < 2 π> j kn x [ n ] = x [ n + N ] = X ( e j Ω ) j Ω n d Ω e x [ n ] = a k e k = <N> CT Fourier Series CT Fourier transform kt dt X ( ) = x ( t ) e jωt dt 1 2 π j x ( t ) = T a k e T T −∞ 2 1 π x ( t ) = x ( t + T ) = a k e j T kt x ( t ) = X ( ) e jωt k = −∞ 2 π −∞
Four Types of “Time” discrete vs. continuous ( ) and periodic vs aperiodic ( ) DT Fourier Series DT Fourier transform n n CT Fourier Series CT Fourier transform t t

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Four Types of “Frequency” discrete vs. continuous ( ) and periodic vs aperiodic ( ) DT Fourier Series DT Fourier transform 2 π k Ω N CT Fourier Series CT Fourier transform ω 2 π k T
Relations among Fourier Representations Different Fourier representations are related because they apply to signals that are related. DTFS (discrete-time Fourier series): periodic DT DTFT (discrete-time Fourier transform): aperiodic DT CTFS (continuous-time Fourier series): periodic CT CTFT (continuous-time Fourier transform): aperiodic CT periodic DT DTFS aperiodic DT DTFT periodic CT CTFS aperiodic CT CTFT N → ∞ periodic extension T → ∞ periodic extension interpolate sample interpolate sample

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Relation between Fourier Series and Transform A periodic signal can be represented by a Fourier series or by an equivalent Fourier transform. Series: represent periodic signal as weighted sum of harmonics x ( t ) = x ( t + T ) = a k e 0 kt ; ω 0 = 2 π T k = −∞ The Fourier transform of a sum is the sum of the Fourier transforms: X ( ) = 2 πa k δ ( ω 0 ) k = −∞ Therefore periodic signals can be equivalently represented as Fourier transforms (with impulses!).
Relation between Fourier Series and Transform A periodic signal can be represented by a Fourier series or by an equivalent Fourier transform.

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MIT6_003S10_lec20 - 6.003 Signals and Systems Relations...

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