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MIT6_003S10_lec20_handout

# MIT6_003S10_lec20_handout - 6.003 Signals and Systems...

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± 6.003: Signals and Systems Lecture 20 April 22, 2010 6.003: Signals and Systems Relations among Fourier Representations April 22, 2010 Mid-term Examination #3 Wednesday, 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. π N π N π T π T Four Fourier Representations We have discussed four closely related Fourier representations. DT Fourier Series DT Fourier transform a k = a k + N = 1 x [ n ] e j 2 kn X ( e j Ω )= x [ n ] e j Ω n N n = <N> n = −∞ x [ n ]= x [ n + N a k e j 2 kn x [ n 1 X ( e j Ω ) e j Ω n d Ω k = <N> 2 π < 2 π> CT Fourier Series CT Fourier transform a k = 1 ± x ( t ) e j 2 kt dt X ( ± x ( t ) e jωt dt T T −∞ ± x ( t x ( t + T a k e j 2 kt x ( t 1 X ( ) e 2 π k = −∞ −∞ 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 Four Types of “Frequency” discrete vs. continuous ( ) and periodic vs aperiodic ( ± ) DT Fourier Series DT Fourier transform 2 π k N Ω CT Fourier Series CT Fourier transform 2 π k ω T 1

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6.003: Signals and Systems Lecture 20 April 22, 2010 Relation between Fourier Series and Transform 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 Relation between Fourier Series and Transform A periodic signal can be represented by a Fourier series or by an equivalent Fourier transform. x ( t )= x ( t + T k = −∞ a k e 0 kt Fourier Transform 0 ω 0 k Fourier Series 01 a 0 a 1 a 1 a 2 a 2 a 3 a 3 a 4 a 4 A periodic signal can be represented by a Fourier series or by an equivalent Fourier transform.
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MIT6_003S10_lec20_handout - 6.003 Signals and Systems...

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