MIT6_003S10_lec19_handout

MIT6_003S10_lec19_handout - 6.003: Signals and Systems...

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± ± ± ± ± ± 6.003: Signals and Systems Lecture 19 April 15, 2010 6.003: Signals and Systems DT Fourier Representations April 15, 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. Review: DT Frequency Response The frequency response of a DT LTI system is the value of the system function evaluated on the unit circle. H ( z ) cos(Ω n ) | H ( e j Ω ) | cos ² Ω n + H ( e j Ω ) ³ H ( e j Ω )= H ( z ) | z = e j Ω Comparision of CT and DT Frequency Responses CT frequency response: H ( s ) on the imaginary axis, i.e., s = . DT frequency response: H ( z ) on the unit circle, i.e., z = e j Ω . ω s -plane z -plane σ | H ( ) | H ( e j Ω ) 1 Ω 0 ω π 0 π Check Yourself A system H ( z 1 az z a has the following pole-zero diagram. z -plane Classify this system as one of the following filter types. 1. high pass 2. low pass 3. band pass 4. all pass 5. band stop 0. none of the above Effects of Phase 1
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6.003: Signals and Systems Lecture 19 April 15, 2010 Effects of Phase Effects of Phase http://public.research.att.com/~ttsweb/tts/demo.php Effects of Phase x [ n ] ??? y [ n ]= x [ n ] artificial speech synthesized by Robert Donovan Review: Periodicity DT frequency responses are periodic functions of Ω , with period 2 π . If Ω 2 1 +2 πk where k is an integer then H ( e j Ω 2 )= H ( e j 1 +2 πk ) H ( e j Ω 1 e j 2 πk H ( e j Ω 1 ) The periodicity of H ( e j Ω ) results because H ( e j Ω ) is a function of e j Ω , which is itself periodic in Ω . Thus DT complex exponentials have many “aliases.” e j Ω 2 = e j 1 +2 πk ) = e j Ω 1 e j 2 πk = e j Ω 1 Because of this aliasing, there is a “highest” DT frequency: Ω= π . artificial speech synthesized by Robert Donovan Effects of Phase x [ n ] ??? y [ n x [ n ] How are the phases of X and Y related? Review: Periodic Sinusoids There are N distinct DT complex exponentials with period N . If e j Ω n is periodic in N then e j Ω n = e j Ω( n + N ) = e j Ω n e j Ω N and e j Ω N must be 1, and Ω must be one of the N th roots of 1 . Example: N =8 z -plane 2
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± ± ± ± ± ± ± ² 6.003: Signals and Systems Lecture 19 April 15, 2010 Review: DT Fourier Series DT Fourier series represent DT signals in terms of the amplitudes and phases of harmonic components. DT Fourier Series a k = a k + N = 1 x [ n ] e j Ω 0 n 0 = 2 π (“analysis” equation) N N n = <N> x [ n ]= x [ n + N ]= a k e jk Ω 0 n (“synthesis” equation) k = <N> DT Fourier Series π N π = x [3] 1 j 1 j a 3 DT Fourier series have simple matrix interpretations. 2 ± ± Ω kn [ ]= [ +4]= 0 n n 4 = = j xn a e a e a k k k 4 4 4 k = k = k = < > < > < > [0] 1 1 1 1 0 x a [1] 1 1 j j 1 x a [2] 1 1 1 1 2 x a 1 1 1 2 Ω kn [ ] [ ] 0 n n = = = = j +4 a a xne e k k 4 4 4 4 4 4 = = = < > < > < > n n n 1 1 [1] j j 1 a x These matrices are inverses of each other.
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MIT6_003S10_lec19_handout - 6.003: Signals and Systems...

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