MIT6_003S10_lec19

# MIT6_003S10_lec19 - 6.003: Signals and Systems DT Fourier...

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6.003: Signals and Systems DT Fourier Representations April 15, 2010

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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.
± ² 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 ( e j Ω ) | cos Ω n + H ( e j Ω ) H ( z ) cos(Ω n ) H ( e j Ω )= H ( z ) | j Ω z = e

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± ± ± ± ± ± Comparision of CT and DT Frequency Responses CT frequency response: H ( s ) on the imaginary axis, i.e., s = . j Ω DT frequency response: H ( z ) on the unit circle, i.e., z = e . ω s -plane z -plane σ H ( e j Ω ) | H ( ) | 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

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± ± ± ± Check Yourself Classify the system ... 1 az H ( z )= z a Find the frequency response: j Ω j Ω a 1 ae j Ω e complex H ( e j Ω = e e j Ω a e j Ω a conjugates Because complex conjugates have equal magnitudes, H ( e j Ω )=1 . all-pass ﬁlter
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. 4 1. high pass 2. low pass 3. band pass 4. all pass 5. band stop 0. none of the above

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Eﬀects of Phase
Eﬀects of Phase

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Eﬀects of Phase http://public.research.att.com/~ttsweb/tts/demo.php
Eﬀects of Phase artificial speech synthesized by Robert Donovan

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Eﬀects of Phase x [ n ] ??? y [ n ]= x [ n ] artificial speech synthesized by Robert Donovan
Eﬀects of Phase x [ n ] y [ n ]= x [ n ] ??? How are the phases of X and Y related?

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Eﬀects of Phase How are the phases of X and Y related? a k = ± jk Ω 0 n x [ n ] e n b k = ± Ω 0 n x [ n ] e = ± Ω 0 m x [ m ] e = a k n m Flipping x [ n ] about n =0 ﬂips a k about k . Because x [ n ] is real-valued, a k is conjugate symmetric: a k = a k . b k = a k = a k j a k = | a k | e The angles negated at all frequencies.
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 ) H ( e j Ω 1 e j 2 H ( e j Ω 1 ) j Ω The periodicity of H ( e j Ω ) results because H ( e j Ω ) is a function of e , which is itself periodic in Ω . Thus DT complex exponentials have many “aliases.” j Ω 2 j 1 +2 ) j Ω 1 e j 2 j Ω 1 e = e = e = e Because of this aliasing, there is a “highest” DT frequency: Ω= π .

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Review: Periodic Sinusoids There are N distinct DT complex exponentials with period N .
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## This note was uploaded on 12/14/2011 for the course EE 6.003 taught by Professor Freeman during the Fall '11 term at MIT.

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MIT6_003S10_lec19 - 6.003: Signals and Systems DT Fourier...

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