MIT6_003S10_lec20

MIT6_003S10_lec20 - 6.003: Signals and Systems Relations...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
6.003: Signals and Systems Relations among Fourier Representations April 22, 2010
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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. .
Background image of page 2
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.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
± ± ± ± 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 x [ n ]= x [ n + N X ( e j Ω ) j Ω n d Ω e x [ n a k e k = 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 x ( t X ( ) e k = −∞ 2 π −∞
Background image of page 4
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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 6
Relations among Fourier Representations Different Fourier representations are related because they apply to signals that 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
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Relation between Fourier Series and Transform A periodic signal can be represented by a Fourier series or 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 equivalently represented as Fourier transforms (with impulses!).
Background image of page 8
Relation between Fourier Series and Transform A periodic signal can be represented by a Fourier series or an equivalent Fourier transform.
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 42

MIT6_003S10_lec20 - 6.003: Signals and Systems Relations...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online