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# lecture_11 - MIT OpenCourseWare http/ocw.mit.edu 2.161...

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MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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F m m Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing - Continuous and Discrete Fall Term 2008 Lecture 11 1 Reading: Class Handout: Sampling and the Discrete Fourier Transform Class Handout: The Fast Fourier Transform Proakis and Manolakis (4th Ed.) Ch. 7 Oppenheim, Schafer & Buck (2nd Ed.) Chs. 8 & 9 1 The Discrete Fourier Transform continued In Lecture 10 the DFT pair associated with a sample set { f n } of length N was defined as N 1 F m = f n e j 2 πmn/N n =0 N 1 1 f n = F m e j 2 πmn/N N m =0 The value F m was interpreted as F (j Ω) evaluated at Ω = 2 π/N Δ T . 1.1 Organization of the DFT The N components in a DFT represent one period of a periodic spectrum. The first N/ 2 lines in the spectrum represent physical frequencies 0 . . . ( π/ Δ T ) radians/second. The components in the upper half of the sequence, F N/ 2+1 . . . F N 1 , may be considered to be the negative frequency components F N/ 2+1 . . . F 1 in the spectrum. It is common to translate the upper half of the record to the the left side of a plot to enhance the physical meaning. F N e q u i - s p m i n o n e p e N - 1 a c e d s a m p l e s r i o d o f F m i n t e r p r e t t o p N / 2 v a l u e s a s r e p r e s e n t i n g n e g a t i v e f r e q u e n c i e s i n F * ( j 9 ) . 0 N / 2 N - 1 N / 2 - 1 0 1 copyright c D.Rowell 2008 11–1
1.2 Spectral Resolution of the DFT The DFT pair provide a transform relationship between a pair of (complex) data sets { f n } and { F m } , each of length N . If the sampling interval associated with { f n } is Δ T units, the record duration is T = N Δ T. The frequency resolution, or line spacing, ΔΩ, in the DFT is 2 π 2 π 1 ΔΩ = = rad/s, or Δ F = Hz. (1) N Δ T T T and the frequency range spanned by the N lines in the DFT is 2 π 1 N ΔΩ = rad/s, or N Δ F = Hz. (2) Δ T Δ T The sequence { F m } represents both the positive and negative frequencies in a two-sided spectrum. The highest (positive) frequency component in the spectrum is half of this range (the Nyquist frequency), that is π 1 Ω max = rad/s, or F max = Hz. (3) Δ T T We conclude therefore, that the resolution within the DFT depends on the duration T of the data record, and the maximum frequency depends on the sampling interval Δ T . 1.3 Properties of the Discrete Fourier Transform Because the DFT is derived directly as a sampled continuous Fourier transform, it inherits most of the properties of the Fourier transform. We repeat some of the important properties here. In addition other properties are based on the assumed periodicity of { f n } and { F m } : 1. Linearity: If { f n } and { g n } are both length N , and { f n } DFT ⇐⇒ { F m } and { g n } DFT ⇐⇒ { G m } then DFT a { f n } + b { g n } ⇐⇒ a { F m } + b { G m } 2. Symmetry Properties of the DFT: If { f n } is a real-valued sequence then { F m } = F m from which it follows that {{ F m }} is an even function of m and {{ F m }} is an odd function of m .

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