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# lecture_21 - 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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Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing - Continuous and Discrete Fall Term 2008 Lecture 21 1 Reading: Class Handout: Interpolation (Up-sampling) and Decimation (Down-sampling) Proakis and Manolakis: Secs. 11.1 11.5, 12.1 Oppenheim, Schafer, and Buck: Sec. 4.6, Appendix A Stearns and Hush, Ch. 9 1 Interpolation and Decimation 1.1 Up-Sampling (Interpolation) by an Integer Factor Consider a data set { f n } of length N , where f n = f ( n Δ T ), n = 0 . . . N 1 and Δ T is the sampling interval. The task is to resample the data at a higher rate so as to create a new data set { f ˆ n } , of length KN , representing samples of the same continuous waveform f ( t ), sampled at intervals Δ T/K . The following figure shows a cosinusoidal data set with N = 8 samples, and resampled with the same data set interpolated by a factor K = 4. 1 copyright c D.Rowell 2008 2 4 6 8 −1 −0.5 0 0.5 1 10 20 30 −1 −0.5 0 0.5 1 (a) original data set (8 samples) (b) interpolated data set (32 samples) 21–1
1.1.1 A Frequency Domain Method This method is useful for a finite-sized data record. Consider the DFTs { F m } and { F ˆ m } of a pair of sample sets { f n } and { f ˆ n } , both recorded from f ( t ) from 0 t < T , but with sampling intervals Δ T and Δ T/K respectively. Let N and KN be the corresponding sample sizes. It is assumed that Δ T has been chosen to satisfy the Nyquist criterion: Let F ( j Ω) = F { f ( t ) } be the Fourier transform of f ( t ), and let f ( t ) be sampled at intervals Δ T to produce f ( t ). Then F ( j Ω) = 1 F j Ω 2 πn (1) Δ T Δ T n =0 is periodic with period 2 π/ Δ T , and consists of scaled and shifted replicas of F ( j Ω). Let the total sampling interval be T to produce N = T/ Δ T samples. If the same waveform f ( t ) is sampled at intervals Δ T/K to produce f ˆ ( t ) the period of its Fourier transform F ˆ ( j Ω) is 2 πK/ Δ T and F ˆ ( j Ω) = K F j Ω 2 πKn (2) Δ T Δ T n =0 which differs only by a scale factor, and an increase in the period. Let the total sampling period be T as above, to generate KN samples. We consider the DFTs to be sampled representations of a single period of F ( j Ω) and F ˆ ( j Ω). The equivalent line spacing in the DFT depends only on the total duration of the sample set T , and is ΔΩ = 2 π/T in each case: 2 πm F m = F j , m = 0 , 1 , . . . N 1 T 2 πm ˆ F m = ˆ F j , m = 0 , 1 , . . . KN 1 . T From Eqs. (1) and (2) the two DFTs { F m } and F ˆ m are related: KF m m = 0 , 1 , . . . , N/ 2 1 F ˆ m = 0 m = N/ 2 , . . . , NK N/ 2 1 KF m ( K 1) N m = NK N/ 2 , . . . , KN 1 The effect of increasing N (or decreasing Δ T ) in the sample set, while maintaining T = N Δ T constant, is to increase the length of the DFT by raising the effective Nyquist frequency Ω N .

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