Aliasing in Fourier Analysis

Transcription

1 / Aliasing in Fourier Analysis Optional Assessment; Practically Important Rubin H Landau Sally Haerer, Producer-Director Based on A Survey of Computational Physics by Landau, Páez, & Bordeianu with Support from the National Science Foundation Course: Computational Physics II

2 Outline 2 /

3 What is Aliasing? Signal contains 2 functions sin(πt/2) & sin(2πt) Distinguish? Interfere? - Finite Sampling Ambiguity Sample at t =, 2, 4, 6, 8,: y Sample at t =, 2, 4 3,... ( ): sin(πt/2) = sin(2πt) Finite sample high-ω between the cracks 3 /

4 What is Aliasing? Signal contains 2 functions sin(πt/2) & sin(2πt) Distinguish? Interfere? - Finite Sampling Ambiguity Sample at t =, 2, 4, 6, 8,: y Sample at t =, 2, 4 3,... ( ): sin(πt/2) = sin(2πt) Finite sample high-ω between the cracks 4 /

5 What is Aliasing? Signal contains 2 functions sin(πt/2) & sin(2πt) Distinguish? Interfere? - Finite Sampling Ambiguity Sample at t =, 2, 4, 6, 8,: y Sample at t =, 2, 4 3,... ( ): sin(πt/2) = sin(2πt) Finite sample high-ω between the cracks 5 /

6 What is Aliasing? Signal contains 2 functions sin(πt/2) & sin(2πt) Distinguish? Interfere? - Finite Sampling Ambiguity Sample at t =, 2, 4, 6, 8,: y Sample at t =, 2, 4 3,... ( ): sin(πt/2) = sin(2πt) Finite sample high-ω between the cracks 6 /

7 What is Aliasing? Signal contains 2 functions sin(πt/2) & sin(2πt) Distinguish? Interfere? - Finite Sampling Ambiguity Sample at t =, 2, 4, 6, 8,: y Sample at t =, 2, 4 3,... ( ): sin(πt/2) = sin(2πt) Finite sample high-ω between the cracks 7 /

8 What is Aliasing? Signal contains 2 functions sin(πt/2) & sin(2πt) Distinguish? Interfere? - Finite Sampling Ambiguity Sample at t =, 2, 4, 6, 8,: y Sample at t =, 2, 4 3,... ( ): sin(πt/2) = sin(2πt) Finite sample high-ω between the cracks 8 /

9 What is Aliasing? Signal contains 2 functions sin(πt/2) & sin(2πt) Distinguish? Interfere? - Finite Sampling Ambiguity Sample at t =, 2, 4, 6, 8,: y Sample at t =, 2, 4 3,... ( ): sin(πt/2) = sin(2πt) Finite sample high-ω between the cracks 9 /

10 / Consequences of Aliasing (Wikipedia) High-ω contaminates low Moiré distortion in synthesis High-ω aliased by low Math: for sampling rate s = N/T ω, ω 2s Same DFT if s = N T ω 2 ()

11 / Consequences of Aliasing (Wikipedia) High-ω contaminates low Moiré distortion in synthesis High-ω aliased by low Math: for sampling rate s = N/T ω, ω 2s Same DFT if s = N T ω 2 ()

12 2 / Consequences of Aliasing (Wikipedia) High-ω contaminates low Moiré distortion in synthesis High-ω aliased by low Math: for sampling rate s = N/T ω, ω 2s Same DFT if s = N T ω 2 ()

13 3 / Consequences of Aliasing (Wikipedia) High-ω contaminates low Moiré distortion in synthesis High-ω aliased by low Math: for sampling rate s = N/T ω, ω 2s Same DFT if s = N T ω 2 ()

14 4 / Consequences of Aliasing (Wikipedia) High-ω contaminates low Moiré distortion in synthesis High-ω aliased by low Math: for sampling rate s = N/T ω, ω 2s Same DFT if s = N T ω 2 ()

15 5 / Consequences of Aliasing (Wikipedia) High-ω contaminates low Moiré distortion in synthesis High-ω aliased by low Math: for sampling rate s = N/T ω, ω 2s Same DFT if s = N T ω 2 ()

16 6 / Consequences of Aliasing (Wikipedia) High-ω contaminates low Moiré distortion in synthesis High-ω aliased by low Math: for sampling rate s = N/T ω, ω 2s Same DFT if s = N T ω 2 ()

17 7 / Eliminating Aliasing Recall: s = N/T = sampling rate Nyquist criterion: no frequency > s/2 in input signal Filter out high ω (e.g. sinc filter) good low ω Good High ω Can t do high-ω this sampling rate Need more sampling, higher s higher ω in spectrum middle (ends = error prone) Recall: padding with s (larger T ) smoother Y (ω) Y( ) 2 ω 4

18 8 / Eliminating Aliasing Recall: s = N/T = sampling rate Nyquist criterion: no frequency > s/2 in input signal Filter out high ω (e.g. sinc filter) good low ω Good High ω Can t do high-ω this sampling rate Need more sampling, higher s higher ω in spectrum middle (ends = error prone) Recall: padding with s (larger T ) smoother Y (ω) Y( ) 2 ω 4

19 9 / Eliminating Aliasing Recall: s = N/T = sampling rate Nyquist criterion: no frequency > s/2 in input signal Filter out high ω (e.g. sinc filter) good low ω Good High ω Can t do high-ω this sampling rate Need more sampling, higher s higher ω in spectrum middle (ends = error prone) Recall: padding with s (larger T ) smoother Y (ω) Y( ) 2 ω 4

20 2 / Eliminating Aliasing Recall: s = N/T = sampling rate Nyquist criterion: no frequency > s/2 in input signal Filter out high ω (e.g. sinc filter) good low ω Good High ω Can t do high-ω this sampling rate Need more sampling, higher s higher ω in spectrum middle (ends = error prone) Recall: padding with s (larger T ) smoother Y (ω) Y( ) 2 ω 4

21 2 / Eliminating Aliasing Recall: s = N/T = sampling rate Nyquist criterion: no frequency > s/2 in input signal Filter out high ω (e.g. sinc filter) good low ω Good High ω Can t do high-ω this sampling rate Need more sampling, higher s higher ω in spectrum middle (ends = error prone) Recall: padding with s (larger T ) smoother Y (ω) Y( ) 2 ω 4

22 22 / Eliminating Aliasing Recall: s = N/T = sampling rate Nyquist criterion: no frequency > s/2 in input signal Filter out high ω (e.g. sinc filter) good low ω Good High ω Can t do high-ω this sampling rate Need more sampling, higher s higher ω in spectrum middle (ends = error prone) Recall: padding with s (larger T ) smoother Y (ω) Y( ) 2 ω 4

23 23 / Eliminating Aliasing Recall: s = N/T = sampling rate Nyquist criterion: no frequency > s/2 in input signal Filter out high ω (e.g. sinc filter) good low ω Good High ω Can t do high-ω this sampling rate Need more sampling, higher s higher ω in spectrum middle (ends = error prone) Recall: padding with s (larger T ) smoother Y (ω) Y( ) 2 ω 4

24 24 / Eliminating Aliasing Recall: s = N/T = sampling rate Nyquist criterion: no frequency > s/2 in input signal Filter out high ω (e.g. sinc filter) good low ω Good High ω Can t do high-ω this sampling rate Need more sampling, higher s higher ω in spectrum middle (ends = error prone) Recall: padding with s (larger T ) smoother Y (ω) Y( ) 2 ω 4

25 25 / Eliminating Aliasing Recall: s = N/T = sampling rate Nyquist criterion: no frequency > s/2 in input signal Filter out high ω (e.g. sinc filter) good low ω Good High ω Can t do high-ω this sampling rate Need more sampling, higher s higher ω in spectrum middle (ends = error prone) Recall: padding with s (larger T ) smoother Y (ω) Y( ) 2 ω 4

26 26 / Assessment of Aliasing Perform DFT on y(t) = sin ( π 2 t) + sin(2πt). 2 True TF peaks at ω = π/2 & ω = 2π. 3 Look for aliasing at low sample rate. 4 Verify that aliasing vanishes at high sampling rate. 5 Verify the Nyquist criterion computationally.

27 27 / Assessment of Aliasing Perform DFT on y(t) = sin ( π 2 t) + sin(2πt). 2 True TF peaks at ω = π/2 & ω = 2π. 3 Look for aliasing at low sample rate. 4 Verify that aliasing vanishes at high sampling rate. 5 Verify the Nyquist criterion computationally.

28 28 / Assessment of Aliasing Perform DFT on y(t) = sin ( π 2 t) + sin(2πt). 2 True TF peaks at ω = π/2 & ω = 2π. 3 Look for aliasing at low sample rate. 4 Verify that aliasing vanishes at high sampling rate. 5 Verify the Nyquist criterion computationally.

29 29 / Assessment of Aliasing Perform DFT on y(t) = sin ( π 2 t) + sin(2πt). 2 True TF peaks at ω = π/2 & ω = 2π. 3 Look for aliasing at low sample rate. 4 Verify that aliasing vanishes at high sampling rate. 5 Verify the Nyquist criterion computationally.

30 3 / Assessment of Aliasing Perform DFT on y(t) = sin ( π 2 t) + sin(2πt). 2 True TF peaks at ω = π/2 & ω = 2π. 3 Look for aliasing at low sample rate. 4 Verify that aliasing vanishes at high sampling rate. 5 Verify the Nyquist criterion computationally.

31 3 / Assessment of Aliasing Perform DFT on y(t) = sin ( π 2 t) + sin(2πt). 2 True TF peaks at ω = π/2 & ω = 2π. 3 Look for aliasing at low sample rate. 4 Verify that aliasing vanishes at high sampling rate. 5 Verify the Nyquist criterion computationally.

32 32 / Summary If sampling rate is low, some high frequency components can contaminate the deduced low-frequency components. The reconstructed signal will show distortions. Nyquist criterion to eliminate aliasing: no frequency > (N/T )/2 in input signal.