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Aliasing in Fourier Analysis Optional Assessment; Practically - - PowerPoint PPT Presentation

May 22, 2023 859 likes •1.2k views

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

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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

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Outline

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What is Aliasing?

  • 1

1 2 4 6

Signal contains 2 functions sin(πt/2) & sin(2πt) Distinguish? Interfere? Finite Sampling Ambiguity Sample at t = 0, 2, 4, 6, 8,: y ≡ 0 Sample at t = 0, 12

10, 4 3, . . . (•):

sin(πt/2) = sin(2πt) Finite sample ⇒ high-ω “between the cracks”

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SLIDE 4

What is Aliasing?

  • 1

1 2 4 6

Signal contains 2 functions sin(πt/2) & sin(2πt) Distinguish? Interfere? Finite Sampling Ambiguity Sample at t = 0, 2, 4, 6, 8,: y ≡ 0 Sample at t = 0, 12

10, 4 3, . . . (•):

sin(πt/2) = sin(2πt) Finite sample ⇒ high-ω “between the cracks”

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SLIDE 5

What is Aliasing?

  • 1

1 2 4 6

Signal contains 2 functions sin(πt/2) & sin(2πt) Distinguish? Interfere? Finite Sampling Ambiguity Sample at t = 0, 2, 4, 6, 8,: y ≡ 0 Sample at t = 0, 12

10, 4 3, . . . (•):

sin(πt/2) = sin(2πt) Finite sample ⇒ high-ω “between the cracks”

5 / 1

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SLIDE 6

What is Aliasing?

  • 1

1 2 4 6

Signal contains 2 functions sin(πt/2) & sin(2πt) Distinguish? Interfere? Finite Sampling Ambiguity Sample at t = 0, 2, 4, 6, 8,: y ≡ 0 Sample at t = 0, 12

10, 4 3, . . . (•):

sin(πt/2) = sin(2πt) Finite sample ⇒ high-ω “between the cracks”

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SLIDE 7

What is Aliasing?

  • 1

1 2 4 6

Signal contains 2 functions sin(πt/2) & sin(2πt) Distinguish? Interfere? Finite Sampling Ambiguity Sample at t = 0, 2, 4, 6, 8,: y ≡ 0 Sample at t = 0, 12

10, 4 3, . . . (•):

sin(πt/2) = sin(2πt) Finite sample ⇒ high-ω “between the cracks”

7 / 1

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SLIDE 8

What is Aliasing?

  • 1

1 2 4 6

Signal contains 2 functions sin(πt/2) & sin(2πt) Distinguish? Interfere? Finite Sampling Ambiguity Sample at t = 0, 2, 4, 6, 8,: y ≡ 0 Sample at t = 0, 12

10, 4 3, . . . (•):

sin(πt/2) = sin(2πt) Finite sample ⇒ high-ω “between the cracks”

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SLIDE 9

What is Aliasing?

  • 1

1 2 4 6

Signal contains 2 functions sin(πt/2) & sin(2πt) Distinguish? Interfere? Finite Sampling Ambiguity Sample at t = 0, 2, 4, 6, 8,: y ≡ 0 Sample at t = 0, 12

10, 4 3, . . . (•):

sin(πt/2) = sin(2πt) Finite sample ⇒ high-ω “between the cracks”

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SLIDE 10

Consequences of Aliasing

  • 1

1 2 4 6

(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 (1)

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SLIDE 11

Consequences of Aliasing

  • 1

1 2 4 6

(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 (1)

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SLIDE 12

Consequences of Aliasing

  • 1

1 2 4 6

(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 (1)

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SLIDE 13

Consequences of Aliasing

  • 1

1 2 4 6

(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 (1)

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SLIDE 14

Consequences of Aliasing

  • 1

1 2 4 6

(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 (1)

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SLIDE 15

Consequences of Aliasing

  • 1

1 2 4 6

(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 (1)

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SLIDE 16

Consequences of Aliasing

  • 1

1 2 4 6

(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 (1)

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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 right @ this sampling rate Need more sampling, higher s → higher ω in spectrum middle (ends = error prone) Recall: padding with 0s (larger T) → smoother Y(ω)

20 1 Y( )

ω

40

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SLIDE 18

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 right @ this sampling rate Need more sampling, higher s → higher ω in spectrum middle (ends = error prone) Recall: padding with 0s (larger T) → smoother Y(ω)

20 1 Y( )

ω

40

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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 right @ this sampling rate Need more sampling, higher s → higher ω in spectrum middle (ends = error prone) Recall: padding with 0s (larger T) → smoother Y(ω)

20 1 Y( )

ω

40

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SLIDE 20

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 right @ this sampling rate Need more sampling, higher s → higher ω in spectrum middle (ends = error prone) Recall: padding with 0s (larger T) → smoother Y(ω)

20 1 Y( )

ω

40

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SLIDE 21

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 right @ this sampling rate Need more sampling, higher s → higher ω in spectrum middle (ends = error prone) Recall: padding with 0s (larger T) → smoother Y(ω)

20 1 Y( )

ω

40

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SLIDE 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 right @ this sampling rate Need more sampling, higher s → higher ω in spectrum middle (ends = error prone) Recall: padding with 0s (larger T) → smoother Y(ω)

20 1 Y( )

ω

40

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SLIDE 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 right @ this sampling rate Need more sampling, higher s → higher ω in spectrum middle (ends = error prone) Recall: padding with 0s (larger T) → smoother Y(ω)

20 1 Y( )

ω

40

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SLIDE 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 right @ this sampling rate Need more sampling, higher s → higher ω in spectrum middle (ends = error prone) Recall: padding with 0s (larger T) → smoother Y(ω)

20 1 Y( )

ω

40

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SLIDE 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 right @ this sampling rate Need more sampling, higher s → higher ω in spectrum middle (ends = error prone) Recall: padding with 0s (larger T) → smoother Y(ω)

20 1 Y( )

ω

40

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Assessment of Aliasing

1

Perform DFT on y(t) = sin π

2t

  • + 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.

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SLIDE 27

Assessment of Aliasing

1

Perform DFT on y(t) = sin π

2t

  • + 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.

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SLIDE 28

Assessment of Aliasing

1

Perform DFT on y(t) = sin π

2t

  • + 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.

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SLIDE 29

Assessment of Aliasing

1

Perform DFT on y(t) = sin π

2t

  • + 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.

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SLIDE 30

Assessment of Aliasing

1

Perform DFT on y(t) = sin π

2t

  • + 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.

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SLIDE 31

Assessment of Aliasing

1

Perform DFT on y(t) = sin π

2t

  • + 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.

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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.

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