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

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 1 / 1

Outline 2 / 1

What is Aliasing? 1 Signal contains 2 functions sin ( π t / 2 ) & sin ( 2 π t ) 2 4 6 0 Distinguish? Interfere? -1 Finite Sampling Ambiguity Sample at t = 0, 2, 4, 6, 8,: y ≡ 0 Sample at t = 0 , 12 10 , 4 3 , . . . ( • ): sin ( π t / 2 ) = sin ( 2 π t ) Finite sample ⇒ high- ω “between the cracks” 3 / 1

What is Aliasing? 1 Signal contains 2 functions sin ( π t / 2 ) & sin ( 2 π t ) 2 4 6 0 Distinguish? Interfere? -1 Finite Sampling Ambiguity Sample at t = 0, 2, 4, 6, 8,: y ≡ 0 Sample at t = 0 , 12 10 , 4 3 , . . . ( • ): sin ( π t / 2 ) = sin ( 2 π t ) Finite sample ⇒ high- ω “between the cracks” 4 / 1

What is Aliasing? 1 Signal contains 2 functions sin ( π t / 2 ) & sin ( 2 π t ) 2 4 6 0 Distinguish? Interfere? -1 Finite Sampling Ambiguity Sample at t = 0, 2, 4, 6, 8,: y ≡ 0 Sample at t = 0 , 12 10 , 4 3 , . . . ( • ): sin ( π t / 2 ) = sin ( 2 π t ) Finite sample ⇒ high- ω “between the cracks” 5 / 1

What is Aliasing? 1 Signal contains 2 functions sin ( π t / 2 ) & sin ( 2 π t ) 2 4 6 0 Distinguish? Interfere? -1 Finite Sampling Ambiguity Sample at t = 0, 2, 4, 6, 8,: y ≡ 0 Sample at t = 0 , 12 10 , 4 3 , . . . ( • ): sin ( π t / 2 ) = sin ( 2 π t ) Finite sample ⇒ high- ω “between the cracks” 6 / 1

What is Aliasing? 1 Signal contains 2 functions sin ( π t / 2 ) & sin ( 2 π t ) 2 4 6 0 Distinguish? Interfere? -1 Finite Sampling Ambiguity Sample at t = 0, 2, 4, 6, 8,: y ≡ 0 Sample at t = 0 , 12 10 , 4 3 , . . . ( • ): sin ( π t / 2 ) = sin ( 2 π t ) Finite sample ⇒ high- ω “between the cracks” 7 / 1

What is Aliasing? 1 Signal contains 2 functions sin ( π t / 2 ) & sin ( 2 π t ) 2 4 6 0 Distinguish? Interfere? -1 Finite Sampling Ambiguity Sample at t = 0, 2, 4, 6, 8,: y ≡ 0 Sample at t = 0 , 12 10 , 4 3 , . . . ( • ): sin ( π t / 2 ) = sin ( 2 π t ) Finite sample ⇒ high- ω “between the cracks” 8 / 1

What is Aliasing? 1 Signal contains 2 functions sin ( π t / 2 ) & sin ( 2 π t ) 2 4 6 0 Distinguish? Interfere? -1 Finite Sampling Ambiguity Sample at t = 0, 2, 4, 6, 8,: y ≡ 0 Sample at t = 0 , 12 10 , 4 3 , . . . ( • ): sin ( π t / 2 ) = sin ( 2 π t ) Finite sample ⇒ high- ω “between the cracks” 9 / 1

Consequences of Aliasing 1 2 4 6 0 -1 (Wikipedia) ω , ω − 2 s High- ω contaminates low Same DFT if Moiré distortion in synthesis “High- ω aliased by low” s = N T ≤ ω (1) Math: for sampling rate s = N / T 2 10 / 1

Consequences of Aliasing 1 2 4 6 0 -1 (Wikipedia) ω , ω − 2 s High- ω contaminates low Same DFT if Moiré distortion in synthesis “High- ω aliased by low” s = N T ≤ ω (1) Math: for sampling rate s = N / T 2 11 / 1

Consequences of Aliasing 1 2 4 6 0 -1 (Wikipedia) ω , ω − 2 s High- ω contaminates low Same DFT if Moiré distortion in synthesis “High- ω aliased by low” s = N T ≤ ω (1) Math: for sampling rate s = N / T 2 12 / 1

Consequences of Aliasing 1 2 4 6 0 -1 (Wikipedia) ω , ω − 2 s High- ω contaminates low Same DFT if Moiré distortion in synthesis “High- ω aliased by low” s = N T ≤ ω (1) Math: for sampling rate s = N / T 2 13 / 1

Consequences of Aliasing 1 2 4 6 0 -1 (Wikipedia) ω , ω − 2 s High- ω contaminates low Same DFT if Moiré distortion in synthesis “High- ω aliased by low” s = N T ≤ ω (1) Math: for sampling rate s = N / T 2 14 / 1

Consequences of Aliasing 1 2 4 6 0 -1 (Wikipedia) ω , ω − 2 s High- ω contaminates low Same DFT if Moiré distortion in synthesis “High- ω aliased by low” s = N T ≤ ω (1) Math: for sampling rate s = N / T 2 15 / 1

Consequences of Aliasing 1 2 4 6 0 -1 (Wikipedia) ω , ω − 2 s High- ω contaminates low Same DFT if Moiré distortion in synthesis “High- ω aliased by low” s = N T ≤ ω (1) Math: for sampling rate s = N / T 2 16 / 1

ω 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 ) 1 Y( ) 0 0 20 40 17 / 1

ω 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 ) 1 Y( ) 0 0 20 40 18 / 1

ω 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 ) 1 Y( ) 0 0 20 40 19 / 1

ω 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 ) 1 Y( ) 0 0 20 40 20 / 1

ω 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 ) 1 Y( ) 0 0 20 40 21 / 1

ω 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 ) 1 Y( ) 0 0 20 40 22 / 1

ω 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 ) 1 Y( ) 0 0 20 40 23 / 1

ω 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 ) 1 Y( ) 0 0 20 40 24 / 1

ω 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 ) 1 Y( ) 0 0 20 40 25 / 1

Assessment of Aliasing � π � Perform DFT on y ( t ) = sin 2 t + sin ( 2 π t ) . 1 True TF peaks at ω = π/ 2 & ω = 2 π . 2 Look for aliasing at low sample rate. 3 Verify that aliasing vanishes at high sampling rate. 4 Verify the Nyquist criterion computationally. 5 26 / 1

Assessment of Aliasing � π � Perform DFT on y ( t ) = sin 2 t + sin ( 2 π t ) . 1 True TF peaks at ω = π/ 2 & ω = 2 π . 2 Look for aliasing at low sample rate. 3 Verify that aliasing vanishes at high sampling rate. 4 Verify the Nyquist criterion computationally. 5 27 / 1

Assessment of Aliasing � π � Perform DFT on y ( t ) = sin 2 t + sin ( 2 π t ) . 1 True TF peaks at ω = π/ 2 & ω = 2 π . 2 Look for aliasing at low sample rate. 3 Verify that aliasing vanishes at high sampling rate. 4 Verify the Nyquist criterion computationally. 5 28 / 1

Assessment of Aliasing � π � Perform DFT on y ( t ) = sin 2 t + sin ( 2 π t ) . 1 True TF peaks at ω = π/ 2 & ω = 2 π . 2 Look for aliasing at low sample rate. 3 Verify that aliasing vanishes at high sampling rate. 4 Verify the Nyquist criterion computationally. 5 29 / 1

Assessment of Aliasing � π � Perform DFT on y ( t ) = sin 2 t + sin ( 2 π t ) . 1 True TF peaks at ω = π/ 2 & ω = 2 π . 2 Look for aliasing at low sample rate. 3 Verify that aliasing vanishes at high sampling rate. 4 Verify the Nyquist criterion computationally. 5 30 / 1