Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform Foundations of Digital Signal Processing
Outline • Review of Sampling • The Nyquist-Shannon Sampling Theorem • Continuous-time Reconstruction / Interpolation • Aliasing and anti-Aliasing • Deriving Transforms from the Fourier Transform • Discrete-time Fourier Transform, Fourier Series, Discrete-time Fourier Series • The Discrete Fourier Transform
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 1 News
Homework #5 . Due this week . Submit via canvas
Coding Problem #4 . Due this week . Submit via canvas
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 2 Exam 1 Grades
The class did exceedingly well . Mean: 89.3 . Median: 91.5
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 3 Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform Foundations of Digital Signal Processing
Outline • Review of Sampling • The Nyquist-Shannon Sampling Theorem • Continuous-time Reconstruction / Interpolation • Aliasing and anti-Aliasing • Deriving Transforms from the Fourier Transform • Discrete-time Fourier Transform, Fourier Series, Discrete-time Fourier Series • The Discrete Fourier Transform
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 4 Sampling
Discrete-Time Fourier Transform
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 5 Sampling
Sampling
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 6 Discrete-Time Fourier Transform
Example: Consider the DTFT signal
= ∞ + + 𝑗𝑗𝑗𝑗𝑗𝑗 2 4 − 8 𝜋𝜋 𝜋𝜋 𝑋𝑋 𝜔𝜔 𝑒𝑒 �+𝑢𝑢 𝜔𝜔 − 2𝜋𝜋𝜋𝜋 − 𝑢𝑢 𝜔𝜔 − 2𝜋𝜋𝜋𝜋 𝑘𝑘=−∞ 2 4 𝜋𝜋 𝜋𝜋 . Sketch the magnitude𝑢𝑢 of 𝜔𝜔 − .− 2𝜋𝜋𝜋𝜋 − 𝑢𝑢 𝜔𝜔 − − 2𝜋𝜋𝜋𝜋
𝑋𝑋 𝜔𝜔
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 7 Discrete-Time Fourier Transform
Example: Consider the DTFT signal
= ∞ + + 𝑗𝑗𝑗𝑗𝑗𝑗 2 4 − 8 𝜋𝜋 𝜋𝜋 𝑋𝑋 𝜔𝜔 𝑒𝑒 �+𝑢𝑢 𝜔𝜔 − 2𝜋𝜋𝜋𝜋 − 𝑢𝑢 𝜔𝜔 − 2𝜋𝜋𝜋𝜋 𝑘𝑘=−∞ 2 4 𝜋𝜋 𝜋𝜋 . Sketch the phase of 𝑢𝑢 𝜔𝜔. − − 2𝜋𝜋𝜋𝜋 − 𝑢𝑢 𝜔𝜔 − − 2𝜋𝜋𝜋𝜋
𝑋𝑋 𝜔𝜔
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 8 Sampling
2𝜋𝜋 𝛿𝛿𝑇𝑇𝒔𝒔 𝑡𝑡 𝛿𝛿Ω𝑠𝑠 Ω 𝑇𝑇𝑠𝑠
ℱ 𝛿𝛿𝑇𝑇𝒔𝒔 𝑡𝑡 𝛿𝛿𝑇𝑇𝒔𝒔 𝑡𝑡 1 1 1 1 1 1 1 2𝜋𝜋 2𝜋𝜋 2𝜋𝜋 2𝜋𝜋 2𝜋𝜋 2𝜋𝜋 2𝜋𝜋 𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠
𝑡𝑡 3 2 1 1 2 3 Ω
−3𝑇𝑇𝑠𝑠 −2𝑇𝑇𝑠𝑠 −𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 2𝑇𝑇𝑠𝑠 3𝑇𝑇𝑠𝑠 − Ωs − Ωs − Ωs Ωs Ωs Ωs Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 9 Sampling
2𝜋𝜋 𝛿𝛿𝑇𝑇𝒔𝒔 𝑡𝑡 𝛿𝛿Ω𝑠𝑠 Ω 1 𝑇𝑇𝑠𝑠
2𝜋𝜋 𝑥𝑥 𝑛𝑛 𝛿𝛿𝑇𝑇𝒔𝒔 𝑡𝑡 𝑋𝑋 Ω ∗ 𝛿𝛿Ω𝑠𝑠 Ω 2𝜋𝜋 𝑇𝑇𝑠𝑠
ℱ 𝛿𝛿𝑇𝑇𝒔𝒔 𝑡𝑡 𝛿𝛿𝑇𝑇𝒔𝒔 𝑡𝑡 1 1 1 1 1 1 1 2𝜋𝜋 2𝜋𝜋 2𝜋𝜋 2𝜋𝜋 2𝜋𝜋 2𝜋𝜋 2𝜋𝜋 𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠
𝑡𝑡 3 2 1 1 2 3 Ω
−3𝑇𝑇𝑠𝑠 −2𝑇𝑇𝑠𝑠 −𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 2𝑇𝑇𝑠𝑠 3𝑇𝑇𝑠𝑠 − Ωs − Ωs − Ωs Ωs Ωs Ωs Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 10 Sampling
2𝜋𝜋 𝛿𝛿𝑇𝑇𝒔𝒔 𝑡𝑡 𝛿𝛿Ω𝑠𝑠 Ω 1 𝑇𝑇𝑠𝑠 ∞
𝒔𝒔 𝑥𝑥 𝑛𝑛 𝛿𝛿𝑇𝑇 𝑡𝑡 𝑠𝑠 𝑠𝑠 � 𝑋𝑋 Ω − Ω 𝑘𝑘 𝑇𝑇 𝑘𝑘=−∞
ℱ 𝛿𝛿𝑇𝑇𝒔𝒔 𝑡𝑡 𝛿𝛿𝑇𝑇𝒔𝒔 𝑡𝑡 1 1 1 1 1 1 1 2𝜋𝜋 2𝜋𝜋 2𝜋𝜋 2𝜋𝜋 2𝜋𝜋 2𝜋𝜋 2𝜋𝜋 𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠
𝑡𝑡 3 2 1 1 2 3 Ω
−3𝑇𝑇𝑠𝑠 −2𝑇𝑇𝑠𝑠 −𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 2𝑇𝑇𝑠𝑠 3𝑇𝑇𝑠𝑠 − Ωs − Ωs − Ωs Ωs Ωs Ωs Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 11 Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform Foundations of Digital Signal Processing
Outline • Review of Sampling • The Nyquist-Shannon Sampling Theorem • Continuous-time Reconstruction / Interpolation • Aliasing and anti-Aliasing • Deriving Transforms from the Fourier Transform • Discrete-time Fourier Transform, Fourier Series, Discrete-time Fourier Series • The Discrete Fourier Transform
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 12 Sampling
Question: Can I preserve all information when I sample? . Yes!
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 13 Sampling
Question: How fast do I sample to preserve information?
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 14 Sampling
Question: How fast do I sample to preserve information?
= 1
Ωmax
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 15 Sampling
Question: How fast do I sample to preserve information?
= 1
Ωmax
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 16 Sampling
Question: How fast do I sample to preserve information?
= 1
Ωmax
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 17 Sampling
Question: How fast do I sample to preserve information?
= 1
Ωmax
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 18 Sampling
Question: How fast do I sample to preserve information? . We need to sample twice as fast as the maximum frequency > 2
Ωs Ωmax
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 19 Sampling
Question: How fast do I sample to preserve information? . We need to sample twice as fast as the maximum frequency > 2 Nyquist-Shannon f > 2f Sampling Theorem Ωs Ωmax s max
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 20 Sampling
Question: How fast do I sample to preserve information? . We need to sample twice as fast as the maximum frequency > 2 Nyquist-Shannon f > 2f Sampling Theorem Ωs Ωmax s max = 2 <- Nyquist Rate
ΩN Ωmax
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 21 Example
Example: Consider the signal = cos . . What is the Nyquist rate? 𝑥𝑥 𝑡𝑡 5𝜋𝜋𝜋𝜋 . Sketch the Fourier transform of the sampled signal when = 12 𝑋𝑋𝑠𝑠 Ω Ωs 𝜋𝜋
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 22 Example
Example: Consider the frequency signal = + . What is the Nyquist rate? 𝑋𝑋 Ω 𝑢𝑢 𝜔𝜔 5𝜋𝜋 − 𝑢𝑢 𝜔𝜔 − 5𝜋𝜋 . Sketch the Fourier transform of the sampled signal when = 20 𝑋𝑋𝑠𝑠 Ω Ωs 𝜋𝜋
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 23 Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform Foundations of Digital Signal Processing
Outline • Review of Sampling • The Nyquist-Shannon Sampling Theorem • Continuous-time Reconstruction / Interpolation • Aliasing and anti-Aliasing • Deriving Transforms from the Fourier Transform • Discrete-time Fourier Transform, Fourier Series, Discrete-time Fourier Series • The Discrete Fourier Transform
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 24 Sampling
Question: How do I return to continuous–time?
> 2 Nyquist-Shannon f > 2f Sampling Theorem Ωs Ωmax s max = 2 <- Nyquist Rate
ΩN Ωmax
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 25 Sampling
Question: How do I return to continuous–time? . Filter: Low pass filter to keep /2 /2
. Amplify: Amplify signal by Ω𝑠𝑠 ≤ Ω ≤ Ω𝑠𝑠 𝑇𝑇𝑠𝑠
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 26 Sampling
Question: How do I return to continuous–time? . Apply a low-pass reconstruction filter ◊ Cut-off frequency: /2 ◊ Gain: Ω𝑠𝑠 𝑇𝑇𝑠𝑠
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 27 Sampling
Question: How do I return to continuous–time? . Apply a low-pass reconstruction filter ◊ Cut-off frequency: /2 ◊ Gain: Ω𝑠𝑠 𝑇𝑇𝑠𝑠
= = 1.5708 2𝜋𝜋 𝑇𝑇𝑠𝑠 Ω𝑠𝑠
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 28 Sampling
Question: How do I return to continuous–time? . Apply a low-pass reconstruction filter ◊ Cut-off frequency: /2 ◊ Gain: Ω𝑠𝑠 𝑇𝑇𝑠𝑠
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 29 Sampling
Question: What is happening when I multiply in frequency?
+ /2 /2
𝑋𝑋 Ω 𝑇𝑇𝑠𝑠 𝑢𝑢 Ω Ω𝑠𝑠 − 𝑢𝑢 Ω − Ω𝑠𝑠
= = 1.5708 2𝜋𝜋 𝑇𝑇𝑠𝑠 Ω𝑠𝑠
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 30 Sampling
Question: What is happening when I multiply in frequency?
rect Ω 𝑋𝑋 Ω 𝑇𝑇𝑠𝑠𝑋𝑋 Ω Ω𝑠𝑠
= = 1.5708 2𝜋𝜋 𝑇𝑇𝑠𝑠 Ω𝑠𝑠
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 31 Sampling
Question: What is happening when I multiply in frequency?
rect 2 /2 Ω 𝑇𝑇𝑠𝑠𝑋𝑋 Ω Ω𝑠𝑠
/2 sinc /2 𝑇𝑇𝑠𝑠 Ω𝑠𝑠 𝑥𝑥 𝑛𝑛 ∗ Ω𝑠𝑠 𝑡𝑡 𝜋𝜋
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 32 Sampling
Question: What is happening when I multiply in frequency?
rect 2 /2 Ω 𝑇𝑇𝑠𝑠𝑋𝑋 Ω Ω𝑠𝑠
/2 sinc /2 = sinc /2 𝑇𝑇𝑠𝑠 Ω𝑠𝑠 𝑥𝑥 𝑛𝑛 ∗ Ω𝑠𝑠 𝑡𝑡 𝑥𝑥 𝑛𝑛 ∗ Ω𝑠𝑠 𝑡𝑡 𝜋𝜋
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 33 Sampling
Question: What is happening when I multiply in frequency?
rect 2 /2 Ω 𝑇𝑇𝑠𝑠𝑋𝑋 Ω Ω𝑠𝑠
sinc /2
𝑥𝑥 𝑛𝑛 ∗ Ω𝑠𝑠 𝑡𝑡
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 34 Sampling
Question: What is happening when I multiply in frequency?
rect 2 /2 Ω 𝑇𝑇𝑠𝑠𝑋𝑋 Ω Ω𝑠𝑠
sinc /2
𝑥𝑥 𝑛𝑛 ∗ Ω𝑠𝑠 𝑡𝑡
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 35 Reconstruction
Consider a cosine
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 36 Reconstruction
Consider a cosine – Sampled at =
𝑻𝑻𝒔𝒔 𝟏𝟏
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 37 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 38 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 39 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 40 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 41 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 42 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 43 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 44 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 45 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 46 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 47 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 48 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 49 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 50 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 51 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 52 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 53 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 54 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 55 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 56 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 57 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 58 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 59 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 60 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 61 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 62 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 63 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 64 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 65 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 66 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 67 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 68 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 69 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 70 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 71 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 72 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 73 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 74 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 75 Reconstruction
Consider a cosine – Low pass filter to keep / to /
−𝛀𝛀𝒔𝒔 𝟐𝟐 𝛀𝛀𝒔𝒔 𝟐𝟐
Low Pass Reconstruction Filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 76 Reconstruction
Consider a cosine – After the reconstruction filter
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 77 Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform Foundations of Digital Signal Processing
Outline • Review of Sampling • The Nyquist-Shannon Sampling Theorem • Continuous-time Reconstruction / Interpolation • Aliasing and anti-Aliasing • Deriving Transforms from the Fourier Transform • Discrete-time Fourier Transform, Fourier Series, Discrete-time Fourier Series • The Discrete Fourier Transform
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 78 Sampling
Aliasing occurs when we do not satisfy the sampling theorem
Question: What can happen when there is aliasing?
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 79 Example 1
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 80 Aliasing
Example: Consider = cos . Sample this at a rate of = 𝑥𝑥 𝑡𝑡 2𝜋𝜋𝑡𝑡 ◊ What is the NyquistΩ rate?𝑠𝑠 8𝜋𝜋 ◊ What is the cutoff frequency for the low-pass filter?
𝑋𝑋 Ω
𝜋𝜋
−10𝜋𝜋 −8𝜋𝜋 −6𝜋𝜋 −4𝜋𝜋 −2𝜋𝜋 2𝜋𝜋 4𝜋𝜋 6𝜋𝜋 8𝜋𝜋 10𝜋𝜋 Ω Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 81 Aliasing
Example: Consider = cos . Sample this at a rate of = 𝑥𝑥 𝑡𝑡 2𝜋𝜋𝑡𝑡 > ◊ What is the NyquistΩ rate?𝑠𝑠 8𝜋𝜋 ◊ What is the cutoff frequency for the low-pass filter? = 4 Ω𝑠𝑠 4𝜋𝜋 Ω𝑐𝑐 𝜋𝜋
𝑋𝑋 Ω
𝜋𝜋
−10𝜋𝜋 −8𝜋𝜋 −6𝜋𝜋 −4𝜋𝜋 −2𝜋𝜋 2𝜋𝜋 4𝜋𝜋 6𝜋𝜋 8𝜋𝜋 10𝜋𝜋 Ω Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 82 Aliasing
Example: Consider = cos . Sample this at a rate of = 𝑥𝑥 𝑡𝑡 2𝜋𝜋𝑡𝑡 > ◊ What is the NyquistΩ rate?𝑠𝑠 8𝜋𝜋 ◊ What is the cutoff frequency for the low-pass filter? = Ω𝑠𝑠 4𝜋𝜋 Ω𝑐𝑐 4𝜋𝜋
𝑋𝑋𝑠𝑠 Ω 8 = = = 4 … 𝜋𝜋 Ω𝑠𝑠 … 𝜋𝜋 𝑇𝑇𝑠𝑠 2𝜋𝜋 2𝜋𝜋
−10𝜋𝜋 −8𝜋𝜋 −6𝜋𝜋 −4𝜋𝜋 −2𝜋𝜋 2𝜋𝜋 4𝜋𝜋 6𝜋𝜋 8𝜋𝜋 10𝜋𝜋 Ω Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 83 Aliasing
Example: Consider = cos . Sample this at a rate of = 𝑥𝑥 𝑡𝑡 2𝜋𝜋𝑡𝑡 > ◊ What is the NyquistΩ rate?𝑠𝑠 8𝜋𝜋 ◊ What is the cutoff frequency for the low-pass filter? = Ω𝑠𝑠 4𝜋𝜋 Ω𝑐𝑐 4𝜋𝜋
𝑋𝑋𝑠𝑠 Ω 8 = = = 4 … 𝜋𝜋 Ω𝑠𝑠 … 𝜋𝜋 𝑇𝑇𝑠𝑠 2𝜋𝜋 2𝜋𝜋
−10𝜋𝜋 −8𝜋𝜋 −6𝜋𝜋 −4𝜋𝜋 −2𝜋𝜋 2𝜋𝜋 4𝜋𝜋 6𝜋𝜋 8𝜋𝜋 10𝜋𝜋 Ω Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 84 Aliasing
Example: Consider = cos . Sample this at a rate of = 𝑥𝑥 𝑡𝑡 2𝜋𝜋𝑡𝑡 > ◊ What is the NyquistΩ rate?𝑠𝑠 8𝜋𝜋 ◊ What is the cutoff frequency for the low-pass filter? = Ω𝑠𝑠 4𝜋𝜋 Ω𝑐𝑐 4𝜋𝜋
𝑋𝑋𝑠𝑠 Ω Gain: 1 Reconstruction Filter = = 4 2𝜋𝜋 4 𝑇𝑇𝑠𝑠 … Ω𝑠𝑠 … 𝜋𝜋
−10𝜋𝜋 −8𝜋𝜋 −6𝜋𝜋 −4𝜋𝜋 −2𝜋𝜋 2𝜋𝜋 4𝜋𝜋 6𝜋𝜋 8𝜋𝜋 10𝜋𝜋 Ω Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 85 Aliasing
Example: Consider = cos . Sample this at a rate of = 𝑥𝑥 𝑡𝑡 2𝜋𝜋𝑡𝑡 > ◊ What is the NyquistΩ rate?𝑠𝑠 8𝜋𝜋 ◊ What is the cutoff frequency for the low-pass filter? = Ω𝑠𝑠 4𝜋𝜋 ◊ Reconstructed Signal: = cos Ω𝑐𝑐 4𝜋𝜋 𝑥𝑥 𝑡𝑡 2𝜋𝜋𝜋𝜋 𝑋𝑋𝑠𝑠 Ω Gain: 1 Reconstruction Filter = = 4 2𝜋𝜋 𝑇𝑇𝑠𝑠 Ω𝑠𝑠 𝜋𝜋
−10𝜋𝜋 −8𝜋𝜋 −6𝜋𝜋 −4𝜋𝜋 −2𝜋𝜋 2𝜋𝜋 4𝜋𝜋 6𝜋𝜋 8𝜋𝜋 10𝜋𝜋 Ω Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 86 Example 2
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 87 Aliasing
Example: Consider = cos . Sample this at a rate of = 𝑥𝑥 𝑡𝑡 2𝜋𝜋𝑡𝑡 > ◊ What is the NyquistΩ rate?𝑠𝑠 3𝜋𝜋 ◊ What is the cutoff frequency for the low-pass filter? = 1. Ω𝑠𝑠 4𝜋𝜋 Ω𝑐𝑐 5𝜋𝜋
𝑋𝑋 Ω
𝜋𝜋
−10𝜋𝜋 −8𝜋𝜋 −6𝜋𝜋 −4𝜋𝜋 −2𝜋𝜋 2𝜋𝜋 4𝜋𝜋 6𝜋𝜋 8𝜋𝜋 10𝜋𝜋 Ω Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 88 Aliasing
Example: Consider = cos . Sample this at a rate of = 𝑥𝑥 𝑡𝑡 2𝜋𝜋𝑡𝑡 > ◊ What is the NyquistΩ rate?𝑠𝑠 3𝜋𝜋 ◊ What is the cutoff frequency for the low-pass filter? = 1. Ω𝑠𝑠 4𝜋𝜋 Ω𝑐𝑐 5𝜋𝜋
𝑋𝑋 Ω 3 = = 2 𝜋𝜋 3𝜋𝜋 𝜋𝜋 𝑇𝑇𝑠𝑠 2𝜋𝜋
−10𝜋𝜋 −8𝜋𝜋 −6𝜋𝜋 −4𝜋𝜋 −2𝜋𝜋 2𝜋𝜋 4𝜋𝜋 6𝜋𝜋 8𝜋𝜋 10𝜋𝜋 Ω Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 89 Aliasing
Example: Consider = cos . Sample this at a rate of = 𝑥𝑥 𝑡𝑡 2𝜋𝜋𝑡𝑡 > ◊ What is the NyquistΩ rate?𝑠𝑠 3𝜋𝜋 ◊ What is the cutoff frequency for the low-pass filter? = 1.5 Ω𝑠𝑠 4𝜋𝜋 Ω𝑐𝑐 𝜋𝜋
Gain: 𝑋𝑋 Ω 2 3 = = = = 3 2 𝑠𝑠 2𝜋𝜋 𝜋𝜋 3𝜋𝜋 𝑇𝑇 𝑠𝑠 𝜋𝜋 Ω 𝑇𝑇𝑠𝑠 2𝜋𝜋
−10𝜋𝜋 −8𝜋𝜋 −6𝜋𝜋 −4𝜋𝜋 −2𝜋𝜋 2𝜋𝜋 4𝜋𝜋 6𝜋𝜋 8𝜋𝜋 10𝜋𝜋 Ω Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 90 Aliasing
Example: Consider = cos . Sample this at a rate of = 𝑥𝑥 𝑡𝑡 2𝜋𝜋𝑡𝑡 > ◊ What is the NyquistΩ rate?𝑠𝑠 3𝜋𝜋 ◊ What is the cutoff frequency for the low-pass filter? = 1.5 Ω𝑠𝑠 4𝜋𝜋 ◊ Reconstructed Signal: = cos Ω𝑐𝑐 𝜋𝜋 𝑥𝑥 𝑡𝑡 𝜋𝜋𝜋𝜋 𝑋𝑋 Ω Frequency: 1
𝜋𝜋
𝜋𝜋
−10𝜋𝜋 −8𝜋𝜋 −6𝜋𝜋 −4𝜋𝜋 −2𝜋𝜋 2𝜋𝜋 4𝜋𝜋 6𝜋𝜋 8𝜋𝜋 10𝜋𝜋 Ω Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 91 Example 3
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 92 Aliasing
Consider = sin . Sample this at a rate of = 𝑥𝑥 𝑡𝑡 2𝜋𝜋𝑡𝑡 > ◊ What is the NyquistΩ rate?𝑠𝑠 3𝜋𝜋 ◊ What is the cutoff frequency for the low-pass filter? = 1.5 Ω𝑠𝑠 4𝜋𝜋 Ω𝑐𝑐 𝜋𝜋
𝑋𝑋 Ω
j
𝜋𝜋
−10𝜋𝜋 −8𝜋𝜋 −6𝜋𝜋 −4𝜋𝜋 −2𝜋𝜋 2𝜋𝜋-j 4𝜋𝜋 6𝜋𝜋 8𝜋𝜋 10𝜋𝜋 Ω Foundations of Digital Signal Processing Lecture 12:𝜋𝜋 Sampling, Aliasing, and the Discrete Fourier Transform 93 Aliasing
Consider = sin . Sample this at a rate of = 𝑥𝑥 𝑡𝑡 2𝜋𝜋𝑡𝑡 > ◊ What is the NyquistΩ rate?𝑠𝑠 3𝜋𝜋 ◊ What is the cutoff frequency for the low-pass filter? = 1. Ω𝑠𝑠 4𝜋𝜋 Ω𝑐𝑐 5𝜋𝜋
1 3 𝑋𝑋=Ω = 2 3𝜋𝜋 j j j j j j j 𝑇𝑇𝑠𝑠 2𝜋𝜋 𝜋𝜋 𝜋𝜋 𝜋𝜋 𝜋𝜋 𝜋𝜋 𝜋𝜋 𝜋𝜋
−10𝜋𝜋 −8𝜋𝜋 −-j6𝜋𝜋 −4𝜋𝜋-j −2𝜋𝜋 -j 2𝜋𝜋-j 4𝜋𝜋 -j6𝜋𝜋 8𝜋𝜋-j 10𝜋𝜋 -j Ω Foundations𝜋𝜋 of Digital Signal𝜋𝜋 Processing𝜋𝜋 Lecture 12:𝜋𝜋 Sampling, Aliasing,𝜋𝜋 and the Discrete𝜋𝜋 Fourier Transform𝜋𝜋 94 Aliasing
Consider = sin . Sample this at a rate of = 𝑥𝑥 𝑡𝑡 2𝜋𝜋𝑡𝑡 > ◊ What is the NyquistΩ rate?𝑠𝑠 3𝜋𝜋 ◊ What is the cutoff frequency for the low-pass filter? = 1. Ω𝑠𝑠 4𝜋𝜋 Ω𝑐𝑐 5𝜋𝜋
1 3 𝑋𝑋=Ω = 2 3𝜋𝜋 j j j j j j j 𝑇𝑇𝑠𝑠 2𝜋𝜋 𝜋𝜋 𝜋𝜋 𝜋𝜋 𝜋𝜋 𝜋𝜋 𝜋𝜋 𝜋𝜋
−10𝜋𝜋 −8𝜋𝜋 −-j6𝜋𝜋 −4𝜋𝜋-j −2𝜋𝜋 -j 2𝜋𝜋-j 4𝜋𝜋 -j6𝜋𝜋 8𝜋𝜋-j 10𝜋𝜋 -j Ω Foundations𝜋𝜋 of Digital Signal𝜋𝜋 Processing𝜋𝜋 Lecture 12:𝜋𝜋 Sampling, Aliasing,𝜋𝜋 and the Discrete𝜋𝜋 Fourier Transform𝜋𝜋 95 Aliasing
Consider = sin . Sample this at a rate of = 𝑥𝑥 𝑡𝑡 2𝜋𝜋𝑡𝑡 > ◊ What is the NyquistΩ rate?𝑠𝑠 3𝜋𝜋 ◊ What is the cutoff frequency for the low-pass filter? = 1. Ω𝑠𝑠 4𝜋𝜋 ◊ Reconstructed Signal: = sin = sin Ω𝑐𝑐 5𝜋𝜋 𝑥𝑥 𝑡𝑡 −𝜋𝜋𝜋𝜋 − 𝜋𝜋𝜋𝜋 𝑋𝑋 Ω Time-reversal
j
𝜋𝜋
−10𝜋𝜋 −8𝜋𝜋 −6𝜋𝜋 −4𝜋𝜋 −2𝜋𝜋 -j 2𝜋𝜋 4𝜋𝜋 6𝜋𝜋 8𝜋𝜋 10𝜋𝜋 Ω Foundations of Digital Signal Processing𝜋𝜋 Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 96 Example 4
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 97 Aliasing with a sinusoid
= cos ( = 1/5)
𝑥𝑥 𝑡𝑡 10𝜋𝜋𝜋𝜋 𝑇𝑇
2 2 Ω −30𝜋𝜋 − 0𝜋𝜋 −10𝜋𝜋 10𝜋𝜋 0𝜋𝜋 30𝜋𝜋 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 98 Aliasing with a sinusoid
= cos ( = 1/5) . Sample with a sampling rate of = ( = 1/20) 𝑥𝑥 𝑡𝑡 10𝜋𝜋𝜋𝜋 𝑇𝑇 Ω𝑠𝑠 40𝜋𝜋 𝑇𝑇𝑠𝑠
3 5 Ω −50𝜋𝜋 −30𝜋𝜋 −10𝜋𝜋 10𝜋𝜋 0𝜋𝜋 0𝜋𝜋 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 99 Aliasing with a sinusoid
= cos ( = 1/5) . Sample with a sampling rate of = ( = 1/20) 𝑥𝑥 𝑡𝑡 10𝜋𝜋𝜋𝜋 𝑇𝑇 Ω𝑠𝑠 40𝜋𝜋 𝑇𝑇𝑠𝑠
3 5 Ω −50𝜋𝜋 −30𝜋𝜋 −10𝜋𝜋 10𝜋𝜋 0𝜋𝜋 0𝜋𝜋 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 100 Aliasing with a sinusoid
= cos ( = 1/5) . Sample with a sampling rate of = ( = 1/15) 𝑥𝑥 𝑡𝑡 10𝜋𝜋𝜋𝜋 𝑇𝑇 Ω𝑠𝑠 30𝜋𝜋 𝑇𝑇𝑠𝑠
3 5 Ω −50𝜋𝜋 −30𝜋𝜋 −10𝜋𝜋 10𝜋𝜋 0𝜋𝜋 0𝜋𝜋 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 101 Aliasing with a sinusoid
= cos ( = 1/5) . Sample with a sampling rate of = ( = 1/15) 𝑥𝑥 𝑡𝑡 10𝜋𝜋𝜋𝜋 𝑇𝑇 Ω𝑠𝑠 30𝜋𝜋 𝑇𝑇𝑠𝑠
3 5 Ω −50𝜋𝜋 −30𝜋𝜋 −10𝜋𝜋 10𝜋𝜋 0𝜋𝜋 0𝜋𝜋 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 102 Aliasing with a sinusoid
= cos ( = 1/5) . Sample with a sampling rate of = ( = 1/10) 𝑥𝑥 𝑡𝑡 10𝜋𝜋𝜋𝜋 𝑇𝑇 Ω𝑠𝑠 20𝜋𝜋 𝑇𝑇𝑠𝑠
3 5 Ω −50𝜋𝜋 −30𝜋𝜋 −10𝜋𝜋 10𝜋𝜋 0𝜋𝜋 0𝜋𝜋 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 103 Aliasing with a sinusoid
= cos ( = 1/5) . Sample with a sampling rate of = ( = 1/10) 𝑥𝑥 𝑡𝑡 10𝜋𝜋𝜋𝜋 𝑇𝑇 Ω𝑠𝑠 20𝜋𝜋 𝑇𝑇𝑠𝑠
3 5 Ω −50𝜋𝜋 −30𝜋𝜋 −10𝜋𝜋 10𝜋𝜋 0𝜋𝜋 0𝜋𝜋 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 104 Aliasing with a sinusoid
= cos ( = 1/5) . Sample with a sampling rate of = ( = 2/15) 𝑥𝑥 𝑡𝑡 10𝜋𝜋𝜋𝜋 𝑇𝑇 Ω𝑠𝑠 15𝜋𝜋 𝑇𝑇𝑠𝑠
3 5 Ω −50𝜋𝜋 −30𝜋𝜋 −10𝜋𝜋 10𝜋𝜋 0𝜋𝜋 0𝜋𝜋 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 105 Aliasing with a sinusoid
= cos ( = 1/5) . Sample with a sampling rate of = ( = 2/15) 𝑥𝑥 𝑡𝑡 10𝜋𝜋𝜋𝜋 𝑇𝑇 Ω𝑠𝑠 15𝜋𝜋 𝑇𝑇𝑠𝑠
3 5 Ω −50𝜋𝜋 −30𝜋𝜋 −10𝜋𝜋 10𝜋𝜋 0𝜋𝜋 0𝜋𝜋 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 106 Aliasing with a sinusoid
= cos ( = 1/5) . Sample with a sampling rate of = ( = 1/5) 𝑥𝑥 𝑡𝑡 10𝜋𝜋𝜋𝜋 𝑇𝑇 Ω𝑠𝑠 10𝜋𝜋 𝑇𝑇𝑠𝑠
3 5 Ω −50𝜋𝜋 −30𝜋𝜋 −10𝜋𝜋 10𝜋𝜋 0𝜋𝜋 0𝜋𝜋 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 107 Aliasing with a sinusoid
= cos ( = 1/5) . Sample with a sampling rate of = ( = 1/5) 𝑥𝑥 𝑡𝑡 10𝜋𝜋𝜋𝜋 𝑇𝑇 Ω𝑠𝑠 10𝜋𝜋 𝑇𝑇𝑠𝑠
3 5 Ω −50𝜋𝜋 −30𝜋𝜋 −10𝜋𝜋 10𝜋𝜋 0𝜋𝜋 0𝜋𝜋 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 108 Aliasing with a sinusoid
= cos ( = 1/5) . Sample with a sampling rate of = ( = 2/5) 𝑥𝑥 𝑡𝑡 10𝜋𝜋𝜋𝜋 𝑇𝑇 Ω𝑠𝑠 5𝜋𝜋 𝑇𝑇𝑠𝑠
3 5 Ω −50𝜋𝜋 −30𝜋𝜋 −10𝜋𝜋 10𝜋𝜋 0𝜋𝜋 0𝜋𝜋 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 109 Aliasing with a sinusoid
= cos ( = 1/5) . Sample with a sampling rate of = ( = 2/5) 𝑥𝑥 𝑡𝑡 10𝜋𝜋𝜋𝜋 𝑇𝑇 Ω𝑠𝑠 5𝜋𝜋 𝑇𝑇𝑠𝑠
3 5 Ω −50𝜋𝜋 −30𝜋𝜋 −10𝜋𝜋 10𝜋𝜋 0𝜋𝜋 0𝜋𝜋 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 110 Aliasing with a sinusoid
= cos ( = 1/5) . Sample with a sampling rate of = ( = 2/5) 𝑥𝑥 𝑡𝑡 10𝜋𝜋𝜋𝜋 𝑇𝑇 Ω𝑠𝑠 5𝜋𝜋 𝑇𝑇𝑠𝑠
3 5 Ω −50𝜋𝜋 −30𝜋𝜋 −10𝜋𝜋 10𝜋𝜋 0𝜋𝜋 0𝜋𝜋 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 111 Anti-Aliasing Filter
Question: How do we reduce the effects of aliasing?
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 112 Anti-Aliasing Filter
Question: How do we reduce the effects of aliasing? . Apply a low-pass anti-aliasing filter ◊ Cut-off frequency: /2 ◊ Gain: 1 Ω𝑠𝑠
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 113 Anti-Aliasing Filter
Example: Consider the following signal. . What is the Nyquist rate?
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 114 Anti-Aliasing Filter
Example: Consider the following signal. . What is the Nyquist rate?
40𝜋𝜋
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 115 Anti-Aliasing Filter
Example: Consider the following signal. . What is the Nyquist rate?
. Sketch the Fourier transform40𝜋𝜋 after sampling at = 20 . . Use no anti-aliasing filter Ω𝑠𝑠 𝜋𝜋
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 116 Anti-Aliasing Filter
Example: Consider the following signal. . What is the Nyquist rate?
. Sketch the Fourier transform40𝜋𝜋 after sampling at = 20 . . Use no anti-aliasing filter Ω𝑠𝑠 𝜋𝜋
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 117 Anti-Aliasing Filter
Example: Consider the following signal. . What is the Nyquist rate?
. Sketch the Fourier transform40𝜋𝜋 after sampling at = 20 . . Use an anti-aliasing filter Ω𝑠𝑠 𝜋𝜋
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 118 Anti-Aliasing Filter
Example: Consider the following signal. . What is the Nyquist rate?
. Sketch the Fourier transform40𝜋𝜋 after sampling at = 20 . . Use an anti-aliasing filter Ω𝑠𝑠 𝜋𝜋
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 119 Anti-Aliasing Filter
Example: Consider the following signal. . What is the Nyquist rate?
. Sketch the Fourier transform40𝜋𝜋 after sampling at = 20 . . Use an anti-aliasing filter Ω𝑠𝑠 𝜋𝜋
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 120 Sampling and Aliasing in Real Life
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 121 Aliasing
Where have we seen such effects before in real life?
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 122 Sampling
Sampling a line . We sample a continuous image at each point
Illustrations from Cornell CS465 Spring 2006 Slides
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 123 Sampling
Sampling a line . We sample a continuous image at each point
Aliasing . Creates undesired high frequency information
Illustrations from Cornell CS465 Spring 2006 Slides
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 124 Anti-Aliasing
Sampling a line . We sample a continuous image at each point
Aliasing . Creates undesired high frequency information
Anti-aliasing . Create a smooth image Illustrations from Cornell CS465 Spring 2006 Slides
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 125 Anti-Aliasing
Box Filter . Convolve the continuous (high resolution) image with a box
Illustrations from Cornell CS465 Spring 2006 Slides
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 126 Anti-Aliasing
Gaussian Filter . Convolve the continuous (high resolution) image with a Gaussian
Illustrations from Cornell CS465 Spring 2006 Slides
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 127 Anti-Aliasing
Sampling of lines
Illustrations from Cornell CS465 Spring 2006 Slides Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 128 Anti-Aliasing
Sampling of lines . Anti-aliasing with a box filter
Illustrations from Cornell CS465 Spring 2006 Slides Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 129 Anti-Aliasing
Sampling of lines . Anti-aliasing with a Gaussian filter
Illustrations from Cornell CS465 Spring 2006 Slides Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 130 Aliasing Example
A Wheel
From: https://www.youtube.com/watch?v=bI8lrqBBAXQ
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 131 Aliasing Example
Making water float
From: https://www.youtube.com/watch?v=mODqQvlrgIQ
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 132 Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform Foundations of Digital Signal Processing
Outline • Review of Sampling • The Nyquist-Shannon Sampling Theorem • Continuous-time Reconstruction / Interpolation • Aliasing and anti-Aliasing • Deriving Transforms from the Fourier Transform • Discrete-time Fourier Transform, Fourier Series, Discrete-time Fourier Series • The Discrete Fourier Transform
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 133 Deriving Transforms
Consider the Fourier Transform…. . What happens if we sample ?
= ∞𝑥𝑥 𝑡𝑡 −𝑗𝑗Ωt 𝑋𝑋 Ω � 𝑥𝑥 𝑡𝑡 𝑒𝑒 𝑑𝑑𝑑𝑑 −∞
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 134 Deriving Transforms
Consider the Fourier Transform…. . What happens if we sample ?
= ∞ 𝑥𝑥 𝑡𝑡 −𝑗𝑗Ωt 𝑋𝑋 Ω � 𝑥𝑥 𝑡𝑡 𝛿𝛿𝑇𝑇𝒔𝒔 𝑡𝑡 𝑒𝑒 𝑑𝑑𝑑𝑑 −∞
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 135 Deriving Transforms
Consider the Fourier Transform…. . What happens if we sample ?
= ∞ 𝑥𝑥 𝑡𝑡 −𝑗𝑗Ωt 𝑋𝑋 Ω � 𝑥𝑥 𝑡𝑡 𝛿𝛿𝑇𝑇𝒔𝒔 𝑡𝑡 𝑒𝑒 𝑑𝑑𝑑𝑑 −∞ = ∞ ∞ −𝑗𝑗Ωt � 𝑥𝑥 𝑡𝑡 � 𝛿𝛿 𝑡𝑡 − 𝑛𝑛𝑇𝑇𝑠𝑠 𝑒𝑒 𝑑𝑑𝑑𝑑 −∞ 𝑛𝑛=−∞ = ∞ ∞ −𝑗𝑗Ωt � � 𝑥𝑥 𝑡𝑡 𝛿𝛿 𝑡𝑡 − 𝑛𝑛𝑇𝑇𝑠𝑠 𝑒𝑒 𝑑𝑑𝑑𝑑 𝑛𝑛=−∞ −∞
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 136 Deriving Transforms
Consider the Fourier Transform…. . What happens if we sample ?
= ∞ 𝑥𝑥 𝑡𝑡 −𝑗𝑗Ωt 𝑋𝑋 Ω � 𝑥𝑥 𝑡𝑡 𝛿𝛿𝑇𝑇𝒔𝒔 𝑡𝑡 𝑒𝑒 𝑑𝑑𝑑𝑑 −∞ = ∞ ∞ −𝑗𝑗Ωt � 𝑥𝑥 𝑡𝑡 � 𝛿𝛿 𝑡𝑡 − 𝑛𝑛𝑇𝑇𝑠𝑠 𝑒𝑒 𝑑𝑑𝑑𝑑 −∞ 𝑛𝑛=−∞ = ∞ −𝑗𝑗Ω𝑛𝑛𝑇𝑇𝑠𝑠 � 𝑥𝑥 𝑛𝑛𝑇𝑇𝑠𝑠 𝑒𝑒 𝑛𝑛=−∞
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 137 Deriving Transforms
Consider the Fourier Transform…. . What happens if we sample ?
= ∞ 𝑥𝑥 𝑡𝑡 −𝑗𝑗Ωt 𝑋𝑋 Ω � 𝑥𝑥 𝑡𝑡 𝛿𝛿𝑇𝑇𝒔𝒔 𝑡𝑡 𝑒𝑒 𝑑𝑑𝑑𝑑 −∞ = ∞ ∞ −𝑗𝑗Ωt � 𝑥𝑥 𝑡𝑡 � 𝛿𝛿 𝑡𝑡 − 𝑛𝑛𝑇𝑇𝑠𝑠 𝑒𝑒 𝑑𝑑𝑑𝑑 −∞ 𝑛𝑛=−∞ = ∞ Choose = 1, = −𝑗𝑗Ω𝑛𝑛𝑇𝑇𝑠𝑠 � 𝑥𝑥 𝑛𝑛𝑇𝑇𝑠𝑠 𝑒𝑒 𝑠𝑠 𝑛𝑛=−∞ 𝑇𝑇 Ω 𝜔𝜔
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 138 Deriving Transforms
Consider the Fourier Transform…. . What happens if we sample ?
= ∞𝑥𝑥 𝑡𝑡 −𝑗𝑗𝜔𝜔𝑛𝑛 𝑋𝑋 𝜔𝜔 � 𝑥𝑥 𝑛𝑛 𝑒𝑒 𝑛𝑛=−∞
. The Fourier Transform becomes the DTFT
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 139 Deriving Transforms
Consider the Inverse Fourier Transform…. . What happens if we sample X ?
t = ∞ Ω +𝑗𝑗Ωt 𝑥𝑥 � 𝑋𝑋 Ω 𝑒𝑒 𝑑𝑑Ω −∞
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 140 Deriving Transforms
Consider the Inverse Fourier Transform…. . What happens if we sample X ?
t = ∞ Ω +𝑗𝑗Ωt 𝑥𝑥 � 𝑋𝑋 Ω 𝑒𝑒 𝑑𝑑Ω −∞ = ∞ ∞ +𝑗𝑗Ωt � 𝑋𝑋 Ω � 𝛿𝛿 Ω − 𝑘𝑘Ω𝑠𝑠 𝑒𝑒 𝑑𝑑Ω −∞ 𝑘𝑘=−∞ = ∞ Choose = +𝑗𝑗𝑗𝑗Ω𝑠𝑠t 𝑠𝑠 � 𝑋𝑋 𝑘𝑘Ω 𝑒𝑒 𝑠𝑠 𝑘𝑘 𝑘𝑘=−∞ 𝑋𝑋 𝑘𝑘Ω 𝑐𝑐
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 141 Deriving Transforms
Consider the Inverse Fourier Transform…. . What happens if we sample X ?
t = ∞ Ω +𝑗𝑗𝑗𝑗Ω𝑠𝑠t 𝑥𝑥 � 𝑐𝑐𝑘𝑘𝑒𝑒 𝑘𝑘=−∞
. The Fourier Transform becomes the Fourier Series
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 142 Deriving Transforms
Consider the Inverse Discrete-Time Fourier Transform…. . What happens if we sample X ? 1 = 𝜔𝜔 +𝑗𝑗𝑗𝑗n 𝑥𝑥 𝑛𝑛 � 𝑋𝑋 𝜔𝜔 𝑒𝑒 𝑑𝑑𝑑𝑑 2𝜋𝜋 2𝜋𝜋
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 143 Deriving Transforms
Consider the Inverse Discrete-Time Fourier Transform…. . What happens if we sample X ? 1 = 𝜔𝜔 +𝑗𝑗𝑗𝑗n 𝑥𝑥 𝑛𝑛 � 𝑋𝑋 𝜔𝜔 𝑒𝑒 𝑑𝑑𝑑𝑑 1 2𝜋𝜋 2𝜋𝜋 = ∞ +𝑗𝑗𝑗𝑗n � 𝑋𝑋 𝜔𝜔 2𝜋𝜋 � 𝛿𝛿 𝜔𝜔 − 𝑘𝑘𝜔𝜔𝑠𝑠 𝑒𝑒 𝑑𝑑𝑑𝑑 2𝜋𝜋 2𝜋𝜋 𝑘𝑘=−∞
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 144 Deriving Transforms
Consider the Inverse Discrete-Time Fourier Transform…. . What happens if we sample X ? 1 = 𝜔𝜔 +𝑗𝑗𝑗𝑗n 𝑥𝑥 𝑛𝑛 � 𝑋𝑋 𝜔𝜔 𝑒𝑒 𝑑𝑑𝑑𝑑 1 2𝜋𝜋 2𝜋𝜋 = 2𝜋𝜋 ∞ +𝑗𝑗𝑗𝑗n � 𝑋𝑋 𝜔𝜔 2𝜋𝜋 � 𝛿𝛿 𝜔𝜔 − 𝑘𝑘𝜔𝜔𝑠𝑠 𝑒𝑒 𝑑𝑑𝑑𝑑 2𝜋𝜋 0 𝑘𝑘=−∞ 0 < 0 𝑘𝑘𝜔𝜔𝑠𝑠 ≥ 𝑘𝑘𝜔𝜔𝑠𝑠 < 2𝜋𝜋 𝑘𝑘 ≥ 2𝜋𝜋 𝑘𝑘 𝜔𝜔𝑠𝑠 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 145 Deriving Transforms
Consider the Inverse Discrete-Time Fourier Transform…. . What happens if we sample X ? 1 = 𝜔𝜔 +𝑗𝑗𝑗𝑗n 𝑥𝑥 𝑛𝑛 � 𝑋𝑋 𝜔𝜔 𝑒𝑒 𝑑𝑑𝑑𝑑 1 2𝜋𝜋 2𝜋𝜋 = 2𝜋𝜋 ∞ +𝑗𝑗𝑗𝑗n � 𝑋𝑋 𝜔𝜔 2𝜋𝜋 � 𝛿𝛿 𝜔𝜔 − 𝑘𝑘𝜔𝜔𝑠𝑠 𝑒𝑒 𝑑𝑑𝑑𝑑 2𝜋𝜋 0 𝑘𝑘=−∞ 2𝜋𝜋 Let 2 / = −1 = 𝜔𝜔𝑠𝑠 = 2 / +𝑗𝑗𝑗𝑗𝜔𝜔𝑠𝑠n 𝑠𝑠 � 𝑋𝑋 𝑘𝑘𝜔𝜔 𝑒𝑒 𝜋𝜋 𝜔𝜔 𝑁𝑁 𝑘𝑘=0 𝜔𝜔𝑠𝑠 𝜋𝜋 𝑁𝑁 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 146 Deriving Transforms
Consider the Inverse Discrete-Time Fourier Transform…. . What happens if we sample X ? 1 = 𝜔𝜔 +𝑗𝑗𝑗𝑗n 𝑥𝑥 𝑛𝑛 � 𝑋𝑋 𝜔𝜔 𝑒𝑒 𝑑𝑑𝑑𝑑 1 2𝜋𝜋 2𝜋𝜋 = 2𝜋𝜋 ∞ +𝑗𝑗𝑗𝑗n � 𝑋𝑋 𝜔𝜔 2𝜋𝜋 � 𝛿𝛿 𝜔𝜔 − 𝑘𝑘𝜔𝜔𝑠𝑠 𝑒𝑒 𝑑𝑑𝑑𝑑 2𝜋𝜋 0 𝑘𝑘=−∞ = 𝑁𝑁−1 Let 2 / = 2𝜋𝜋𝜋𝜋 +𝑗𝑗 𝑁𝑁 n = 2 / � 𝑋𝑋 𝑘𝑘𝜔𝜔 𝑒𝑒 𝑠𝑠 𝑘𝑘=0 𝜋𝜋 𝜔𝜔 𝐾𝐾 𝜔𝜔𝑠𝑠 𝜋𝜋 𝐾𝐾 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 147 Deriving Transforms
Consider the Inverse Discrete-Time Fourier Transform…. . What happens if we sample X ?
= 𝑁𝑁−1 𝜔𝜔 2𝜋𝜋 +𝑗𝑗 𝑁𝑁 kn 𝑥𝑥 𝑛𝑛 � 𝑋𝑋 𝑘𝑘 𝑒𝑒 𝑘𝑘=0
. The DTFT becomes the Discrete-Time Fourier Series
Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 148