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 = ∞ + + 2 4 − 8 �+ − 2 − − 2 =−∞ 2 4 . Sketch the magnitude of − .− 2 − − − 2 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 7 Discrete-Time Fourier Transform Example: Consider the DTFT signal = ∞ + + 2 4 − 8 �+ − 2 − − 2 =−∞ 2 4 . Sketch the phase of . − − 2 − − − 2 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 8 Sampling 2 Ω Ω ℱ 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 2 1 1 2 3 Ω −3 −2 − 2 3 − Ωs − Ωs − Ωs Ωs Ωs Ωs Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 9 Sampling 2 Ω Ω 1 2 Ω ∗ Ω Ω 2 ℱ 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 2 1 1 2 3 Ω −3 −2 − 2 3 − Ωs − Ωs − Ωs Ωs Ωs Ωs Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 10 Sampling 2 Ω Ω 1 ∞ � Ω − Ω =−∞ ℱ 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 2 1 1 2 3 Ω −3 −2 − 2 3 − 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? 5 . Sketch the Fourier transform of the sampled signal when = 12 Ω Ωs 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? Ω 5 − − 5 . Sketch the Fourier transform of the sampled signal when = 20 Ω Ωs 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 Ω ≤ Ω ≤ Ω 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: Ω 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: Ω = = 1.5708 2 Ω 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: Ω 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 Ω Ω Ω − Ω − Ω = = 1.5708 2 Ω 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 Ω Ω Ω = = 1.5708 2 Ω 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 Ω Ω /2 sinc /2 Ω ∗ Ω 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 Ω Ω /2 sinc /2 = sinc /2 Ω ∗ Ω ∗ Ω 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 Ω Ω sinc /2 ∗ Ω 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 Ω Ω sinc /2 ∗ Ω 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 = 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 / − Low Pass Reconstruction Filter Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier