6.003: Signal Processing Sampling and Reconstruction September 13, 2018 Importance of Discrete Representations Our goal is to develop signal processing tools that help us under- stand and manipulate the world. analyze Model Result (math, computation) make model interpret results World New Understanding analysis System Behavior design The increasing power and decreasing cost of computation makes the use of computation increasingly attractive. However, many impor- tant signals are naturally described with continuous domain. → understand the relation between CT and sampled signals Sampling Converting continuous-time (CT) signals to discrete time (DT). x(t) x[n] = x(nT ) t n 0T 2T 4T 6T 8T 10T 0 2 4 6 8 10 T = sampling interval How does sampling affect the information contained in a signal? Sampling We would like to sample in a way that preserves information. However, information is clearly lost in the sampling process. x(t) t From this example, it's clear that uniformly spaced samples tell us nothing about the values of x(t) between the samples. Sampling Even worse, information can be distorted, becoming misinformation. 7π Samples of an original signal x(t) = cos 3 n 7π π cos 3 n? cos 3 n? t π are just as easily interpreted as coming from cos 3 n. Effects of Undersampling Are Easily Heard Sampling Music 1 f = s T • fs = 44.1 kHz • fs = 22 kHz • fs = 11 kHz • fs = 5.5 kHz • fs = 2.8 kHz J.S. Bach, Sonata No. 1 in G minor Mvmt. IV. Presto Nathan Milstein, violin Effects of Undersampling are Easily Seen Sampling Images original: 2048 × 1536 Effects of Undersampling are Easily Seen Sampling Images downsampled: 1024 × 768 Effects of Undersampling are Easily Seen Sampling Images downsampled: 512x384 Effects of Undersampling are Easily Seen Sampling Images downsampled: 256 × 192 Effects of Undersampling are Easily Seen Sampling Images downsampled: 128 × 96 Effects of Undersampling are Easily Seen Sampling Images downsampled: 64 × 48 Effects of Undersampling are Easily Seen Sampling Images downsampled: 32 × 24 Effects of Undersampling If we hope to use sampled signals to understand behaviors of con- tinuous signals, we must understand effects of sampling. Effects of Undersampling If we hope to use sampled signals to understand behaviors of con- tinuous signals, we must understand effects of sampling. Some of the effects of undersampling arise from fundamental prop- erties of discrete time. CT: x(t) = cos ωt DT: x[n] = cos Ωn These signals differ in important ways. Check Yourself Compare two signals: 3πn x [n] = cos 1 4 5πn x [n] = cos 2 4 How many of the following statements are true? x1[n] has period N=8. x2[n] has period N=8. x1[n] = x2[n]. Check Yourself 3πt cos 4 = cos ω1t t 5πt cos 4 = cos ω2t t ω1 6= ω2 but x1[n] = x2[n]! ω1 + ω2 = 2π cos ω1n = cos(2π − ω2)n = cos(2πn − ω2n) = cos(−ω2n) = cos ω2n Check Yourself Compare two signals: 3πn x [n] = cos 1 4 5πn x [n] = cos 2 4 How many of the following statements are true? 3 √ x1[n] has period N=8. √ x2[n] has period N=8. √ x1[n] = x2[n]. Check Yourself Consider 3 CT signals: x1(t) = cos(4000t) ; x2(t) = cos(5000t) ; x3(t) = cos(6000t) Each of these is sampled so that x1[n] = x1(nT ) ; x2[n] = x2(nT ) ; x3[n] = x3(nT ) where T = 0.001. Which list goes from lowest to highest DT frequency? 0. x1[n] x2[n] x3[n] 1. x1[n] x3[n] x2[n] 2. x2[n] x1[n] x3[n] 3. x2[n] x3[n] x1[n] 4. x3[n] x1[n] x2[n] 5. x3[n] x2[n] x1[n] Check Yourself The CT signals are simple sinusoids: x1(t) = cos(4000t) ; x2(t) = cos(5000t) ; x3(t) = cos(6000t) The DT signals are sampled versions (T = 0.001): x1[n] = cos(4n) ; x2[n] = cos(5n) ; x3[n] = cos(6n) How does the shape of cos(Ωn) depend on Ω? Check Yourself As the frequency Ω increases, the shapes of the sampled signals deviate from those of the underlying CT signals. Ω = 1 : x[n] = cos(n) n Ω = 2 : x[n] = cos(2n) n Ω = 3 : x[n] = cos(3n) n Check Yourself Worse and worse representation.ΩL Ω = 4 : x[n] = cos(4n) = cos(2π − 4n) ≈ cos(2.283n) n Ω = 5 : x[n] = cos(5n) = cos(2π − 5n) ≈ cos(1.283n) n Ω = 6 : x[n] = cos(6n) = cos(2π − 6n) ≈ cos(0.283n) n The same DT sequence represents many different values of Ω. Check Yourself For Ω > π, a lower frequency ΩL has the same sample values as Ω. Ω = 4 : x[n] = cos(4n) = cos(2π − 4n) ≈ cos(2.283n) n Ω = 5 : x[n] = cos(5n) = cos(2π − 5n) ≈ cos(1.283n) n Ω = 6 : x[n] = cos(6n) = cos(2π − 6n) ≈ cos(0.283n) n The same DT sequence represents many different values of Ω. Check Yourself Although there are multiple frequencies Ω with the same samples, only one of these frequencies is in the range 0 ≤ Ω ≤ π. Thus we can remove the ambiguity of which frequency is represented by a set of samples by choosing the one in the range 0 ≤ Ω ≤ π. We call that range of frequencies the base band of frequencies. If we limit our attention to frequencies in the base band, then there is a maximum possible discrete frequency Ωmax = π. Note this difference from CT, where there is no maximum frequency. Check Yourself Consider 3 CT signals: x1(t) = cos(4000t) ; x2(t) = cos(5000t) ; x3(t) = cos(6000t) Each of these is sampled so that x1[n] = x1(nT ) ; x2[n] = x2(nT ) ; x3[n] = x3(nT ) where T = 0.001. Which list goes from lowest to highest DT frequency? 5 0. x1[n] x2[n] x3[n] 1. x1[n] x3[n] x2[n] 2. x2[n] x1[n] x3[n] 3. x2[n] x3[n] x1[n] 4. x3[n] x1[n] x2[n] 5. x3[n] x2[n] x1[n] Relation Between CT and DT Frequencies If a CT signal x(t) = cos ωt is sampled at times t = nT , the resulting DT signal is x[n] = cos Ωn where Ω = ωT . If we restrict DT frequencies to the range 0 ≤ Ω ≤ π, then the corresponding CT frequencies are in the range 0 ≤ ω ≤ ωN where π ω = N T and ω 1 1 f = N = = f N 2π 2T 2 s where fN is the Nyquist frequency, which is half the sample rate fs. Base Band Reconstruction If all frequencies in a CT signal are less than the Nyquist frequency, then we can reconstruct the original CT signal perfectly by discarding frequencies in the reconstruction that exceed the Nyquist frequency. This is called base-band reconstruction. Anti-Aliasing If there are frequencies in the CT signal that are greater than the Nyquist frequency, then these will alias to frequencies in the base band that have nothing to do with the original frequency. These should be removed before sampling. Anti-Aliasing Demonstration Sampling Music 2π ω = = 2πf s T s • fs = 11 kHz without anti-aliasing • fs = 11 kHz with anti-aliasing • fs = 5.5 kHz without anti-aliasing • fs = 5.5 kHz with anti-aliasing • fs = 2.8 kHz without anti-aliasing • fs = 2.8 kHz with anti-aliasing J.S. Bach, Sonata No. 1 in G minor Mvmt. IV. Presto Nathan Milstein, violin Quantization The information content of a signal scales not only with sample rate but also with the number of bits used to represent each sample. 1 e 2 bits10 3 bits 4 bits g a t l o 01 v 0 t u p t u O ­1 00 ­1 0 1 ­1 0 1 ­1 0 1 Input voltage Input voltage Input voltage 1 0 ­1 0 0.5 1 0 0.5 1 0 0.5 1 Time (second) Time (second) Time (second) Bit rate = (# bits/sample)×(# samples/sec) Check Yourself We hear sounds that range in amplitude from 1,000,000 to 1. How many bits are needed to represent this range? 1. 5 bits 2. 10 bits 3. 20 bits 4. 30 bits 5. 40 bits Check Yourself How many bits are needed to represent 1,000,000:1? bits range 1 2 2 4 3 8 4 16 5 32 6 64 7 128 8 256 9 512 10 1, 024 11 2, 048 12 4, 096 13 8, 192 14 16, 384 15 32, 768 16 65, 536 17 131, 072 18 262, 144 19 524, 288 20 1, 048, 576 Check Yourself We hear sounds that range in amplitude from 1,000,000 to 1. How many bits are needed to represent this range? 3 1. 5 bits 2. 10 bits 3. 20 bits 4. 30 bits 5. 40 bits Quantization Demonstration Quantizing Music • 16 bits/sample • 8 bits/sample • 6 bits/sample • 4 bits/sample • 3 bits/sample • 2 bit/sample J.S. Bach, Sonata No. 1 in G minor Mvmt. IV. Presto Nathan Milstein, violin Quantization Demonstration Quantizing Music • 16 bits/sample • 8 bits/sample • 6 bits/sample • 4 bits/sample • 3 bits/sample • 2 bit/sample J.S. Bach, Sonata No. 1 in G minor Mvmt. IV. Presto Nathan Milstein, violin Quantization Demonstration Quantizing Music • 16 bits/sample • 8 bits/sample • 6 bits/sample • 4 bits/sample • 3 bits/sample • 2 bit/sample J.S. Bach, Sonata No. 1 in G minor Mvmt. IV.