EE 451: Digital Signal Processing
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1 EE 451: Digital Signal Processing Power Spectral Density Estimation Aly El-Osery Electrical Engineering Department, New Mexico Tech Socorro, New Mexico, USA December 4, 2017 Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
2 Motivation Sensors suffer from noise effects that can not be removed through calibration, consquently, we need to understand the nature of the noise be able to extract parameters from actual data develop models to mimic noise in simulation to provide performance capabilities Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
3 Purpose Estimate the distribution of power in a signal. Unfortunately, truth and what is practical cause a problem. Truth Infinitely long. Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
4 Purpose Estimate the distribution of power in a signal. Unfortunately, truth and what is practical cause a problem. Truth Infinitely long. Practice Finite length. Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
5 Purpose Estimate the distribution of power in a signal. Unfortunately, truth and what is practical cause a problem. Truth Infinitely long. Continuous in time and value. Practice Finite length. Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
6 Purpose Estimate the distribution of power in a signal. Unfortunately, truth and what is practical cause a problem. Truth Infinitely long. Continuous in time and value. Practice Finite length. Discrete in time and value. Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
7 Purpose Estimate the distribution of power in a signal. Unfortunately, truth and what is practical cause a problem. Truth Infinitely long. Continuous in time and value. Provides true distribution of power. Practice Finite length. Discrete in time and value. Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
8 Purpose Estimate the distribution of power in a signal. Unfortunately, truth and what is practical cause a problem. Truth Infinitely long. Continuous in time and value. Provides true distribution of power. Practice Finite length. Discrete in time and value. Only approximation of distribution of power. Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
9 Purpose Estimate the distribution of power in a signal. Unfortunately, truth and what is practical cause a problem. Truth Infinitely long. Continuous in time and value. Provides true distribution of power. Practice Finite length. Discrete in time and value. Only approximation of distribution of power. Let s make it more interesting Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
10 Purpose Estimate the distribution of power in a signal. Unfortunately, truth and what is practical cause a problem. Truth Infinitely long. Continuous in time and value. Provides true distribution of power. Practice Finite length. Discrete in time and value. Only approximation of distribution of power. Let s make it more interesting The signal is stochastic in nature. Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
11 Energy and Power Assume the voltage across a resistor R is e(t) and is producing a current i(t). The instantaneous power per ohm is p(t) = e(t)i(t)/r = i 2 (t). Total Energy Average Power T E = lim i 2 (t)dt (1) T T 1 T P = lim i 2 (t)dt (2) T 2T T Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
12 Arbitrary signal x(t) Total Normalized Energy E lim T T T x(t) 2 dt = x(t) 2 dt (3) Normalized Power 1 T P lim x(t) 2 dt (4) T 2T T Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
13 Correlation For Energy Signals φ(τ) = x(t)x(t +τ)dt (5) For Power Signals 1 T R(τ) = lim x(t)x(t +τ)dt (6) T 2T T For Periodic Signals R(τ) = 1 T 0 T 0 x(t)x(t +τ)dt (7) Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
14 Energy Spectral Density Rayleigh s Energy Theorem or Parseval s theorem E = Energy Spectral Density x(t) 2 dt = X(F) 2 df (8) G(F) X(F) 2 (9) with units of volts 2 -sec 2 or, if considered on a per-ohm basis, watts-sec/hz=joules/hz Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
15 Power Spectral Density P = 1 T S(F)dF = lim x(t) 2 dt (10) T 2T T where we define S(F) as the power spectral density with units of watts/hz. Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
16 Basic Definitions Define an experiment with random outcome. Mapping of the outcome to a variable random variable. Mapping of the outcome to a function random function. Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
17 Probability (Cumulative) Distribution Function (cdf) F X (x) = probability that X x = P(X x) (11) Describes the manner random variables take different values. Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
18 Probability Density Function (pdf) and f X (x) = df X(x) dx P(x 1 < X x 2 ) = F X (x 2 ) F X (x 1 ) = x2 (12) x 1 f X (x)dx (13) PDF P(x 1 < X < x 2 ) (Compute Area) µ x 1 x 2 Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
19 PDF of Discrete Random Variables If the random variable X takes a set of discrete values x i with probability p i, the pdf of X is expressed in terms of Dirac delta functions, i.e., f X (x) = p i δ(x x i ) (14) i Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
20 Gaussian Distribution f X (x) = For example if σ x = σ and µ x = 0 0.1% [ 1 exp x µ ] x σ x 2π 2σx 2 ( ) 1 x 2 σ 2π exp 2σ 2 σ 34% 34% 2% 14% 14% 2% 0.1% 3σ 2σ σ σ 2σ 3σ x (15) Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
21 PDF of White Noise Random Signals Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
22 PDF of White Noise Random Signals Histogram and Pdf of random samples Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
23 Mean and Variance Mean of a Discrete RV X = E[X] = M x j P j (16) j=1 Mean of a Continuous RV X = E[X] = Variance of a RV xf X (x)dx (17) σ 2 X E{ [X E(X)] 2} = E[X 2 ] E 2 [X] (18) Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
24 Covariance and Autocorrelation Given a two random variables X and Y. Covariance µ XY = E { [X x][y Ȳ] } = E[XY] E[X]E[Y] (19) Correlation Coefficient ρ XY = µ XY σ X σ Y (20) Autocorrelation Γ X (τ) = E[X(t)X(t +τ)] (21) Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
25 Terminology X(t,ζ1) t X(t,ζ2)... X(t,ζM) t t X(t,ζ i ): sample function. The governing experiment: random or stochastic process. All sample functions: ensemble. X(t j,ζ): random variable. t1 t2 Figure: Sample functions of a random process Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
26 Strict Sense Stationarity If the joint pdfs depend only on the time difference regardless of the time origin, then the random process is known as stationary. For stationary process means and variances are independent of time and the covariance depends only on the time difference. Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
27 Wide Sense Stationarity If the joint pdfs depends on the time difference but the mean and variances are time-independent, then the random process is known as wide-sense-stationary. Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
28 Ergodicity If the time statistics equals ensemble statistics, then the random process is known as ergodic. Any statistic calculated by averaging of all members of an ergodic ensemble at a fixed time can also be calculated by using a single representative waveform and averging over all time. Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
29 Power Spectral Density Given a sample function X(t,ζ i ) of a random process, we obtain the power spectral density by S(F) F Γ(τ) (22) i.e., for a wide sense stationary signal, the power spectral density and autocorrelation are Fourier transform pairs. Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
30 Input-Output Relationship of Linear Systems x(t) H(F) y(t) S Y (F) = H(F) 2 S X (F) (23) Noise Shaping If x(t) is white noise, we can design the filter h(t) to shape the noise. Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
31 Big Picture t T Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
32 Big Picture CTFT t F T 1/T Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
33 Big Picture T t Sample 1/T F t T s Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
34 Big Picture t F T 1/T DTFT t F T s F s = 1/T s Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
35 Big Picture t F T 1/T Sample t F T s F s = 1/T s F F = 1/T o Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
36 Big Picture t F T 1/T t F T s F s = 1/T s IDFT t F T o F = 1/T o Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
37 Sampling Remarks Must sample more than twice bandwidth to avoid aliasing. FFT represents a periodic version of the time domain signal could have time domain aliasing. Number of points in FFT is the same as number of points in time domain signal. Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
38 Obtaining PSD for Discrete Signals What we want is For infinitely long signals. Γ X (τ) = E[X(t)X(t +τ)] CT FT S X (F) Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
39 Obtaining PSD for Discrete Signals What we want is For infinitely long signals. What we can compute is For finite length signals. Γ X (τ) = E[X(t)X(t +τ)] CT FT S X (F) γ X (m) = E[X(n)X(n+m)] DFT P X (f) Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
40 What do we need in an estimate As N and in the mean squared sense Unbiased Asymptotically the mean of the estimate approaches the true power. Variance Variance of the estimate approaches zero. Resulting in a consistent estimate of the power spectrum. Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
41 Possible PSD Options Periodogram computed using 1/N times the magnitude squared of the FFT lim E[P X(f)] = S X (f) N lim var[p X(f)] = SX 2 (f) N Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
42 Possible PSD Options Periodogram computed using 1/N times the magnitude squared of the FFT lim E[P X(f)] = S X (f) N lim var[p X(f)] = SX 2 (f) N Welch Method computed by segmenting the data (allowing overlaps), windowing the data in each segment then computing the average of the resultant priodogram E[P X (f)] = 1 2πMU S X(f) W(f) var[p X (f)] 9 8L S2 X (f) Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
43 Welch Method Assuming data length N, segment length M, Bartlett window, and 50% overlap FFT length = M = 1.28/ f = 1.28F s / F Resulting number of segments = L = 2N M Length of data collected in sec. = 1.28L 2 F Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
44 pwelch Function [Pxx,f] = pwelch(x,window,noverlap,... nfft,fs, range ) You can use [] in fields that you want the default to be used. Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
45 pwelch Function - WGN signal Fs = 1000; x = sqrt(0.1*fs)*randn(1,100000); [Pxx,f] = pwelch(x,1024,[],[],fs, onesided ); Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
46 pwelch Function - WGN signal Fs = 1000; x = sqrt(0.1*fs)*randn(1,100000); [Pxx,f] = pwelch(x,1024,[],[],fs, onesided ); PSD Frequency (Hz) Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
47 pwelch Function - WGN signal Fs = 1000; x = sqrt(0.1*fs)*randn(1,100000); [Pxx,f] = pwelch(x,1024,[],[],fs, onesided ); PSD Variance to high Frequency (Hz) Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
48 pwelch Function - WGN signal [Pxx,f] = pwelch(x,128,[],[],fs, onesided ) Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
49 pwelch Function - WGN signal [Pxx,f] = pwelch(x,128,[],[],fs, onesided ) PSD Frequency (Hz) Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
50 pwelch Function - WGN signal [Pxx,f] = pwelch(x,128,[],[],fs, onesided ) PSD Reduced window size. Variance is now smaller Frequency (Hz) Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
51 pwelch Function - cos + WGN signal Fs = 100; t = 0:1/Fs:5; x = cos(2*pi*10*t)+cos(2*pi*11*t)+... sqrt(0.1*fs)*randn(1,length(t)); [Pxx,f] = pwelch(x,1024,[],[],fs, onesided ); Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
52 pwelch Function - cos + WGN signal Fs = 100; t = 0:1/Fs:5; x = cos(2*pi*10*t)+cos(2*pi*11*t)+... sqrt(0.1*fs)*randn(1,length(t)); [Pxx,f] = pwelch(x,1024,[],[],fs, onesided ); PSD Frequency (Hz) Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
53 pwelch Function - cos + WGN signal Fs = 100; t = 0:1/Fs:5; x = cos(2*pi*10*t)+cos(2*pi*11*t)+... sqrt(0.1*fs)*randn(1,length(t)); [Pxx,f] = pwelch(x,1024,[],[],fs, onesided ); 1.00 PSD Window larger than length of data. Frequency components can t be resolved. Variance high. Frequency (Hz) Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
54 pwelch Function - cos + WGN signal Fs = 100; t = 0:1/Fs:5; x = cos(2*pi*10*t)+cos(2*pi*11*t)+... sqrt(0.1*fs)*randn(1,length(t)); [Pxx,f] = pwelch(x,1024,[],4096,fs, onesided ); Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
55 pwelch Function - cos + WGN signal Fs = 100; t = 0:1/Fs:5; x = cos(2*pi*10*t)+cos(2*pi*11*t)+... sqrt(0.1*fs)*randn(1,length(t)); [Pxx,f] = pwelch(x,1024,[],4096,fs, onesided ); PSD Frequency (Hz) Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
56 pwelch Function - cos + WGN signal Fs = 100; t = 0:1/Fs:5; x = cos(2*pi*10*t)+cos(2*pi*11*t)+... sqrt(0.1*fs)*randn(1,length(t)); [Pxx,f] = pwelch(x,1024,[],4096,fs, onesided ); PSD As expected increasing nfft does not help Frequency (Hz) Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
57 pwelch Function - cos + WGN signal Fs = 100; t = 0:1/Fs:5; x = cos(2*pi*10*t)+cos(2*pi*11*t)+... sqrt(0.1*fs)*randn(1,length(t)); [Pxx,f] = pwelch(x,128,[],4096,fs, onesided ); Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
58 pwelch Function - cos + WGN signal Fs = 100; t = 0:1/Fs:5; x = cos(2*pi*10*t)+cos(2*pi*11*t)+... sqrt(0.1*fs)*randn(1,length(t)); [Pxx,f] = pwelch(x,128,[],4096,fs, onesided ); PSD Frequency (Hz) Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
59 pwelch Function - cos + WGN signal Fs = 100; t = 0:1/Fs:5; x = cos(2*pi*10*t)+cos(2*pi*11*t)+... sqrt(0.1*fs)*randn(1,length(t)); [Pxx,f] = pwelch(x,128,[],4096,fs, onesided ); 1.00 PSD Frequency (Hz) Decreasing the window size decreases the variance. Still can t resolve the two frequencies. Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
60 pwelch Function - cos + WGN signal Fs = 100; t = 0:1/Fs:50; x = cos(2*pi*10*t)+cos(2*pi*11*t)+... sqrt(0.1*fs)*randn(1,length(t)); [Pxx,f] = pwelch(x,128,[],4096,fs, onesided ); Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
61 pwelch Function - cos + WGN signal Fs = 100; t = 0:1/Fs:50; x = cos(2*pi*10*t)+cos(2*pi*11*t)+... sqrt(0.1*fs)*randn(1,length(t)); [Pxx,f] = pwelch(x,128,[],4096,fs, onesided ); PSD Frequency (Hz) Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
62 pwelch Function - cos + WGN signal Fs = 100; t = 0:1/Fs:50; x = cos(2*pi*10*t)+cos(2*pi*11*t)+... sqrt(0.1*fs)*randn(1,length(t)); [Pxx,f] = pwelch(x,128,[],4096,fs, onesided ); 1.00 PSD Length of data sequence must be increased. Still can t resolve the two frequencies as the window size is too small. Frequency (Hz) Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
63 pwelch Function - cos + WGN signal Fs = 100; t = 0:1/Fs:50; x = cos(2*pi*10*t)+cos(2*pi*11*t)+... sqrt(0.1*fs)*randn(1,length(t)); [Pxx,f] = pwelch(x,256,[],4096,fs, onesided ); Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
64 pwelch Function - cos + WGN signal Fs = 100; t = 0:1/Fs:50; x = cos(2*pi*10*t)+cos(2*pi*11*t)+... sqrt(0.1*fs)*randn(1,length(t)); [Pxx,f] = pwelch(x,256,[],4096,fs, onesided ); PSD Frequency (Hz) Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
65 pwelch Function - cos + WGN signal Fs = 100; t = 0:1/Fs:50; x = cos(2*pi*10*t)+cos(2*pi*11*t)+... sqrt(0.1*fs)*randn(1,length(t)); [Pxx,f] = pwelch(x,256,[],4096,fs, onesided ); PSD Now we can resolve the two frequencies Frequency (Hz) Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
66 Spectral Estimation - Remarks The length of the data sequence determines the maximum resolution that can be observed. Increasing the window length of each segment in the data increases the resolution. Decreasing the window length of each segment in the data decreases the variance of the estimate. nfft only affects the amount of details shown and not the resolution. Aly El-Osery (NMT) EE 451: Digital Signal Processing December 4, / 37
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