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MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing - Continuous and Discrete Fall Term 2008 Lecture 221 Reading: ² Proakis and Manolakis: Secs. 12,1 – 12.2 ² Oppenheim, Schafer, and Buck: ² Stearns and Hush: Ch. 13 1 The Correlation Functions (continued) In Lecture 21 we introduced the auto-correlation and cross-correlation functions as measures of self- and cross-similarity as a function of delay ¿ . We continue the discussion here. 1.1 The Autocorrelation Function There are three basic definitions (a) For an infinite duration waveform: Z T=2 1 Áff (¿) = lim f(t)f(t + ¿) dt T !1 T ¡T=2 which may be considered as a “power” based definition. (b) For an finite duration waveform: If the waveform exists only in the interval t1 · t · t2 Z t2 ½ff (¿ ) = f(t)f(t + ¿ ) dt t1 which may be considered as a “energy” based definition. (c) For a periodic waveform: If f(t) is periodic with period T Z t0+T 1 Áff (¿) = f(t)f(t + ¿) dt T t0 for an arbitrary t0, which again may be considered as a “power” based definition. 1copyright �c D.Rowell 2008 22–1 Example 1 Find the autocorrelation function of the square pulse of amplitude a and duration T as shown below. f ( t ) a t 0 T The wave form has a finite duration, and the autocorrelation function is Z T ½ff (¿) = f(t)f(t + ¿) dt 0 The autocorrelation function is developed graphically below f ( t ) a t 0 T f ( t + t ) a t - t 0 T - t r f f ( t ) a 2 a 2 T r f f ( t ) = ( - t ) -T 0 T t Z T ¡¿ 2 ½ff (¿ ) = a dt 0 = a2(T ¡ j¿j) ¡ T · ¿ · T = 0 otherwise. 22–2 Example 2 Find the autocorrelation function of the sinusoid f(t) = sin(Ωt + Á). Since f(t) is periodic, the autocorrelation function is defined by the average over one period Z t0+T 1 Áff (¿) = f(t)f(t + ¿) dt: T t0 and with t0 = 0 Z 2¼=­ Ω Áff (¿) = sin(Ωt + Á) sin(Ω(t + ¿) + Á) dt 2¼ 0 1 = cos(Ωt) 2 and we see that Áff (¿) is periodic with period 2¼=Ω and is independent of the phase Á. 1.1.1 Properties of the Auto-correlation Function (1) The autocorrelation functions Áff (¿) and ½ff (¿) are even functions, that is Áff (¡¿) = Áff (¿); and ½ff (¡¿ ) = ½ff (¿): (2) A maximum value of ½ff (¿) (or Áff (¿ ) occurs at delay ¿ = 0, j½ff (¿)j · ½ff (0); and jÁff (¿)j · Áff (0) and we note that Z 1 2 ½ff (0) = f (d) dt ¡1 is the “energy” of the waveform. Similarly Z 1 1 2 Áff (0) = lim f (t) dt T !1 T ¡1 is the mean “power” of f(t). (3) ½ff (¿) contains no phase information, and is independent of the time origin. (4) If f(t) is periodic with period T , Áff (¿) is also periodic with period T . (5) If (1) f(t) has zero mean (¹ = 0), and (2) f(t) is non-periodic, lim ½ff (¿) = 0: ¿ !1 22–3 1.1.2 The Fourier Transform of the Auto-Correlation Function Consider the transient case Z 1 ¡j ­¿ Rf f (j Ω) = ½f f (¿ ) e d¿ ¡1 Z 1 �Z 1 ¶ ¡j ­¿ = f(t)f(t + ¿) dt e d¿ ¡1 ¡1 Z 1 Z 1 j ­t ¡j ­º = f(t) e dt: f(º) e dº ¡1 ¡1 = F (¡j Ω)F (j Ω) = jF (j Ω)j2 or F 2 ½ff (¿) Ã! Rff (j Ω) = jF (j Ω)j where Rff (Ω) is known as the energy density spectrum of the transient waveform f(t). Similarly, the Fourier transform of the power-based autocorrelation function, Áff (¿) Z 1 ¡j ­¿ Φf f (j Ω) = F fÁf f (¿)g = Áff (¿) e d¿ ¡1 à ! Z 1 Z T =2 1 ¡j ­¿ = lim f(t)f(t + ¿) dt e d¿ ¡1 T !1 T ¡T =2 is known as the power density spectrum of an infinite duration waveform. From the properties of the Fourier transform, because the auto-correlation function is a real, even function of ¿, the energy/power density spectrum is a real, even function of Ω, and contains no phase information. 1.1.3 Parseval’s Theorem From the inverse Fourier transform Z 1 Z 1 2 1 ½ff (0) = f (t) dt = Rff (j Ω) dΩ 1 2¼ ¡1 or Z 1 Z 1 2 1 2 f (t) dt = jF (j Ω)j dΩ; 1 2¼ ¡1 which equates the total waveform energy in the time and frequency domains, and which is known as Parseval's theorem. Similarly, for infinite duration waveforms Z T=2 Z 1 2 1 lim f (t) dt = Φ(j Ω) dΩ T !1 ¡T=2 2¼ ¡1 equates the signal power in the two domains. 22–4 1.1.4 Note on the relative “widths” of the Autocorrelation and Power/Energy Spectra As in the case of Fourier analysis of waveforms, there is a general reciprocal relationship between the width of a signals spectrum and the width of its autocorrelation function. ² A narrow autocorrelation function generally implies a “broad” spectrum ff f ( t ) F f f ( j W) broad autocorrelation narrow spectrum t W ² and a “broad” autocorrelation function generally implies a narrow-band waveform. ff f ( t ) F f f ( j W) narrow autocorrelation broad spectrum t W In the limit, if Áff (¿) = ±(¿), then Φff (j Ω) = 1, and the spectrum is defined to be “white”. ff f ( t ) F f f ( j W) "w hite" spectrum im pulse autocorrelation 1 t W 1.2 The Cross-correlation Function The cross-correlation function is a measure of self-similarity between two waveforms f(t) and g(t). As in the case of the auto-correlation functions we need two definitions: Z T=2 1 Áfg(¿) = lim f(t)g(t + ¿) d¿ T !1 T ¡T=2 in the case of infinite duration waveforms, and Z 1 ½fg(¿ ) = f(t)g(t + ¿) d¿ ¡1 for finite duration waveforms. 22–5 Example 3 Find the cross-correlation function between the following two functions f ( t ) g ( t ) T T a a t t 0 T 0 1 T 2 In this case g(t) is a delayed version of f(t). The cross-correlation is rf g ( t ) a 2 0 t T2 - T 1 where the peak occurs at ¿ = T2 ¡ T1 (the delay between the two signals). 1.2.1 Properties of the Cross-Correlation Function (1) Áfg(¿) = Ágf (¡¿ ), and the cross-correlation function is not necessarily an even function. (2) If Áfg(¿) = 0 for all ¿, then f(t) and g(t) are said to be uncorrelated. (3) If g(t) = af(t ¡ T ), where a is a constant, that is g(t) is a scaled and delayed version of f(t), then Áff (¿) will have its maximum value at ¿ = T . Cross-correlation is often used in optimal estimation of delay, such as in echolocation (radar, sonar), and in GPS receivers. 22–6 Example 4 In an echolocation system, a transmitted waveform s(t) is reflected off an object at a distance R and is received a time T = 2R=c sec. later. The received signal r(t) = ®s(t¡T )+n(t) is attenuated by a factor ® and is contaminated by additive noise n(t). R transm itted s ( t ) w a v e f o r m reflecting object s t - T r e c e i v e d a ( ) velocity of propagation: c r ( t ) 2 R w a v e f o r m + delay T = c n ( t ) Z 1 Ásr(¿) = s(t)r(t + ¿) dt 1 Z 1 = s(t)(n(t + ¿) + ®s(t ¡ T + ¿)) dt 1 = Ásn(¿) + ®Áss(¿ ¡ T ) and if the transmitted waveform s(t) and the noise n(t) are uncorrelated, that is Ásn(¿ ) ´ 0, then Ásr(¿) = ®Áss(¿ ¡ T ) that is, a scaled and shifted version of the auto-correlation function of the trans­ mitted waveform – which will have its peak value at ¿ = T , which may be used to form an estimator of the range R. 1.2.2 The Cross-Power/Energy Spectrum We define the cross-power/energy density spectra as the Fourier transforms of the cross- correlation functions: Z 1 ¡j ­¿ Rfg(j Ω) = ½fg(¿) e d¿ ¡1 Z 1 ¡j ­¿ Φfg(j Ω) = Áfg(¿) e d¿: ¡1 Then Z 1 ¡j ­¿ Rfg(j Ω) = ½fg(¿) e d¿ ¡1 Z 1 Z 1 ¡j ­¿ = f(t)g(t + ¿) e dt d¿ ¡1 ¡1 Z 1 Z 1 j ­t ¡j ­º = f(t) e dt g(º) e dº ¡1 ¡1 22–7 or Rfg(j Ω) = F (¡j Ω)G(j Ω) Note that although Rff (j Ω) is real and even (because ½ff (¿) is real and even, this is not the case with the cross-power/energy spectra, Φfg(j Ω) and Rfg(j Ω), and they are in general complex. 2 Linear System Input/Output Relationships with Random In­ puts: Consider a linear system H(j Ω) with a random input f(t). The output will also be random y t f ( t ) ( ) H ( j W ) t t f f f ( ) f y y ( ) Then Y (j Ω) = F (j Ω)H(j Ω); Y (j Ω)Y (¡j Ω) = F (j Ω)H(j Ω)F (¡j Ω)H(¡j Ω) or 2 Φyy(j Ω) = Φff (j Ω) jH(j Ω)j : Also F (¡j Ω)Y (j Ω) = F (¡j Ω)F (j Ω)H(j Ω); or Φfy(j Ω) = Φff (j Ω)H(j Ω): Taking the inverse Fourier transforms © ª ¡1 2 Áyy(¿ ) = Áff (¿) ­ F jH(j Ω)j Áfy(¿ ) = Áff (¿) ­ h(¿ ): 3 Discrete-Time Correlation Define the correlation functions in terms of summations, for example for an infinite length sequence Áfg(n) = E ffmgm+ng N 1 X = lim fmgm+n; N !1 2N + 1 m=¡N and for a finite length sequence N X ½fg(n) = fmgm+n: m=¡N 22–8 The following properties are analogous to the properties of the continuous correlation func­ tions: (1) The auto-correlation functions Á(ff(n) and ½ff (n) are real, even functions.
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  • STAT 830 the Basics of Nonparametric Models The

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    STAT 830 The basics of nonparametric models The Empirical Distribution Function { EDF The most common interpretation of probability is that the probability of an event is the long run relative frequency of that event when the basic experiment is repeated over and over independently. So, for instance, if X is a random variable then P (X ≤ x) should be the fraction of X values which turn out to be no more than x in a long sequence of trials. In general an empirical probability or expected value is just such a fraction or average computed from the data. To make this precise, suppose we have a sample X1;:::;Xn of iid real valued random variables. Then we make the following definitions: Definition: The empirical distribution function, or EDF, is n 1 X F^ (x) = 1(X ≤ x): n n i i=1 This is a cumulative distribution function. It is an estimate of F , the cdf of the Xs. People also speak of the empirical distribution of the sample: n 1 X P^(A) = 1(X 2 A) n i i=1 ^ This is the probability distribution whose cdf is Fn. ^ Now we consider the qualities of Fn as an estimate, the standard error of the estimate, the estimated standard error, confidence intervals, simultaneous confidence intervals and so on. To begin with we describe the best known summaries of the quality of an estimator: bias, variance, mean squared error and root mean squared error. Bias, variance, MSE and RMSE There are many ways to judge the quality of estimates of a parameter φ; all of them focus on the distribution of the estimation error φ^−φ.