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.