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: Proakis and Manolakis: Secs. 12,1 – 12.2 • • Oppenheim, Schafer, and Buck: 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 τ . We continue the discussion here. 1.1 The Autocorrelation Function There are three basic definitions (a) For an infinite duration waveform: � T/21 φff (τ) = lim f(t)f(t + τ) d t T →∞ 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 t2 ρff (τ ) = f(t)f(t + τ ) dt t1 which may be considered as a “energy” based definition. (c) For a periodic waveform: If f(t) is periodic with period T t0+T1 �φff (τ) = f(t)f(t + τ) d t T t0 for an arbitrary t0, which again may be considered as a “power” based definition. 1copyright cD.Rowell 2008 22–1f ( t )tT0af ( t +t)t0a-tT -tr (t)tT0af f- T2r (t) = a ( T - t )f f2Example 1 Find the autocorrelation function of the square pulse of amplitude a and duration T as shown below. f ( t )tT0aThe wave form has a finite duration, and the autocorrelation function is � T ρff (τ) = f(t)f(t + τ) dt 0 The autocorrelation function is developed graphically below � T −τ ρff (τ ) = a 2 dt 0 = a τ )2(T − | | − T ≤ τ ≤ T = 0 otherwise. 22–2Example 2 Find the autocorrelation function of the sinusoid f(t) = sin(Ωt + φ). Since f(t) is periodic, the autocorrelation function is defined by the average over one period t0+T1 �φff (τ) = f(t)f(t + τ) dt. T t0 and with t0 = 0 � 2π/ΩΩ φff (τ) = 2π 0 sin(Ωt + φ) sin(Ω(t + τ) + φ) dt 1 = cos(Ωt)2 and we see that φff (τ) is periodic with period 2π/Ω and is independent of the phase φ. 1.1.1 Properties of the Auto-correlation Function (1) The autocorrelation functions φff (τ) and ρff (τ) are even functions, that is φff (−τ) = φff (τ), and ρff (−τ ) = ρff (τ). (2) A maximum value of ρff (τ) (or φff (τ ) occurs at delay τ = 0, |ρff (τ)| ≤ ρff (0), and |φff (τ)| ≤ φff (0) and we note that � ∞ρff (0) = f2(d) dt −∞ is the “energy” of the waveform. Similarly 1 � ∞ φff (0) = lim f2(t) dt T →∞ T −∞ is the mean “power” of f(t). (3) ρff (τ) contains no phase information, and is independent of the time origin. (4) If f(t) is periodic with period T , φff (τ) is also periodic with period T . (5) If (1) f (t) has zero mean (µ = 0), and (2) f(t) is non-periodic, lim ρff (τ) = 0. τ →∞ 22–31.1.2 The Fourier Transform of the Auto-Correlation Function Consider the transient case Rf f (j Ω) = � ∞ ρf f (τ ) e−j Ωτ dτ = −∞� ∞ �� ∞ f(t)f(t + τ) dt � e−j Ωτ dτ = −∞ −∞� ∞ f(t) ej Ωt dt. � ∞ f(ν) e−j Ων dν = −∞ F (−j Ω)F (j Ω) −∞ = |F (j Ω)| 2 or ρff (τ) F Rff (j Ω) = F (j Ω)2 ←→ | | 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, φff (τ) � ∞Φf f (j Ω) = F {φf f (τ)} = φff (τ) e−j Ωτ dτ = � ∞ � lim 1 −∞� T /2 f(t)f(t + τ) dt � e−j Ωτ dτ −∞ T →∞ 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 τ, the energy/p ower 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 � ∞ 1 � ∞ρff (0) = f2(t) dt = Rff (j Ω) dΩ 2π∞ −∞ or 1� ∞ � ∞ 2f2(t) dt =2π |F (j Ω)| dΩ, ∞ −∞ 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 � T/2 1 � ∞lim f2(t) dt = Φ(j Ω) dΩ −T/2T →∞ 2π −∞ equates the signal power in the two domains. 22–4f ( t )f ftF (jW )f fWb r o a d a u t o c o r r e l a t i o nn a r r o w s p e c t r u mf ( t )f ftF (jW )f fWn a r r o w a u t o c o r r e l a t i o nb r o a d s p e c t r u mf ( t )f ftF (jW )f fW" w h i t e " s p e c t r u mi m p u l s e a u t o c o r r e l a t i o n11.1.4 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. • A narrow autocorrelation function generally implies a “broad” spectrum • and a “broad” autocorrelation function generally implies a narrow-band waveform. In the limit, if φff (τ) = δ(τ), then Φff (j Ω) = 1, and the spectrum is defined to be “white”. 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: � T/21 φfg(τ) = lim f(t)g(t + τ) dτ T →∞ T −T/2 in the case of infinite duration waveforms, and � ∞ρfg(τ ) = f(t)g(t + τ ) dτ −∞ for finite duration waveforms. 22–5Example 3 Find the cross-correlation function between the following two functions f ( t )tT0ag ( t )tT0aT1T2In this case g(t) is a delayed version of f (t). The cross-correlation is r (t)tT - T0af g22 1where the peak occurs at τ = T2 − T1 (the delay between the two signals). 1.2.1 Properties of the Cross-Correlation Function (1) φfg(τ) = φgf (−τ ), and the cross-correlation function is not necessarily an even function. (2) If φfg(τ) = 0 for all τ, 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 …