lecture_23

# lecture_23 - MIT OpenCourseWare http/ocw.mit.edu 2.161...

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Unformatted text preview: 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 23 1 Reading: Proakis and Manolakis: Secs. 14.1 – 14.2 • • Oppenheim, Schafer, and Buck: 10.6 – 10.8 Stearns and Hush: 15.4, 15.6 • 1 Non-Parametric Power Spectral Density Estimation In Lecture 22 we defined the power-density spectrum Φ ff (j Ω) of an infinite duration, real function f ( t ) as the Fourier transform of its autocorrelation function φ ff ( τ ) 1 T/ 2 Φ ff (jΩ) = F lim f ( t ) f ( t + τ )d t . T →∞ T − T/ 2 with units of (physical-units) 2 .s or (physical-units) 2 /Hz, where physical-units are the units of f ( t ), for example volts. The waveform power contained in the spectral region between Ω 1 < Ω < Ω 2 is | | 1 − Ω 1 Ω 2 P = Φ ff (jΩ)dΩ + Φ ff (jΩ)dΩ 2 π Ω 1 Ω 2 − Ω 2 1 = Φ ff (jΩ)dΩ π Ω 1 since Φ ff (j Ω) is a real, even function. Similarly, we defined the energy-density spectrum R ff (j Ω) of a real finite duration wave- form f ( t ) of duration T as the Fourier transform of its energy based autocorrelation function ρ ff ( τ ) T R ff (jΩ) = F f ( t ) f ( t + τ )d t . with units of (physical-units) 2 .s 2 or (physical-units) 2 .s/Hz, again where physical-units are the units of f ( t ). In this lecture we address the issue of estimating the PSD (power spectral density) Φ ff (j Ω) of an infinite process using a finite length sample of the process. PSD analysis is an important tool in engineering analysis. The practical problem is to form reliable estimates of 1 copyright c D.Rowell 2008 23–1 the PSD from finite records of an infinite random process. For example, the following figure shows a stationary random process with three possible finite records. An estimator Φ ˆ f f (j Ω) of Φ f f (j Ω) is to made from one of the finite-length records. f ( t ) R e c o r d 1 R e c o r d 2 R e c o r d 3 We ask ourselves about the statistics of estimators derived from the different records, in particular, (1) the bias in the estimator B Φ ˆ f f (j Ω) = Φ f f (j Ω) − E Φ ˆ f f (j Ω) (2) the variance of the estimator 2 V Φ ˆ f f (j Ω) = E Φ f f (j Ω) − Φ ˆ f f (j Ω) 1.1 The Periodogram 1.1.1 The Continuous Periodogram If f ( t ) is a stationary, real, random process, its autocorrelation function is defined by the ensemble statistics φ f f ( τ ) = E { f ( t ) f ( t + τ ) } . For an ergodic process, if we have a single record of duration T we can compute an estimator, φ ˆ f f ( τ ) based on the time average: 1 T / 2 φ ˆ f f ( τ ) = f ( t ) f ( t + τ ) d t, T − T / 2 and φ f f ( τ ) = lim φ ˆ f f ( τ ) ....
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lecture_23 - MIT OpenCourseWare http/ocw.mit.edu 2.161...

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