If drops quic kly with then pro cess hanges quic kly with time its time samples ecome uncorrelated er short erio of time Con ersely when drops slo wly with samples are highly correlated er long time Th us is measure of the rate of hange of with time ID: 26575 Download Pdf

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If drops quic kly with then pro cess hanges quic kly with time its time samples ecome uncorrelated er short erio of time Con ersely when drops slo wly with samples are highly correlated er long time Th us is measure of the rate of hange of with time

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opic 8: er sp ectral densit and TI systems The er sp ectral densit of WSS random pro cess Resp onse of an TI system to random signals Linear MSE estimation ES150 Ha rva rd SEAS The auto correlation function and the rate of hange Consider WSS random pro cess with the auto correlation function ). If drops quic kly with then pro cess hanges quic kly with time: its time samples ecome uncorrelated er short erio of time. Con ersely when drops slo wly with samples are highly correlated er long time. Th us is measure of the rate of hange of with time and hence is related to the fr

quency esp onse of ). or example, sin usoidal eform sin(2 will ary rapidly with time if it is at high frequency (large ), and ary slo wly at lo frequency (small ). In fact, the ourier transform of is the erage er densit er the frequency domain. ES150 Ha rva rd SEAS

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The er sp ectral densit of WSS pro cess The ower sp ctr al density (psd) of WSS random pro cess is giv en the ourier transform (FT) of its auto correlation function `1 d or discrete-time pro cess the psd is giv en the discrete-time FT (DTFT) of its auto correlation sequence `1 Since the DTFT is erio dic in with erio

1, only need to consider )) can reco ered from taking the in erse FT `1 ES150 Ha rva rd SEAS Prop erties of the er sp ectral densit is real and ev en The area under is the erage er of `1 (0) is the erage er densit hence the erage er of in the frequency band is is nonnegativ e: for all (sho wn later) In general, an function that is real, ev en, nonnegativ and has ˇnite area can psd function. ES150 Ha rva rd SEAS

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White noise Band-limited white noise: zero-mean WSS pro cess whic has the psd as constan within and zero elsewhere. Similar to white ligh con taining all

frequencies in equal amoun ts. Its erage er is Its auto-correlation function is sin(2 sinc (2 or an the samples for are uncorrelated. −W N/2 (f) f 1/(2W) 2/(2W) )=NWsinc(2W ) ES150 Ha rva rd SEAS White-noise pro cess: Letting obtain white noise pr ess whic has for all or white noise pro cess, all samples are uncorrelated. The pro cess has inˇnite er and hence not ph ysically realizable. It is an idealization of ph ysical noises. Ph ysical systems usually are band-limited and are a˛ected the noise within this band. If the white noise is Gaussian random pro cess, then ha ES150 Ha

rva rd SEAS

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Gaussian white noise (GWN) 100 200 300 400 500 600 700 800 900 1000 −4 −3 −2 −1 Time Magnitude GN results from taking the deriv ativ of the Bro wnian motion (or the Wiener pro cess). All samples of GWN pro cess are indep enden and iden tically Gaussian distributed. ery useful in mo deling broadband noise, thermal noise. ES150 Ha rva rd SEAS Cross-p er sp ectral densit Consider join tly-WSS random pro cesses and ): Their cr oss-c orr elation function is deˇned as )] Unlik the auto-correlation ), the cross-correlation is not necessarily

ev en. Ho ev er ;X The cr oss-p ower sp ctr al density is deˇned as In general, is complex ev en if the pro cesses are real-v alued. ES150 Ha rva rd SEAS

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Example: Signal plus white noise Let the observ ation where is the an ted signal and is white noise. and are zero-mean uncorrelated WSS pro cesses. is also WSS pro cess )] )] gf The psd of is the sum of the psd of and and are join tly-WSS )] )] Th us ). ES150 Ha rva rd SEAS Review of TI systems Consider system that maps )] The system is linear if )] )] )] The system is time-in arian if )] )] An TI system can completely

haracterized its impulse resp onse )] The input-output relation is obtained through con olution `1 d In the frequency domain: The system tr ansfer function is the ourier transform of )] `1 dt ES150 Ha rva rd SEAS 10

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Resp onse of an TI system to WSS random signals Consider an TI system h(t) X(t) Y(t) Apply an input whic is WSS random pro cess The output then is also WSS )] `1 d (0) )] `1 `1 dsdr Tw pro cesses and are join tly WSS `1 ds rom these, also obtain `1 dr ES150 Ha rva rd SEAS 11 The results are similar for discrete-time systems. Let the impulse resp onse The resp onse

of the system to random input pro cess is The system transfer function is nf With WSS input the output is also WSS (0) and are join tly WSS ES150 Ha rva rd SEAS 12

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requency domain analysis aking the ourier transforms of the correlation functions, ha where is the complex conjugate of ). The output-input psd relation Example: White noise as the input. Let ha the psd as for all then the psd of the output is Th us the transfer function completely determines the shap of the output psd. This also sho ws that an psd ust nonne gative ES150 Ha rva rd SEAS 13 er in WSS random pro cess

Some signals, suc as sin( ), ma not ha ˇnite energy but can ha ˇnite erage er. The erage er of random pro cess is deˇned as lim !1 dt or WSS pro cess, this ecomes lim !1 dt (0) But (0) is related to the FT of the psd ). Th us ha three ys to express the er of WSS pro cess (0) `1 The area under the psd function is the erage er of the pro cess. ES150 Ha rva rd SEAS 14

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Linear estimation Let zero-mean WSS random pro cess, whic are in terested in estimating. Let the observ ation, whic is also zero-mean random pro cess join tly WSS with or example, could noisy observ

ation of ), or the output of system with as the input. Our goal is to design linear, time-in arian ˇlter that pro cesses to pro duce an estimate of ), whic is denoted as h(t) X(t) Y(t) Assuming that kno the auto- and cross-correlation functions ), ), and ). estimate eac sample ), use an observ ation windo on as If and 0, this is (causal) ˇltering problem: estimating ES150 Ha rva rd SEAS 15 from the past and presen observ ations. If this is an inˇnite smo othing problem: reco ering from the en tire set of noisy observ ations. The linear estimate is of the form d Similarly for

discrete-time pro cessing, the goal is to design the ˇlter co ecien ts to estimate as Next consider the optim um linear ˇlter based on the MMSE criterion. ES150 Ha rva rd SEAS 16

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Optim um linear MMSE estimation The MMSE linear estimate of based on is the signal that minimizes the MSE MSE By the orthogonalit principle, the MMSE estimate ust satisfy )] h The error is orthogonal to all observ ations ). Th us for )] d (1) ˇnd need to solv an inˇnite set of in tegral equations. Analytical solution is usually not ossible in general. But it can solv ed in

imp ortan sp ecial case: inˇnite smo othing ), and ˇltering 0). ES150 Ha rva rd SEAS 17 urthermore, the error is orthogonal to the estimate )] d The MSE is then giv en i )] h (0) d (2) or the discrete-time case, ha (3) (0) (4) rom (3), one can design the ˇlter co ecien ts ES150 Ha rva rd SEAS 18

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Inˇnite smo othing When a; ha `1 d aking the ourier transform giv es the transfer function for the optim um ˇlter ES150 Ha rva rd SEAS 19

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