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Unformatted text preview: 79 Lecture 1920 3. Least Squares Filtering: Consider the example of acoustic echo cancellation in tele conferencing applications. Input speech signal f [ n ] enters the system. It is converted into an acoustic signal which is radiated by a loudspeaker into a teleconferencing room. The acoustic signals get reflected by the surroundings and enter the microphone along with noise. This is converted to electrical signals by the microphone. See Fig. 61 for an illustration. Microphone Loud Speaker I/P signal O/P signal f[n] d[n] Figure 61: Acoustic Echo Cancellation Assumption: We model this system as an LSI system that takes f [ n ] as input and pro duces an output y [ n ] and then adds noise to it to get d [ n ] . Let h [ n ] denotes the impulse response of the LSI system. We observe N samples of f [ n ] and N samples of d [ n ] . Objective: Find h [ n ] that best fits the input f [ n ] and noisy output d [ n ] . This is schemat ically shown in Fig. 62. This problem is also referred to as system identification. + f[n] z[n] d[n] h[n] LSI Figure 62: Least Squares Filtering Method 1 (Covariance method): In this method we use only data that is explicitly avail able without making assumptions about data not available. Assumption: 1. We are given f [1] , f [2] , ··· f [ N ] and d [1] , d [2] , ··· d [ N ] 80 2. Length of h [ n ] is m , i.e., h [ n ] = 0 for n < and n ≥ m . We would like to minimize N summationdisplay n = m ( d [ n ] − y [ n ]) 2 − − − ( A ) where y [ n ] = ∑ m 1 k =0 h [ k ] f [ n − k ] . Hilbert Space problem formulation: S = R N m +1 , ( x, y ) = x T y , V = span { p , p 1 , ··· , p m 1 } x = d [ m ] d [ m + 1] . . . d [ N ] , c = h [0] h [1] . . . h [ m − 1] , p i = f [ m − i ] f [ m − i + 1] . . . f [ m − i + N − m ] ⇒ A = [ p p 1 ··· p m 1 ] = f [ m ] f [ m − 1] ··· f [1] f [ m + 1] f [ m ] ··· f [2] . . . . . . . . . . . . f [ N ] f [ N − 1] ··· f [ N − m + 1] ⇒ h [0] h [1] . . . h [ m − 1] = ( A H A ) 1 A H x Observe that we started the index ’n’ from ’m’ in equation A so that the matrix A can be filled in without making assumption about f[n] and d[n] for n ≤ or n > N ....
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 Spring '11
 J
 Probability theory, probability density function, Hilbert space, σx, hilbert space problem

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