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Project3 - ECE 5650/4650 Computer Project#3 Adaptive Filter...

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Introduction 1 ECE 5650/4650 Computer Project #3 Adaptive Filter Simulation This project is to be treated as a take-home exam, meaning each student is to due his/her own work without consulting others. The grading for this third computer project will be handled differ- ently than the first two. A separate grade category exits for this project, thus allowing this project to count up to 20% percent of the final grade (details given below). The project due date is 5:00 PM Thursday, December 15, 2011 (Final Exam week) . The grading options are as follows: (1) (current syllabus) Computer project #3 20%, final exam 25%; (2) Computer project #3 15%, final exam 30%. The other grade distribution percentages remain the same as the syllabus sheet discussed the first day of class. Introduction In this project you will be investigating adaptive noise cancellation (correlation cancellation) tech- niques using adaptive FIR and adaptive IIR filters. The input signal will be modeled as a real sinu- soid(s) in additive white noise. Specifically the systems investigated are called adaptive line enhancers or ALEs. A rework of the ALE for adaptive interference cancellation, is also consid- ered using a real speech waveform. Background Theory A common estimation theory problem is to estimate a random process of interest (signal) from an observed random process (e.g. signal plus noise). A solution is to use a causal minimum mean- square error (MMSE) filter (Wiener filter) to process the observations. A discrete-time Wiener fil- ter takes the form of an FIR filter with M + 1 weights. Let be the observations, be the signal to estimate, and be the filter weights. The MMSE estimate is of the form (1) The weights an, are chosen such that (2) is minimized. A block diagram of the filter is shown in Figure 1. The optimal weights are found by setting y n   x n   a m x ˆ n   a m y n m m 0 = M = a m m 0 1 M = E e 2 n   E x n   x ˆ n   2 =
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Background Theory 2 (3) The solution is a result of the projection theorem or orthogonality principle 1 , which says that we choose constants such that the error is orthogonal to the observations ( data ), i.e., (4) The resulting system of equations (5) are known as the normal equations , or in Papoulis the Yule-Walker equations. The function is the autocorrelation sequence corresponding to and is the cross correla- tion sequence between and . In an adaptive Wiener filter the error signal is fed back to the filter weights to adjust them using a steepest-descent algorithm . Note that the error surface generated by over the parameter space is convex cup (i.e. a bowl shape) as shown in Figure 2. The filter decorrelates the output error so that signals in common to both and in a correla- tion sense are cancelled. A block diagram of an adaptive FIR filter is shown in Figure 3.
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