zz-35 - ECE 6010 Programming Assignment#2 System and...

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ECE 6010 Programming Assignment #2 System and Autoregressive Identification 1 Introduction System identification is the means by which systems are modeled mathematically based on measured data. It is often a precursor to other engineering tasks, such as control or signal processing. The system models are frequently described parametrically. System identification is sometimes done using only measurements of the output of the system. In this exercise, however, it is assumed that both input and output data are available. (Usually having both input and output data significantly simplifies the system identification, often resulting in straightforward linear equations to solve.) It is usually assumed that the system measurements are made in the presence of noise. A common assumption is that the noise signals are white. However, to make things more interesting, in this assignment the noise is assumed to be from an AR process, whose parameters are also to be identified. Your task is to take measured data and from it, determine the system and noise parameters. 2 System Model For brevity, we will employ an operator notation frequently used in the literature. If G ( z ) = L g X i =0 g i z - i . we will use the notation s ( t ) = G ( z ) u ( t ) as a shorthand for the filtering operation s ( t ) = L g X i =0 g i u ( t - i ) . A known discrete-time input signal u ( t ) is applied to a system which is assumed to be FIR, having transfer function G ( z ) = L g X i =0 g i z - i , where we will also assume that the order of the filter L g is known. The filtered signal s ( t ) = G ( z ) u ( t ) is corrupted by an additive noise signal which is assumed to be autoregressive (AR): ν ( t ) = H ( z ) e ( t ) where e ( t ) is a zero-mean, stationary, ergodic, white-noise signal with variance σ 2 e , and H ( z ) has the all-pole form H ( z ) = 1 1 - L H i =1 a i z - i . (1) The measured output signal is thus y ( t ) = s ( t ) + ν ( t ) = G ( z ) u ( t ) + ν ( t ) = G ( z ) u ( t ) + H ( z ) e ( t ) . G ( z ) ν ( t ) y ( t ) u ( t ) s ( t ) 1
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The system identification problem to be addressed here is this: given the input/output measurements { u ( t ) , y ( t ) } , estimate G ( z ) and H ( z ). 2.1 The AR Model and Its Prediction The model for the noise ν ( t ) can be written another way. The IIR filter H ( z ) could be written (say, using long division) as H ( z ) = X i =0 h i z - i . with h 0 = 1. (That is why we call it IIR — it has an infinite number of coefficients in its impulse response.) So ν ( t ) = X i =0 h i z - i ! e ( t ) = e ( t ) + X i =1 h i e ( t - i ) . For the development below, we will find it convenient to develop a predictor for ν ( t ). Given the sequence of measurements ν ( s ) for s t - 1, what is the best estimate of ν ( t )? We will denote this estimate by ˆ ν ( t | t - 1). If by “best” we mean best in the minimum mean-squared error sense, we want to find an estimator ˆ ν ( t | t - 1) which minimizes E [( ν ( t ) - ˆ ν ( t | t - 1)) 2 ] . We know that this will be the conditional mean: ˆ ν ( t | t - 1) = E [ ν ( t ) | ν ( s ) , s = t - 1 , t - 2 , . . . ] .
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