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Unformatted text preview: Lecture #6 Nonparametric Identification with Random Signals Summary This lecture covers the basic method of determining the frequency response of discretetime LTI systems using input signals that have random characteristics. We do not use the theory of stochastic processes explicitly, instead, the second order statistics of signals and their relations to LTI systems are developed. This is a very convenient method to avoid straying too far into the technicalities of stochastic processes. We do this in three stages: 1. Second order properties of signals; Expectations, Correlations and Spectra 2. Relations between a system’s frequency and impulse response on the one hand and second order statistics of input and output signals on the other 3. Identification of frequency reponses The development is done for discretetime signals and systems though it can also be done in the continuous time case as well. Contents 6.1 Second order properties of signals . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.2 Connection with LTI systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.3 Identification using correlation data . . . . . . . . . . . . . . . . . . . . . . . 51 6.1 Second order properties of signals Consider a discretetime signal u ( k ) defined for all time∞ < k < ∞ . The following development applies whether the signal u is either random or deterministic with only a 45 46 Jo˜ao P. Hespanha minor change in language. The basic operation we will need is that of taking averages, specifically the asymptotic average of u defined by E { u } := lim N →∞ 1 2 N + 1 N k = N u ( k ) . This operation can be understood as follows. Take the data in the signal u over a time window of length 2 N + 1, find the average, and then take the limit as the size of the window goes to infinity ( N → ∞ ). In practice, one has only finite data, so the underlying assumption in this method is that with N “large enough” one approximates the above defined quantity. If u ( k ) is the result of the k ’th trial of a random variable, then the above quantity is just asymptotic average of all the trials. We use the symbol E {} to denote the asymptotic average because if u is a stochastic process that is ergodic , then the above time average would be the same as the expectation of u (which is also referred to as the “sample average”). We will not need to investigate these concepts here since we assume that the signals we deal with are deterministic. The operator E {} has the important property of linearity. If u 1 and u 2 are two signals, then E { αu 1 + βu 2 } = α E { u 1 } + β E { u 2 } . This property is easily verifiable from the definition....
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 Spring '07
 RODWELL
 Frequency, Signal Processing, Autocorrelation, LTI system theory, White Gaussian noise

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