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Unformatted text preview: Stochastic Process 11/24/2006 Lecture 9 Random Sequence (2) NCTUEE Summary In this lecture, I will discuss: • Linear Time Invariant System • WSS in LTI • Power Spectral Density • Markov Chain Notation We will use the following notation rules, unless otherwise noted, to represent symbols during this course. • Boldface upper case letter to represent MATRIX • Boldface lower case letter to represent vector • Superscript ( · ) T and ( · ) H to denote transpose and hermitian (conjugate transpose), respectively • Upper case italic letter to represent RANDOM VARIABLE 91 1 Linear Time Invariant System (1) Definition of a system A system is a mapping that transforms an arbitrary input sequence into an output sequence. (2) A system is linear if to a linear combination of inputs corresponds the same linear combination of outputs. That is, suppose we know x 1 [ n ] → L {} → L { x 1 [ n ] } and x 2 [ n ] → L {} → L { x 2 [ n ] } . Then, the linear system with operator L {} guarantees a 1 x 1 [ n ] + a 2 x 2 [ n ] → L {} → a 1 L { x 1 [ n ] } + a 2 L { x 2 [ n ] } . (3) The impulse response h [ n ] of a linear system is the output sequence when the input is an impulse δ [ n ], h [ n ] , L { δ [ n ] } . For any input sequence x [ n ], we can write x [ n ] = ∞ X k =∞ x [ k ] δ [ n k ] . If the system is linear, then the output sequence y [ n ] is y [ n ] = L { x [ n ] } = L ( ∞ X k =∞ x [ k ] δ [ n k ] ) = ∞ X k =∞ x [ k ] L { δ [ n k ] } = ∞ X k =∞ x [ k ] h [ n,k ] , where we define h [ n,k ] , L { δ [ n k ] } as the output response at time n to an impulse applied at time k . 92 (4) A system is called timeinvariant if shifting the input by k time units results in shifting the output by k time unit. That is, suppose we know x [ n ] → L {} → y [ n ] with L {·} being a timeinvariant operator. Then, the timeinvariant property tells us x [ n k ] → L {} → y [ n k ] . So, for a linear timeinvariant (LTI) system with impulse response h [ n ], the output sequence is y [ n ] = ∞ X k =∞ x [ k ] L { δ [ n k ] } = ∞ X k =∞ x [ k ] h [ n k ] = x [ n ] * h [ n ] , which is the convolution between the input and the impulse response. 93 (5) ( Fourier Transform ) The Fourier transform, or more precisely the discretetime Fourier transform, for a sequence x [ n ] is defined by X ( ω ) , ∞ X k =∞ x [ n ] e jωn for π ≤ ω ≤ π. The inverse Fourier transform is x [ n ] = 1 2 π Z π π X ( ω ) e jωn dω. (6) The Fourier transform of y [ n ] = x [ n ] * h [ n ] is Y ( ω ) = X ( ω ) H ( ω ) . Proof: Y ( ω ) = X n y [ n ] e jωn = X n ‡ x [ n ] * h [ n ] · e jωn = X n X k x [ k ] h [ n k ] e jωn = X n X k x [ k ] h [ n k ] e jω ( n k + k ) = X k x [ k ] e jωk ˆ X n h [ n k ] e jω ( n k ) !...
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 Winter '10
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