Lect9 - Stochastic Process Lecture 9 Random Sequence(2...

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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 9-1
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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 . 9-2
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(4) A system is called time-invariant 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 time-invariant operator. Then, the time-invariant property tells us x [ n - k ] L {} y [ n - k ] . So, for a linear time-invariant (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. 9-3
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(5) ( Fourier Transform ) The Fourier transform, or more precisely the discrete-time 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 - ( n - k + k ) = X k x [ k ] e - jωk ˆ X n h [ n - k ] e - ( n - k ) ! = X ( ω ) H ( ω ) . ¥ 9-4
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2 Random Sequences in LTI Systems (1) When the input X [ n ] to a linear system is a random sequence, the output Y [ n ] = L { X [ n ] } is also a random sequence in the sense that for each outcome ε Ω , the deterministic sample X [ n, ε ] is mapped to Y [ n, ε ] = L { X [ n, ε ] } . That is, overall, Y [ n, ε ] is yet another mapping from each outcome ε of the sample space to a new sample sequence .
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