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In the single-input, single-output case where y and w are both scalar random variables, eqn (448) reduces to Sy (j ) = |H(j )|2 Sw (j ) (449)

Eqns (448) and (449) are useful for nding a linear system realization H(j ) appropriate for producing a given colored noise process y from a white noise input w Example 414 A linear system with transfer function 2 H(s) = +s

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1 6 11 16 21 26 31 36 41 46 a b c b d c c b a c 2 7 12 17 22 27 32 37 42 47 a a c a d d d d d d 3 8 13 18 23 28 33 38 43 48 c e b a d a e d c a 4 9 14 19 24 29 34 39 44 49 b e c e c c d c e e 5 10 15 20 25 30 35 40 45 50 a c d e a c e a c c

2 is excited by white noise w(t) with PSD de ned by Sw (j ) = w , where w , > 0 The spectral density of the output y is calculated by eqn (449) as

Sy (j ) = = =

(450)

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The inverse transform of eqn (450) (which can be found using partial fractions) yields the autocorrelation function for the system output

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The previous example discussed a natural use of stochastic process theory to calculate the PSD and correlation of the output of a linear system output driven by white noise In at least two cases, the reverse process is of interest: 1 When the output autocorrelation function for a linear system with white noise inputs is known, and it is desired to nd the linear system transfer function 2 When the actual noise driving a process is colored, it may be more convenient for analysis to treat the colored noise as the output of a linear system driven by white noise Finding the appropriate linear system model is equivalent to solving Item 1 In either case, an autocorrelation function is given for a WSS stochastic process and the objective is to determine a linear system (represented by a transfer function G(s)) and a white noise process (represented by a PSD 2 w ) such that the output of the linear system driven by the white noise process yields the speci ed autocorrelation function Example 415 In the single channel error model of eqn (349), assume that the additive gyro measurement error g is determined to have the autocorrelation function R g ( ) = 10 4 e 10| | rad s

(451)

Find a causal linear system with a white noise input process so that the output of the linear system when driven by the white noise process has the

correlation function speci ed in eqn (451) Specify the augmented error state model for the system The spectral density of the output process v is

S g (j ) = =

10 4 20 10 4 10 4 + = = 10 + j 10 j 100 + 2 20 20 = (01)2 (10 + s) (10 s)

(452)

Therefore, by comparison with eqn (449), a suitable linear system gyro error model is 20 (453) G(s) = 10 + s and the PSD of the corresponding white driving noise process wg is Swg (j ) = 0012 rad/s2 Hz

ie, Rwg ( ) = 0012 ( )

(454)

Appending this state space realization model of eqn (349) yields x 0 1 0 0 v 0 0 g 0 = 0 1 0 20 R g 0 0 0 10 b

of the gyro error model to the error x 0 v 0 + 0 bg 1 wg (455)

1 6 11 16 21 26 31 36 41 46 e d c e b c a e b d 2 7 12 17 22 27 32 37 42 47 e c a e d b a c c e 3 8 13 18 23 28 33 38 43 48 a d d d c a a d c e 4 9 14 19 24 29 34 39 44 49 a a c c c e a c c a 5 10 15 20 25 30 35 40 45 50 e e c b e c b b d a

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