Therefore, this section considers the statistical properties of the output of a linear system when the system model is known and the input is white.
The results of this section are often used to model error components with significant time correlation via the state augmentation approach. In such situations, linear stochastic systems forced by Gaussian white noise are of special interest. Careful selection of the linear system and white noise statistics can often yield an output stochastic process with statistics matching those of the time correlated error process. Examples of such processes are further discussed in Section 4.6.3.
The output y(t) of a (deterministic) time-invariant, causal, linear system with (stochastic) input w(t) is
where h represents the impulse response matrix of the linear system. Because w(t) is a random variable, so is y(t). The Fourier transform of the impulse response yields the system transfer function
The Energy Spectrum of a time-invariant linear system is
where
represents the complex conjugate transpose of
If
denotes the mean of the system input, then the mean of the system output is
where the transition from the second to the third line uses the following facts: expectation and time-integration are linear operators and the impulse response h is deterministic. Eqn. (4.41) shows that the mean of the output process is the convolution of the impulse response with the mean of the input process. In the case where the system is causal and the mean of the input process is constant, the mean of the output reduces to
The output autocorrelation function is slightly more complicated to derive. Its derivation is addressed in two steps. First, from eqn. (4.29), we find an expression for the cross correlation between y and w,
Because the linear system is causal, this can be expressed as
Second, we derive the expression for the autocorrelation of y,
If w is wide sense stationary, then eqns. (4.42-4.43) with
reduce to
The Fourier transforms of eqns. (4.44-4.45) provide the cross power spectral density of y and w and the power spectral density of y
In the single-input, single-output case where y and w are both scalar random variables, eqn. (4.48) reduces to
Eqns. (4.48) and (4.49) are useful for finding a linear system realization 
appropriate for producing a given colored noise process y from a white noise input w.
Example 4.14 A linear system with transfer function
is excited by white noise w(t) with PSD defined by
where
The spectral density of the output y is calculated by eqn. (4-4-9) as is excited by white noise w(t) with PSD defined by
where
The spectral density of the output y is calculated by eqn. (4-4-9)
The inverse transform of eqn.. (4-50) (which can be found using partial fractions) yields the autocorrelation function for the system output
The parameter
is the correlation time of the random variable y.
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 find 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 aW) such that the output of the linear system driven by the white noise process yields the specified autocorrelation function.
Example 4.15 In the single channel error model of eqn. (3-49), assume that the additive gyro measurement error eg is determined to have the autocorrelation function
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 specified in eqn. (4.51). Specify the augmented error state model for the system.
The spectral density of the output process v is
Therefore, by comparison with eqn. (4.49), a suitable linear system gyro error model is
and the PSD of the corresponding white driving noise process wg is
A state space realization of the gyro error transfer function G(s) is
This state space realization of the gyro error model to the error model of eqn. (3.49) yields
Note that the noise process and linear system must be specified jointly, since there are more degrees of freedom than design constraints. Also, the selection of G(s) from the two factors of eqn. (4.52) is not arbitrary. The transfer function G(s) is selected so that its poles and zeros lie in the strict left half of the complex s-plane. This results in a causal, stable linear system model. Such a choice is always possible, when the spectral density is the ratio of polynomials containing only even powers of s. The interested reader is referred to the subject of spectral factorization.
When the driving noise is white and Gaussian, the process specified in eqn. (4.54) is an example of a scalar Gauss-Markov process. Higher order vector Gauss-Markov processes, such as the system of eqn. (4.55), result when the correlation function is more complex. Additional examples of simple Gauss-Markov process that are often useful in navigation system error model are discussed in Section 4.6.3.
