Estimate the autospectral density SXX(ωc) based only on a single long time history of the process {X(t)} and the results obtained from an ideal bandpass filter that transmits only those Fourier components with ω ≈ ± ωc. In particular, the harmonic transfer function can be written as

Question

Estimate the autospectral density [latex] S_{XX}(\omega _{c}) [/latex] based only on a single long time history of the process [latex] \left\{X\left(t\right) \right\} [/latex] and the results obtained from an ideal bandpass filter that transmits only those Fourier components with [latex] \omega \approx \pm \omega _{c} [/latex] . In particular, the harmonic transfer function can be written as [latex] H(\omega )=U(\left|\omega \right| -\omega _{c}+\epsilon )U(\omega _{c}+\epsilon -\left|\omega \right| ) [/latex] .

Accepted Answer

Let [latex] \left\{Z\left(t\right) \right\} [/latex] be the output from the filter when [latex] \left\{X\left(t\right) \right\} [/latex] is the input. This then gives us [latex] S_{ZZ}(\omega )=S_{XX}(\omega ) [/latex] within the pass band and [latex] S_{ZZ}(\omega )=0 [/latex] otherwise, as shown in the accompanying sketch. The [latex] \left\{Z\left(t\right) \right\} [/latex] } process will be mean-zero because the stationary mean value of [latex] \left\{X\left(t\right) \right\} [/latex] is a component at frequency zero and this component is blocked by the filter. Thus, the variance and mean-squared value of [latex] \left\{Z\left(t\right) \right\} [/latex] are both given by

[latex] E\left[Z^{2}(t)\right] =\int_{-\infty }^{\infty }{S_{ZZ}(\omega )}d\omega =2\int_{0}^{\infty }{S_{ZZ}(\omega )}d\omega =2\int_{\omega _{c}{-\epsilon }}^{\omega _{c}{+\epsilon} }{S_{XX}(\omega )} d\omega [/latex]

and if [latex] \epsilon [/latex] is small enough that [latex] S_{XX}(\omega ) [/latex] is nearly linear across the range of integration, then we have

[latex] E\left[Z^{2}(t)\right] \approx 4\epsilon S_{XX}(\omega _{c}) [/latex]

Thus, [latex] E\left[Z^{2}(t)\right] / (4\epsilon) [/latex] can be used as an estimate of the unknown [latex] S_{XX}(\omega _{c}) [/latex] .

Assuming covariance ergodicity of [latex] \left\{Z\left(t\right) \right\} [/latex] , the value of [latex] E\left[Z^{2}(t)\right] [/latex] can be estimated from a time average, giving

[latex] S_{XX}(\omega _{c})\approx \frac{1}{4\epsilon T}\int_{0}^{T}{Z^{2}(s)}ds [/latex]

for a large value of [latex] T [/latex] . Note that the finite bandwidth [latex] 2\epsilon [/latex] tends to smooth the measured autospectral density, which results in a peak in [latex] S_{XX}(\omega ) [/latex] being almost lost if it is not wider than [latex] 2\epsilon [/latex] . Based on this fact, it may appear that one would want to choose [latex] \epsilon [/latex] as small as possible to minimize this error due to variation of [latex] S_{XX}(\omega ) [/latex] across the pass band. On the other hand, it can be shown that the convergence of the time average of [latex] Z^{2}(t) [/latex] to its expected value is slower when the bandwidth is smaller. Thus, if one uses a very small [latex] \epsilon [/latex] value, then it is necessary also to have a very large [latex] T [/latex] value, which may not be feasible in some situations.

To estimate the entire [latex] S_{XX}(\omega ) [/latex] autospectral density function in this way, one must have a filter with a variable center frequency [latex] \omega _{c} [/latex] (and possibly a variable [latex] \epsilon [/latex] ), and repeat the procedure for different [latex] \omega _{c} [/latex] values until the function is defined with sufficient resolution. This technique for estimating [latex] S_{XX}(\omega ) [/latex] has been quite important in the past, but it would not be commonly used now, in an age of inexpensive digital computation. Section 6.9 discusses the approach that is usually used now to estimate [latex] S_{XX}(\omega ) [/latex] from recorded data.

Question 6.5: Estimate the autospectral density SXX(ωc) based only on a si...

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