8.9 Aliasing  8.8 Examples  8.8.4 Effect of a

8.8.5 The autoregressive process

Consider a sensor with a characteristic response time t_R, driven by a varying signal Z(t). The sensor for example might be a thermometer that changes temperature toward the air temperature X(t) to which it is exposed, so that if X(t) is the temperature of the sensor then

$${{dX(t)}\over{dt}} = {{1}\over{T}}[Z(t)-X(t)] \ .$$ (8.44)

 

The solution is

$$X(t) = {{1}\over{t_R}} \int_{-\infty}^t Z(t^\prime)e^{-(t-t^\prime)/t_R} dt^\prime$$ (8.45)

 

as can be verified by differentiation. X(t) is thus the convolution of Z(t) with a transfer function h(u) given by

$$= {{1}\over{t_R}} e^{-u/t_R} ,\ 0\le u\le\infty$$ h(u)

$$= 0 ,\ u < 0$$ (8.46)

 

The effect is the same as applying a filter to the input signal Z(t), and the variance spectrum of X(t) can then be determined as in the preceding example.

In the case where Z(t) is a random variable or "white noise" source, the variance spectrum from this source would be constant with frequency:

$$\Gamma_{ZZ} = C.$$ (8.47)

 

The transform of the decaying exponential function h(t) is

$$\tilde h(\nu)={{1}\over{1+i2\pi\nu t_R}}$$ (8.48)

 

The variance spectrum is given by (8.42) as

$$\Gamma_{ZZ}(\nu) = {{C}\over{1+(2\pi\nu t_R)^2}}$$ (8.49)

 

and therefore the autocorrelation function, obtained from the Fourier transform of (8.49), is a decaying exponential:

$$\rho(\tau) = e^{-\tau/t_R}.$$ (8.50)

 

The effect of the filter function is to cause the autocorrelation function to assume an exponential form if the input is random, and the effect on the variance spectrum is to give a spectrum that for large frequency decays as $\nu^{-2}$. These are important models for instrumental response to noise.

   
 8.9 Aliasing  8.8 Examples  8.8.4 Effect of a 

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