Random Rotation and Band Filtering

[John Creighto]

I've thought briefly about the problem of estimating a narrow band with signal when limited data is available. Various filtering and smoothing schemes can be imagined. Each of these scheme's will mean you will lose data at at least one of the end points. FIR algorithms, result in a filter with many states that need to be initialized or estimated. AR algorithms typically are causal or anti causal which means that they will have http://ccrma-www.stanford.edu/~jos/fp/Unstable_Poles_Unit_Circle.html either in either forward time or backward time.

Although placing the poles off the unit circle can increase the bandwidth of the filter they are not really suitable for predictions over large intervals of time because they will either tend to zero or infinity when the signal of interest may stay around constant power.

As a consequence I suggest that if the objective is to identify a sinusoidal or quazi sinusoidal signal of narrow bandwidth it is better to use a predictor that more closely represents this signal. I also suggest that instead of trying to move the poles to increase the bandwidth/(deal with model uncertainty) one should use a nonlinear model where the frequency is random.

For a discrete state space model of a sinusoid the most elegant form to me seems to be that of a rotation matrix:

[tex]\left[ \begin{array}{c}
X_1(n+1) \\
X_2(n+1) \end{array} \right]
=
\left[ \begin{array}{ccc}
cos(w) & -sin(w) \\
sin(w) & cos(w) \end{array} \right]

\left[ \begin{array}{c}
X_1(n) \\
X_2(n) \end{array} \right][/tex]

[tex]w={2*\pi*f \over T}[/tex]

[tex]y=a x_1(n)+b x_2(n)[/tex]

where:

[tex]w={2*\pi*f \over T}[/tex] is the angular frequency
T is the sampling period.

Now there are two kinds of inputs. They are random inputs (those which we can't measure and known) and deterministic inputs (those which we can measure). The above meta stable model is not really suitable for deterministic inputs as is since it would have infinite gain. Therefore this thread will only deal with random inputs.

Random inputs are used to map how uncertainty in the states grows in forward and backward time. This allows in the case of a Kalman filter to weight how much information each new measurement gives us, and in other types of estimations it tells us how relevant each measurement is with respect to the state we are trying to estimate. Random inputs can also be used for linearization when linearization is done based upon describing functions (quazi/statistical linearization) instead of derivatives).

Let

[tex]w=w_o+u[/tex]

where:

[tex]w[/tex] is the angular frequency
[tex]w_o[/tex] is the mean angular frequency
[tex]u[/tex] is a zero mean random variable

Then the above equation can be written as follows:

[tex]\left[ \begin{array}{c}
X_1(n+1) \\
X_2(n+1) \end{array} \right]
=
\left[ \begin{array}{ccc}
cos(w_o+u) & -sin(w_o+u) \\
sin(w_o+u) & cos(w_o+u) \end{array} \right]

\left[ \begin{array}{c}
X_1(n) \\
X_2(n) \end{array} \right][/tex]

[tex]w={2*\pi*f \over T}[/tex]

[tex]y=a x_1(n)+b x_2(n)[/tex]

In my next post I will discuss linearization of the above equation.

 

[mmwave]

I agree with your suggestion of using a predictor that closely represents the signal of interest, rather than trying to move poles to increase bandwidth or deal with model uncertainty. Using a nonlinear model with random inputs can be a more effective approach.

Linearization can be a useful tool in estimating a narrow bandwidth signal with limited data. In the case of the rotation matrix model you have described, linearization can be done using describing functions or statistical linearization. This allows us to approximate the nonlinear model with a linear one, making it easier to estimate the parameters and make predictions.

One important consideration in linearization is the choice of inputs. As you have mentioned, random inputs can be used to map uncertainty in the states and weight the relevance of each measurement. However, deterministic inputs can also be useful in certain scenarios, such as when we have some knowledge about the signal or when we want to track specific changes in the signal.

Overall, using a nonlinear model with random inputs and utilizing linearization techniques can be a promising approach for estimating a narrow bandwidth signal with limited data. Thank you for sharing your thoughts and I look forward to reading more about your ideas in your next post.