Finding constants to get a required frequency from a discrete signal processing function

Question

I use a simple filter in my programing for many situations. Mathematically I think its called a progression.

The function takes the form.

func (v){
   c = (c + (v - r) * a) * b;
   r = r + c;
   return r;
}

where

• $v$ is any number and is the input value, each time step it may be a different.
• $a$ and $b$ are constants within the range $0 < a$ and $0 < b < 1$
• $c$ is the change over the time step, it is static (programing lingo for it remembers its last value). Its initial value is normally $c = 0$
• $r$ is the result, it is static. It initial value is set to $v$ on the first time step.

It can be thought of as a signal processing function. The function has the effect of smoothing out a signal.

The following images show the response of $c$ (green) and $r$ (red) to the input $v$ (blue) over time

As you can see the value of $r$ approaches $v$ over time, the signal is smoothed out.

For some values of $a$,$b$ I get a the following response.

The value of $r$ and $c$ oscillate over time. As long as $b < 1$ the oscillations seem to flatten out.

I am very interested in the period of the oscillations. From experimentation, for the same values of $a$ and $b$ and no matter the input value $v$ the oscillations seem to have a constant period (to within my ability to measure it)

What I would like to do is find what values of $a$,$b$ I can use to get a specific frequency.

My current and ugly solution is a statistical solution, where I sample the frequency of $c$ over many different values of $a$,$b$ and for many different input signals $v$. I can then select a value for $b$ and use the table to find what value $a$ should be for a required frequency.

The following image show a some of the sampled data and hints to me that there is a formula that will fit this data.

$b$ is on the x axis, $a$ is the colour coded index and the y axies is increasing period going up (high frequencies are down with infinity at $y = 0$)

My solution is not accurate especially at higher frequencies.

Is there a way to compute possible values of $a$ and $b$ given a required period/frequency?

My best attempt is this

$f(b)\ =0.8+\left(1.2-a\right)\frac{\left(1.2-a+\frac{\left(b-0.4\right)}{2}\right)^{2\left(1.2-a\right)}}{\frac{\left(b-0.4\right)}{2}}$

$f(b)$ returns the period. It is just a guess on my part but its close as it has a similar look the the sampled data when I change the value $a$ but I have given up on playing with the constants to find a fit.

It looks like this

Sorry I have no idea what tags this should come under, hope I got it correct.

Any help is appreciated if just to point me in the right direction or tell me it's not possible. Thank you. :)

Answer 1

In signal processing terms you've implemented a recursive filter. The difference equations corresponding to your code are

$$\begin{align}c[n]&=b\cdot (c[n-1]+a\cdot(x[n]-y[n]))\\ y[n]&=y[n-1]+c[n]\end{align}\tag{1}$$

where I've followed the convention to use $x[n]$ for the input signal, and $y[n]$ for the output signal.

Such recursions can be solved using the Z-transform. Applying the Z-transform to $(1)$ gives:

$$\begin{align}C(z)&=b\cdot [C(z)z^{-1}+a\cdot(X(z)-Y(z))]\\ Y(z)&=Y(z)z^{-1}+C(z)\end{align}\tag{2}$$

which can be solved for $Y(z)$ in terms of $X(z)$:

$$H(z)=\frac{Y(z)}{X(z)}=\frac{ab}{1+(ab-b-1)z^{-1}+bz^{-2}}\tag{3}$$

The function $H(z)$ is called the system's transfer function. It completely characterizes the system.

The roots of the denominator of $(3)$ determine whether the system's response to a step oscillates or not. If the roots are real-valued there is no oscillation, and if there are complex-conjugated roots there will be oscillations.

Let us define $d=ab-b-1$. Then the roots of the denominator of $(3)$ are

$$z_{\infty}=-\frac{d}{2}\pm\sqrt{\frac{d^2}{4}-b}\tag{4}$$

So there are only oscillations if $b>\frac{d^2}{4}$, i.e., if there are complex roots. The frequency of the oscillations is determined by the angle of the complex roots. Assuming $b>\frac{d^2}{4}$, we have

$$\theta=\arg\{z_{\infty}\}=\begin{cases}\arctan\left(-2\frac{\sqrt{b-\frac{d^2}{4}}}{d}\right),&d<0\\ \arctan\left(-2\frac{\sqrt{b-\frac{d^2}{4}}}{d}\right)+\pi,&d>0\end{cases}\tag{5}$$

The oscillation frequency is simply $f_0=\theta/(2\pi)$.