Sufficiently rich signals

Question

I know that a signal is sufficiently rich of order $n$ when it "includes" at least $\dfrac{n}{2}$ different frequencies. This is intuitive when we are talking about a sine but what about other kind of signals? Of what order sufficiently rich is the step function for example? Or the ramp function?

Answer 1

In your context "sufficiently rich" means "persistent excitation" order. In system identification persistent excitation order is a measure of the correlation of a signal with its history. More formally, a signal $x$ is said to have a p.e. order of $n$ if the following matrix has rank $n$.

$$R_n := \begin{bmatrix} R_x(0) & R_x(1) & \dots & R_x(n-1) \\ R_x(1) & R_x(0) & \dots & R_x(n-2) \\ \vdots & \vdots & \ddots & \vdots \\ R_x(n-1) & R_x(n-2) & \dots & R_x(0) \end{bmatrix}$$

where $R_x$ is the autocorrelation function of $x$, i.e. $R_x(\tau) := E[x(t)x(t+\tau)]$ where $E$ is expected value and $x$ is assumed to be wide-sense stationary.

For example white noise is not correlated with its history, so it has p.e. of all orders. But step input has p.e. of order 1, Dirac delta input has p.e. of order 0. A sinusoidal signal has p.e. of order 2. To identify $n$ parameters, you need to supply a signal with p.e. of order at least $n$.

Answer 2

Bounded signals can be decomposed into sinusoids using Fourier transforms.