Chirped Signals and the Principle of Unit-Economy

Consider chirped audio signals. Their frequency is a monotonic function of time. There are signals characterized by a down-chirp, like that of a bat using its sonar echolocation ability to track its prey.

Figure 2.18: Phase space representation of a chirped signal occupying $N$ phase space cells. The grey level of each cell expresses the intensity of the corresponding wave packet.

There are also signals characterized by an up-chirp. The phase space representation of a typical example is depicted by the shaded phase space cells in Figure 2.18. An up-chirp signal starts at low frequency and stops at some maximum frequency, say

$$\omega_{max}=N\varepsilon~.$$

Here $N$ is the number of phase space cells which the signal occupies. For illustrative purposes consider a signal with a linear up-chirp,

$$f(t)=\sum_{n=1}^N \alpha_n P_{nn}^\varepsilon(t)~.$$ (273)

This representation is based on the by-now-familiar orthonormal wave packet functions Eq.(2.61),

$$P_{jl}^\varepsilon (t) = {1\over {\sqrt {2\pi\varepsilon}}} \int... ...epsilon}_{j\varepsilon} e^{-2\pi il\omega/\varepsilon} e^{i\omega t} d\omega~.$$

Recall that the constant $\varepsilon$ , which characterizes the shape

$$\Delta\omega=\varepsilon$$

$$\Delta t=\frac{2\pi}{\varepsilon}$$

of each phase space cell, is a parameter which identifies this family of wave packets, $\{ P_{jl}^\varepsilon (t):~j,\ell=0,\pm 1,\cdots \}$ .

However, there are also other families characterized by other parameter values. Consider another set of wave packets whose phase space cells have dimension

$$\Delta\omega=2\varepsilon$$

$$\Delta t=\frac{2\pi}{2\varepsilon}$$

These basis functions are obtained from $P_{jl}^\varepsilon (t)$ by making the replacement $\varepsilon\to 2\varepsilon$ . This amplifies and compresses the wave packets in the time domain. Indeed, the defining integral expression for $P_{jl}^{2\varepsilon} (t)$ yields

$$P_{jl}^{2\varepsilon} (t)=\sqrt{2} P_{jl}^{\varepsilon} (2t)~.$$

What is the representation of the given chirp signal with respect to this new basis? The answer is based on the transformation formula

$$P_{j'l'}^{\varepsilon} (t)=\sum_{j=-\infty}^\infty\sum_{\ell=-\in... ...repsilon} (t) \,\langle P_{jl}^{2\varepsilon},P_{j'l'}^{\varepsilon} \rangle~.$$

It is worth while to do the calculation explicitly because the answer turns out to be fairly simple and informative. The simplicity starts to become evident when one splits the chirped signal into odd and even labelled terms. Assuming without loss of generality that the number of terms is even, $N=2M$ , one has

$$f(t)=\sum_{m=0}^{M-1} \alpha_{2m+1}P^\varepsilon_{(2m+1)~(2m+1)}(t) +\sum_{m=1}^{M} \alpha_{2m}P^\varepsilon_{(2m)~(2m)}(t)~.$$ (274)

The odd and even transformation formulas are

$$P^\varepsilon_{(2m+1)~\ell}(t) =\frac{1}{\sqrt{2}} P^{2\varepsilon}_{m~2\ell}(t)+\frac{1}{\pi\sqrt{2}}\sum_{k=0}^\infty \frac{2i}{2\ell -2k-1} P^{2\varepsilon}_{m~(2k+1)}(t)$$
and

$$P^\varepsilon_{2m~\ell}(t) =\frac{1}{\sqrt{2}} P^{2\varepsilon}_{m~2\ell}(t)-\frac{1}{\pi\sqrt{2}}\sum_{k=0}^\infty \frac{2i}{2\ell -2k-1} P^{2\varepsilon}_{m~(2k+1)}(t)~.$$

Consequently, the chirped signal is given by

$${f(t)=}$$

$$\sum_{m=0}^{M-1}\left[ \alpha_{2m+1}\frac{1}{\sqrt{2}}P^{2\vareps... ...nfty \frac{2\alpha_{2m+1}}{2(2m+1) -2k-1} P^{2\varepsilon}_{m~(2k+1)}(t)\right]$$

$$+ \sum_{m=1}^{M}\left[ \alpha_{2m}\frac{1}{\sqrt{2}}P^{2\varepsilon... ...\infty \frac{2\alpha_{2m}}{2(2m) -2k-1} P^{2\varepsilon}_{m~(2k+1)}(t)\right]~.$$

Compare the two representations, Eq.(2.74) and Eq.(2.75), of the chirped signal. In Eq.(2.74) $f(t)$ is represented by a set of $N=2M$ basis vectors (in physics and engineering also known as ``degrees of freedom''),

$$\{ P_{nn}^\varepsilon (t):~n=1,\cdots,2M\}.$$ (276)

In Eq.(2.75) by contrast, $f(t)$ is represented by the basis

$$\{ P^{2\varepsilon}_{m~(4m+2)}(t):~m=0,\cdots,M-1\}\cup \{ P^{2\varepsilon}_{m~(4m)}(t):~m=1,\cdots,M\}\cup$$

$$\{ P^{2\varepsilon}_{m~(2k+1)}(t):~m=0,\cdots,M;\,k=0,\pm 1,\pm 2,\cdots \}~,$$ (277)

which has a substantially larger number of elements.

Is the choice of basis vectors arbitrary? The principle of unit-economy²¹applied to this example demands that one pick the wave packet basis Eq.(2.76), whose coefficients in Eq.(2.73) express the essential properties of the degrees of freedom of the chirped signal. One should not pick the other basis, Eq.(2.77), whose amplitudes in Eq.(2.75) are nonessential because specifying them might lead to signals which are not chirped at all.

The mathematical implementation of the principle of unit-economy to signal processing consists of the requirement that one pick an optimal basis to represent the set of signals under consideration. This means that one pick a subspace of minimal dimension in order to accomodate these signals.

Footnotes

... unit-economy²¹

The principle of unit-economy[#!Rand:1990!#,#!Rand1967!#], also known informally as the ``crow epistemology'', is the principle that stipulates the formation of a new concept

1. when the description of a set of elements becomes too complex,
2. when the elements of the set are used repeatedly, and
3. when the elements of the set require further study.

It is obvious that the last is the most important because that is the nature of cognition, pushing back the frontier of knowledge.

The principle of unit economy is implemented by a process of conceptualization, which is a method of expanding man's consciousness by reducing the number of its content's units - a systematic means to an unlimited integration of cognitive data.

The principle of unit-economy forbids the formation of a new concept if that formation is based on some nonessential property.

The principle of unit-economy is a statement not only about the structure of mathematics, but also more generally about why one forms concepts in the first place, be they first-order concepts based on perceptual data (``percepts''), or be they higher-level concepts based on already-formed concepts.

The principle of unit-economy is a guiding principle that leads us from an unlimited number of specific units (i.e. members of a class, in our example, signals characterized by a chirp) to a single new concept (in mathematics also known as an ``equivalence class'', in our example, the concept ``chirped signal''). By repeatedly applying this principle to percepts, as well as to the product of such applications, one can reduce a vast amount of information to a minimal number of units. These one's consciousness can readily keep in the forefront of one's mind, digest them, assimilate them, manipulate them, and use them without any danger of information overload.