Energy of a signal

Question

Suppose that $E(k)$ be the energy distribution of a signal in spectral domain with a finite total energy. Is there anyway to concentrates all the energy of signal around a specific point?

Answer 1

Without understanding the context of your question, you might consider a Class C power amplifier used in broad cast radio.

It uses a high Q resonator with nonlinear feedback. It is not a DSP technique but does concentrate energy if you have a loose definition of input.

The best you can do with a linear process is to excite a high Q resonance.

Answer 2

Yes indeed. Bear with me, I am in a mood for a long explanation.

Remember physics of motion, and how one concentrates the information related to the motion of an object on its center of mass, that capture a unique point from a distribution of mass.

A similar reasoning can be applied to positive quantities $q(k)$ (like energy) whose sum is finite. Thus, you can normalize them to unitless and positive quantities $q(k)/\sum_k q(k)$ that sums to $1$, similarly to probability distributions.

There, to characterize or summarize distributions with a few values, you can use the notion of moments (probably borrowed from the same moment notion in physics) or cumulants:

an expression involving the product of a distance and another physical quantity, and in this way it accounts for how the physical quantity is located or arranged

The $\alpha$ moments $m_q(c,\alpha)$ of $q(\cdot)$, centered around $c$, are given by:

$$m_q(c,\alpha) = \sum_l (l-c)^\alpha \frac{q(l)}{\sum_k q(k)} = \frac{\sum_l (l-c)^\alpha q(l)}{\sum_k q(k)} \,.$$

Here, the $q$s can be interpreted as unit-sum weights, affecting the "distance" $(l-c)^\alpha$ (more a power-norm, to be honest).

So, in discrete signal processing, using orthogonal or energy-preserving transformations, it is customary to take $q(\cdot)$ as the energy of signal samples, either in the time or the frequency domain. And $l,k$ are discrete indices. For distribution location, we usually set $\alpha=1$.

So for finite-energy signal $s[n]$, you can define its "center mass" in time with:

$$ \overline{s} =\frac{\sum_n n |s[n]|^2}{\sum_n |s[n]|^2}$$ and in frequency by: $$ \overline{E} =\frac{\sum_k k E[k]}{\sum_k E[k]}$$ but beware in summing on positive frequencies only for a meaninful interpretation for real signals (as spectra are symmetric, and would yield a center of frequency mass at $0$).

To illustrate it with a picture and a Matlab code (using FFTR.m):

clear all;
timeStart = 0;
timeStop = 1;
nPoint = 1024;
time = linspace(timeStart,timeStop,nPoint)';
dataFreqSampling = 1/median(diff(time));
dataFreq = 56;
dataSpread  = 0.2;
data = sin(2*pi*dataFreq*time).*exp(-((time-(timeStop-timeStart)/16)/(dataSpread)).^2);

dataEnergy = data.^2;
timeCenterOfMass = sum(time.*dataEnergy)/sum(dataEnergy);

[fftR,fftAxe] = FFTR(data,1/dataFreqSampling);
dataEnergyFreq = fftR.^2;
freqCenterOfMass = sum(fftAxe.*dataEnergyFreq)/sum(dataEnergyFreq);

figure;clf;
subplot(2,1,1);hold on
plot(time,data);axis tight;
plot(timeCenterOfMass,0,'o');axis tight;grid on
xlabel('Time (s)');ylabel('Amplitude')
legend('Time Signal','Center of mass')

subplot(2,1,2);hold on
plot(fftAxe,fftR);axis tight;
plot(freqCenterOfMass,0,'o');axis tight;grid on
xlabel('Frequency (Hz)');ylabel('Amplitude')
legend('Frequency Spectrum','Center of mass')

More details about:

• moment names in statistics: Higher-order cumulant and moment names beyond variance, skewness and kurtosis
• How to calculate the mean/center frequency of the spectrum?
• What is meant by “spectral moment”?