# Energy of a signal

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?

• Is the "specific point" that you are thinking about in the frequency domain or in the time domain? Are nonlinear methods acceptable or are you thinking about linear filtering only? Feb 23 '19 at 17:07
• Bandpass filter with a very narrow passband plus normalization to the original level? Feb 23 '19 at 18:13

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.

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;

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')