What is meant by “spectral moment”?

Question

I have consulted the almighty oracles of google and wiki, but I cannot seem to find a definition for the phrase "the moment of the spectrum".

A legacy work text I am reading uses it in the following manner, defining the number of zero-crossings per unit time as the following:

$$ N_0 = \frac1{\pi} \left(\frac{m_2}{m_0}\right)^{1/2}$$

It then goes on to further define the number of extrema per unit time as given by:

$$ N_e = \frac{1}{\pi}\left(\frac{m_4}{m_2}\right)^{1/2}$$

where it goes on to finally say, "where $m_i$ is the $i$th moment of the spectrum."

Anyone encounter this before? What is the "moment" of a spectrum? I have never heard of it in the DSP literature before.

Answer 1

Assume low-pass signals throughout.

Since $X(f)$ is usually complex-valued, using the power spectrum $|X(f)|^2$ is probably a better idea, especially if you want to take square roots etc. afterwards. Thus, $m_k$ is defined as $$m_k = \int_{-\infty}^\infty f^k |X(f)|^2 \mathrm df.$$ Note in particular that $m_0$ is the power in the signal, and $m_1 = 0$ Now, the Gabor bandwidth $G$ of a signal is given by $$G = \sqrt{\frac{\displaystyle \int_{-\infty}^\infty f^2 |X(f)|^2 \mathrm df}{\displaystyle\int_{-\infty}^\infty |X(f)|^2 \mathrm df}} = \sqrt{\frac{m_2}{m_0}}.$$ To put this in a slightly different perspective, $|X(f)|^2$ is a nonnegative function, and the "area under the curve $|X(f)|^2$," viz. $m_0$, is the power in the signal. Therefore, $|X(f)|^2/m_0$ is effectively a probability density function of a zero-mean random variable whose variance is $$\sigma^2 = \int_{-\infty}^\infty f^2 \frac{|X(f)|^2}{m_0} \mathrm df = \frac{\displaystyle\int_{-\infty}^\infty f^2 |X(f)|^2 \mathrm df}{\displaystyle\int_{-\infty}^\infty |X(f)|^2 \mathrm df} = G^2$$.

A sinusoid of frequency $G$ Hz has $2G = 2\sqrt{\frac{m_2}{m_0}}$ zero crossings per second. Since Mohammad is reading a legacy book, it may well be doing all this in radian frequency $\omega$, and thus if $G$ is the Gabor bandwidth in radians per second, we need to divide by $2\pi$ giving $$N_0 = \frac{1}{\pi} \sqrt{\frac{m_2}{m_0}} ~ \text{zero crossings per second.}$$

Turning to extrema, the derivative of $x(t)$ has Fourier transform $j2\pi f X(f)$ and power spectrum $|2\pi f X(f)|^2$. Its Gabor bandwidth is $$\begin{align*}G^\prime &= \sqrt{\frac{\displaystyle\int_{-\infty}^\infty f^2 |2\pi f X(f)|^2 \mathrm df}{\displaystyle\int_{-\infty}^\infty |2\pi f X(f)|^2 \mathrm df}}\\ &= \sqrt{\frac{\displaystyle\int_{-\infty}^\infty f^4 |X(f)|^2 \mathrm df}{\displaystyle\int_{-\infty}^\infty f^2 |X(f)|^2 \mathrm df}}\\ &= \sqrt{\frac{m_4}{m_2}}. \end{align*}$$ Using the same arguments as before (two zero-crossings of the derivative per period is the same as two extrema per period), radian versus Hertzian frequency, we get $$N_e = \frac{1}{\pi} \sqrt{\frac{m_4}{m_2}} ~ \text{extrema per second.}$$

Answer 2

I don't know that I've heard that term before, but I would interpret the term "moment" as having an analogous meaning to the physical concepts of center of mass and first and second moments of area:

$$ m_k=\int_{-\infty}^{\infty} f^kX(f)\ df $$

That is, the content at every frequency in the spectrum is weighted by the $k$-th power of the frequency and the result is summed up across the entire spectrum. Not sure if this is what you want, but it's a concept of a moment for a spectrum (or any function of a single variable, for that matter).

Answer 3

The ratios you mention are instances of standardized moments or $L$-moments. Moments in signal processing are similar to moments for physics, and moments in statistics. In physics, the notion of moment is:

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

It can be seems as a generalization of the concept of the center of mass. The mean, the standard deviation, or skewness and kurtosis are derived notions, and they can be computed in any domain, like time or frequency. Basically, the $\alpha$-moment of a function $g$ over domain $D$, around value $c$, is defined in integral form by:

$$m_{g_D}(\alpha,c) = \int_D (t-c)^\alpha g(t)d t$$ or $$m_{g_D}(\alpha,c) = \int_D [t-c|^\alpha g(t)d t$$ when needed. Classically, for real signals $x$, due to symmetry, in the Fourier domain with $X(f)$, spectral moments are defined with respect to the power normalized energy ($g(\cdot) = |X(\cdot)|^2$) $$m_\alpha = \int_{f\ge 0} f^\alpha \frac{|X(f)|^2}{\int_{\nu\ge 0} |X(\nu)|^2d\nu}df$$

See for instance: Efficient computation of spectral moments for determination of random response statistics for spectral moments: "Spectral moments are calculated from the one-sided PSD".

Turned into ratios of $L$-moments, they can become scale-fre, unit-free indicators of functions behaviors, including extrema, zero-crossing or sparsity (with $m_1/m_2$ for instance).