# Origin of term spectrogram

I'm writing up a manuscript and I use the term spectrogram quite a bit. I've defined a spectrogram as the magnitude squared of the short time Fourier transform of my signal. It's an estimator for a time-varying spectrum of a non-stationary process. I'd like to cite a paper for the origin of the term but I'm not sure what to cite. Could anyone give me some advice?

• FWIW Wikipedia cites JL Flanagan, Speech Analysis, Synthesis and Perception, Springer- Verlag, New York, 1972 – Paul R Oct 10 '12 at 22:50
• Though that reference uses the term spectrogram I don't believe it's the best text to cite if you want the origins of the term. – ncRubert Oct 11 '12 at 0:13
• Oxford dictionary calls it, not the spectrum data itself, but the visual representation. So it's the name of a type of picture. Don't "confuse the map with the territory". – hotpaw2 Oct 11 '12 at 0:15
• @ncRubert : Even so, it would NOT be the square of the magnitude. It would be some sort of picture of the square of the magnitude. – hotpaw2 Oct 11 '12 at 17:26
• The term spectrogram as it widely used in the time-frequency signal processing literature is a quadratic time-frequency distribution. For a window h and a signal f, it is given by: $S_h(t_0,\omega) = | \int f(t)h^*(t-t_0) exp(-i 2 \pi t \omega) dt | ^2$ . You could refer to the plot of this as a spectrogram but it is a mathematical object in its own right irrespective of how I plot it. – ncRubert Oct 11 '12 at 18:18

IIRC, etymology of "Spectro" is "of radiant energy". "Gram" is derived from the Greek word gramma, γράμμα which is a written learning.

In Gabor's "Theory of Communication" (1947) these time-frequency plots are called "information diagrams" or "time/frequency diagrams". "Spectrogram" is another name for the diagram, not necessarily the signal; the wikipedia article says it is a "representation". Maybe time-varying spectrum would be a better term for the actual signal.

In any case, it would be better to reference the origin of the concept, rather than the origin of the word. The aforementioned paper may be a good one to start with. e.g. see Figure 1.6.

As many mentionned before, the term "Spectogram" refers to a representation of some data, with a similar etymology as the word "diagram".

The squared magnitude of the Fourier transform of a signal, short term or not, would be called a "Power Spectrum".

In fact, the power spectrum is defined as the Fourier transform of a signal multiplied by the complex conjugate of that Fourier transform. $$P(k)=F(x)\cdot F(x)^*=Re(F(x))^2+Im(F(x))^2$$ Where $P(k)$ is the power spectrum of your signal, with respect to the wavenumber $k$, and $F(x)$ denotes the Fourier transform of your signal ($x$ being your signal).

If your signal is purely symmetric, then there will be no imaginary part and the power spectrum would be equivalent to the squared magnitude of the Fourier transform.

You could then refer to your figures as "diagrams of the Power Spectrum", or do like me and refer to them as "Figure 1.5"!

I guess that some variations exist in different fields, as Wikipedia defines my mathematical construct as the "Energy Spectral Density" (http://en.wikipedia.org/wiki/Spectral_density) In a field where you work with actual physical density (mass/volume), we try to avoid using that word for anything else...

• A stationary signal possesses a power spectrum. A non-stationary signal does not have a well-defined power spectrum. The spectrogram is calculated in the context of a non-stationary signal, and is slightly different from the squared magnitude of the Fourier transform of the signal. There is a time dependence in the spectrogram. See my comment above. – ncRubert Oct 11 '12 at 22:08
• A "spectrogram" calculated using the STFT is effectively performing a Fourier transform of a finite period of the signal, as though that portion is periodic. So the values of the spectrogram at that point show the power spectrum of that portion of this signal, assuming it is periodic. – geometrikal Oct 11 '12 at 23:23
• In general, a non-stationary signal is not periodic. The windowed portions of the signal don't share the same statistics. That is why we are computing a spectrogram and not a periodogram. – ncRubert Oct 12 '12 at 0:28
• Hehe, when I talked about different fields having different standards... I worked mainly on turbulence, and astrophysics in general, and most signals were periodic. The cloud would be large enough so the portion we would see was considered periodic. At the limit, the whole sky is circular so is is considered periodic! My answer should be modified to take windowing into account. – PhilMacKay Oct 12 '12 at 20:56