# What is the real meaning of “frequency” for a non periodic signal?

For a non periodic signal, in the first one second the frequency of a particular amplitude (value) may be of 4 Hz, and then in the next one second it may or may not be 4 Hz. Then which value of frequency should you consider?

• The notion of " frequency of a particular amplitude (value)" is not fully clear to me. Could you please elaborate? – Laurent Duval Aug 11 '16 at 19:41
• Please specify your question. Do you refer to DFT amplitudes? – M529 Aug 11 '16 at 20:04
• There no "real" meaning. Instead there are a lot of different meanings that depend on context. Fourier says that any (non-pathological) signal can be broken down into the sum of nice clean periodic sinusoids. Consider that one. – hotpaw2 Aug 11 '16 at 23:39
• A side note, a non periodic signal can contain only pure sine waves. For example $\sin(t)+\sin(\pi t)$ is not periodic, because the ratio of the periods of the two frequencies is irrational. – fibonatic Aug 13 '16 at 1:24

There are two (among many other) ways of looking at frequency of periodic signals. One is how often a periodic signal repeats the same waveform. For example, a square wave which is 1 for 0.5 second and -1 for 0.5 second (and repeats) can be viewed as a waveform that repeats the same pattern every second. In that sense, it has a frequency of 1 cycle/second. The second way of looking at frequency is in terms of sinusoidal waves. According to Fourier series, many periodic signals can be expressed as a sum of infinite number of sine/cosine waves (called harmonics) . In that sense, the same square wave is expressed a sum of infinite sinusoidal waves, with frequencies 1Hz, 2Hz, 3Hz, ...

Frequency of non periodic signals cannot be expressed in either of the above mentioned ways. However, there is a way to split certain aperiodic signals into infinite sinusoidal signals using a technique called Fourier transform (which is similar to Fourier series mentioned above). Using that technique, an aperiodic signal can be represented using a continuous band of frequencies. Some signals can be represented using a finite band of frequencies (called its bandwidth). For example, some special aperiodic signal may be represented by a frequency band of 5-13Hz (whose bandwidth is 13-5 = 8Hz). Some other aperiodic signals may need infinite bandwidth to completely represent them.

Many signals may not have a way to represent either in terms of fourier series or fourier transform. It is difficult to talk about frequency for such signals.

The notion of frequency can be made very broad. For a cisoid (a complex sinusoidal function), a pure real sine or a cosine, defined with one specific frequency, you can safely say that the signal has "a frequency".

Outside this simple case, a finite-length, a non-periodic, a sampled signal, provided the mathematical definitions make sense, can be equipped with definitions of objects that extend the notion of "frequency" defined by a single number (4 Hz in your case): Fourier transforms, frequency spectrum, power spectral density, taking globally, or instantaneous frequency, that attempts to define a notion of frequency at each sample.

Nota: while looking for the origin of the word "cisoid" (abbreviated as $\mathrm{cis}$), I just found Cisoidal Oscillations, George A. Campbell, 1911:

The oscillations here defined as " cisoidal oscillations " are those of the form $C \mathrm{cis} pt = C (\cos pt+i \sin pt) = Ce^{ipt}$

• I've never heard the word cisoid. Thanks for that :) – M529 Aug 11 '16 at 20:05
• @M529 My pleasure, I like that compact name. – Laurent Duval Aug 11 '16 at 20:23