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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?

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    $\begingroup$ The notion of " frequency of a particular amplitude (value)" is not fully clear to me. Could you please elaborate? $\endgroup$ – Laurent Duval Aug 11 '16 at 19:41
  • $\begingroup$ Please specify your question. Do you refer to DFT amplitudes? $\endgroup$ – M529 Aug 11 '16 at 20:04
  • $\begingroup$ 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. $\endgroup$ – hotpaw2 Aug 11 '16 at 23:39
  • $\begingroup$ 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. $\endgroup$ – fibonatic Aug 13 '16 at 1:24
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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.

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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} $

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

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