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I've been involved in Signal processing for quiet a time but I'm still so confused of what frequency could possibly is, as it has different meanings in different scenarios, for example,

According to Wikipedia,

Frequency is the number of occurrences of a repeating event per unit time.

Calculating the frequency of a repeating event is accomplished by counting the number of times that event occurs within a specific time period, then dividing the count by the length of the time period. For example, if 71 events occur within 15 seconds the frequency is:

enter image description here

This is the most basic definition of the frequency that everybody knows. But what is the definition of Frequency in case of Digital Images and Sounds ?

For example, if a newborn baby's heart beats at a frequency of 120 times a minute, its period (the interval between beats) is half a second.

make sense so far.

Now here is the Sine wave of various frequencies,

enter image description here

the bottom waves have higher frequencies than those above. The horizontal axis represents time.

The above signal makes sense to me that it has a frequency but how about a non periodic signals like a human voice, ?

Have a look, enter image description here

this signal is not repeated on any instance so how one could say what is its frequency and how one could count the number of cycles repeated ?

aforesaid,

Frequency is the number of occurrences of a repeating event per unit time.

How does this statement can be true in case of the frequency of human voice ? because when we speak we do not repeat anything than how the voice can have some frequency ?

and in case of images,

the rapid change in the color is the part of the High frequency of the image

How this can be counted as frequency ? if the image has all the different pixel values how there could be any frequency ?

I am so curious to know deep in detail about the exact definition of Frequency that holds for everything mentioned above.

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The key insight that Fourier had when he developed Fourier analysis is that any absolutely integrable (thanks Jason R) function can be represented as the weighted sum of sines and cosines. Explaining why this is true is way beyond the scope of this answer. I suggest you study Fourier theory to understand this better.

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    $\begingroup$ +1 for the succinct answer. It's hard to come up with a detailed-enough answer to address all of the OP's concerns. One nitpick: rigorous mathematicians (not too many of them around here) would point out that the Fourier transform (or Fourier series) can't be applied to any arbitrary function. One sufficient condition for a function's Fourier transform to exist is that it is absolutely integrable: $\int_{-\infty}^{\infty} |x(t)| dt < \infty$. This is often the case. And, for Fourier series, the function $x(t)$ must be periodic (also with some provisos to ensure that the series converges). $\endgroup$ – Jason R Oct 21 '12 at 1:22
  • $\begingroup$ so would that be correct to say that human voice is not based on one frequency there are unlimited number of frequencies on each human voice ? $\endgroup$ – Sufiyan Ghori Oct 21 '12 at 9:11
  • $\begingroup$ Yes, human voice is not single-frequency (if it was, it would sound like a sinusoidal tone). Strictly speaking, any finite-duration signal does have infinite bandwidth. But, most energy in human voice is concentrated in a band just a few kHz wide. In that band, there is an unlimited number of frequencies in the sense that frequency is continuous and not discretely-valued, but again, that's just a detail of the math that isn't really important on a practical basis. If you're interested in learning more about the spectrum of human voice for various sounds, that's a whole topic in itself. $\endgroup$ – Jason R Oct 21 '12 at 14:00
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Words mean different things to different people. Sometimes approximate things. Such as that the repeated events might not be exactly identical, but only "approximately" or partially identical. Or that the repetition rate varies "slightly". Where the words approximately and slightly may vary in meaning as well.

In terms of signal processing, one might look at your voice signal as composed of the sum of pure periodic signals and wildly non-periodic signals, so that the repeated events looks hidden to you, but can be extracted by various forms of analysis (such as a DFT/FFT).

Same with images.

Furthermore the term frequency is often used for both the repetition of pure sinusoidal components, or for larger very non-sinusoidal looking patterns that the human ear is good at detecting very approximate (sometimes almost hidden) repetitions thereof, called the "pitch".

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I think the definition that frequency is no. of occurrences of a repeating event is good only for periodic events. In other cases, we can say that the frequency is something to do with the change in the rate of something. If something changes rapidly, we say that it is of high frequency, whereas if this variable does not change rapidly, i.e., it changes smoothly, we say that it is of low frequency. And as others also said, there are ways to quantitively interpret it using FT for stationary signals or Wavelet Transform for non-stationary signals.

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Frequency rather then taking as cycles/sec if you take it as the rate of change of signal then you can understand ,in image frequency is the change in intensity(or color) value like frequency near the edges is high because there is sharp change in intensity values.

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