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What is difference between angular frequency ($\omega$) and regular frequency($f$)? As far as i am able to understand is that electric power signal has $f=60Hz$ and many other cases where i see "$f$" when discussion is about frequency but where is application of $\omega$?

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@Man. Hi. Measuring frequency in terms of $\omega$ (radians/second) or in terms of $f$ (cycles/second, Hz) is the same as measuring speed in miles/hour or kilometers/hour. People measure frequency in terms that are convenient for them. In algebra equations it's convenient to represent frequency in terms of $\omega$ because it's easier to write the single $\omega$ character than to write the three $2{\pi}f$ characters. People also sometimes prefer using $\omega$ because the trigonometric functions in algebra, and in most signal processing software, expect angles to be measured in radians. In the laboratory, working engineers (I mean the guys who know which end of the soldering iron is hot) usually prefer to measure frequency in terms of Hz because their oscilloscopes, frequency counters, spectrum analyzers, and network analyzers measure frequency in Hz.

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Many signals we observe are functions of time, i.e., their value changes with an independent time variable, which can be measured in seconds. Frequency is a measure of how fast a signal changes and thus measured in "per second" which is the same as Hertz: 1 Hz = 1/s.

An oscillation is an excellent example: a harmonic wave can be described as $e^{\jmath 2\pi f t}$. Here $f$ is the frequency, which tells us how many cycles the complex phasor carries out in one second. That's the frequency in Hertz, think of it as cycles per second.

Another way of thinking about the complex phasor is as $e^{\jmath \varphi(t)}$, where $\varphi(t)$ is the angle the phasor has with the positive real axis. For a harmonic wave, from above we have $\varphi(t) = 2\pi f t$, i.e., the angle increases linearly in time. Now one may wonder at what speed it is increasing. The answer is obviously its first derivative, i.e., $\frac{{\rm d}\varphi(t)}{{\rm d}t} = 2\pi f$. This is the angular frequency, often described as $\omega$. It tells us how fast the angle grows and is thus measured in radian per second [rad/s] (though of course if you prefer your angle in degrees, you could convert it to [deg/s], just as you like). This is particularly important if we care about the phase a lot. One example would be digital communications, where the phase often contains the actual information and rates of change (due to things like clock drift or Doppler) can be limiting factors.

That's one reason for using them at least. Another one could be that people are just too lazy to always write the $2\pi$ and prefer $e^{\jmath \omega t}$ over $e^{\jmath 2\pi f t}$. This will vary across disciplines a bit.

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It's a matter of the units used to measure frequency as events which happen per unit of time.

For the regular frequency f, we ask: How any wavelengths pass a specific point per second?\

For the angular frequency $\omega$, we ask: How many radians are traversed per second?

The easiest way to visualise the angular frequency is by understanding the relationship between the unit circle and sine waves. This is a good video to gain that understanding. What you'll notice from the video is that a full wavelength of a sine wave traverses an angle of $2\pi$ radians, so when we say that $\omega = 2\pi f$, we are effectively saying "$f$ is the number of wavelengths that pass a point every second, and since each wavelength traverses an angle of $2\pi$, the angular frequency, the angle traversed per second, is equal to the wavelengths multiplied by $2\pi$.

I hope this helps a little!

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In simple terms your regular frequency is how many times in a second, the signal goes through the same cycle. This is not restricted to a sinusoid alone (because any real world signal can be represented by sum of sinusoids). When you say regular frequency is 60Hz, the signal cycles itself 60 times in a second. Each period is 1/60s. A sinewave is a fundamental periodic signal (sinusoid) having a single frequency.

$x = sin(2\pi f*t)$

$f$ is your frequency, and $t$ is the time. As time increases you can see the value of $x$ repeat itself every $1/f$ seconds.

Now, see the term $2\pi f t$. As $t$ increases from $n/f$ to $(n+1)/f$, the term $2\pi f t$ increases from $2\pi n$ till $2\pi n+2\pi$. Corresponding value of $x$ varies from 1 and comes back towards 1 (it passes through 1 cycle). So basically what varied here is the angle of $sin()$ function. It increased by $2\pi$ in an interval of $1/f$ seconds. So what is the angular frequency? It is $2\pi /(1/f) = 2\pi f$.

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