The Relationship Between Angular and Cyclic Frequency in Fourier Transform

[Kenny_L]

Hi everybody. There has been a thread about this on physics forums, where the Fourier transform X(w) of x(t) volts (with time units in seconds) could be considered as volt second, or volt per Hz. So when we see tables of Fourier transform pairs, we might see Fourier transform plots associated with cos(wt) having impulses with values 'pi' for the negative and positive frequency components. The 'pi' volt per Hz values are seen in angular frequency plots (of Fourier transform). However, for plots of cyclic frequency "f" (in Hz), the Fourier transform pairs table present values of '1/2' (ie. 0.5) for the impulses. I assumed that the vertical axis values should turn out the same value for angular frequency and cyclic frequency plots (and that it would be just up to us to know which horizontal-axis units we are dealing with).

For example, if I plot a bunch of values as a function of time (in seconds), where the vertical values have units of volt. And if I plot the same bunch of values as a function of some other related horizontal axis parameter, then the vertical values would still have the same units (and same values!). But, for Fourier transforms of a sinusoid, it appears that the vertical axis value itself can change - depending on whether it is plotted against angular frequency or cyclic frequency (ie. pi impulses for angular frequency plots and 1/2 impulses for cyclic frequency plots). In this case, what's the recommendation? Would the recommendation be to treat the angular frequency Fourier transform domain to be different from the cyclic frequency Fourier transform domain?

It is known that whatever the value we have for the angular frequency components, a division by '2pi' generally yields the values for the cyclic frequency plots. But I've always been ---- let's say confused -- about why the vertical axis values themselves for 'w' and 'f' plots are different (instead of being the same values). I'm thinking...if that's how it pans out mathematically, then that's how it is.

But... if we have 'pi' volt per Hz for the angular frequency plot, then what are the units for the '1/2' in the cyclic frequency plots?

Thanks for helping out!

 

[scottdave]

Thanks is an interesting question. In transmitters, I often see how power is distributed across a frequency spectrum. The "height" of the graph in this case has dimension of power/frequency and the horizontal is frequency. The area under this graph is the total power output.

 

[ADDA]

int freq_index(float freq) {
    return (int)( (freq * (float)FRAME_SIZE) / (float)SAMPLE_RATE );
}

float index_freq(int index) {
    return ( ((float)index / (float)FRAME_SIZE) * (float)SAMPLE_RATE );
}

Easy enough to translate for me, at least, in code.

I am using finite and discrete transforms. It seems you are in the time invariant section.

I do not really understand your question, yet if you were to perhaps substitute the velocity's inverse as the period for the sample rate, and then you would need somehow to substitute the frame size variable as a observation section of the data. For example, I use this with Audio Processing, and the sample rate variable is 44100 where as I vary between a sampling window and frame rate. So 44100 = 2 * 2 * 3 * 3 * 5 * 5 * 7 * 7; So if I were to choose a rate of 60 = 2 * 2 * 3 * 5; Then the remaining factors would be the frame size variable, 3 * 5 * 7 * 7 = 735;

Again I'm thinking finite.