Understanding what the complex cosine spectrum is showing

[Natalie Johnson]

The complex exponential form of cosine

cos(k omega t) = 1/2 * e^(i k omega t) + 1/2 * e^(-i k omega t)

The trigonometric spectrum of cos(k omega t) is single amplitude of the cosine function at a single frequency of k on the real axis which is using the basis function of cosine, right?

The complex exponential spectrum of cos(k omega t) has two amplitudes at 1/2, one at k and -k.

I am confused what this x-axis is representing, i get its the called the frequency domain but it is infact the index k multiplied by omega where omega is fundamental frequency which is constant. And also its not technically the frequency domain because negative frequencies don't exist, but its commonly called freq domain.

I also get this x-y plane shows amplitudes but amplitudes of what? Its not amplitudes of a cosine because these exponentials are actually made up of sines and cosines by eulers formula which are orthogonal and can't be on the same axis. So this x-axis is of +- infinity of k times fundamental frequency but what exactly is this and its size.

Are the exponentials of cosine, which have opposite signs, basis functions that completely represent the space and they together show the rotational direction that can be represented by a cosine?

Im all muddled up.

Appreciate your advice, thanks

 

[olivermsun]

I am confused what this x-axis is representing, i get its the called the frequency domain but it is infact the index k multiplied by omega where omega is fundamental frequency which is constant. And also its not technically the frequency domain because negative frequencies don't exist, but its commonly called freq domain.

You may think of either $k$ or $\omega$ as the "fundamental" frequency and the other as a scaling factor. It doesn't really matter.

For complex numbers, the frequency can be signed because the phase angle can progress in either the counter-clockwise or clockwise directions.

I also get this x-y plane shows amplitudes but amplitudes of what? Its not amplitudes of a cosine because these exponentials are actually made up of sines and cosines by eulers formula which are orthogonal and can't be on the same axis. So this x-axis is of +- infinity of k times fundamental frequency but what exactly is this and its size.

Sines and cosines are equivalently represented by a complex exponential. Notice in particular that the expression you have for the cosine in terms of complex exponentials is exactly the sum of some complex number and its complex conjugate, which by Euler's formula eliminates the sine and leaves you only with the cosine.

Are the exponentials of cosine, which have opposite signs, basis functions that completely represent the space and they together show the rotational direction that can be represented by a cosine?

I'm not entirely sure what you mean. Could you please re-state the question?