# Understanding what the complex cosine spectrum is showing

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• Natalie Johnson
In summary, a complex cosine spectrum is a graphical representation of the frequency components in a signal obtained by taking the Fourier transform and plotting the magnitude and phase of the complex numbers. It differs from a regular cosine spectrum by including phase information, allowing for more accurate analysis and processing. The peaks and valleys in the spectrum represent the signal's dominant and lowest frequencies, which can be used for filtering and analyzing the signal. While it can be used for non-periodic signals, it may not be as effective as other methods of analysis.
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.

Natalie Johnson said:
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?

## 1. What is a complex cosine spectrum?

A complex cosine spectrum is a graphical representation of the frequency components present in a signal. It is obtained by taking the Fourier transform of the signal and plotting the magnitude and phase of the complex numbers obtained. It is commonly used in signal processing and analysis to understand the frequency content of a signal.

## 2. How is a complex cosine spectrum different from a regular cosine spectrum?

A regular cosine spectrum only shows the magnitude of the frequency components, while a complex cosine spectrum also includes the phase information. This means that a complex cosine spectrum provides a more complete picture of the signal's frequency content, allowing for more accurate analysis and processing.

## 3. What do the peaks and valleys in a complex cosine spectrum represent?

The peaks in a complex cosine spectrum represent the frequency components with the highest magnitude in the signal, while the valleys represent the frequencies with the lowest magnitude. The location and height of these peaks and valleys can provide valuable information about the signal's characteristics, such as its dominant frequencies and any harmonics present.

## 4. How can a complex cosine spectrum be used to filter a signal?

A complex cosine spectrum can be used to identify and isolate specific frequency components in a signal. By selectively removing or attenuating certain peaks or valleys in the spectrum, unwanted frequencies can be filtered out, resulting in a cleaner, more focused signal. This is known as frequency domain filtering and is commonly used in audio and image processing.

## 5. Can a complex cosine spectrum be used to analyze non-periodic signals?

Yes, a complex cosine spectrum can be used to analyze non-periodic signals, although it may not provide as clear or accurate of a representation as it would for periodic signals. This is because the frequency components in non-periodic signals are constantly changing, making it more difficult to identify specific peaks and valleys in the spectrum. In these cases, other methods of signal analysis may be more suitable.

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