# Power from a Fourier transform

• neil.thompson
In summary, the speaker has been playing around with FFTs and noticed that the maximum amplitude of the FFT output decreases when the frequency increases, even though there are more cycles. They are wondering how the energy is related to the FT and if it is usual to normalize it in some way. The other person suggests different ways of normalizing and mentions that taking a finer grid may reveal a wider/narrower curve and affect the peak amplitude.

#### neil.thompson

So I have been away from education for a little while now and I'm going through some refresher stuff - in particular I have been playing around with FFTs.

If i take (with MATLAB notation):

time = 0:0.01:10
y = fft(sin(2*pi*f*time))

with f = 5
then the maximum amplitude of the fft output is about 498.

with f = 10
the maximum amplitude of fft output is 492.

I understand the amplitude is 'halved' in both cases because this fft is ambiguous so the energy is spread over two peaks. But why is the energy less when the frequency increases? I have more cycles in the case with more frequency, but I suppose this means I have less samples. Also, is it usual to normalise this in some way? It seems like this is something you wouldn't want if you were dealing were plotting energy return from doppler shifts.

neil.thompson said:
But why is the energy less when the frequency increases?
How is the energy related to the FT? Have you checked the width of the peaks?

neil.thompson said:
Also, is it usual to normalise this in some way? It seems like this is something you wouldn't want if you were dealing were plotting energy return from doppler shifts.
There are different ways of normalizing the FFT, but it is customary that doing FFT-1(FFT(f)) will return N × f, although MATLAB returns the original result.

DrClaude said:
How is the energy related to the FT? Have you checked the width of the peaks?
.

Good point - my language was imprecise. What I meant was peak power, or peak amplitude -- I guess what I'm asking is: if I have a frequency resolution on the FFT such that all of the signal should be confined to a signal frequency bin (e.g., each bin spans is 2kHz, my input signal is a single tone at 4.5kHz so all of the signal should fold into the third bin) so why does the height of the amplitude peak depend on the frequency of the signal?

edit: and into the bargain, related to your point, if I have enough data to get a very fine resolution FT then I guess this will reveal that there will be some spread across bins so the peak isn't the same because the curve is wider/narrower.

DrClaude said:
There are different ways of normalizing the FFT, but it is customary that doing FFT-1(FFT(f)) will return N × f, although MATLAB returns the original result.

Right, thanks.

Have you tried to take a finer grid an see what you get then?

The Fourier transform is a powerful tool in signal processing that allows us to decompose a signal into its individual frequency components. In your example, you are using the Fast Fourier Transform (FFT) to analyze a sinusoidal signal with a frequency of either 5 or 10 Hz.

The FFT works by breaking down a signal into smaller segments, known as bins, and analyzing the frequency content of each bin. The maximum amplitude of the FFT output represents the energy at that particular frequency. In your case, you are seeing a decrease in the maximum amplitude as the frequency increases from 5 to 10 Hz.

This decrease in energy is due to the fact that as the frequency increases, the signal becomes more "compressed" in time. This means that there are fewer samples available to capture the signal, resulting in a decrease in energy at that particular frequency. This is also why you see two peaks in the FFT output - the energy is spread over two bins.

In terms of normalization, it is common to normalize the FFT output to account for the number of samples used. This allows for a more accurate comparison of the energy at different frequencies. However, it is important to note that the normalization factor may vary depending on the specific application and signal being analyzed.

In conclusion, the decrease in energy at higher frequencies in your FFT output is a result of the signal becoming more compressed in time, leading to fewer samples available to capture the signal. Normalization may be necessary for accurate comparisons between different frequencies.

## 1. How is power calculated from a Fourier transform?

The power from a Fourier transform is calculated by taking the squared amplitude of each frequency component and summing them together. This is known as the power spectrum.

## 2. What does power from a Fourier transform represent?

The power from a Fourier transform represents the amount of energy at each frequency in a given signal. It can also be interpreted as the contribution of each frequency to the overall signal.

## 3. Can power from a Fourier transform be negative?

No, power from a Fourier transform is always a positive value. This is because it is calculated by squaring the amplitude of each frequency component, which ensures that the result is always positive.

## 4. How is the power spectrum graphically represented?

The power spectrum is typically graphed on a frequency vs. power scale, with the frequency on the x-axis and the power on the y-axis. The resulting graph is often called a power spectral density plot.

## 5. What is the relationship between the power spectrum and the original signal?

The power spectrum is mathematically related to the original signal through the Fourier transform. It can be used to analyze the frequency content and power distribution of the original signal, and can also be used for signal processing and filtering.