# physical meaning of fourier transform

I have a question regarding the fourier transform. If I plot the frequency against the fourier transform for a periodic signal and I get a peak, What is the physics behind it? I want to know the physical meaning of it.

Why we get a peak and what does it signify?

Thanks

• The peak signifies a periodic component in the input data. The physical interpretation depends on what the input data is. For example if you transform several years of average daily temperature at the south pole, it will be unsurprising to find a peak at a frequency of 1/365 days. This indicates that there is an annual periodicity in the temperature. It is colder there in summer months than winter months. – John Jan 15 '14 at 12:02
• An FFT peak is not necessarily at a waveforms periodicity frequency. Common example is a bass tone with a missing fundamental. – hotpaw2 Jan 15 '14 at 15:18

I read your question a little differently than the other people who have answered. The role of the Fourier transform in physics is as a bi-directional one-to-one mapping between conjugate variables. The most famous example of a conjugate variable pair is position/momentum, but there are many others. Some examples of conjugate variable pairs:

• position and momentum over Planck's constant $[x,p/\hbar]$
• time and energy $[t,E]$
• aperture and the sine of the diffraction angle divided by the wavelength $\left[x,\frac{\sin\theta}{\lambda}\right]$

One of the consequences in quantum mechanics is that the operators (quantum mechanics uses differential operators which are related to the classical variables) do not commute. So, using the position/momentum example; $xp\neq px$. This leads to an uncertainty relation between conjugate variables.

The usual statement of the Heisenberg uncertianty principle is that the more well one knows the position of a quantum particle, the less one can know about the momentum. Mathematically; $$\sigma_x\sigma_ p\geq\frac{\hbar}{2}.$$ However, the more general statement of the uncertainty principle is that the uncertainty between any pair of conjugate variables is given by $$\sigma_1\sigma_2\geq\frac12.$$ This uncertainty relation leads to some rather strange effects in quantum physics. For example, the uncertainty relation between energy and time allow for the violation of energy conservation on very brief timescales, a law which is sacred in classical physics.

If you are looking for more information on the uses of the Fourier transform in physics and engineering with many worked examples, have a look at this free textbook by J. F. James.

• I came across a paper on particle trapping using Paul traps. The trapped particle exhibits periodic motion. So, knowing the position of the particle over a range of time and taking the fourier transform of this periodic motion of the particle, three peaks are obtained. So I am finding it difficult as to how to analyse this result. What does this peak obtained denote. – nichith Jan 16 '14 at 4:29
• The Paul trap (quadrupole ion trap) traps ions in a harmonic well using two DC fields and one RF field. The harmonic potentials of the trap exhibit two natural frequencies, one in the radial direction and one in the z direction. That is to say, the ions are oscillating back and forth at two different frequencies. Perhaps the third frequency is your RF frequency. Have a look at this reference if you have more questions. I would also suggest posting this to the physics or chemistry stack exchange with a more descriptive title. – Chris Mueller Jan 16 '14 at 5:42

The peak represents the most dominant frequency in your periodic signal. The Fourier transform represents the energy at each frequency in your time-domain signal. You only get peaks when specific frequencies are particularly strong. If you took the FT of white noise you would get a fairly flat line.

Any non-pathological exactly periodic signal can be decomposed into the sum of a bunch of sinewaves (overtones or harmonics). The weight of each harmonic component determines the shape of the waveform. A peak, assuming it is above the noise floor, would merely represent the largest weighted harmonic component.

The physics of many common simple systems with a (nearly) linear restoring force can be approximated by a differential equation whose solution is a series of complex exponentials, which can be broken down into a bunch of sinewaves at different frequencies and starting phases. Thus the usefulness of the FT in studying or controlling such systems.

That component signifies the magnitude of that signal in the frequency domain. Is something similar with taking the peak (higher point of it in time domain) of a sinusoidal and placing it in a graph,where the x axis will represent the frequencies,and y the magnitude,on x on the spot where the f=1/T of that sinusoidal is located.