# Quick question on Fourier transform

• semc
In summary, as a physics student, you can decompose a periodic function into sine functions with different frequencies using Fourier transform. However, when generating ultra-short pulses, you need many frequencies, which is why Fourier transform is applicable in this case. Passing monochromatic light through an optical chopper introduces higher harmonics and follows the uncertainty principle, resulting in a spread of wavelengths. This can be further understood through the concept of Gibbs phenomenon and the relationship between time and frequency domains in Fourier transforms.

#### semc

Hi all, as a physics student, I seldom use Fourier transform but from my understanding, given a periodic function you can decompose the function into sine function with different frequencies. Also, to get a ultra short pulse in time domain, this would require mixing many frequencies. I would therefore like to ask what happens if I pass my monochromatic light through a optical chopper. Is Fourier transform not applicable in this case since ultra short pulse generation does not require many frequencies?

semc said:
I would therefore like to ask what happens if I pass my monochromatic light through a optical chopper. Is Fourier transform not applicable in this case since ultra short pulse generation does not require many frequencies?

Ultra short pulses DO require many frequencies. Fourier transform theory applies

Of course when analyzing pulses, it may be advantageous to look at the wavelet transform instead depending on what you actually hope to get out of the analysis.

Fouier theory says that the spectrum Z of the product of two signals z(t) = x(t) y(t) is the convolution of their spectra, Z = X * Y. Your signal consists of a monochromatic wave times a periodic square wave, so its spectrum is the convolution of their spectra. You can easily figure it out from there.

A single frequency like a pure sine wave is only a single frequency if it starts at the beginning of time and goes on forever. As soon as you start or stop it you introduce a step which is composed of higher frequencies. The sharper the step the higher are the included frequencies. So chopping introduces harmonics.

dauto said:
Ultra short pulses DO require many frequencies. Fourier transform theory applies

I am aware that you need many frequencies to get short pulse. That's what Fourier transform tells us. In my example, I specifically said I only have single wavelength as source.

A single frequency like a pure sine wave is only a single frequency if it starts at the beginning of time and goes on forever.

So if I turn on and off my laser I introduce higher harmonics as well?

Fouier theory says that the spectrum Z of the product of two signals z(t) = x(t) y(t) is the convolution of their spectra, Z = X * Y. Your signal consists of a monochromatic wave times a periodic square wave, so its spectrum is the convolution of their spectra. You can easily figure it out from there.

Ain't this just the math? I am confused how higher frequencies are generated.

The rising and falling edges of the "chopper" are very short, and therefore contain high frequency components. An ideal square wave modulation has infinite slopes, thus having a spectrum that is non-zero to infinite frequencies.

marcusl said:
The rising and falling edges of the "chopper" are very short, and therefore contain high frequency components. An ideal square wave modulation has infinite slopes, thus having a spectrum that is non-zero to infinite frequencies.

Don't really understand how this comes about. Is there any reference to explain in detail about this?

semc said:
I am confused how higher frequencies are generated.

That's an application of the uncertainty relation. When you chop down the beam you are making a measurement of the photon location. By the uncertainty principle the photon momentum becomes uncertain creating a spread on the wavelength. In other words, the uncertainty principle is the physical principle that expresses the spread in wavelength required by the mathematics of waves which can be studied and understood through Fourier transforms.

I second what dauto said. There is an uncertainty relation for Fourier transforms which is the analog of Heisenberg's Uncertainty Principle, and that applies to common classical signals. You can find some information on Wikipedia, and more in any text on Fourier transforms. I like Bracewell's text, which has an excellent chapter discussing relationships between the "two domains" (time and frequency).

"Quote by dauto
Ultra short pulses DO require many frequencies. Fourier transform theory applies

I am aware that you need many frequencies to get short pulse. That's what Fourier transform tells us. In my example, I specifically said I only have single wavelength as source."

Once you modulate the carrier (pure sine) with an envelope, you no longer have a pure sine of single wavelength. You have a time-varying envelope that may have a rich spectrum, as I indicated earlier.

1 person
A square wave (& therefore loosly speaking a square pulse) is the sum of all the odd harmonics.

Fourier decomposition of a (periodic) signal will give you a long string of harmonics along with their amplitudes that when added together will give you your original signal back. If the wave is square then you will only get odd harmonics. In practice you may well get low amplitude even harmonics as well.

semc said:
Don't really understand how this comes about. Is there any reference to explain in detail about this?
You might start here http://en.wikipedia.org/wiki/Gibbs_phenomenon

A square wave has an infinite spread of frequencies. Gibbs ringing is what happens when you try to approximate a square wave with a finite spread of frequencies.

I think the key point here is that chopping/modulating a signal introduces extra frequency components.

Claude.

## What is a Fourier transform?

A Fourier transform is a mathematical technique that is used to decompose a complex signal into its individual frequency components. It converts a signal from its original time or space domain into a frequency domain representation.

## Why is a Fourier transform useful?

A Fourier transform is useful because it allows us to analyze and manipulate complex signals in terms of their frequency components. This can help in various applications such as signal processing, image and sound analysis, and data compression.

## What is the difference between a Fourier transform and a Fourier series?

A Fourier transform is used for continuous signals while a Fourier series is used for periodic signals. The Fourier series decomposes a periodic signal into a sum of sinusoidal components, while the Fourier transform decomposes a non-periodic signal into a continuous spectrum of frequencies.

## What are some common applications of Fourier transform in science?

Fourier transform has many applications in science, including signal processing, image and sound analysis, data compression, and solving differential equations in physics and engineering. It is also used in chemistry to identify and analyze the composition of molecules.

## Are there any limitations to using Fourier transform?

While Fourier transform is a powerful tool, it has some limitations. It assumes that the signal is periodic or infinitely long, which may not always be the case in real-world applications. Additionally, it cannot accurately capture sudden changes or discontinuous signals.