# How Fourier Transform creates the filters?

I would like to know how the fourier transform creates filters to extract the constituent signals? I did learn from a book it extracts the spike information and then analyze and combine those informations?

But i have a doubt like if there are more than two constituent signal which cancel each other then how the spikes be useful? Many thanks, i am just beginning to understand this.

• This question sounds too broad. In two words, with Fourier transform you can filter signals in time-domain by decomposing them in sums of sines and cosine. The other side of the coin is that with this kind of analysis you can filter only stationary signals. As a suggestion, study more before asking questions like this one :) – james42 Sep 2 '17 at 4:29
• A warm welcome to Quant.SE and thank you for that question. Please see my answer below. – vonjd Sep 3 '17 at 6:41

A filter is a mathematical operator that serves to convert an original time-series into another time-series or function. The purpose is to remove some particular features (e.g. trends, business cycle, seasonalities and noise) that are associated with specific frequency components.

To get the basic idea think of a moving average which is nothing but a filter to remove (supposed) noise.

In the case of the fourier transform the time series is transformed from the time domain into the frequency domain.

Mathematically the filter is applied in both cases by convolution of the original series with a coefficient vector (basically nothing but the dot product). In case of the moving average the coefficient vector is a rectangular function, in case of the fourier transform you basically use some kind of trigonometric functions (via the complex exponential function) to extract the frequencies.

You can find more here:
Analysis of Financial Time-Series Using Fourier and Wavelet Methods by Philippe Masset

To get a general intuition about the fourier transform you can find many excellent answers here:
https://math.stackexchange.com/questions/1002/fourier-transform-for-dummies

• In general the fourier transform $X(f)$ at a frequency $f$ is a complex number: the real part is equal to the integral of the product of $x$ with a cosine wave of frequency $f$, the imaginary part is also an integral over all time, of the product of the signal $x$ and a sine wave of frequency $f$. So basically it "filters" or identifies each frequency component by multiplication with a sine/cosine wave of that frequency. – Alex C Sep 2 '17 at 16:09
• @AlexC: Indeed, thank you, I edited my answer to reflect your comment. – vonjd Sep 2 '17 at 16:16