How Fourier Filters Isolate Frequency Components in Time Series
Summary
The answer explains a filter as an operator that transforms a time series to remove or emphasize selected features. Trends, cycles, seasonal patterns, and noise may be associated with different frequency components. A moving average provides a familiar example: it smooths a series using a set of coefficients, often with the aim of reducing noise.
Fourier analysis first represents a series in the frequency domain, where trigonometric components can be used to identify frequencies. The answer relates filtering to convolution: the original series is combined with a coefficient sequence, with moving averages using rectangular weights and Fourier-based approaches using trigonometric functions. This overview does not provide a specific filter design, address sampling or edge effects, or explain how to separate components that overlap or cancel. It is an introductory intuition rather than a complete signal-processing method.
Key ideas
- A filter transforms a time series to alter selected features associated with frequency components.
- A moving average is a filter that smooths a series using a coefficient sequence.
- Fourier analysis represents a time series in the frequency domain using trigonometric components.
- Filtering can be described mathematically as convolution of the series with coefficients.
- The explanation is introductory and does not specify a complete filter design.
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# How Fourier Transform creates the filters? # 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. ## Answer by vonjd (score 5, accepted) https://quant.stackexchange.com/a/35886 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
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