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Fast Fourier Transform (FFT) FilterDear friends!
I'm happy to present an implementation of the Fast Fourier Transform (FFT) algorithm. The script uses the FFT procedure to decompose the input time series into its cyclical constituents, in other words, its frequency components , and convert it back to the time domain with modified frequency content, that is, to filter it.
Input Description and Usage
Source and Length :
Indicates where the data comes from and the size of the lookback window used to build the dataset.
Standardize Input Dataset :
If enabled, the dataset is preprocessed by subtracting its mean and normalizing the result by the standard deviation, which is sometimes useful when analyzing seasonalities. This procedure is not recommended when using the FFT filter for smoothing (see below), as it will not preserve the average of the dataset.
Show Frequency-Domain Power Spectrum :
When enabled, the results of Fourier analysis (for the last price bar!) are plotted as a frequency-domain power spectrum , where “power” is a measure of the significance of the component in the dataset. In the spectrum, lower frequencies (longer cycles) are on the right, higher frequencies are on the left. The graph does not display the 0th component, which contains only information about the mean value. Frequency components that are allowed to pass through the filter (see below) are highlighted in magenta .
Dominant Cycles, Rows :
If this option is activated, the periods and relative powers of several dominant cyclical components that is, those that have a higher power, are listed in the table. The number of the component in the power spectrum (N) is shown in the first column. The number of rows in the table is defined by the user.
Show Inverse Fourier Transform (Filtered) :
When enabled, the reconstructed and filtered time-domain dataset (for the last price bar!) is displayed.
Apply FFT Filter in a Moving Window :
When enabled, the FFT filter with the same parameters is applied to each bar. The last data point of the reconstructed and filtered dataset is used to build a new time series. For example, by getting rid of high-frequency noise, the FFT filter can make the data smoother. By removing slowly evolving low-frequency components (including non-periodic constituents), one can reveal and analyze shorter cycles. Since filtering is done in real-time in a moving window (similar to the moving average), the modified data can potentially be used as part of a strategy and be subjected to other technical indicators.
Lowest Allowed N :
Indicates the number of the lowest frequency component used in the reconstructed time series.
Highest Allowed N :
Indicates the number of the highest frequency component used in the reconstructed time series.
Filtering Time Range block:
Specifies the time range over which real-time FFT filtering is applied. The reason for the presence of this block is that the FFT procedure is relatively computationally intensive. Therefore, the script execution may encounter the time limit imposed by PulseWire when all historical bars are processed.
As always, I look forward to your feedback!
Also, leave a comment if you'd be interested in the tutorial on how to use this tool and/or in seeing the FFT filter in a strategy.
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Fourier Analysis and Filtering [tbiktag]This tool uses Fourier transform to decompose the input time series into its periodic constituents and seasonalities , in other words, its frequency components . It also can reconstruct the time-domain data while using only the frequency components within a user-defined range (band-pass filtering). Thereby, this tool can reveal the cyclical characteristics of the studied market.
USAGE
The source and the size of the input data can be chosen by the “ Dataset Source ” and “ Dataset Size ” options. Price, volume, or some technical indicator (e.g., RSI, MACD, etc.) can serve as a source of the input data.
“ Action ” defines the type of the plot that will be displayed. Two options are available:
- Fourier Analysis
If selected, the frequency spectrum of the squares of the Fourier coefficient magnitudes is displayed. The zero-frequency component is on the right. Since the magnitudes of half of the coefficients are repeated, the graph displays only half of the frequency components.
The squared magnitude of a given frequency component is a measure of its power , that is, its contribution to the total variance of the dataset. Thus, by analyzing the frequency-domain spectrum, one can identify the most prominent seasonalities and then visualize them by using the " Band-pass Filtering " option (see below). Note that the zero component stores information about the amount of data, so it is naturally higher when the data is not centered at zero.
By activating the " Info about Frequency Component " option, the user can display information about the power and frequency of the selected Fourier component.
- Band-pass Filter
This option reconstructs and plots the dataset in the time domain, blocking frequency components outside of the cutoff frequencies (defined by the input parameters “ Upper Cutoff ” and “ Lower Cutoff ” input parameters in the “ Band-pass Filter Properties ” section).
FURTHER READING
In general, Fourier analysis has a long history of attempted applications for analyzing price data and estimating market cycles. For example, see the paper by John Ehlers
www.mesasoftware.com
and also some tools available here on PulseWire, such as:
“Function: Discrete Fourier Transform” by @RicardoSantos
“Fourier series Model Of The Market” by @e2e4mfck
“Ehlers Discrete Fourier Transform” by @cheatcountry
Thus, I tried to make this tool versatile and user-friendly so you all can experiment with your own analysis.
Enjoy and don't hesitate to leave your feedback in the comments below! Indicator

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Exotic SMA Explorations Treasure TroveThis is my "Exotic SMA Explorations Treasure Trove" intended for educational purposes, yet these functions will also have utility in special applications with other algorithms. Firstly, the Pine built-in sma() is exceedingly more efficient computationally on TV servers than these functions will be. I just wanted to make that very crystal clear. My notes elaborate on this in the code blatantly.
Anyhow, the simple moving average(SMA) is one of the most common averaging filters used in a wide variety of algorithms. "Simply put," it's name says a lot about it. The purpose of this script, is to demonstrate variations of it's calculation in a multitude of exotic forms. In certain scenarios our algorithms may require a specific mathemagical touch that is pertinent to our intended goals. Like screwdrivers, we often need different types depending on the objective we are trying to attain. The SMA also serves as the most basic of finite impulse response(FIR) algorithms. For example, things like weighted moving averages can be constructed by using the foundational code of SMA.
One other intended demonstration of this script, is running multiple functions for comparison. I have had to use this from time to time for my own comparisons of performance. Also, imbedded into this code is a method to generically and recklessly in this case, adapt an algorithm. I will warn you, RSI was NEVER intended to adapt an algorithm. It only serves as a crude method to display the versatility of these different algorithms, whether it be a benefit or hinderance concerning dynamic adaptability.
Lastly, this script shows the versatility of TV's NEW additions input(group=) and input(inline=) upgrades in action. The "Immense Power of Pine" is always evolving and will continue to do so, I assure you of that. We can now categorize our input()s without using the input(type=input.bool) hackTrick. Although, that still will have it's enduring versatility, at least for myself.
NOTICE: You have absolute freedom to use this source code any way you see fit within your new Pine projects. You don't have to ask for my permission to reuse these functions in your published scripts, simply because I have better things to do than answer requests for the reuse of these functions. Sufficient accreditation regarding this script and compliance with "TV's House Rules" regarding code reuse, is as easy as copying the functions in their entirety as is. Fair enough? Good!
When available time provides itself, I will consider your inquiries, thoughts, and concepts presented below in the comments section, should you have any questions or comments regarding this indicator. When my indicators achieve more prevalent use by TV members, I may implement more ideas when they present themselves as worthy additions. Have a profitable future everyone!
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