Fourier For Loop [BackQuant]Fourier For Loop
PLEASE Read the following, as understanding an indicator's functionality is essential before integrating it into a trading strategy. Knowing the core logic behind each tool allows for a sound and strategic approach to trading.
Introducing BackQuant's Fourier For Loop (FFL) — a cutting-edge trading indicator that combines Fourier transforms with a for-loop scoring mechanism. This innovative approach leverages mathematical precision to extract trends and reversals in the market, helping traders make informed decisions. Let's break down the components, rationale, and potential use-cases of this indicator.
Understanding Fourier Transform in Trading
The Fourier Transform decomposes price movements into their frequency components, allowing for a detailed analysis of cyclical behavior in the market. By transforming the price data from the time domain into the frequency domain, this indicator identifies underlying patterns that traditional methods may overlook.
In this script, Fourier transforms are applied to the specified calculation source (defaulted to HLC3). The transformation yields magnitude values that can be used to score market movements over a defined range. This scoring process helps uncover long and short signals based on relative strength and trend direction.
Why Use Fourier Transforms?
Fourier Transforms excel in identifying recurring cycles and smoothing noisy data, making them ideal for fast-paced markets where price movements may be erratic. They also provide a unique perspective on market volatility, offering traders additional insights beyond standard indicators.
Calculation Logic: For-Loop Scoring Mechanism
The For Loop Scoring mechanism compares the magnitude of each transformed point in the series, summing the results to generate a score. This score forms the backbone of the signal generation system.
Long Signals: Generated when the score surpasses the defined long threshold (default set at 40). This indicates a strong bullish trend, signaling potential upward momentum.
Short Signals: Triggered when the score crosses under the short threshold (default set at -10). This suggests a bearish trend or potential downside risk.'
Thresholds & Customization
The indicator offers customizable settings to fit various trading styles:
Calculation Periods: Control how many periods the Fourier transform covers.
Long/Short Thresholds: Adjust the sensitivity of the signals to match different timeframes or risk preferences.
Visualization Options: Traders can visualize the thresholds, change the color of bars based on trend direction, and even color the background for enhanced clarity.
Trading Applications
This Fourier For Loop indicator is designed to be versatile across various market conditions and timeframes. Some of its key use-cases include:
Cycle Detection: Fourier transforms help identify recurring patterns or cycles, giving traders a head-start on market direction.
Trend Following: The for-loop scoring system helps confirm the strength of trends, allowing traders to enter positions with greater confidence.
Risk Management: With clearly defined long and short signals, traders can manage their positions effectively, minimizing exposure to false signals.
Final Note
Incorporating this indicator into your trading strategy adds a layer of mathematical precision to traditional technical analysis. Be sure to adjust the calculation start/end points and thresholds to match your specific trading style, and remember that no indicator guarantees success. Always backtest thoroughly and integrate the Fourier For Loop into a balanced trading system.
Thus following all of the key points here are some sample backtests on the 1D Chart
Disclaimer: Backtests are based off past results, and are not indicative of the future .
INDEX:BTCUSD
INDEX:ETHUSD
BINANCE:SOLUSD

# FFT

SpectrumLibrary "Spectrum"
This library includes spectrum analysis tools such as the Fast Fourier Transform (FFT).
method toComplex(data, polar)
Creates an array of complex type objects from a float type array.
Namespace types: array
Parameters:
data (array) : The float type array of input data.
polar (bool) : Initialization coordinates; the default is false (cartesian).
Returns: The complex type array of converted data.
method sAdd(data, value, end, start, step)
Performs scalar addition of a given float type array and a simple float value.
Namespace types: array
Parameters:
data (array) : The float type array of input data.
value (float) : The simple float type value to be added.
end (int) : The last index of the input array (exclusive) on which the operation is performed.
start (int) : The first index of the input array (inclusive) on which the operation is performed; the default value is 0.
step (int) : The step by which the function iterates over the input data array between the specified boundaries; the default value is 1.
Returns: The modified input array.
method sMult(data, value, end, start, step)
Performs scalar multiplication of a given float type array and a simple float value.
Namespace types: array
Parameters:
data (array) : The float type array of input data.
value (float) : The simple float type value to be added.
end (int) : The last index of the input array (exclusive) on which the operation is performed.
start (int) : The first index of the input array (inclusive) on which the operation is performed; the default value is 0.
step (int) : The step by which the function iterates over the input data array between the specified boundaries; the default value is 1.
Returns: The modified input array.
method eMult(data, data02, end, start, step)
Performs elementwise multiplication of two given complex type arrays.
Namespace types: array
Parameters:
data (array type from RezzaHmt/Complex/1) : the first complex type array of input data.
data02 (array type from RezzaHmt/Complex/1) : The second complex type array of input data.
end (int) : The last index of the input arrays (exclusive) on which the operation is performed.
start (int) : The first index of the input arrays (inclusive) on which the operation is performed; the default value is 0.
step (int) : The step by which the function iterates over the input data array between the specified boundaries; the default value is 1.
Returns: The modified first input array.
method eCon(data, end, start, step)
Performs elementwise conjugation on a given complex type array.
Namespace types: array
Parameters:
data (array type from RezzaHmt/Complex/1) : The complex type array of input data.
end (int) : The last index of the input array (exclusive) on which the operation is performed.
start (int) : The first index of the input array (inclusive) on which the operation is performed; the default value is 0.
step (int) : The step by which the function iterates over the input data array between the specified boundaries; the default value is 1.
Returns: The modified input array.
method zeros(length)
Creates a complex type array of zeros.
Namespace types: series int, simple int, input int, const int
Parameters:
length (int) : The size of array to be created.
method bitReverse(data)
Rearranges a complex type array based on the bit-reverse permutations of its size after zero-padding.
Namespace types: array
Parameters:
data (array type from RezzaHmt/Complex/1) : The complex type array of input data.
Returns: The modified input array.
method R2FFT(data, inverse)
Calculates Fourier Transform of a time series using Cooley-Tukey Radix-2 Decimation in Time FFT algorithm, wikipedia.org
Namespace types: array
Parameters:
data (array type from RezzaHmt/Complex/1) : The complex type array of input data.
inverse (int) : Set to -1 for FFT and to 1 for iFFT.
Returns: The modified input array containing the FFT result.
method LBFFT(data, inverse)
Calculates Fourier Transform of a time series using Leo Bluestein's FFT algorithm, wikipedia.org This function is nearly 4 times slower than the R2FFT function in practice.
Namespace types: array
Parameters:
data (array type from RezzaHmt/Complex/1) : The complex type array of input data.
inverse (int) : Set to -1 for FFT and to 1 for iFFT.
Returns: The modified input array containing the FFT result.
method DFT(data, inverse)
This is the original DFT algorithm. It is not suggested to be used regularly.
Namespace types: array
Parameters:
data (array type from RezzaHmt/Complex/1) : The complex type array of input data.
inverse (int) : Set to -1 for DFT and to 1 for iDFT.
Returns: The complex type array of DFT result.

loxxfftLibrary "loxxfft"
This code is a library for performing Fast Fourier Transform (FFT) operations. FFT is an algorithm that can quickly compute the discrete Fourier transform (DFT) of a sequence. The library includes functions for performing FFTs on both real and complex data. It also includes functions for fast correlation and convolution, which are operations that can be performed efficiently using FFTs. Additionally, the library includes functions for fast sine and cosine transforms.
Reference:
www.alglib.net
fastfouriertransform(a, nn, inversefft)
Returns Fast Fourier Transform
Parameters:
a (float ) : float , An array of real and imaginary parts of the function values. The real part is stored at even indices, and the imaginary part is stored at odd indices.
nn (int) : int, The number of function values. It must be a power of two, but the algorithm does not validate this.
inversefft (bool) : bool, A boolean value that indicates the direction of the transformation. If True, it performs the inverse FFT; if False, it performs the direct FFT.
Returns: float , Modifies the input array a in-place, which means that the transformed data (the FFT result for direct transformation or the inverse FFT result for inverse transformation) will be stored in the same array a after the function execution. The transformed data will have real and imaginary parts interleaved, with the real parts at even indices and the imaginary parts at odd indices.
realfastfouriertransform(a, tnn, inversefft)
Returns Real Fast Fourier Transform
Parameters:
a (float ) : float , A float array containing the real-valued function samples.
tnn (int) : int, The number of function values (must be a power of 2, but the algorithm does not validate this condition).
inversefft (bool) : bool, A boolean flag that indicates the direction of the transformation (True for inverse, False for direct).
Returns: float , Modifies the input array a in-place, meaning that the transformed data (the FFT result for direct transformation or the inverse FFT result for inverse transformation) will be stored in the same array a after the function execution.
fastsinetransform(a, tnn, inversefst)
Returns Fast Discrete Sine Conversion
Parameters:
a (float ) : float , An array of real numbers representing the function values.
tnn (int) : int, Number of function values (must be a power of two, but the code doesn't validate this).
inversefst (bool) : bool, A boolean flag indicating the direction of the transformation. If True, it performs the inverse FST, and if False, it performs the direct FST.
Returns: float , The output is the transformed array 'a', which will contain the result of the transformation.
fastcosinetransform(a, tnn, inversefct)
Returns Fast Discrete Cosine Transform
Parameters:
a (float ) : float , This is a floating-point array representing the sequence of values (time-domain) that you want to transform. The function will perform the Fast Cosine Transform (FCT) or the inverse FCT on this input array, depending on the value of the inversefct parameter. The transformed result will also be stored in this same array, which means the function modifies the input array in-place.
tnn (int) : int, This is an integer value representing the number of data points in the input array a. It is used to determine the size of the input array and control the loops in the algorithm. Note that the size of the input array should be a power of 2 for the Fast Cosine Transform algorithm to work correctly.
inversefct (bool) : bool, This is a boolean value that controls whether the function performs the regular Fast Cosine Transform or the inverse FCT. If inversefct is set to true, the function will perform the inverse FCT, and if set to false, the regular FCT will be performed. The inverse FCT can be used to transform data back into its original form (time-domain) after the regular FCT has been applied.
Returns: float , The resulting transformed array is stored in the input array a. This means that the function modifies the input array in-place and does not return a new array.
fastconvolution(signal, signallen, response, negativelen, positivelen)
Convolution using FFT
Parameters:
signal (float ) : float , This is an array of real numbers representing the input signal that will be convolved with the response function. The elements are numbered from 0 to SignalLen-1.
signallen (int) : int, This is an integer representing the length of the input signal array. It specifies the number of elements in the signal array.
response (float ) : float , This is an array of real numbers representing the response function used for convolution. The response function consists of two parts: one corresponding to positive argument values and the other to negative argument values. Array elements with numbers from 0 to NegativeLen match the response values at points from -NegativeLen to 0, respectively. Array elements with numbers from NegativeLen+1 to NegativeLen+PositiveLen correspond to the response values in points from 1 to PositiveLen, respectively.
negativelen (int) : int, This is an integer representing the "negative length" of the response function. It indicates the number of elements in the response function array that correspond to negative argument values. Outside the range , the response function is considered zero.
positivelen (int) : int, This is an integer representing the "positive length" of the response function. It indicates the number of elements in the response function array that correspond to positive argument values. Similar to negativelen, outside the range , the response function is considered zero.
Returns: float , The resulting convolved values are stored back in the input signal array.
fastcorrelation(signal, signallen, pattern, patternlen)
Returns Correlation using FFT
Parameters:
signal (float ) : float ,This is an array of real numbers representing the signal to be correlated with the pattern. The elements are numbered from 0 to SignalLen-1.
signallen (int) : int, This is an integer representing the length of the input signal array.
pattern (float ) : float , This is an array of real numbers representing the pattern to be correlated with the signal. The elements are numbered from 0 to PatternLen-1.
patternlen (int) : int, This is an integer representing the length of the pattern array.
Returns: float , The signal array containing the correlation values at points from 0 to SignalLen-1.
tworealffts(a1, a2, a, b, tn)
Returns Fast Fourier Transform of Two Real Functions
Parameters:
a1 (float ) : float , An array of real numbers, representing the values of the first function.
a2 (float ) : float , An array of real numbers, representing the values of the second function.
a (float ) : float , An output array to store the Fourier transform of the first function.
b (float ) : float , An output array to store the Fourier transform of the second function.
tn (int) : float , An integer representing the number of function values. It must be a power of two, but the algorithm doesn't validate this condition.
Returns: float , The a and b arrays will contain the Fourier transform of the first and second functions, respectively. Note that the function overwrites the input arrays a and b.
█ Detailed explaination of each function
Fast Fourier Transform
The fastfouriertransform() function takes three input parameters:
1. a: An array of real and imaginary parts of the function values. The real part is stored at even indices, and the imaginary part is stored at odd indices.
2. nn: The number of function values. It must be a power of two, but the algorithm does not validate this.
3. inversefft: A boolean value that indicates the direction of the transformation. If True, it performs the inverse FFT; if False, it performs the direct FFT.
The function performs the FFT using the Cooley-Tukey algorithm, which is an efficient algorithm for computing the discrete Fourier transform (DFT) and its inverse. The Cooley-Tukey algorithm recursively breaks down the DFT of a sequence into smaller DFTs of subsequences, leading to a significant reduction in computational complexity. The algorithm's time complexity is O(n log n), where n is the number of samples.
The fastfouriertransform() function first initializes variables and determines the direction of the transformation based on the inversefft parameter. If inversefft is True, the isign variable is set to -1; otherwise, it is set to 1.
Next, the function performs the bit-reversal operation. This is a necessary step before calculating the FFT, as it rearranges the input data in a specific order required by the Cooley-Tukey algorithm. The bit-reversal is performed using a loop that iterates through the nn samples, swapping the data elements according to their bit-reversed index.
After the bit-reversal operation, the function iteratively computes the FFT using the Cooley-Tukey algorithm. It performs calculations in a loop that goes through different stages, doubling the size of the sub-FFT at each stage. Within each stage, the Cooley-Tukey algorithm calculates the butterfly operations, which are mathematical operations that combine the results of smaller DFTs into the final DFT. The butterfly operations involve complex number multiplication and addition, updating the input array a with the computed values.
The loop also calculates the twiddle factors, which are complex exponential factors used in the butterfly operations. The twiddle factors are calculated using trigonometric functions, such as sine and cosine, based on the angle theta. The variables wpr, wpi, wr, and wi are used to store intermediate values of the twiddle factors, which are updated in each iteration of the loop.
Finally, if the inversefft parameter is True, the function divides the result by the number of samples nn to obtain the correct inverse FFT result. This normalization step is performed using a loop that iterates through the array a and divides each element by nn.
In summary, the fastfouriertransform() function is an implementation of the Cooley-Tukey FFT algorithm, which is an efficient algorithm for computing the DFT and its inverse. This FFT library can be used for a variety of applications, such as signal processing, image processing, audio processing, and more.
Feal Fast Fourier Transform
The realfastfouriertransform() function performs a fast Fourier transform (FFT) specifically for real-valued functions. The FFT is an efficient algorithm used to compute the discrete Fourier transform (DFT) and its inverse, which are fundamental tools in signal processing, image processing, and other related fields.
This function takes three input parameters:
1. a - A float array containing the real-valued function samples.
2. tnn - The number of function values (must be a power of 2, but the algorithm does not validate this condition).
3. inversefft - A boolean flag that indicates the direction of the transformation (True for inverse, False for direct).
The function modifies the input array a in-place, meaning that the transformed data (the FFT result for direct transformation or the inverse FFT result for inverse transformation) will be stored in the same array a after the function execution.
The algorithm uses a combination of complex-to-complex FFT and additional transformations specific to real-valued data to optimize the computation. It takes into account the symmetry properties of the real-valued input data to reduce the computational complexity.
Here's a detailed walkthrough of the algorithm:
1. Depending on the inversefft flag, the initial values for ttheta, c1, and c2 are determined. These values are used for the initial data preprocessing and post-processing steps specific to the real-valued FFT.
2. The preprocessing step computes the initial real and imaginary parts of the data using a combination of sine and cosine terms with the input data. This step effectively converts the real-valued input data into complex-valued data suitable for the complex-to-complex FFT.
3. The complex-to-complex FFT is then performed on the preprocessed complex data. This involves bit-reversal reordering, followed by the Cooley-Tukey radix-2 decimation-in-time algorithm. This part of the code is similar to the fastfouriertransform() function you provided earlier.
4. After the complex-to-complex FFT, a post-processing step is performed to obtain the final real-valued output data. This involves updating the real and imaginary parts of the transformed data using sine and cosine terms, as well as the values c1 and c2.
5. Finally, if the inversefft flag is True, the output data is divided by the number of samples (nn) to obtain the inverse DFT.
The function does not return a value explicitly. Instead, the transformed data is stored in the input array a. After the function execution, you can access the transformed data in the a array, which will have the real part at even indices and the imaginary part at odd indices.
Fast Sine Transform
This code defines a function called fastsinetransform that performs a Fast Discrete Sine Transform (FST) on an array of real numbers. The function takes three input parameters:
1. a (float array): An array of real numbers representing the function values.
2. tnn (int): Number of function values (must be a power of two, but the code doesn't validate this).
3. inversefst (bool): A boolean flag indicating the direction of the transformation. If True, it performs the inverse FST, and if False, it performs the direct FST.
The output is the transformed array 'a', which will contain the result of the transformation.
The code starts by initializing several variables, including trigonometric constants for the sine transform. It then sets the first value of the array 'a' to 0 and calculates the initial values of 'y1' and 'y2', which are used to update the input array 'a' in the following loop.
The first loop (with index 'jx') iterates from 2 to (tm + 1), where 'tm' is half of the number of input samples 'tnn'. This loop is responsible for calculating the initial sine transform of the input data.
The second loop (with index 'ii') is a bit-reversal loop. It reorders the elements in the array 'a' based on the bit-reversed indices of the original order.
The third loop (with index 'ii') iterates while 'n' is greater than 'mmax', which starts at 2 and doubles each iteration. This loop performs the actual Fast Discrete Sine Transform. It calculates the sine transform using the Danielson-Lanczos lemma, which is a divide-and-conquer strategy for calculating Discrete Fourier Transforms (DFTs) efficiently.
The fourth loop (with index 'ix') is responsible for the final phase adjustments needed for the sine transform, updating the array 'a' accordingly.
The fifth loop (with index 'jj') updates the array 'a' one more time by dividing each element by 2 and calculating the sum of the even-indexed elements.
Finally, if the 'inversefst' flag is True, the code scales the transformed data by a factor of 2/tnn to get the inverse Fast Sine Transform.
In summary, the code performs a Fast Discrete Sine Transform on an input array of real numbers, either in the direct or inverse direction, and returns the transformed array. The algorithm is based on the Danielson-Lanczos lemma and uses a divide-and-conquer strategy for efficient computation.
Fast Cosine Transform
This code defines a function called fastcosinetransform that takes three parameters: a floating-point array a, an integer tnn, and a boolean inversefct. The function calculates the Fast Cosine Transform (FCT) or the inverse FCT of the input array, depending on the value of the inversefct parameter.
The Fast Cosine Transform is an algorithm that converts a sequence of values (time-domain) into a frequency domain representation. It is closely related to the Fast Fourier Transform (FFT) and can be used in various applications, such as signal processing and image compression.
Here's a detailed explanation of the code:
1. The function starts by initializing a number of variables, including counters, intermediate values, and constants.
2. The initial steps of the algorithm are performed. This includes calculating some trigonometric values and updating the input array a with the help of intermediate variables.
3. The code then enters a loop (from jx = 2 to tnn / 2). Within this loop, the algorithm computes and updates the elements of the input array a.
4. After the loop, the function prepares some variables for the next stage of the algorithm.
5. The next part of the algorithm is a series of nested loops that perform the bit-reversal permutation and apply the FCT to the input array a.
6. The code then calculates some additional trigonometric values, which are used in the next loop.
7. The following loop (from ix = 2 to tnn / 4 + 1) computes and updates the elements of the input array a using the previously calculated trigonometric values.
8. The input array a is further updated with the final calculations.
9. In the last loop (from j = 4 to tnn), the algorithm computes and updates the sum of elements in the input array a.
10. Finally, if the inversefct parameter is set to true, the function scales the input array a to obtain the inverse FCT.
The resulting transformed array is stored in the input array a. This means that the function modifies the input array in-place and does not return a new array.
Fast Convolution
This code defines a function called fastconvolution that performs the convolution of a given signal with a response function using the Fast Fourier Transform (FFT) technique. Convolution is a mathematical operation used in signal processing to combine two signals, producing a third signal representing how the shape of one signal is modified by the other.
The fastconvolution function takes the following input parameters:
1. float signal: This is an array of real numbers representing the input signal that will be convolved with the response function. The elements are numbered from 0 to SignalLen-1.
2. int signallen: This is an integer representing the length of the input signal array. It specifies the number of elements in the signal array.
3. float response: This is an array of real numbers representing the response function used for convolution. The response function consists of two parts: one corresponding to positive argument values and the other to negative argument values. Array elements with numbers from 0 to NegativeLen match the response values at points from -NegativeLen to 0, respectively. Array elements with numbers from NegativeLen+1 to NegativeLen+PositiveLen correspond to the response values in points from 1 to PositiveLen, respectively.
4. int negativelen: This is an integer representing the "negative length" of the response function. It indicates the number of elements in the response function array that correspond to negative argument values. Outside the range , the response function is considered zero.
5. int positivelen: This is an integer representing the "positive length" of the response function. It indicates the number of elements in the response function array that correspond to positive argument values. Similar to negativelen, outside the range , the response function is considered zero.
The function works by:
1. Calculating the length nl of the arrays used for FFT, ensuring it's a power of 2 and large enough to hold the signal and response.
2. Creating two new arrays, a1 and a2, of length nl and initializing them with the input signal and response function, respectively.
3. Applying the forward FFT (realfastfouriertransform) to both arrays, a1 and a2.
4. Performing element-wise multiplication of the FFT results in the frequency domain.
5. Applying the inverse FFT (realfastfouriertransform) to the multiplied results in a1.
6. Updating the original signal array with the convolution result, which is stored in the a1 array.
The result of the convolution is stored in the input signal array at the function exit.
Fast Correlation
This code defines a function called fastcorrelation that computes the correlation between a signal and a pattern using the Fast Fourier Transform (FFT) method. The function takes four input arguments and modifies the input signal array to store the correlation values.
Input arguments:
1. float signal: This is an array of real numbers representing the signal to be correlated with the pattern. The elements are numbered from 0 to SignalLen-1.
2. int signallen: This is an integer representing the length of the input signal array.
3. float pattern: This is an array of real numbers representing the pattern to be correlated with the signal. The elements are numbered from 0 to PatternLen-1.
4. int patternlen: This is an integer representing the length of the pattern array.
The function performs the following steps:
1. Calculate the required size nl for the FFT by finding the smallest power of 2 that is greater than or equal to the sum of the lengths of the signal and the pattern.
2. Create two new arrays a1 and a2 with the length nl and initialize them to 0.
3. Copy the signal array into a1 and pad it with zeros up to the length nl.
4. Copy the pattern array into a2 and pad it with zeros up to the length nl.
5. Compute the FFT of both a1 and a2.
6. Perform element-wise multiplication of the frequency-domain representation of a1 and the complex conjugate of the frequency-domain representation of a2.
7. Compute the inverse FFT of the result obtained in step 6.
8. Store the resulting correlation values in the original signal array.
At the end of the function, the signal array contains the correlation values at points from 0 to SignalLen-1.
Fast Fourier Transform of Two Real Functions
This code defines a function called tworealffts that computes the Fast Fourier Transform (FFT) of two real-valued functions (a1 and a2) using a Cooley-Tukey-based radix-2 Decimation in Time (DIT) algorithm. The FFT is a widely used algorithm for computing the discrete Fourier transform (DFT) and its inverse.
Input parameters:
1. float a1: an array of real numbers, representing the values of the first function.
2. float a2: an array of real numbers, representing the values of the second function.
3. float a: an output array to store the Fourier transform of the first function.
4. float b: an output array to store the Fourier transform of the second function.
5. int tn: an integer representing the number of function values. It must be a power of two, but the algorithm doesn't validate this condition.
The function performs the following steps:
1. Combine the two input arrays, a1 and a2, into a single array a by interleaving their elements.
2. Perform a 1D FFT on the combined array a using the radix-2 DIT algorithm.
3. Separate the FFT results of the two input functions from the combined array a and store them in output arrays a and b.
Here is a detailed breakdown of the radix-2 DIT algorithm used in this code:
1. Bit-reverse the order of the elements in the combined array a.
2. Initialize the loop variables mmax, istep, and theta.
3. Enter the main loop that iterates through different stages of the FFT.
a. Compute the sine and cosine values for the current stage using the theta variable.
b. Initialize the loop variables wr and wi for the current stage.
c. Enter the inner loop that iterates through the butterfly operations within each stage.
i. Perform the butterfly operation on the elements of array a.
ii. Update the loop variables wr and wi for the next butterfly operation.
d. Update the loop variables mmax, istep, and theta for the next stage.
4. Separate the FFT results of the two input functions from the combined array a and store them in output arrays a and b.
At the end of the function, the a and b arrays will contain the Fourier transform of the first and second functions, respectively. Note that the function overwrites the input arrays a and b.
█ Example scripts using functions contained in loxxfft
Real-Fast Fourier Transform of Price w/ Linear Regression
Real-Fast Fourier Transform of Price Oscillator
Normalized, Variety, Fast Fourier Transform Explorer
Variety RSI of Fast Discrete Cosine Transform
STD-Stepped Fast Cosine Transform Moving Average

Spectral Gating (SG)The Spectral Gating (SG) Indicator is a technical analysis tool inspired by music production techniques. It aims to help traders reduce noise in their charts by focusing on the significant frequency components of the data, providing a clearer view of market trends.
By incorporating complex number operations and Fast Fourier Transform (FFT) algorithms, the SG Indicator efficiently processes market data. The indicator transforms input data into the frequency domain and applies a threshold to the power spectrum, filtering out noise and retaining only the frequency components that exceed the threshold.
Key aspects of the Spectral Gating Indicator include:
Adjustable Window Size: Customize the window size (ranging from 2 to 6) to control the amount of data considered during the analysis, giving you the flexibility to adapt the indicator to your trading strategy.
Complex Number Arithmetic: The indicator uses complex number addition, subtraction, and multiplication, as well as radius calculations for accurate data processing.
Iterative FFT and IFFT: The SG Indicator features iterative FFT and Inverse Fast Fourier Transform (IFFT) algorithms for rapid data analysis. The FFT algorithm converts input data into the frequency domain, while the IFFT algorithm restores the filtered data back to the time domain.
Spectral Gating: At the heart of the indicator, the spectral gating function applies a threshold to the power spectrum, suppressing frequency components below the threshold. This process helps to enhance the clarity of the data by reducing noise and focusing on the more significant frequency components.
Visualization: The indicator plots the filtered data on the chart with a simple blue line, providing a clean and easily interpretable representation of the results.
Although the Spectral Gating Indicator may not be a one-size-fits-all solution for all trading scenarios, it serves as a valuable tool for traders looking to reduce noise and concentrate on relevant market trends. By incorporating this indicator into your analysis toolkit, you can potentially make more informed trading decisions.

FFT Strategy Bi-Directional Stop/Profit/Trailing + VMA + AroonThis strategy uses the Fast Fourier Transform inspired from the source code of @tbiktag for the Fast Fourier Transform & @lazybear for the VMA filter.
If you are not familiar with the Fast Fourier transform it is a variation of the Discrete Fourier Transform. Veritasium on youtube has a great video on it with a follow up recommendation from 3brown1blue. In short it will extract all the frequencies from a set of data. @tbiktag laid the groundwork for creating the indicator which will allow you to isolate only those signals which are the most relevant and remove the noise. I recommend having @tbiktag's FFT Transform indicator side by side with this to understand what my variation is doing by setting similar settings .
Using this idea, you can then optimize a strategy to the frequencies that are best. The main entry signal is when the FFT Signal crosses above or below the 0 line .
Included with this strategy is the ability to optionally bi-directionally set:
Stop Loss
Trailing Stop Loss
Take Profit
Trailing Take Profit
Entries are optionally further filtered by use of the VMA using the algorithm from LazyBear which allows you to adjust a variable moving average with 3 market trend detections. Green represents upwards momentum; Blue sideways trading and Red downwards momentum. The idea being to filter out buy or sell entries unless the market is moving in that direction, and this makes a big difference as you can see for yourself when you turn it off or on. Turning it off will change the color of the FFT signal to orange instead of the green, blue, red colors .
I have added 2 custom stop loss types as well for experimentation:
1. VMA Filter stop loss to exit the trade if the VMA detects a market trend direction change matching the rules you have set. I have set this to off by default, but it is there so you can see what affect it may have on other tickers. It can increase the profit factor but usually at a cost of net profit.
2. The Aroon Filter stop loss with different lengths for the short or long direction. For the Aroon strategy (which is a trend change detector) it is considered bullish if the upper line (green in my code) is above 70 and the lower line (red in my code) is below 30 and the opposite for the bearish case. With this in mind, I have set it to filter by default only the extreme ends (99 and 1) to increase profit factor and net profit but I encourage you to try different settings and see how it affects things. Turning this off yields much higher net profit but at the cost of the profit factor and drawdown . To disable this just uncheck the 'Use Aroon Filter Long' (or short) and it will also hide the aroon graphics and crosses on the plot.
I will be adding more features in an attempt to lower the drawdown on this strategy but I hope you enjoy what I have so far!

Function: Discrete Fourier TransformExperimental:
function for inverse and discrete fourier transform in one, if you notice errors please let me know! use at your own risk...

Low Frequency Fourier TransformThis Study uses the Real Discrete Fourier Transform algorithm to generate 3 sinusoids possibly indicative of future price.
I got information about this RDFT algorithm from "The Scientist and Engineer's Guide to Digital Signal Processing" By Steven W. Smith, Ph.D.
It has not been tested thoroughly yet, but it seems that that the RDFT isn't suited for predicting prices as the Frequency Domain Representation shows that the signal is similar to white noise, showing no significant peaks, indicative of very low periodicity of price movements.