loxxfft

Library "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/fasttransforms/

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