SciPy - fhtoffset() Function

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The fhtoffset() method belongs to the FHT (Fast Hankel Transform) toolkit in SciPy. It figures out the best logarithmic offset to align the data points in log space for effective Hankel transformations. This offset makes sure the input data fits the grid that the FHT algorithm needs.

When undergoing a Fourier or Hankel transform data must be logarithmically spaced and fall within a specified grid for accurate calculations . If the data is not aligned correctly with this grid, incorrect results may occur. To more closely align with this required grid, the data is shifted using a calculated adjustment called an offset.

This method helps to boost the accuracy of the Fast Hankel Transform, which is widely used for applications in signal processing, physics, and astronomy. Without using an offset, there might be errors or inaccuracies in the converted data.

People often use the fhtoffset() function with the ifht (inverse FHT) and fht (fast hankel transform) techniques. When used together, these methods allow for quick and accurate Hankel transformations on log-spaced data, making them valuable for solving tough math problems.

Syntax

Following is the syntax of the SciPy fhtoffset() method −

.fhtoffset(dln, mu, initial=0.0, bias=0.0)

Parameters

This method accepts the following parameters −

• dln (float): The logarithmic spacing of the input array.

• mu (float): The order of the Hankel transform.

• initial (float, optional, default=0.0): An initial guess for the logarithmic offset, typically 0.

• bias (float, optional, default=0.0): A bias adjustment to the logarithmic offset.

Return Value

offset (float): The ideal logarithmic offset for calculating the Fast Hankel Transform.

Example 1: Calculating Optimal Logarithmic Offset

This example shows how to calculate the ideal logarithmic offset for the Fast Hankel Transform(FHT) using fhtoffset() method. This offset aligns logarithmically spaced data points to the grid required for the Fast Hankel Transform, improving the accuracy of the results.

In this example, we specify the hankel transform order (mu = 0.5) and logarithmic spacing (dln = 0.1). By using these inputs, the ideal offset is calculated via the fhtoffset() method and the result is printed. To minimize the mistakes when applying the Fast Hankel Transform to logarithmically spaced data, this offset, -0.027642801500698512 is crucial. Following is the code −

import numpy as np
from scipy.fft import fhtoffset

# Parameters
dln = 0.1  
mu = 0.5   
offset = fhtoffset(dln, mu)
print("Optimal offset:",offset)

Following is an output of the above code −

Optimal offset: -0.027642801500698512

Example 2: Using the bias parameter in fhtoffset

In this example let us see how to modify the calculated logarithmic offset for specific requirements using the bias parameter in fhtoffset. This parameter manually adjusts the calculatd offset, in applications involving the Fast Hankel Trasform.

In this example, we calculate the offset with and without the bias parameter. The bias introduces a user-defined adjustment to the offset value, which can be useful for fine-tuning the transform in specific scenarios.

import numpy as np
from scipy.fft import fhtoffset

# Parameters
dln = 0.1  
mu = 0.5   
bias = 0.05  # Custom bias adjustment

# Compute the offset without bias
offset_no_bias = fhtoffset(dln, mu)

# Compute the offset with bias
offset_with_bias = fhtoffset(dln, mu, bias=bias)

print(f"Offset without bias: {offset_no_bias}")
print(f"Offset with bias: {offset_with_bias}")

The output of the code is as follows −

Offset without bias: -0.027642801500698512
Offset with bias: -0.027644068121934852

Example 3: Using fhtoffset with ifht(Inverse Fast Hankel Transform)

In this example we will see a real world application, we have a data related to astronomy about star brightness or distance this is expressed in logarithmic terms because space observations span a large range of distances or intensities.

Before implementing the Fast Hankel Transform (FHT), the offset value makes sure the data is properly aligned. When processing logarithmically spaced data, like measurements, we can prevent misalignments and get more accurate results by correcting the data with the offset.

Now let us calculate the offset value using fhtoffset() method, which we then apply it to reconstruct the astronomical data representing star brightness, using ihft(Inverse Fast Hankel Transform). The offset corrects this spacing for accurate transformation. This reconstructed data is then used for further trasformation or analysis and we will get the best and precise results.

import numpy as np
from scipy.fft import fhtoffset, ifht

dln = 0.1
mu = 0.5 
initial_data = np.logspace(0, 1, 10)  # Example data in logarithmic space
offset = fhtoffset(dln, mu)

# Apply the inverse Fast Hankel Transform using the offset
reconstructed_data = ifht(initial_data, dln, mu, offset=offset)
print(f"Optimal offset: {offset}")
print("Reconstructed data:", reconstructed_data)

Output of the above code is as follows −

Optimal offset: -0.027642801500698512
Reconstructed data: [6.03694565 9.1576778  6.06496283 7.7729893  1.69796113 0.80792169
 1.14000591 3.9575659  2.89501008 1.33848583]

Depending on the SciPy version and numerical precision, the code's exact output values may vary.