SciPy - dst() Function

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The scipy.fft.dst() method gives the Discrete Sine Transform, which is the mathematical procedure that uses sines to transform spatial data into the frequency domain. This transform is used typically to analyze data with odd symmetry by transforming it into frequency components for processing.

The scipy.fft.dst() method supports four types of DST, each useful for a different application. Type 1 is equivalent to an FFT for real, odd-symmetric inputs. Type 2 is the most commonly used transform and therefore the standard. Type 3 is the inverse transform, useful for reconstructing data. Type 4 is for advanced applications, where the symmetry needs to be more precise.

DST performs well with DCT or FFT for full signal analysis and processing, meaning both symmetric and odd-symmetric data sets can be processed efficiently.

DST is an important technique used for signal processing, image compression, and for problems of partial differential equations. It efficiently isolates the sine wave components especially in cases of unusual boundary conditions.

Syntax

The syntax for the SciPy dst() method is as follows −

.dst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False, workers=None, orthogonalize=None)

Parameters

This method accepts the following parameters −

• x − The input array (spatial-domain data to be transformed).

• type − The type parameter in scipy.fft.dst defines the transform variant: Type 1 for FFT-like odd-symmetric inputs, Type 2 as standard, Type 3 as its inverse, and Type 4 for advanced symmetry.

• n − Length of the transform; pads or truncates input data if specified.

• axis − Axis for the DST in multi-dimensional arrays (default: last axis).

• norm − The normalization mode allows control over scaling: "backward" (default, no scaling), "ortho" (unitary scaling for energy preservation), and "forward" (scales the output by 1/n, where n is the transform length).

• overwrite_x − Modifies input to save memory if True. Fact: IDCT, when paired with DCT, provides perfect reconstruction of the original data if no coefficients are modified.

• workers − Number of threads to use for parallel computation. Default is 1.

• orthogonalize − Allows explicit orthogonalization for advanced use cases. Default is None.

Return Value

dst (ndarray of reals) − The DST-transformed array, representing the frequency-domain components of the input.

Example 1: Basic DST Transformation

This example demonstrates how the DST employs sine functions to transform spatial-domain data into frequency-domain representation. The output indicates the sine wave components of the input data.

The below code, is the basic example of using the dst() method on a simple array [1, 2, 3, 4]. The result displays frequency components represented as sine coefficients.

import numpy as np
import scipy.fft
from scipy.fft import dst

data = [1, 2, 3, 4]
result = scipy.fft.dst(data)
print("DST of data:", result)

When we run above program, it produces following result

DST of data: [13.06562965 -5.65685425  5.411961   -4.        ]

Example 2: DST on a 2D Array

This example applies DST on a 2D array of rows, transforming each row independently into its frequency domain representation.

The dst() function is applied row-wise to the array [[3, 5, 7], [8, 6, 4]] using the default axis (axis=-1). The output is a two-dimensional array with frequency coefficients for each row.

import numpy as np
import scipy.fft
from scipy.fft import dst

data = [1, 3, 5, 7]
result = dst(data, type=1)
print("DST Type 1 of data:", result)

Following is an output of the above code

DST Type 1 of data: [ 24.6214683  -13.7638192    5.81234022  -3.24919696]

Example 3: Difference Between DST and DCT

DST and DCT are two transforms which change data from spatial domain into a frequency domain but they do have different basis functions, DST uses sine waves while assuming odd symmetry thus applies to odd boundary conditions; DCT uses cosine waves and assumes even symmetry-thus perfect for smooth data or image compression.

This code passes the same input array to both dst() and dct(). DST emphasizes odd-symmetric components with sine waves, whereas DCT emphasizes even-symmetric components with cosine waves, putting an emphasis on their unique frequency representation.

import numpy as np
from scipy.fft import dst, dct

data = [1, 3, 5, 3]  # Symmetrical-like data

# Applying DST
dst_result = dst(data, norm='ortho')
print("DST result:", dst_result)

# Applying DCT
dct_result = dct(data, norm='ortho')
print("DCT result:", dct_result)

Output of the above code is as follows

DST result: [ 6.30864406 -2.          0.44834153  0.        ]
DCT result: [ 6.         -1.84775907 -2.          0.76536686]

Example 4:

This example shows how dst() and fft can work together to analyze signals. While DST isolates the odd-symmetric sine wave components, FFT captures the whole spectrum of frequencies, which is ideal for high-resolution analysis in audio processing or in vibration analysis.

import numpy as np
from scipy.fft import dst, fft

# Create a composite signal: sine wave + cosine wave
t = np.linspace(0, 1, 100, endpoint=False)
signal = np.sin(2 * np.pi * 5 * t) + 0.5 * np.cos(2 * np.pi * 10 * t)

# Apply DST (focuses on sine components)
dst_result = dst(signal, norm='ortho')
print("DST result (odd-symmetric sine components):", dst_result[:10]) 

# Apply FFT (captures both sine and cosine components)
fft_result = fft(signal)
print("FFT result (complete frequency spectrum):", np.abs(fft_result[:10]))  

Output of the above code is as follows

DST result (odd-symmetric sine components): [ 3.62017374e-03 -2.88114656e-16  1.39539925e-02  1.73632957e-16
  3.74298480e-02  2.38782243e-15  1.08797404e-01  2.20833496e-16
  5.47468662e-01  6.98401123e+00]
FFT result (complete frequency spectrum): [1.55431223e-14 1.00830176e-14 2.14580936e-14 1.97535756e-14
 2.68418810e-15 5.00000000e+01 1.44296599e-14 1.42000103e-14
 7.63415081e-15 3.87440389e-15]