"""
dct_dictionary.py - Utility functions for generating a Discrete Cosine Transform (DCT)
orthonormal basis matrix and testing matrix properties.
This module provides functionality to:
1. Generate a Discrete Cosine Transform (DCT) orthonormal basis matrix.
2. Use the resulting matrix for orthogonal transformations in signal processing.
The DCT basis is widely used in data compression (such as JPEG image compression)
and signal processing due to its energy compaction properties.
Example usage:
--------------
>>> from dct_dictionary import dct_dictionary
>>> dct_matrix = dct_dictionary(4)
>>> print(dct_matrix)
array([[ 0.5 , 0.5 , 0.5 , 0.5 ],
[ 0.65328148, 0.27059805, -0.27059805, -0.65328148],
[ 0.5 , -0.5 , -0.5 , 0.5 ],
[ 0.27059805, -0.65328148, 0.65328148, -0.27059805]])
"""
import numpy as np
import scipy.fftpack as fftpack
[docs]
def dct_dictionary(N):
"""
Generates a Discrete Cosine Transform (DCT) orthonormal basis matrix.
The DCT basis transforms a signal into a sum of cosine functions oscillating at different frequencies.
This matrix is useful in signal processing, data compression, and other areas where an orthogonal transformation
is needed to represent signals in terms of their frequency components.
Parameters
----------
N : int
The size of the dictionary (i.e., the length of the signal).
Returns
-------
dict_matrix : numpy.ndarray
The generated DCT dictionary matrix of shape (N, N), where each column represents
a DCT basis vector.
Raises
------
ValueError
If the input size N is not positive.
Notes
-----
- The DCT basis is energy compaction-efficient, meaning most of the signal's energy is concentrated
in the first few coefficients, making it useful for compression.
- The DCT basis is orthonormal, meaning its columns are orthogonal vectors of unit length.
Example
-------
>>> dct_dictionary(4)
array([[ 0.5 , 0.5 , 0.5 , 0.5 ],
[ 0.65328148, 0.27059805, -0.27059805, -0.65328148],
[ 0.5 , -0.5 , -0.5 , 0.5 ],
[ 0.27059805, -0.65328148, 0.65328148, -0.27059805]])
"""
# Input validation
if N <= 0:
raise ValueError("N must be a positive integer.")
# Generate the DCT orthonormal basis matrix
dict_matrix = fftpack.dct(np.eye(N), norm='ortho')
return dict_matrix