A small Python script to test a claim from the StackExchange answer https://dsp.stackexchange.com/a/53947/1464 about the coherence of a matrix

coherence.py

"""
This is a small script investigating the coherence discussed in the
Stackexchange Signal Processing answer here:
https://dsp.stackexchange.com/a/53947/1464

"""

import numpy as np
from scipy.fftpack import dct
from scipy.linalg import circulant

def coherence(mat):
    """
    Calculate the coherence of a matrix.

    """
    
    gram_mat = mat.conj().T @ mat
    gram_mat[np.diag_indices_from(gram_mat)] = 0
    return np.max(np.abs(gram_mat))

# Construct the 2D DCT matrix
dct1_mat = dct(np.eye(32))
B = np.kron(dct1_mat, dct1_mat)
B /= np.linalg.norm(B, axis=0)

# Construct a circulant 200 x 1024 matrix (from random Gaussian
# samples)
circ_vec = np.random.randn(1, 1024)
A = circulant(circ_vec)[:200,:]
A /= np.linalg.norm(A, axis=0)

# Construct and normalise the product of A and B
prod_mat = A @ B
prod_mat /= np.linalg.norm(prod_mat, axis=0)

# Just out of curiosity, try a completely random Gaussian matrix in
# place of A
C = np.random.randn(200,1024)
C /= np.linalg.norm(C)
prod_mat2 = C @ B
prod_mat2 /= np.linalg.norm(prod_mat2)

print(f"Coherence of A: {coherence(A)}")
print(f"Coherence of B: {coherence(B)}")
print(f"Coherence of A @ B: {coherence(prod_mat)}")
print(f"Coherence of C: {coherence(C)}")
print(f"Coherence of C @ B: {coherence(prod_mat2)}")