• Understand how scale and translation control time–frequency resolution

• State the admissibility condition ensuring perfect reconstruction

• Build intuition for why wavelets localize energy in time and scale

Populat Continuous Wavelets

Python Code

Step1: List your installed continuous wavelets:

import pywt
print(pywt.wavelist(kind='continuous'))

# wavelet_experiment.py

import numpy as np
import matplotlib.pyplot as plt
import pywt
from scipy.signal import chirp

# 1. Generate a toy non‐stationary signal: a linear chirp + noise
fs = 1000               # sampling frequency (Hz)
T  = 1.0                # duration (seconds)
t  = np.linspace(0, T, int(fs * T), endpoint=False)
signal = chirp(t, f0=20, f1=200, t1=T, method='linear') \
         + 0.3 * np.random.randn(len(t))

# 2. Choose scales and wavelets for CWT
scales   = np.arange(1, 128)
wavelets = ['morl', 'mexh', 'gaus1', 'cgau1']   # replaced 'paul1' with 'cgau1'

# 3. Compute and plot the scalogram for each wavelet
for wname in wavelets:
    coefs, freqs = pywt.cwt(signal, scales, wname, sampling_period=1/fs)

    plt.figure(figsize=(10, 4))
    plt.imshow(
        np.abs(coefs),
        extent=[t[0], t[-1], scales[-1], scales[0]],
        cmap='jet',
        aspect='auto'
    )
    plt.title(f'CWT Scalogram – {wname}')
    plt.xlabel('Time [s]')
    plt.ylabel('Scale')
    plt.tight_layout()
    plt.show()

---

# signal is 1-D
length = signal.shape[0]
# or simply
length = len(signal)

print(f"signal length = {length}")

Output

# Assuming `coefs` is a 2D NumPy array:
import numpy as np

# Example: coefs = np.random.randn(127, 1000)

# Get its shape
rows, cols = coefs.shape

print(f"Number of rows:    {rows}")
print(f"Number of columns: {cols}")

# Or in one line:
print(f"coefs has shape {coefs.shape[0]}×{coefs.shape[1]} (rows×columns)")

Output

# signal is 1-D
length = signal.shape[0]
# or simply
length = len(signal)
length1 = len(freqs)
length2 = len(t)

print(f"signal length = {length}")
print(f"freqs length = {length1}")
print(f"t length = {length2}")

Output

---

Change the scale

# wavelet_experiment.py

import numpy as np
import matplotlib.pyplot as plt
import pywt
from scipy.signal import chirp

# 1. Generate a toy non‐stationary signal: a linear chirp + noise
fs = 1000               # sampling frequency (Hz)
T  = 1.0                # duration (seconds)
t  = np.linspace(0, T, int(fs * T), endpoint=False)
signal = chirp(t, f0=20, f1=200, t1=T, method='linear') \
         + 0.3 * np.random.randn(len(t))

# 2. Choose scales and wavelets for CWT
scales   = np.arange(1, 228)
wavelets = ['morl', 'mexh', 'gaus1', 'cgau1']   # replaced 'paul1' with 'cgau1'

# 3. Compute and plot the scalogram for each wavelet
for wname in wavelets:
    coefs, freqs = pywt.cwt(signal, scales, wname, sampling_period=1/fs)

    plt.figure(figsize=(10, 4))
    plt.imshow(
        np.abs(coefs),
        extent=[t[0], t[-1], scales[-1], scales[0]],
        cmap='jet',
        aspect='auto'
    )
    plt.title(f'CWT Scalogram – {wname}')
    plt.xlabel('Time [s]')
    plt.ylabel('Scale')
    plt.tight_layout()
    plt.show()

---

# wavelet_experiment.py

import numpy as np
import matplotlib.pyplot as plt
import pywt
from scipy.signal import chirp

# 1. Generate a toy non‐stationary signal: a linear chirp + noise
fs = 1000               # sampling frequency (Hz)
T  = 1.0                # duration (seconds)
t  = np.linspace(0, T, int(fs * T), endpoint=False)
signal = chirp(t, f0=20, f1=200, t1=T, method='linear') \
         + 0.3 * np.random.randn(len(t))

# 2. Choose scales and wavelets for CWT
scales   = np.arange(1, 1000)
wavelets = ['morl', 'mexh', 'gaus1', 'cgau1']   # replaced 'paul1' with 'cgau1'

# 3. Compute and plot the scalogram for each wavelet
for wname in wavelets:
    coefs, freqs = pywt.cwt(signal, scales, wname, sampling_period=1/fs)

    plt.figure(figsize=(10, 4))
    plt.imshow(
        np.abs(coefs),
        extent=[t[0], t[-1], scales[-1], scales[0]],
        cmap='jet',
        aspect='auto'
    )
    plt.title(f'CWT Scalogram – {wname}')
    plt.xlabel('Time [s]')
    plt.ylabel('Scale')
    plt.tight_layout()
    plt.show()

Output