Signal and noise - Data to Decision

Any measured signal may be tainted by noise.
Recovering the true signal may be impossible.
However, we will look at some methods use various strategies for
- suppressing noise,
- estimating derivatives,
- decomposing into different frequencies,
- decomposing into trend, seasonal changes and residual, and
- exploratory data analysis.

Signal-to-Noise Ratio (SNR)¶

Signal-to-noise is a ratio that can be caulculated in various ways.
- A measure of how easy it is to extract the underlying information.
A common definition is decibel (dB) based.
- 0 decibel means the signal and noise are of equal magnitude.
- Every 10 decibel increase means 10 times higher power of the signal compared to the noise.
Example: HiFi amplifier:
- >100 dB SNR, i.e., 10¹⁰ times higher power of the signal than the noise.
$SNR = 10 \cdot log_{10} \frac{\sum s_i^2}{\sum n_i^2}$ , where $s$ is the signal and $n$ is the noise.
If the true signal and noise are lost, estimates of these can be made.

import numpy as np
def SNR(signal, noise):
    return 10*np.log10(np.sum(signal**2)/np.sum(noise**2))

# Plot a cosine curve from 0 to 4pi
import matplotlib.pyplot as plt
import numpy as np
x = np.linspace(0, 4*np.pi, 500)
y_cos = np.cos(x)
noise = 0.2*np.random.normal(size=(len(x)))

# Plot the cosine curve with noise
sig = [1, 0.2]
noi = [1, 5]
plt.figure(figsize=(8, 4))
for i in range(4):
    plt.subplot(221+i)
    plt.plot(x, y_cos*sig[np.mod(i,2)] + noise*noi[i//2], label='Noisy cosine curve')
    plt.plot(x, y_cos*sig[np.mod(i,2)], label='Cosine curve')
    plt.xlabel('radians')
    plt.ylabel('amplitude')
    plt.title('SNR: {:.2f} dB'.format(SNR(y_cos*sig[np.mod(i,2)], noise*noi[i//2])))
    plt.xlim((0,4*np.pi))
    plt.ylim((-3,3))
#    plt.legend()
plt.tight_layout()
plt.show()