4.8. Signal Temporal Mapping¶
4.8.1. Linear mapping¶
4.8.1.1. Quantization¶
Quantize signal into discreet levels
Before quantization, signal is equalise using either of three methods (1) A-Law, (2) Mu-Law, or (3) Cummulative Distribution Function
import numpy as np
import matplotlib.pyplot as plt
import spkit as sp
x,fs,_ = sp.data.ecg_sample_12leads(sample=3)
x = x[:1000,0]
t = np.arange(len(x))/fs
y1, yint = sp.quantize_signal(x.copy(),n_levels=5)
y2, yint = sp.quantize_signal(x.copy(),n_levels=11)
y3, yint = sp.quantize_signal(x.copy(),n_levels=31)
m1 = np.mean((x-y1)**2)
m2 = np.mean((x-y2)**2)
m3 = np.mean((x-y3)**2)
plt.figure(figsize=(12,3))
plt.plot(t,x,alpha=0.8 ,label='x')
plt.plot(t,y1,alpha=0.8,label=f'y1: L=5, MSE={m1:,.4f}')
plt.plot(t,y2,alpha=0.8,label=f'y2: L=11, MSE={m2:,.4f}')
plt.plot(t,y3,alpha=0.8,label=f'y3: L=31, MSE={m3:,.4f}')
plt.xlim([t[0],t[-1]])
plt.xlabel('time (s)')
plt.ylabel('x')
plt.legend()
plt.grid()
plt.tight_layout()
plt.show()
4.8.1.2. Low resolution¶
Reducing the time-resolution of signal \(x\)
Reducing the time-resolution of signal : x. It is similar as downsampling (or decimation) of a signal, except it is averaged around removed samples.
import numpy as np
import matplotlib.pyplot as plt
import spkit as sp
x,fs,_ = sp.data.ecg_sample_12leads(sample=3)
x = x[1000:2000,0]
x = x + 0.1*np.random.randn(len(x))
t = np.arange(len(x))/fs
N = 5
y1 = x[::N] # decimation by factor of N
y2 = sp.low_resolution(x.copy(),scale=N)
4.8.2. Non-linear mapping¶
4.8.2.1. A-Law¶
A-Law for companding and expanding (Non-linear mapping)
A-Law for companding and expanding - A-Law is one of the ways to map gaussian, or laplacian distribuated signal to uniformly distributed one - for companding - smaller amplitude values are stretched (since they have high-probabilty)
and large amplitutde values are compressed (since they have low probabiltity)
for expending - it reverts the mapping
for -1 ≤ x ≤ 1 and 1 < A
- Companding:
expanding:
\[ \begin{align}\begin{aligned}y(n) = sign(x)* \frac{|x|(1+log(A))} {A} if |x|<1/(1+log(A))\\ = sign(x)* \frac{(exp(-1 + |x|(1+log(A))))} {A} else\end{aligned}\end{align} \]
An alternative to μ-Law
The μ-law algorithm provides a slightly larger dynamic range than the A-law at the cost of worse proportional distortions for small signals.
4.8.2.2. μ-Law¶
μ-Law for companding and expanding (Non-linear mapping)
μ-Law is one of the ways to map gaussian, or laplacian distribuated signal to uniformly distributed one - for companding - smaller amplitude values are stretched (since they have high-probabilty)
and large amplitutde values are compressed (since they have low probabiltity)
for expending - it reverts the mapping
for -1 ≤ x ≤ 1
Companding:
\[y(n) = sign(x)* \frac{log(1 + μ|x|)}{log(1 + μ)}\]expanding:
\[y(n) = sign(x)* \frac{((1 + μ)**|x| - 1)}{μ}\]
An alternative to A-Law
The μ-law algorithm provides a slightly larger dynamic range than the A-law at the cost of worse proportional distortions for small signals.
4.8.2.3. CDF Mapping¶
CDF: Cumulative Distribution Function Mapping (Non-linear mapping)
Map the signal x to y from into CDF of x, y will be uniformly disctribuated anf ranges from 0 to 1. [1]
CDF: Cumulative Distribution Function
\[y(n) = P(X<=x(n))\]
References
Click for more details
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