Delta Preprocessor¶

The delta preprocessor function shifts the input signal by a fixed amount. The is defined as:

\[ x_{shifted_{i}} = x_{i} - \delta, \quad \forall i \in \{1, \dots, N\}, \quad \delta \in \mathbb{R} \]

Preprocess the signal by shifting the signal by some delta value.

Parameters:

Name	Type	Description	Default
`x`	`ndarray`	The signal to preprocess.	required
`delta`	`Union[int, float, int_, float_]`	The value to shift the signal by.	required
`where`	`Callable[[Union[int, float, int_, float_]], Union[bool, bool_]]`	A function that takes a value and returns `True` or `False`. Default is `lambda x: not np.isnan(x)` i.e. a measurement is valid if it is not a `NaN` value.	`lambda : not numpy.isnan(x)`

Returns:

Type	Description
`ndarray`	The shifted signal.

Examples¶

Consider a system that feeds in to a half-wave rectifier. Half-wave rectifiers are used to remove the negative portion of a signal and are use in many applications such as power supplies and AM radios. The half-wave rectifier is defined as:

\[ y_{i} = \max(x_{i}, 0) = \begin{cases} x_{i}, & \text{if } x_{i} \geq 0 \\ 0, & \text{if } x_{i} < 0 \end{cases} , \quad \forall i \in \{1, \dots, N\} \]

where \(x_{i}\) represents an element of the input signal, \(y_{i}\) represents an element of the output signal, and \(N\) is the number of elements in the signal.

Suppose we want to apply the half-wave rectifer to just the AC part of a signal containing both AC and DC components, we can use the delta preprocessor function to eliminate the DC component.

Define Signal, HWR System, and Preprocessor¶

Define a signal with the following parameters:

\[ signal_{i} = DC_{offset} + AC_{amp} \cdot \sin(2 \pi \cdot freq \cdot time_{i}) \]

\(DC_{offset} = 5\) V

\(AC_{amp} = 2\) V

\(freq = 500\) Hz

\(N = 1000\) samples

\(T = 10\) s

import numpy as np
import autonfeat.preprocess.functional as PF

# Define signal
time = np.linspace(0, 10, 1000) # secs
freq = 500                      # Hz
dc_offset = 5                   # V
ac_amp = 2                      # V

signal = dc_offset + ac_amp * np.sin(2 * np.pi * freq * time)

# Define half-wave rectifier
half_wave_rectifier = lambda x_i: np.maximum(x_i, 0)

Transform Signal¶

Transform the signal by the delta transform preprocessor and apply the half-wave rectifier. This will remove the DC component by setting \(\delta = DC_{offset} = 5\) V.

delta = dc_offset # Amount to shift by

# Preprocess signal
signal_transformed = PF.delta_tf(signal, delta=delta)

# Apply half-wave rectifier
system_output = half_wave_rectifier(signal_transformed)

Visualize Transform¶

Visualize the signal, the transformed signal, and the output of the half-wave rectifier with and without the delta transform preprocessor.

import matplotlib.pyplot as plt

# Plot results
fig, (ax1, ax2) = plt.subplots(2, 2, figsize=(10, 6))

# Plot signal and output of half-wave rectifier (before delta transform)
ax1[0].plot(time, signal, label='Signal')
ax1[0].set_xlabel('Time (s)')
ax1[0].set_ylabel('Voltage (V)')
ax1[0].set_title('Original Signal')
ax1[0].grid(True)
ax1[0].legend()

ax1[1].plot(time, half_wave_rectifier(signal), color='orange', label='Output')
ax1[1].set_xlabel('Time (s)')
ax1[1].set_ylabel('Voltage (V)')
ax1[1].set_title('Output of Half-Wave Rectifier (Before Preprocessing)')
ax1[1].grid(True)
ax1[1].legend()

# Plot signal and output of half-wave rectifier (after delta transform)
ax2[0].plot(time, signal_transformed, label='Signal')
ax2[0].set_xlabel('Time (s)')
ax2[0].set_ylabel('Voltage (V)')
ax2[0].set_title('Signal After Delta Transform')
ax2[0].grid(True)
ax2[0].legend()

ax2[1].plot(time, system_output, color='orange', label='Output')
ax2[1].set_xlabel('Time (s)')
ax2[1].set_ylabel('Voltage (V)')
ax2[1].set_title('Output of Half-Wave Rectifier (After Preprocessing)')
ax2[1].grid(True)
ax2[1].legend()

plt.tight_layout()
plt.show()

We can observe how with the help of the delta preprocessor function, shifting the signal by the DC offset of the signal eliminates the DC component of the signal and allows the half-wave rectifier to only act on the AC component of the signal.

Delta

Fun Fact¶

Half-wave rectifiers are equivalent to a rectified linear unit i.e. the \(ReLU\) activation function used in neural networks.

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