Introduction to Electronics - Filters: Impulse response of first-order filters

The impulse response describes how a filter reacts to a sudden, instantaneous input—an impulse. In theory, an impulse is represented by a Dirac delta function, which has an infinitely high peak at time zero and an area of 1. In practical terms, the impulse response provides a complete characterisation of a linear time-invariant system. For first-order filters, such as an RC low-pass filter, the impulse response reveals the filter's inherent dynamics and is useful for predicting the output for any arbitrary input through . The impulse response is closely related to the step response; in fact, for causal systems, the impulse response is the derivative of the step response.

Circuit Diagram

Consider the same RC low-pass filter used in previous articles:

When an impulse is applied at the input, the output voltage of an RC low-pass filter is determined by how the capacitor charges through the resistor.

Mathematical Underpinning

For an RC low-pass filter subjected to a step input of amplitude [math]V_{in}[/math], the step response is

[math]V_{out}(t)=V_{in}\left(1-e^{-t/RC}\right).[/math]

The impulse response, [math]h(t)[/math], is the time derivative of the step response (assuming zero initial conditions). Taking the derivative, we obtain

[math]h(t)=\frac{d}{dt}\left[V_{in}\left(1-e^{-t/RC}\right)\right] = V_{in}\left(\frac{1}{RC}\,e^{-t/RC}\right).[/math]

For a normalized impulse with area 1, we set [math]V_{in}=1[/math], so

[math]h(t)=\frac{1}{RC}\,e^{-t/RC} \quad \text{for } t\ge 0,[/math]

and

[math]h(t)=0 \quad \text{for } t<0.[/math]

This exponential decay function defines the behaviour of the filter in response to an impulse.

Demo Python Script

The following Python script simulates the impulse response of an RC low-pass filter. The script computes [math]h(t)[/math] using the formula above and plots the impulse response over a suitable time range. In this example, we assume [math]R=1\,\text{k}\Omega[/math] and [math]C=1\,\mu\text{F}[/math].

import numpy as np
import matplotlib.pyplot as plt

# Time vector
t = np.linspace(0, 5, 1000) # 0 to 5 ms

# Parameters
tau = 1 # Time constant in milliseconds
impulse_time = 2 # Impulse occurs at 2 ms
impulse_magnitude = 1000

# Create input impulse signal
input_impulse = np.zeros_like(t)
input_impulse[np.argmin(np.abs(t - impulse_time))] = impulse_magnitude

# Create impulse response: zero before impulse_time, then exponential decay
impulse_response = np.zeros_like(t)
impulse_response[t >= impulse_time] = (
impulse_magnitude * np.exp(-(t[t >= impulse_time] - impulse_time) / tau)
)

# Plot
plt.figure(figsize=(6, 4))
plt.plot(t,
impulse_response,
label='Impulse Response h(t)',
color='blue')
plt.plot(t,
input_impulse,
label='Input Impulse',
color='red',
linestyle='--')
plt.axvline(x=impulse_time,
color='red',
linestyle='--',
alpha=0.5) # Visual marker for impulse time
plt.title('Impulse Response of RC Low-Pass Filter (Impulse at 2 ms)')
plt.xlabel('Time (ms)')
plt.ylabel('Amplitude')
plt.legend()
plt.grid(True)
plt.xlim(0, 5)
plt.ylim(0, 1100)
plt.tight_layout()
plt.show()

And, here is the plot generated by the script:

The red dashed vertical line is at t = 0 indicates the moment of the input impulse.

Key Takeaways

Here's a few key points to remember from this article:

• The impulse response characterises the complete dynamic behaviour of a filter by showing its output when subjected to an impulse input.
• For an RC low-pass filter, the impulse response is given by [math]h(t)=\frac{1}{RC}\,e^{-t/RC} \quad (t\ge 0),[/math] which reflects an exponential decay governed by the time constant [math]RC[/math].

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