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Fundamentals of Digital Signal Processing

A Unit Impulse Signal

Explore the concept of the unit impulse signal in digital signal processing. Understand its mathematical definition and why it is fundamental for constructing any discrete-time signal. Learn how combining shifted and scaled unit impulses can form more complex signals. This lesson also introduces basic operations on signals such as addition, multiplication, and time shifting, essential for digital signal processing.

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No computer can plot a continuous-time sine wave. When we see such a figure on our screen, it’s in fact an interpolated version of a discrete-time sine wave in which the samples are joined together with straight lines.

A discrete-time sinusoid

To understand this idea, let’s generate another sinusoid but with a low sample rate. Here, we don’t join the individual samples to display the discrete-time signals, as shown below:

A discrete-time sinusoid
A discrete-time sinusoid

Let’s run some code and observe the effect on this discrete-time signal.

Python 3.10.4
import numpy as np
import matplotlib.pyplot as pl
figWidth = 20
figHeight = 10
# Generating a cosine
fs = 10 # sample rate
Ts = 1/fs # sample time
f = 1
T = 1/f
A = 1
phi = 0
t = np.arange(0, 2*T, Ts)
s = A*np.cos(2*np.pi*f*t + phi)
# Plotting the signal
fig, ax = pl.subplots(figsize=(figWidth, figHeight))
ax.vlines(t, ymin=s, ymax=0, color='g', linewidth=2, zorder=10, clip_on=False)
ax.plot(t, s, 'o', color='orange', markersize=10, zorder=10, clip_on=False)
# Axes lines
ax.axhline(y=0, color='k', linewidth=2)
ax.axvline(x=0, color='k', linewidth=2)
ax.set_xlim(-Ts, 2*T+Ts)
ax.set_xticks(np.linspace(0, 2*T-2*Ts, 7))
ax.set_xticklabels(np.arange(0, 21, 3))
ax.set_ylim(-1.1, 1.1)
ax.set_yticks(np.linspace(-1, 1, 5))
ax.tick_params(labelsize=18)
# Hide the right and top spines
ax.spines['right'].set_visible(False)
ax.spines['top'].set_visible(False)
ax.set_title("A Discrete-Time Sinusoid", fontsize=18)
ax.set_xlabel("$n$", fontsize=18)
ax.set_ylabel("$x[n]$", fontsize=18)
ax.grid()
# saving figure
pl.savefig('output/cosine-discrete.png')

As we can see, all discrete-time signals are constructed from individual samples.

A time-domain impulse

For this reason, a unit impulse δ[n]\delta [n] ...