Building Blocks of All Signals

A complex sinusoid is represented by a single impulse in the frequency domain.

• A real sinusoid is formed by the combination of two such complex sinusoids in time—hence, two impulses in frequency.
• To form any other signal, more than two complex sinusoids in time must be combined together.

Similar to a house with bricks, these sinusoids form the basic unit of signal construction regardless of their shape. One tick on the frequency axis implies one complex sinusoid. When many of them combine together, they form a continuous curve in the frequency domain called the spectrum of that signal.

The idea of building blocks

Scientists tend to break down complex objects into fundamental identical parts, e.g., atoms in physics and elements in chemistry. In a similar manner, signal processing scientists discovered thatsignals can also be broken down into the sum of several sinusoids.

Let’s explore this idea with a few examples.

3-D space

Our world is made up of three dimensions, x, y and z. Any point in space is a combination of these three components.

RGB color model

Any color can be represented as a combination of contributions from red, green, and blue, for example:

• White (0xFFFFFF) has a maximum contribution from each of these colors while black (0x000000) has none
• Magenta can be made from the 0xFF00FF combination. This is shown in the figure below:

Fourier decomposition of signals

Sinusoids play a similar role in the construction of signals. A signal $x(t)$ can be written as:

$$\begin{align*} x(t) &= a_0 + a_1 \sin (2\pi F_1 t) + a_2 \sin (2\pi F_2 t) + \cdots \\ &= a_0 + a_1 \sin (2\pi\frac{1}{T_1} t) + a_2 \sin (2\pi \frac{1}{T_2} t) + \cdots \end{align*}$$ ...