Sinusoidal Signals

Here's the most important idea in signal processing, stated in one sentence:

A sharp click? Sinusoids. A violin note? Sinusoids. Your voice? Sinusoids. A photograph? Sinusoids in two dimensions. This is Fourier's insight, and it's the foundation of everything in DSP.

But before we start combining sinusoids, let's make sure we understand a single one perfectly.

Anatomy of a Sinusoid

A sinusoidal signal is fully described by three numbers:

x(t) = A · cos(2πft + φ)

• A = amplitude, how tall the wave is (peak value). Controls loudness in audio, brightness in images, strength in radio.
• f = frequency, how many cycles per second, measured in Hertz (Hz). One Hz means one complete oscillation per second. Controls pitch in audio, color in light.
• φ = phase, where in the cycle the wave starts at t = 0. Two waves with the same frequency can be perfectly in sync or completely opposed depending on their phase difference.

You already know all of this from the earlier lessons. The new piece is that we're now thinking of this as a signal, something that changes over time and carries information.

Frequency, Period, and Angular Frequency

There are three ways to express how fast a sinusoid oscillates:

Engineers switch between these constantly. Textbooks prefer ω (angular frequency) because it simplifies the math. Lab instruments show f (Hz) because it's more intuitive. You need to be fluent in both.

Phase: The Hidden Parameter

Amplitude and frequency are easy to hear and see. Phase is subtler, but it matters enormously.

Two sinusoids at the same frequency can reinforce or cancel each other depending on their phase difference:

Sinusoids in the Wild

Pure sinusoids are rare in nature, but they're the building blocks that everything else is made of:

• Tuning fork: Very nearly a pure sinusoid, which is why musicians use them as a reference.
• Power line: 60 Hz in North America, 50 Hz in Europe. The sinusoidal voltage that powers your devices.
• Radio carrier: An AM radio station at 880 kHz transmits a sinusoid at 880,000 cycles per second, with the audio signal encoded in its amplitude.
• Musical notes: Middle C is 261.6 Hz. Concert A is 440 Hz. But a real instrument produces many sinusoids at once (harmonics), and that's what gives each instrument its unique timbre.

The difference between a piano and a violin playing the same note? Same fundamental frequency, different mix of harmonics. The next lesson shows you how to combine sinusoids to build any shape you want.

The Complex Exponential Connection

From the previous lessons, you know that a real sinusoid is the real part of a spinning complex exponential:

A·cos(2πft + φ) = Re{A · e^{j(2πft + φ)}}

And equivalently, it's the sum of two complex exponentials spinning in opposite directions:

A·cos(2πft + φ) = ^A⁄₂ · e^{j(2πft + φ)} + ^A⁄₂ · e^{−j(2πft + φ)}

This is why every real sinusoid shows up as two peaks in the frequency spectrum: one at +f and one at −f. It's not a mathematical quirk. It's the two spinning arrows that combine to produce the real wave you observe.