1 Lesson 1 of 4

Signals & Sinusoids

Before we can analyze anything with the Fourier Transform, we need to understand what we're analyzing. This lesson covers the two foundations: signals and the sinusoidal waves that make up every signal.

What Is a Signal?

A signal is any quantity that varies over time (or space, or some other independent variable). Signals are everywhere:

The sound pressure at your eardrum as you listen to music
The voltage on an electrical wire carrying a radio broadcast
The brightness of each pixel row in a digital image
Stock prices over a trading day
Your heartbeat recorded by an ECG

In every case, some measurable value changes as a function of time (or position). We typically write a signal as $x(t)$ for continuous time or $x[n]$ for discrete samples.

Continuous vs. Discrete Signals

A continuous-time signal $x(t)$ is defined at every instant. Sound in the real world is continuous — at any fraction of a second, there's a pressure value.

A discrete-time signal $x[n]$ is defined only at integer indices. When a computer records audio, it measures the sound pressure at regular intervals (e.g., 44,100 times per second for CD audio). Each measurement is a sample.

Fourier analysis works on both kinds, but digital tools (including everything on this site) operate on discrete signals. Lesson 3 covers exactly how we go from continuous to discrete through sampling.

The Sinusoid: The Fundamental Building Block

The core idea behind Fourier analysis is that any signal can be decomposed into a sum of sinusoids (sine and cosine waves) at different frequencies. A general sinusoid is written:

x(t) = A \sin(2\pi f t + \varphi)

This formula has three parameters that fully describe the wave:

Amplitude ( $A$ ) — The peak height of the wave. A larger amplitude means a louder sound or a stronger electrical signal.
Frequency ( $f$ ) — How many complete cycles occur per second, measured in Hertz (Hz). A 440 Hz tone completes 440 full oscillations every second — that's the note A above middle C.
Phase ( $\varphi$ ) — The horizontal shift of the wave, measured in radians. Phase determines where in its cycle the wave starts at $t = 0$ . Two waves with the same amplitude and frequency but different phases are shifted copies of each other.

Try adjusting these three parameters in our Signal Generator to build intuition for how they shape a wave.

Euler's Formula & Complex Exponentials

You'll often see sinusoids written using complex exponentials instead of sine and cosine. This is thanks to Euler's formula:

e^{j\theta} = \cos\theta + j\sin\theta

Here $j$ is the imaginary unit (engineers use $j$ instead of $i$ to avoid confusion with electrical current). The complex exponential $e^{j2\pi ft}$ packs both a cosine and a sine into a single, compact expression.

Why bother? Because the math of the DFT and FFT becomes dramatically simpler with complex exponentials. Multiplying, shifting, and summing sinusoids all reduce to straightforward operations on complex numbers.

Don't worry if complex exponentials feel abstract right now — they'll become more concrete in Lesson 2 when we meet the DFT formula.

Why Sinusoids Matter

Sinusoids are special for a fundamental reason: they are the eigenfunctions of linear time-invariant (LTI) systems. In plain English, if you feed a pure sinusoid into any stable linear system (an amplifier, a room, a filter), the output is still a sinusoid at the same frequency — only its amplitude and phase change.

This means that once you decompose a signal into sinusoids, you can analyze how each frequency component is affected independently. That's the entire power of Fourier analysis.

Key Takeaways

A signal is any value that varies over time (or space).
Continuous signals exist at every instant; discrete signals are sampled at regular intervals.
Every signal can be decomposed into sinusoids of different amplitudes, frequencies, and phases.
Euler's formula lets us write sinusoids as complex exponentials, simplifying the math.
Sinusoids are fundamental because LTI systems preserve frequency — they only change amplitude and phase.

Try It Yourself

Open the Signal Generator and experiment with amplitude, frequency, and phase. Watch how the waveform changes in real time.

Open Signal Generator →

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