Complex Exponentials

In the last lesson, Euler's formula gave us a snapshot: e^jθ is a point on the unit circle at angle θ. But a snapshot isn't very interesting.

What if the angle keeps growing? Set θ = ωt, where ω is a constant and t is time:

e^jωt

Now the point spins. As time advances, ωt increases, and the point sweeps around the unit circle at a steady rate. This is the complex exponential, a spinning arrow in the complex plane, and it is the single most important object in signal processing.

The Spinning Arrow

Think of e^jωt as a clock hand. At t = 0, it points to the right (angle = 0). As time ticks forward, it sweeps counterclockwise. The angular frequency ω controls how fast it spins, and more ω means more rotations per second.

Meanwhile, the real part traces out cos(ωt) and the imaginary part traces out sin(ωt). The sine and cosine waves you've been studying are just shadows of this spinning arrow, projected onto the horizontal and vertical axes.

Positive and Negative Frequency

When ω is positive, e^jωt spins counterclockwise. When ω is negative, e^−jωt spins clockwise. Same speed, opposite direction.

This isn't just a mathematical curiosity. Remember from Euler's formula:

cos(ωt) = ½(e^jωt + e^−jωt)

A real cosine wave is the sum of two complex exponentials: one spinning counterclockwise at +ω and one spinning clockwise at −ω. Their imaginary parts cancel; their real parts add up. This is why real signals always have symmetric spectra, and every positive frequency has a mirror image at the negative frequency.

The General Complex Exponential

So far the arrow has length 1. In practice, signals have different amplitudes and don't always start at angle zero. The general form is:

A · e^{j(ωt + φ)}

• A = amplitude, the length of the arrow (radius of the circle it traces)
• ω = angular frequency, how fast it spins (rad/s)
• φ = phase, where the arrow starts at t = 0

Three numbers completely describe a spinning arrow. And the real part of this expression gives you the corresponding real sinusoid:

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

Why Complex Exponentials Are Special

Many functions oscillate. Square waves oscillate. Sawtooth waves oscillate. But complex exponentials have a property that no other signal has:

In math: if you feed e^jωt into an LTI system, the output is H(ω)·e^jωt, where H(ω) is a single complex number.

The system can't change the frequency. It can only change the amplitude and phase. This is what makes complex exponentials eigenfunctions of LTI systems: they pass through unchanged in character, only scaled.

This property is the entire reason the Fourier transform works:

1. Decompose any signal into a sum of complex exponentials (Fourier transform)
2. Each one passes through the system independently, just multiply by H(ω)
3. Add them back up (inverse Fourier transform) to get the output

Without this eigenfunction property, signal processing as we know it wouldn't exist. Complex exponentials aren't just a convenient notation. They are the natural basis for describing what linear systems do.

The Big Picture

Let's trace the path from the beginning of this course to here:

Every concept built on the last. You now have the mathematical vocabulary to understand the Fourier transform, filtering, modulation, sampling, and most of DSP. The next lessons put these tools to work on real signals.