Complex numbers & phasors

Processing Prerequisites

Learning objectives

Read a complex number in rectangular (a + bi) and polar (|z|, arg z) form and switch between them
Interpret Euler's formula e^(iθ) = cos θ + i sin θ geometrically as a point on the unit circle
Explain why a rotating phasor z · e^(iωt) projects to the real cosine |z| cos(ωt + arg z)
Recognize why every seismic amplitude is a phasor and every filter acts on magnitude and phase together

A complex number is a pair of real numbers, but written together so we can multiply them like rotations instead of just scalings. That single change of perspective is what makes them indispensable for processing.

1. Rectangular form

Write z as

z \;=\; a + bi

where a is the real part, b is the imaginary part, and the symbol i obeys one rule and one rule only:

i^{2} = -1

Every law of algebra you already know still holds — associativity, commutativity, distributivity. The one new thing is how squaring i gives you a negative. That is enough to unlock everything that follows.

2. The complex plane

Plot z as a point in a 2D plane: real axis horizontal, imaginary axis vertical. This is called the complex plane (or the Argand diagram). A complex number is now a vector from the origin to a point. Addition is vector addition; subtraction is vector subtraction; multiplication is — we will see in a moment — something geometrically much more interesting.

3. Polar form: magnitude and phase

Instead of Cartesian coordinates, use length and angle:

|z| = \sqrt{a^2 + b^2}\quad\text{(magnitude)} \qquad \arg z = \arctan\!\left(\frac{b}{a}\right)\quad\text{(phase)}

Then

z = |z|\,\bigl(\cos(\arg z) + i\sin(\arg z)\bigr)

This is just the usual conversion between Cartesian and polar coordinates. The point is: a complex number carries two pieces of information at once — an amplitude and an angle. That pairing is why they match oscillating signals so cleanly.

4. Euler’s formula

\boxed{e^{i\theta} = \cos\theta + i\sin\theta}

This is the single most important identity in signal processing. Read it right to left: the point (cos θ, sin θ) on the unit circle can be written as the exponential e^iθ. That means multiplication by e^iθ is a rotation by angle θ — not a stretch, just a twist.

Combining with polar form:

z = |z|\,e^{i\arg z}

Every complex number is a magnitude times a rotation.

5. Rotating phasors

Now hand θ a clock. Let θ = ωt, where ω is angular frequency (radians per second) and t is time. Then

z(t) = |z|\,e^{i(\omega t + \arg z)}

is a vector of fixed length |z| rotating around the origin at angular rate ω. That is a phasor.

The physical, real-world signal is the projection onto the real axis:

\Re[z(t)] = |z|\,\cos(\omega t + \arg z)

A rotating phasor is a cosine when you look at its shadow on the real axis. The widget below lets you drag a phasor and watch this happen.

6. Why processing lives in phasor-land

A monochromatic seismic wave at frequency ω has two degrees of freedom: its amplitude and its phase. Complex numbers carry exactly those two numbers in a single object. When a linear filter acts on a signal, it multiplies each frequency by a complex gain — the filter’s magnitude response scales amplitudes, and the filter’s phase response shifts phases. Magnitude and phase are inseparable, and treating them as a single complex number is what lets us write

Y(\omega) = H(\omega)\,X(\omega)

instead of two coupled equations for real and imaginary parts. Every deconvolution, every migration, every FWI iteration eventually boils down to an expression like this.

7. Multiplying phasors: magnitudes multiply, phases add

If $z_1 = |z_1|,e^{i\theta_1}$ and $z_2 = |z_2|,e^{i\theta_2}$ , then

z_1 z_2 = |z_1||z_2|\,e^{i(\theta_1 + \theta_2)}

The magnitudes multiply; the phases add. When we say a filter applies a +30° phase shift at frequency ω, that is literally a multiplication by e^iπ/6.

8. Complex conjugate

The conjugate of z = a + bi is ̄z = a − bi: flip the sign of the imaginary part. Two useful facts:

$z \cdot \bar z = |z|^{2}$ — multiplying a complex number by its conjugate gives the squared magnitude, a real, nonnegative number.
$\Re[z] = \tfrac{1}{2}(z + \bar z)$ — the real part is the average of z and its conjugate.

These show up constantly when we manipulate real signals in the frequency domain.

**The one sentence to remember**

A complex number is a magnitude and a phase; e^iθ is a rotation by θ; a rotating phasor projected onto the real axis is a cosine. Everything in Part 0 is a consequence of those three facts.

Where this goes next

Section §0.3 introduces convolution, the forward model that turns a wavelet and a reflectivity series into a seismic trace. Section §0.4 will then show that convolution in time becomes simple multiplication in the frequency domain — which is why we keep coming back to phasors.

References

Bracewell, R. N. (1999). The Fourier Transform and Its Applications (3rd ed.). McGraw-Hill.
Oppenheim, A. V., Schafer, R. W. (2009). Discrete-Time Signal Processing (3rd ed.). Prentice Hall.
Strang, G. (2016). Introduction to Linear Algebra (5th ed.). Wellesley-Cambridge.