Complex signals & phase math
A single SLC pixel is one complex signal, I + jQ, whose amplitude holds the backscatter strength and whose phase holds the round-trip timing of the wave (its rotation angle). The real reason for treating phase as a complex number is computation, not storage — thanks to Euler's formula e^(jθ)=cosθ+jsinθ, a rotation reduces to a single multiplication. An interferogram is Master × Slave* (the conjugate of the slave), and this multiplication automatically extracts the phase difference θ₁−θ₂, which equals the path-length difference and thus converts into elevation and displacement. That θ₁−θ₂ is exactly what SNAPHU solves for.
An SLC pixel is a complex signal (I/Q)
- A radar wave is amplitude plus a rotating oscillation, so a received pixel can be written as coordinates (I, Q) or as the complex number I + jQ — two notations for the same information.
- The amplitude A = √(I² + Q²) is the backscatter strength, i.e. how much of the transmitted wave came back.
- The phase θ = atan2(Q, I) holds the round-trip timing, i.e. the rotation angle, and this phase is the heart of InSAR.
- This is precisely where GRD diverges as a product that keeps only amplitude and discards the phase.
Why complex numbers — rotation is multiplication
- The key is not that complex numbers are convenient for storage, but that they reduce a rotation to a single multiplication.
- The x/y coordinate approach must run trigonometry per pixel — x' = x·cosθ − y·sinθ, y' = x·sinθ + y·cosθ — exploding the cost over millions of pixels.
- The complex approach finishes with one multiplication by e^(jθ) and needs no trig calls at all.
- For example, rotating the signal (1,0) by 90° requires the rotation formula in coordinates, whereas in complex form you simply multiply by j to get (0,1).
| x/y coordinates | Complex numbers | |
|---|---|---|
| Rotation op | rotation formula x·cosθ − y·sinθ … | one multiply by e^(jθ) |
| Trig functions | sin/cos explode per pixel | none |
Euler's formula — a rotation as a single number
- A point on the unit circle is (cosθ, sinθ), or cosθ + j·sinθ as a complex number, and Euler's formula states that this remarkably equals e^(jθ).
- So a wave is written compactly as A·e^(jθ) — a single vector of length A and angle θ on the I/Q plane.
- SAR engineers use Euler's formula not for its proof but because it expresses a rotating wave in one line and reduces phase operations to multiplication.
- amplitude (vector length)
- phase (rotation angle)
- imaginary unit (a 90° rotation)
The conjugate and the interferogram = M × S*
- InSAR wants not the absolute phase but the phase difference θ₁ − θ₂ between two acquisitions, yet a plain product adds the phases (θ₁+θ₂) and becomes meaningless.
- So multiplying by the slave's conjugate — e^(−jθ₂) with the phase sign flipped — yields AB·e^(j(θ₁−θ₂)), automatically extracting the phase difference.
- By the clock analogy, with Master 30° and Slave 20°, plain addition gives a meaningless 50°, but flipping the slave to −20° gives the desired relative angle of 10°.
- SNAP's Interferogram Formation step is effectively a per-pixel phase-difference calculator running this M × S*, and what SNAPHU solves is neither the master nor slave phase but exactly this θ₁ − θ₂, which becomes a path-length difference ΔR and thus elevation and displacement.
If you memorize complex numbers as merely a storage format, M×S* never clicks — you must grasp the computational motive that rotation equals multiplication, which makes the conjugate's sign flip feel natural. The sole reason it is M×S* rather than M×S is that we want subtraction (θ₁−θ₂), not addition (θ₁+θ₂), and a sign mistake flips the entire direction of the phase difference.