In-Phase Quadrature Explorer
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About This MicroSim
There is a problem with correlating against a single sine wave: the answer depends on when you started listening. A 500 Hz tone that arrives a quarter cycle late correlates to zero against a sine — not because the tone is absent, but because it is out of phase.
The fix is to correlate against two test waves, a sine and a cosine, and combine the results:
- I (in-phase) is the correlation against the sine.
- Q (quadrature) is the correlation against the cosine.
- Magnitude is \(\sqrt{I^2 + Q^2}\).
Drag the phase slider. The blue and orange bars swing across their full range, one hitting zero exactly where the other peaks. The green bar does not move.
The bottom plot shows all three across the entire phase range at once: I traces a cosine, Q traces a sine, and the magnitude is a flat line. That flat line is the reason every FFT bin is a complex number rather than a single value.
How to Use
- Start at phase 0. Read the three values. I is at its maximum, Q is zero.
- Drag to π/2 (about 1.57). Now I is zero and Q is at maximum. If you had only the I detector, you would conclude the signal vanished.
- Watch the green magnitude readout throughout. Confirm it reads 1.000 at every phase you try.
- Look at the bottom plot. Identify the phases where I crosses zero, and check what Q is doing at those exact moments.
- Ask yourself: what would the magnitude look like if the signal really were absent? (Zero at every phase — a flat line at the bottom, not the top.)
Why This Is Exactly 1.000
The window is 128 samples at 8 kHz, giving a bin spacing of 62.5 Hz. The signal sits at 500 Hz, which is exactly 8 bins. On an exact bin the correlation sums are exact, so the magnitude comes out to 1.000 rather than 0.998. The sim computes this numerically from the actual sums — it is not substituting the analytic answer.
Lesson Plan
Grade Level
Undergraduate (college junior/senior)
Duration
10-12 minutes
Prerequisites
- Multiply-and-sum correlation
- Sine and cosine differ by a quarter cycle
Learning Objective
Students will be able to examine how the in-phase and quadrature components individually rise and fall with phase, and compare that behavior against the combined magnitude, which stays constant.
Activities
- Find the blind spot (4 min): Students find the phase at which I alone reports zero, and explain why a single-sine detector would fail there.
- Trace both (4 min): Using the bottom plot, students describe the shape of each curve and state the phase relationship between them.
- Generalize (4 min): Students explain why an FFT bin must carry two numbers rather than one.
Assessment
Ask: "A detector correlates only against a sine and reports zero. Name two completely different situations that could produce that reading, and describe the measurement that would tell them apart."
Related Resources
References
- In-phase and quadrature components — the standard treatment of I/Q representation.
- Analytic signal — the broader framework in which magnitude and phase are separated.