Why ideal IQ models fail on real hardware

In textbooks, a direct-conversion receiver gives perfectly orthogonal I and Q channels with matched gain and exact 90-degree phase separation. In practice, analog mismatches and mixer feedthrough create deterministic distortion terms:

• Gain mismatch between I and Q paths
• Phase error from imperfect quadrature
• DC offset at baseband
• Local oscillator (LO) leakage and self-mixing

A compact way to model the received complex baseband sample is: r[n] = alpha * s[n] + beta * conj(s[n]) + d + w[n] where:

• alpha is desired complex gain
• beta captures image terms from IQ imbalance
• d is DC offset
• w[n] is noise/interference

The beta term creates mirror-frequency energy. In waterfall views this appears as a mirrored signal around DC.

Practical calibration loop

1. Estimate and remove DC using a slow IIR or block mean.
2. Estimate IQ mismatch from a known pilot or strong CW tone.
3. Solve least-squares for alpha and beta.
4. Apply widely-linear correction:

s_hat[n] = c1 * r[n] + c2 * conj(r[n])

5. Re-estimate periodically because temperature drift changes analog parameters.

Deployment notes

• Keep calibration in fixed-point if running in FPGA fabric.
• Prefer pilot-aided updates in bursty channels.
• If AGC is active, freeze adaptation while gain changes.

The key insight: many SDR demod failures are not from synchronization logic first, but from uncorrected front-end imbalance corrupting constellation geometry before the DSP chain starts.