Lab 8.1 — CFO estimation and correction

Lab 8.1 — Carrier Frequency Offset Estimation and Correction

Goal

Inject carrier frequency offset into a QPSK signal, estimate it, compensate it and evaluate constellation quality before and after synchronization.

The lab answers the practical question:

Why does the constellation rotate, and how can we estimate and remove this rotation before symbol decisions?

Executable files

Environment	File	Output
Python	blocks/block_08_modulation_and_synchronization/python/lab_8_1_cfo_estimation_correction.py	constellation plots, phase plot and metrics JSON in docs/assets

Run from the repository root:

python blocks/block_08_modulation_and_synchronization/python/lab_8_1_cfo_estimation_correction.py

Generated artifacts:

docs/assets/lab81_cfo_constellation_before.png
docs/assets/lab81_cfo_constellation_after.png
docs/assets/lab81_cfo_phase_evolution.png
docs/assets/lab81_cfo_metrics.json

Processing chain

flowchart LR
    BITS[Random bits] --> QPSK[QPSK symbols]
    QPSK --> CFO[Apply CFO + phase offset]
    CFO --> NOISE[Add noise]
    NOISE --> EST[Estimate CFO]
    EST --> CORR[Correct CFO]
    CORR --> PHASE[Correct residual phase]
    PHASE --> DEC[Hard decisions]
    DEC --> METRICS[EVM and BER]

CFO model

Carrier frequency offset rotates each symbol by a linearly increasing phase:

r[n] = s[n] * exp(j * (2*pi*f_cfo*n/Fs + phi0)) + noise[n]

If f_cfo is nonzero, the constellation does not stay fixed. It rotates over time.

4th-power method for QPSK

For QPSK, raising the signal to the 4th power removes the data modulation approximately:

r4[n] = r[n]^4

The phase slope of r4[n] is four times the CFO phase slope. Therefore:

f_cfo_est = slope(angle(r[n]^4)) * Fs / (2*pi*4)

This is a compact educational estimator. In real systems, preamble-based estimators and tracking loops are often more robust.

Metrics

Metric	Meaning
true CFO	intentionally injected frequency offset
estimated CFO	4th-power estimator result
CFO error	estimated minus true CFO
EVM before	constellation error before synchronization
EVM after	constellation error after CFO/phase correction
BER before	hard-decision BER before correction
BER after	hard-decision BER after correction
residual phase	estimated constant phase after CFO correction

Expected plots

• constellation before CFO correction;
• constellation after CFO correction;
• unwrapped phase evolution before/after correction.

Common mistakes

Mistake	Symptom	Fix
CFO sign is wrong	rotation becomes faster	flip correction sign
phase not corrected	constellation remains rotated	estimate residual phase
estimator used on low SNR	CFO estimate noisy	use longer averaging or preamble
wrong sample rate	CFO estimate scaled incorrectly	check metadata
using real-only signal	QPSK symmetry is broken	use complex IQ

Report checklist

• [ ] State modulation type and symbol count.
• [ ] State true CFO and sample rate.
• [ ] Explain the 4th-power estimator.
• [ ] Report estimated CFO and CFO error.
• [ ] Include constellation before correction.
• [ ] Include constellation after correction.
• [ ] Include phase evolution plot.
• [ ] Report EVM and BER before/after.
• [ ] Explain residual limitations.

Engineering conclusion template

The QPSK signal used a true CFO of ____ Hz. The 4th-power estimator measured ____ Hz,
giving an error of ____ Hz. After correction, EVM improved from ____ % to ____ % and
BER changed from ____ to ____. The result confirms / does not confirm that CFO was the
main synchronization impairment because ______.