Lab 8.4 — End-to-End Synchronization Chain¶

Goal¶

Combine timing offset, carrier frequency offset, phase offset and noise in one synthetic QPSK link, then recover the signal using a staged synchronization chain.

The lab answers the practical question:

How do timing recovery, CFO correction and phase correction work together in a complete receiver chain?

Executable files¶

Environment	File	Output
Python	`blocks/block_08_modulation_and_synchronization/python/lab_8_4_end_to_end_sync_chain.py`	stage constellations, EVM/BER summaries and metrics JSON in `docs/assets`

Run from the repository root:

python blocks/block_08_modulation_and_synchronization/python/lab_8_4_end_to_end_sync_chain.py

Generated artifacts:

docs/assets/lab84_sync_constellation_raw.png
docs/assets/lab84_sync_constellation_after_timing.png
docs/assets/lab84_sync_constellation_final.png
docs/assets/lab84_sync_timing_search.png
docs/assets/lab84_sync_evm_stages.png
docs/assets/lab84_sync_ber_summary.png
docs/assets/lab84_sync_chain_metrics.json

Processing chain¶

flowchart LR
    BITS[Random bits] --> QPSK[QPSK symbols]
    QPSK --> OS[Oversampled waveform]
    OS --> IMP[Timing + CFO + phase + noise]
    IMP --> TIMING[Timing phase search]
    TIMING --> CFO[CFO estimation/correction]
    CFO --> PHASE[Phase correction]
    PHASE --> DEC[Hard decisions]
    DEC --> METRICS[EVM and BER]

Staged receiver¶

The educational receiver applies synchronization in this order:

Timing phase search - select the sampling phase with the strongest decimated symbol energy.
CFO estimation/correction - estimate residual frequency slope using known reference symbols.
Phase correction - remove remaining constant constellation rotation.
Hard decisions - convert corrected symbols to bits.
Metrics - compute EVM and BER before and after synchronization.

What to expect¶

The default run (deterministic, as printed by the script) gives:

Timing offset true/estimated: 3/6 samples
CFO true/estimated/error: 1350.000/1349.996/-0.004 Hz
Phase true/estimated/error: 0.550000/0.600344/-0.000550 rad
EVM raw/timing/CFO/final: 3413.163/19524.022/4.857/4.857 %
BER raw/final: 5.141602e-01/0.000000e+00

How to read the stage table:

The first two EVM values (3413 % and 19524 %) are not "errors of 34x and 195x". Both are far above 100 %, which means the constellation is still rotating (CFO not yet removed) and the scalar alignment has nothing stable to lock to. The EVM number only becomes meaningful after the CFO stage (4.857 %). The interesting drop is BER, from 0.514 (a coin flip) to 0.
The estimated timing phase (6) differs from the injected offset (3) and that is fine. As in Lab 8.3, with rectangular pulses every phase inside the flat part of the symbol is correct; the stage only has to avoid the samples belonging to the neighbouring symbol.
CFO error is -0.004 Hz on 1350 Hz: a fit over thousands of known symbols is very precise in a synthetic model with a flat symbol. Real captures will not be this good.
"EVM after CFO" and "EVM final" are identical (4.857 %). These EVM values are computed after a scalar gain/phase alignment to the reference, which absorbs any constant phase, so the phase stage cannot change them. BER, however, is scored on the receiver's own output with no such alignment, so a wrong phase estimate would show up there as bit errors.
The estimated phase is 0.600 rad, not the injected 0.55, and that is correct. The receiver samples at phase 6, and the 1350 Hz CFO rotates the signal by 2*pi*1350*6/1e6 = 0.051 rad over those 6 samples. Referenced to the sampling instant, the phase error is only -0.0006 rad.

Impairments in the model¶

Impairment	Meaning
timing offset	receiver samples at the wrong point inside the symbol interval
CFO	constellation rotates over time
phase offset	constellation has a constant rotation
noise	spreads constellation points and limits EVM/BER

Metrics¶

Metric	Meaning
estimated timing phase	selected sample phase from timing search
estimated CFO	measured frequency offset after timing selection
CFO error	estimated CFO minus true CFO
estimated phase	measured residual phase after CFO correction
EVM raw	error before synchronization
EVM after timing	error after choosing the sampling phase
EVM after CFO	error after frequency correction
EVM final	error after full synchronization
BER raw/final	decision quality before and after the full chain

Why this closes Block 8¶

Labs 8.1-8.3 isolate individual synchronization problems. Lab 8.4 combines them into a receiver chain:

Previous lab	Role in Lab 8.4
Lab 8.1 CFO	frequency correction stage
Lab 8.2 phase	final constellation rotation correction
Lab 8.3 timing	sampling phase selection
Lab 7.3 loopback metrics	EVM/BER reporting style

Limitations¶

This is still an educational model. A production receiver would additionally need:

frame synchronization;
preamble detection;
non-data-aided tracking loops;
sample-rate offset tracking;
pulse shaping with matched filtering;
IQ imbalance correction;
robust operation at lower SNR.

Report checklist¶

State all injected impairments.
Explain the staged receiver order.
Include raw constellation.
Include constellation after timing recovery.
Include final constellation.
Include EVM by stage.
Include BER before/after.
State remaining limitations before real RF use.

Engineering conclusion template¶

The synthetic receiver used timing offset ____ samples, CFO ____ Hz and phase offset ____ rad.
The staged synchronizer estimated timing phase ____, CFO ____ Hz and phase ____ rad.
EVM improved from ____ % to ____ %, while BER changed from ____ to ____.
This closes / does not close the basic synchronization chain because ______.