Multi-Stream Partial Response Signaling (MS-PRS) is a modulation scheme at the Nyquist rate that adds controlled intersymbol interference through short FIR filters across two bipolar sub-streams, instead of compressing the symbol period the way faster-than-Nyquist signaling does. MS-PRS at Rate 2 carries two bits per channel use. This project uses AFF3CT for efficient BER simulations and tests the scheme over the air between two ADALM-Pluto software-defined radios.
The experimental setup: the transmitter (right) and the receiver (left) each feed a log-periodic antenna, and the two antennas face each other. The link runs at 448 MHz, and each radio has its own crystal.
Inside the modulator, the interleaved bits split into two bipolar sub-streams. One runs through
an L0-tap FIR filter h0, which is where the controlled ISI comes from; the other passes a
single tap. The two are scaled and summed into one Nyquist-rate symbol, carrying two bits between
them. The gains set the energy split, eta_0 + eta_1 = Es: the balanced family divides it evenly,
the unbalanced family tilts it toward the single-tap stream.
The chain the simulator runs: encode, interleave, modulate, then a receiver that passes extrinsic information between the equalizer and the decoder. The receiver is a turbo loop: a BCJR equalizer on the 2^(L0-1)-state trellis trades log-likelihood ratios with a rate-1/2 convolutional or LDPC decoder.
Both BCJRs, the equalizer's and the convolutional decoder's, run one of three
algorithms, chosen with --bcjr. MAP works on probabilities and is exact; it
is the default. log-MAP works on their logarithms and reads the Jacobian
correction from a table. max-log-MAP drops the correction. For L0 = 3 with 7
iterations, balanced / unbalanced, on one thread of a Ryzen 5 3600X:
--bcjr |
Eb/N0 at BER 1e-3 | Eb/N0 at BER 1e-4 | Time per frame | Speed vs MAP |
|---|---|---|---|---|
map (default) |
3.89 / 4.06 dB | 4.91 / 5.13 dB | 11.3 / 11.8 ms | 1.00 / 1.00× |
log-map |
3.88 / 4.03 dB | 4.89 / 5.21 dB | 16.8 / 17.8 ms | 0.68 / 0.67× |
max-log-map |
4.38 / 4.06 dB | 4.91 / 5.21 dB | 5.8 / 5.9 ms | 1.95 / 2.00× |
Differences under 0.1 dB are Monte Carlo scatter. max-log-MAP loses half a
decibel where the balanced loop starts to converge, and nothing once it has.
log-MAP is slower than MAP here: on a CPU a multiply-add costs no more than an
add, while every log-domain addition still needs a comparison and a table
lookup. Its advantage belongs to fixed-point hardware. A frame is 4998
information bits and 8 passes of each BCJR; msprs --mode timing --L0 3 --ebn0 4
reproduces the times.
Received symbols on the I/Q plane, measured over the air between the two radios at 15 dB. White crosses mark the ideal points; the scatter around them is what the equalizer works with.
Above, the symbol alphabet and how often each level occurs, against a Gaussian of the same mean and variance. Below, the values reachable in three steps from the all-zero state, coloured by the input pair that produced them.
The BER performance at high-SNR slope is set by d2_min. This is where the error-rate advantage
lives. Every design clears 4-ASK by 1.7 to 3.5 dB.
The equalizer's characteristic against the decoder's, mirrored. The gap between them is the tunnel the turbo loop climbs.
A small network in place of the BCJR, trained on the transmitted bits and run inside the same turbo loop. On the Gaussian channel it trails the BCJR by about 0.9 dB at a BER of 1e-4. It is meant for the radio link, where the received signal departs from the Gaussian model the BCJR assumes.
Build the simulator, then run any mode:
cmake -S . -B build -DCMAKE_PREFIX_PATH="$PWD/.aff3ct/install"
cmake --build build -j
build/bin/msprs --mode coded-msprs --L0 3 --family balanced --iters 7
Full build steps and every mode are in docs/build.md; conventions and the record format in docs/reference.md.
The tap tables in filters/ are the single source of truth for both
filter families and every L0.
The notebooks read what the simulator writes and draw the figures above. They compute nothing themselves:
python -m venv .venv && . .venv/bin/activate
pip install numpy matplotlib jupyter
jupyter lab notebooks/
Apache 2.0, see LICENSE. AFF3CT is a separate project under the MIT licence.