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Digital-link metric calculations

This page fixes the definitions used in the course's hardware BPSK/QPSK reports. A number without a description of the synchronization algorithm, the normalization and the sample size does not count as sufficient evidence.

IQ preparation

A stereo WAV is converted to complex samples

x[n] = I[n] + j*Q[n]

The values are normalized to the ADC full scale. One DC estimate is subtracted from the whole recording:

mu = (1/N) * sum(x[n]), n=0..N-1
x0[n] = x[n] - mu

The mean of an individual burst is not subtracted: a finite deterministic sequence need not have an exactly zero mean, so that operation could alter the wanted signal.

Burst and frame detection

For the energy detector the recording is split into blocks of B = 256 samples:

P[m] = (1/B) * sum(|x0[m*B+n]|^2), n=0..B-1

The threshold is set robustly relative to the median block power:

P_threshold = median(P) + 10 * MAD(P)

A candidate must also peak above median + 15·MAD. Adjacent active blocks are merged into one burst. After the RRC matched filter the frame is checked against the known sequence from the RTL ROM. The normalized correlation is

rho = |sum(r[k] * conj(s[k]))|
      / sqrt(sum(|r[k]|^2) * sum(|s[k]|^2))

In Lab 11.28 the detector accepts a frame at rho >= 0.8. Hardware frames gave 0.898…0.983; control windows between bursts gave about 0.37…0.61.

CFO estimate and complex alignment

For the known QPSK symbols the phase error is

phi[k] = unwrap(arg(r[k] * conj(s[k])))

A linear fit phi[k] ~= phi0 + alpha*k gives the residual frequency offset

delta_f = alpha * symbol_rate / (2*pi)
        = alpha * sample_rate / (2*pi*SPS)

After the CFO correction one complex channel coefficient is estimated by least squares:

g = sum(conj(s[k]) * r[k]) / sum(|s[k]|^2)
r_aligned[k] = r[k] * exp(-j*alpha*k) / g

This alignment removes a constant gain/phase and a linear CFO within the frame. It does not correct nonlinearity, IQ imbalance, timing jitter or a frequency-selective channel.

BER and FER

QPSK decisions are taken from the signs of I/Q. For N_b known bits:

BER = bit_errors / compared_bits

With equal frame lengths the aggregate BER is computed over all detected frames:

BER_aggregate = sum(bit_errors_per_frame)
                / (detected_frames * bits_per_frame)

The frame error rate counts a frame as failed if it has at least one bit error:

FER = frames_with_at_least_one_error / detected_frames

BER = 0 is always reported together with the number of compared bits. For zero errors the approximate 95 % rule-of-three bound BER < 3/N_b is given. It is not a proven BER floor.

EVM

RMS EVM is computed after the CFO correction and the complex scalar alignment:

EVM_RMS_percent = 100 * sqrt(
    sum(|r_aligned[k] - s[k]|^2) / sum(|s[k]|^2)
)

EVM combines noise and residual distortion. A smaller EVM usually improves BER, but a one-to-one relation between them exists only for a fixed modulation, synchronization and channel model.

SNR from EVM

The QPSK OTA analyzer does not measure SNR from a separate calibrated noise-only interval. It publishes a diagnostic estimate

evm_fraction = EVM_RMS_percent / 100
SNR_from_EVM_dB = -20 * log10(evm_fraction)

It equals the SNR only when uncorrelated additive noise dominates and the synchronization is correct. That is why the field is called snr_from_evm_db and not calibrated RF SNR.

Clipping, levels and crest factor

The clipping fraction is computed before the DC correction as the share of complex samples in which at least one axis nearly reaches full scale:

clipping_fraction = count(|I[n]| > 0.999 or |Q[n]| > 0.999) / N

The complex peak/RMS levels in dBFS are stored as well, and

crest_factor_dB = peak_level_dBFS - rms_level_dBFS

A zero clipping fraction rules out digital saturation in the recording, but does not prove the absence of analog compression before the ADC.

Confidence intervals and limits

  • The share of error-free bursts uses a two-sided Wilson 95 % interval.
  • The aggregate BER also gets a Wilson interval, as a descriptive estimate. Errors within one RF burst can be correlated, so the model of independent Bernoulli trials is an approximation.
  • Detection rate, normalized correlation, FER, BER, EVM and CFO must be read together.
  • The best frame does not replace the distribution over all frames; the main Lab 11.28 report uses every detected burst.
  • Absolute SNR, power and an uncertainty budget need a calibrated path and a separate noise measurement.

For k successes in n trials, p_hat = k/n and z = 1.959964, the Wilson bounds are

denom  = 1 + z^2/n
center = (p_hat + z^2/(2*n)) / denom
half   = z * sqrt(p_hat*(1-p_hat)/n + z^2/(4*n^2)) / denom
interval_95 = [center-half, center+half]

For BER the same formula is applied to the number of bit errors, but the result is labelled descriptive because the errors may be correlated.