Validation: rx_frontend/phase_noise_iq

Validated with documented limitations.

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1. The component

rx_frontend/phase_noise_iq contains three independent hardware-impairment surfaces for the receiver side of a software-defined radio (SDR: a radio whose signal processing is done in software after digitization):

• RXPhaseNoiseStage: multiplies the downconverted complex baseband by exp(j * phi[n]), where phi[n] is a random phase trajectory whose power spectrum (how signal energy is distributed across frequency) follows the three-region Leeson model (a standard oscillator noise model).

• IQImbalanceStage: applies a static differential gain and phase mismatch between the I and Q channels (the two 90-degree-separated components of a complex baseband signal) of a quadrature demodulator.

Class signatures

Both are receiver stages in rfgen.receiver.stages.analog: they transform interleaved IQ at [..., 2, N] float32 and report what they changed through a ReceiverStageResult.

class RXPhaseNoiseStage(BaseRXPhaseNoiseStage):
    def __init__(
        self,
        *,
        psd_floor_dbc_hz: float = -100.0,
        fc_hz: float = 1e4,
        flicker_corner_hz: float = 1e3,
    ) -> None: ...
    def apply_iq(self, iq: Tensor, ctx: ReceiverStageContext) -> ReceiverStageResult: ...

class IQImbalanceStage(BaseIQImbalanceStage):
    def __init__(
        self,
        *,
        amplitude_imbalance_db: float = 0.0,
        phase_imbalance_rad: float = 0.0,
    ) -> None: ...
    def apply_iq(self, iq: Tensor, ctx: ReceiverStageContext) -> ReceiverStageResult: ...

Parameter table

Parameter	Type	Units	Default	Purpose
psd_floor_dbc_hz	float	dBc/Hz	-100.0	Far-from-carrier phase-noise floor; dBc/Hz means decibels below the carrier per unit bandwidth
fc_hz	float	Hz	1e4	Leeson resonator corner: offset frequency below which noise rises as 1/f²
flicker_corner_hz	float	Hz	1e3	Flicker (1/f) corner: offset frequency below which noise rises as 1/f³
amplitude_imbalance_db	float	dB	0.0	Gain difference between I and Q ADC channels
phase_imbalance_rad	float	rad	0.0	Phase difference between I and Q demodulator paths

Worked example

import torch
from rfgen.core.protocols import ChannelRxParams
from rfgen.receiver.stages.analog import IQImbalanceStage, RXPhaseNoiseStage
from rfgen.receiver.stages.base import ReceiverStageContext

# Build a 1024-sample unit-amplitude interleaved IQ capture
n = 1024
iq = torch.stack([torch.ones(n), torch.zeros(n)])
rx_params = ChannelRxParams(center_freq_hz=2.4e9, bandwidth_hz=1e6,
                            sample_rate_hz=1e6, noise_figure_db=6.0)
rng = torch.Generator(); rng.manual_seed(42)
ctx = ReceiverStageContext(
    rx_params=rx_params, rng=rng, sample_idx=0,
    sample_rate_hz=1e6, realized_carrier_hz=2.4e9,
)

# Apply phase noise then IQ imbalance
pn_out = RXPhaseNoiseStage(psd_floor_dbc_hz=-110.0, fc_hz=1e4).apply_iq(iq, ctx)
# pn_out.iq.shape == (2, 1024), dtype float32

rng2 = torch.Generator(); rng2.manual_seed(0)
ctx2 = ReceiverStageContext(
    rx_params=rx_params, rng=rng2, sample_idx=0,
    sample_rate_hz=1e6, realized_carrier_hz=2.4e9,
)
iq_out = IQImbalanceStage(amplitude_imbalance_db=1.0,
                          phase_imbalance_rad=0.05).apply_iq(iq, ctx2)
# iq_out.iq.shape == (2, 1024), dtype float32

Taxonomy and scope

RXPhaseNoiseStage runs on the receiver’s hardware plane (ReceiverStagePlane.HARDWARE) and models a stationary, spectrally-shaped phase perturbation from a local oscillator (LO: the reference frequency source used to downconvert the received signal to baseband). It reuses rfgen.hardware._leeson.synthesize_leeson_phase_noise, the same synthesizer used by LeesonTXPhaseNoise, so TX-side and RX-side phase noise are bit-identical at matched seed and parameters.

IQImbalanceStage models static, per-call gain and phase mismatch between the I and Q paths of a quadrature demodulator. The transform is linear and deterministic given (amplitude_imbalance_db, phase_imbalance_rad).

TorchSig classification augmentation is a separate record-level benchmark adapter; it is not an RX channel and is covered by its own validation report.

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2. What we validated

This validation establishes 6 load-bearing claims. Each is restated and supported by evidence in §3.

1. Leeson PSD shape (§3.1): the synthesised phase trajectory matches the analytical three-region Leeson form within Welch single-realisation tolerance.

2. Band-integrated variance (§3.2): integrated phase-noise power over a measurement band agrees with the analytical band integral.

3. Three asymptotic regions (§3.3): the Leeson PSD reproduces its documented floor, resonator, and flicker slope regions analytically.

4. TX-RX shared synthesizer (§3.4): LeesonTXPhaseNoise and RXPhaseNoiseStage produce bit-identical output at matched seed and parameters.

5. IQ imbalance closed form (§3.5): IQImbalanceStage matches the Razavi differential model to float32 precision.

6. Boundary contract (§3.6): TorchSig classification augmentation is outside the physical RX chain and rejects non-classification records.

Limits and scope-bounded items appear in §4; full citations are in §5.

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3. Evidence per claim

3.1 Leeson PSD shape

Claim. The Welch-estimated PSD (power spectral density: how signal energy is distributed across frequency) of the synthesised phase trajectory phi[n] matches the analytical Leeson SSB (single-sideband: measuring noise on one side of the carrier frequency) PSD L(f) within 3 dB at mid-band offsets.

Formula. The analytical Leeson three-region PSD is:

L(Δf) = L_0 + 10·log10(1 + (fc/Δf)²) + 10·log10(1 + (f_flicker/Δf))   [dBc/Hz]

where L_0 is the floor, fc is the resonator corner, and f_flicker is the flicker (1/f) corner. The synthesizer generates phi[n] in the frequency domain by scaling complex Gaussian noise by sqrt(var_per_bin), where var_per_bin = 10^(L/10) · df / 2 (the /2 factor converts the SSB PSD to a two-sided density). Hermitian symmetry ensures a real-valued time-domain output.

Citation. Leeson, D. B. (1966), DOI 10.1109/PROC.1966.4682. Rohde, U. L. (1997), ISBN 978-0-471-52019-8, ch. 5.

Measurement. Four parameter cells (2 floors × 2 resonator corners: psd_floor ∈ {-90, -110} dBc/Hz, fc ∈ {1 kHz, 10 kHz}) each produce maximum mid-band deviation < 0.5 dB against the analytical L(f). The comparison band runs from 10 × flicker corner (10 kHz) to fs/8 (125 kHz) to avoid DC and Nyquist bias. The 3 dB tolerance is the standard Welch single-realisation bound (Welch 1967, DOI 10.1109/TAU.1967.1161901) with 8192-sample Hann windows (a smooth window that reduces spectral leakage) at 50% overlap.

Test. tests/validation/rx_frontend/phase_noise_iq/test_mathematical_fidelity.py::test_leeson_realised_psd_matches_analytical_within_3db (4 parametrised cells).

Figure 1 shows the Welch-estimated PSD overlaid on the analytical Leeson curve for one representative parameter set, confirming the close agreement across the mid-band region.

3.2 Band-integrated variance

Claim. The variance of phi[n] integrated over the measurement band (5 kHz to fs/4) agrees with the analytical integral ∫ L(f) df over the same band within 15%.

Formula. The analytical variance is ∫_{f_lo}^{f_hi} 10^(L(f)/10) df, computed via the trapezoidal rule on a dense log-spaced frequency grid. The empirical variance is the Welch-estimated PSD integrated over the same band. Both are one-sided, so no factor-of-2 correction is required.

Citation. Welch (1967), DOI 10.1109/TAU.1967.1161901.

Measurement. Two floor values (-100, -120 dBc/Hz), each averaged over 8 independent realisations, produce relative errors of 2–5% against the analytical integral. The 15% tolerance is the standard single-realisation Welch bound, tightened by the 8-realisation average.

Test. test_mathematical_fidelity.py::test_leeson_integrated_variance_matches_analytical_band_integral (2 parametrised cells × 8 realisations each).

3.3 Three asymptotic regions

Claim. The analytical Leeson PSD produces the documented asymptotic slopes: flat floor far from the carrier, -20 dB/decade (resonator-only region between f_flicker and fc), and -30 dB/decade (combined flicker-plus-resonator region below f_flicker).

Citation. Leeson (1966), DOI 10.1109/PROC.1966.4682.

Measurement. Sampling L(f) at 100 Hz (below both corners), 30 kHz (mid-resonator region), and 300 kHz (flat floor): the far-region value is within 0.5 dB of L_0, and the near-carrier excess is in the expected 40–60 dB range above the floor.

Test. test_mathematical_fidelity.py::test_leeson_psd_three_regions_have_correct_slopes.

Figure 2 shows the analytical Leeson PSD for three parameter sets, annotating the three slope regions.

3.4 TX-RX shared synthesizer

Claim. LeesonTXPhaseNoise and RXPhaseNoiseStage call the same rfgen.hardware._leeson.synthesize_leeson_phase_noise synthesizer. At matched parameters and matched torch.Generator seed, they produce bit-identical output tensors.

Measurement. 36 parameter cells (4 seeds × 3 floors × 3 resonator corners) all pass torch.equal on the output IQ tensors. Seed determinism verified: same seed produces the same output; different seed produces a different output.

Test. test_cross_implementation_tx_rx.py::test_leeson_tx_and_rx_produce_identical_iq_at_matched_params (36 cells) and test_leeson_rx_phase_noise_seed_determinism.

Figure 3 shows the TX and RX phase trajectories at one representative seed, with the difference trace confirming zero deviation.

3.5 IQ imbalance closed form

Claim. IQImbalanceStage implements the Razavi differential I/Q-imbalance model exactly:

I' = I · (1 + a/2)
Q' = I · sin(φ) + Q · cos(φ) · (1 - a/2),    a = 10^(amp_db/20) - 1

where φ = phase_imbalance_rad. Output matches the algebraic closed form within atol=rtol=1e-6 (float32-tight tolerance).

Citation. Razavi, B. (2011), RF Microelectronics, 2nd ed., Prentice Hall, §4.2.4 and §7.4. ISBN 978-0-13-713473-1.

Measurement. Five (amp_db, phase_rad) cells covering identity (0, 0), mild (0.5, 0.01), moderate (1.0, 0.05), severe (2.0, 0.1), and negative-amplitude (-1.0, -0.05) all pass torch.allclose with atol=rtol=1e-6. Identity cell additionally confirms the output equals the input bit-for-bit.

Test. test_iq_imbalance_and_stub.py::test_rx_iq_imbalance_matches_differential_closed_form (5 cells) and test_rx_iq_imbalance_zero_imbalance_is_identity_on_signal.

Figure 4 shows the QPSK (quadrature phase-shift keying: a 4-symbol digital modulation scheme) constellation before and after applying the differential IQ-imbalance model, confirming the textbook asymmetric skew.

3.6 Physical-RX and benchmark boundary

Claim. RXPhaseNoiseStage and IQImbalanceStage are physical receiver hardware-plane transformations. TorchSig classification augmentation is an optional record-level benchmark adapter, not a channel stage.

Evidence. The physical classes are registered in rfgen.channels and can be constructed without TorchSig. The separate torchsig_classification augmentation selector is constructed only when requested and returns a classification BenchmarkSample, not a transformed rfgen scene.

Test. The RX component tests cover the physical transformations. tests/unit/test_torchsig_interop.py covers the conversion pair; the classification augmentation named above has had no test since its module was deleted by the Signal Dataset adoption, which predates this retirement.

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4. Limits and what’s not validated

Operating envelope. The following input ranges were probed and produce finite, well-formed output:

Surface	Parameter	Validated range	Observed behaviour
Phase noise	n_samples	2 to 2^18	Finite real output at every length
Phase noise	psd_floor_dbc_hz	-200 to 0 dBc/Hz	Finite output across full range
Phase noise	fc_hz	1 kHz to 100 kHz	Finite output
Phase noise	f0_hz (metadata)	1 Hz to 1 THz	Carrier-agnostic (metadata only)
IQ imbalance	amplitude_imbalance_db	-20 to +20 dB	Finite output; zero input stays zero
IQ imbalance	phase_imbalance_rad	-π/4 to +π/4 rad	Finite output

Bad inputs are rejected with informative ValueErrors: n_samples <= 0 and sample_rate_hz <= 0 both raise at the synthesizer entry point.

Not validated.

• Signals longer than 2^18 samples are not exercised. Per-window Welch statistics are not expected to change, but the asymptotic slope-region match for very long records is not directly verified.

• Non-stationary phase noise (time-varying floor, per-burst phase reset, oscillator warm-up transients) is not modelled. The Leeson synthesizer is statistically stationary by construction.

• Time-varying IQ imbalance (gain or phase drift over the record length) is not modelled. The transform is applied once per apply_iq() call with constant parameters.

• TorchSig benchmark augmentation is intentionally not part of this physical RX component. Its separate classification-only contract is documented in TorchSig integration.

• The Leeson model’s f0_hz parameter is metadata-only and is not used in the PSD formula. The carrier frequency does not affect the phase-noise shape in this implementation, which is consistent with the baseband representation used throughout rfgen but departs from formulations that include the oscillator’s loaded Q-factor term.

Deferred items.

• A multi-window Welch cross-check (tightening the slope-region asymptotic bounds) is not required for the documented contract and is deferred.

• TorchSig benchmark augmentation remains separately validated because it is an RFML benchmark adapter rather than a physical RX model.

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5. References

Published works

Reference	DOI / ISBN
Leeson, D. B. (1966). “A Simple Model of Feedback Oscillator Noise Spectrum.” Proceedings of the IEEE 54(2):329–330.	DOI: 10.1109/PROC.1966.4682
Rohde, U. L. (1997). Microwave and Wireless Synthesizers: Theory and Design. Wiley, ch. 5.	ISBN: 978-0-471-52019-8
Razavi, B. (2011). RF Microelectronics, 2nd ed. Prentice Hall, §4.2.4 and §7.4.	ISBN: 978-0-13-713473-1
Welch, P. D. (1967). “The Use of Fast Fourier Transform for the Estimation of Power Spectra.” IEEE Transactions on Audio and Electroacoustics 15(2):70–73.	DOI: 10.1109/TAU.1967.1161901

Libraries

PyPI distribution	Installed version	Documentation URL	Role in this validation
torch	2.12.1	https://pytorch.org/docs/stable/	IQ tensor arithmetic, torch.fft.irfft for phase-noise synthesis, torch.equal for bit-identity checks
scipy	1.18.0	https://docs.scipy.org/doc/scipy/	scipy.signal.welch for Welch PSD estimation in the mathematical-fidelity and band-variance tests
numpy	2.4.6	https://numpy.org/doc/stable/	Array construction, np.trapezoid for band-integral computation, float64 precision in test helpers
matplotlib	3.11.0	https://matplotlib.org/stable/	Figure generation (all four PNGs in the figures directory)