Wideband SNDR: Noise and Distortion Analysis

• Wideband SNDR is a fidelity metric defined as the ratio of output signal power to the sum of noise and distortion power integrated over the signal band.
• It is applied in various systems including quantizers, compressive receivers, notch-based transceiver measurements, and massive-MIMO OFDM, each with a tailored signal decomposition.
• Methodologies such as Bussgang decomposition and perturbative notch filtering enable practical evaluation of SNDR, highlighting trade-offs like noise folding and clipping distortion.

Wideband signal-to-noise-and-distortion ratio (SNDR) is a fidelity metric that extends signal-to-noise ratio by explicitly incorporating nonlinear distortion and by integrating behavior over a signal band rather than only at a single tone or symbol. In the most general form used here, if an input $x=s+n$ is passed through a memoryless nonlinearity $y=f(x)$, then SNDR is defined as the ratio of output signal power to the sum of output noise power and output distortion power, and in the wideband case it admits both a per-frequency form $\mathrm{SNDR}(f)$ and a band-integrated form $\mathrm{SNDR}_{\mathrm{wideband}}$ obtained from the corresponding power spectral densities (Burr et al., 2024). Across recent work, wideband SNDR appears in at least four technically distinct roles: as a Bussgang-based performance metric for quantizers acting on possibly non-Gaussian signals, as a reconstruction-quality metric for compressive wideband receivers subject to noise folding and quantization, as a measurable spectrum-domain profile obtained by notch perturbations in transceivers, and as a distortion-aware link metric in massive-MIMO OFDM systems with clipped power amplifiers (Burr et al., 2024, Davenport et al., 2011, Vaquero-Caballero et al., 2021, Marwaha et al., 8 Sep 2025).

1. Formal definition and wideband decomposition

A general starting point is the decomposition of a real, zero-mean scalar input

$x=s+n$

into a signal component $s$ with arbitrary distribution $p_s(s)$ and variance $\sigma_s^2$, and an independent AWGN component $n$ of variance $\sigma_n^2$. After a memoryless nonlinearity $y=f(x)$0, the output is decomposed into a signal-related linear term, a noise-related linear term, and an uncorrelated distortion term. The instantaneous SNDR is then defined as

$y=f(x)$1

For the noisy case, correlating only with $y=f(x)$2, one defines

$y=f(x)$3

and obtains the exact general formula

$y=f(x)$4

This is Theorem 2 in the quantization analysis of Burr et al. (Burr et al., 2024).

The same work gives a frequency-domain lifting for stationary random processes. Let $y=f(x)$5 and $y=f(x)$6 denote the one-sided PSDs of $y=f(x)$7 and $y=f(x)$8. Then, under sample-wise Bussgang decomposition,

$y=f(x)$9

where $\mathrm{SNDR}(f)$0 is the PSD of the distortion process, uncorrelated with $\mathrm{SNDR}(f)$1 and $\mathrm{SNDR}(f)$2. This yields the per-frequency definition

$\mathrm{SNDR}(f)$3

and the wideband average

$\mathrm{SNDR}(f)$4

with $\mathrm{SNDR}(f)$5 and $\mathrm{SNDR}(f)$6 (Burr et al., 2024).

For jointly Gaussian $\mathrm{SNDR}(f)$7 and $\mathrm{SNDR}(f)$8, the classical Bussgang specialization gives

$\mathrm{SNDR}(f)$9

and therefore

$\mathrm{SNDR}_{\mathrm{wideband}}$0

A key technical point is that the decomposition itself is not restricted to the Gaussian case; what changes is the correct expression for the gain coefficients (Burr et al., 2024).

2. Quantization, non-Gaussian inputs, and sample-wise to wideband lifting

For a general zero-mean stationary input $\mathrm{SNDR}_{\mathrm{wideband}}$1, Burr et al. define the linear gain coefficient

$\mathrm{SNDR}_{\mathrm{wideband}}$2

and the second-moment ratio

$\mathrm{SNDR}_{\mathrm{wideband}}$3

The output can then be written as

$\mathrm{SNDR}_{\mathrm{wideband}}$4

with distortion power

$\mathrm{SNDR}_{\mathrm{wideband}}$5

In the noiseless case $\mathrm{SNDR}_{\mathrm{wideband}}$6, this leads to the signal-to-distortion ratio

$\mathrm{SNDR}_{\mathrm{wideband}}$7

These identities are distribution-general and therefore apply to non-Gaussian symbol alphabets as well as Gaussian inputs (Burr et al., 2024).

For an $\mathrm{SNDR}_{\mathrm{wideband}}$8-level mid-rise uniform quantizer of step $\mathrm{SNDR}_{\mathrm{wideband}}$9, with reconstruction levels

$x=s+n$0

the conditional first and second moments given $x=s+n$1 are

$x=s+n$2

$x=s+n$3

Averaging these against $x=s+n$4 gives

$x=s+n$5

after which $x=s+n$6 follows directly (Burr et al., 2024).

The numerical examples emphasize that the dependence on quantizer step size can be strongly non-monotonic. For a binary input corresponding to the I branch of QPSK, with $x=s+n$7, $x=s+n$8, and $x=s+n$9, the paper reports an unexpected maximum $s$0 at $s$1. For a 4-PAM input corresponding to 16-QAM branches, again with $s$2 and $s$3, the SNDR peaks just above $s$4 (Burr et al., 2024). The paper further notes a counterintuitive outcome: for QPSK, the resulting SNDR as a function of $s$5 can exceed the input SNR because of the interaction between the non-Gaussian amplitude distribution and the truncating quantizer.

Within this framework, the move from scalar-sample analysis to wideband SNDR is technically simple because the coefficients $s$6 and $s$7 depend only on the marginal PDF of $s$8 and on the quantizer nonlinearity, not on the signal spectrum. The memoryless quantizer does not introduce inter-symbol or frequency-dependent coupling, so the scalar decomposition carries directly into the PSD-domain representation (Burr et al., 2024).

3. Compressive wideband acquisition: noise folding, quantization, and net SNDR

In the compressive-sensing receiver analysis of Davenport et al., wideband SNDR arises in a different setting: a real, continuous-time, bandlimited signal $s$9 observed over a $p_s(s)$0 window, with Shannon-Nyquist rate $p_s(s)$1 samples/sec and exactly $p_s(s)$2 nonzero Fourier coefficients. In discrete form, $p_s(s)$3 is $p_s(s)$4-sparse in the Fourier basis. The antenna signal is corrupted by additive white signal noise $p_s(s)$5, $p_s(s)$6, and the $p_s(s)$7 compressive measurements are further corrupted by additive white measurement noise $p_s(s)$8, $p_s(s)$9. With an $\sigma_s^2$0 measurement matrix $\sigma_s^2$1 satisfying the RIP of order $\sigma_s^2$2, and subsampling factor $\sigma_s^2$3, the measurement model is

$\sigma_s^2$4

This provides a concrete setting in which wideband acquisition, sparse recovery, and SNDR must be evaluated jointly (Davenport et al., 2011).

The central penalty identified in the paper is noise folding. When only signal noise is present, each compressive measurement folds together $\sigma_s^2$5 noisy spectrum bins. Under the assumptions that $\sigma_s^2$6 has orthogonal rows of equal norm $\sigma_s^2$7 and satisfies the RIP with constant $\sigma_s^2$8, the system exhibits a $\sigma_s^2$9 SNR loss per octave of subsampling, paralleling the classic noise-folding phenomenon. In the notation of the summary, the SNDR loss is

$n$0

The simulations reinforce this point: oracle-aided CS and bandpass sampling both show exactly $n$1octave RSNR slope until the practical CS limit $n$2, and a practical greedy recovery algorithm, CoSaMP, tracks that behavior until support-identification accuracy degrades near the same limit (Davenport et al., 2011).

The compensating advantage is dynamic range. The quantizer is modeled as a $n$3-bit midrise uniform quantizer $n$4 with step $n$5 and full-scale $n$6, where $n$7. For $n$8,

$n$9

and with appropriate gain scaling $\sigma_n^2$0,

$\sigma_n^2$1

In dB this is roughly $\sigma_n^2$2. For a conventional Nyquist ADC,

$\sigma_n^2$3

In the CS architecture the ADC runs at rate $\sigma_n^2$4 rather than $\sigma_n^2$5, and the paper argues that one can generally employ a quantizer with approximately $\sigma_n^2$6 extra bits for each doubling of sampling time, giving roughly a $\sigma_n^2$7 dynamic-range gain per octave of subsampling, enough to overcome the $\sigma_n^2$8 noise-folding cost. In quantization-only tests with no input noise, starting from a $\sigma_n^2$9-bit or $y=f(x)$00-bit Nyquist ADC, the observed RSNR gain is roughly $y=f(x)$01octave, with about $y=f(x)$02 net gain after four octaves of subsampling (Davenport et al., 2011).

The combined wideband SNDR model adds folded signal noise and quantization noise. Assuming white input noise with variance $y=f(x)$03 per Fourier bin, white quantization noise on $y=f(x)$04 with variance $y=f(x)$05 per measurement, and neglecting RIP constants, the total mean-square error per recovered coefficient is approximately

$y=f(x)$06

Hence

$y=f(x)$07

In dB this becomes

$y=f(x)$08

The paper’s overall conclusion is therefore explicitly dual: compressive wideband acquisition incurs an unavoidable $y=f(x)$09 SNR penalty per octave of subsampling, but lower-rate ADC operation can recover approximately $y=f(x)$10bit of dynamic-range improvement and thereby enable a net SNDR gain in sparse-signal settings (Davenport et al., 2011).

4. Spectrum-domain measurement of wideband SNDR by notch perturbations

Wideband SNDR can also be measured directly along a transceiver chain rather than predicted from a nonlinear model. In the perturbation-based method developed for transceiver noise characterization, the transmitted waveform is modified in frequency according to

$y=f(x)$11

where $y=f(x)$12 inside the notch region $y=f(x)$13 and $y=f(x)$14 outside it. With either a single-notch or dual-notch perturbation, the received spectrum satisfies

$y=f(x)$15

$y=f(x)$16

where $y=f(x)$17 is the bandwidth of interest, $y=f(x)$18 is the total noise-floor spectral density, and

$y=f(x)$19

From these quantities, the per-frequency SNDR profile is defined as

$y=f(x)$20

and the integrated value is

$y=f(x)$21

This gives a direct end-to-end recipe for constructing a wideband SNDR profile over the full signal band (Vaquero-Caballero et al., 2021).

The notch-filter design is simple but application-specific. The paper describes a single-notch measurement, where one narrow notch is moved across the band to stitch out $y=f(x)$22, and a dual-notch measurement with two notches placed symmetrically about the carrier. The reported notch widths are approximately $y=f(x)$23 for an AWG-to-scope experiment and approximately $y=f(x)$24 for the WaveLogic Ai card, with ideally $y=f(x)$25 attenuation in the notch and in practice more than $y=f(x)$26 depth. Implementation uses a real-valued frequency-domain mask applied either inside the TX DSP pre-compensation filter or by multiplying the AWG baseband instruction spectrum (Vaquero-Caballero et al., 2021).

The capture stage can occur at several points in the chain. The paper distinguishes three cases: AWG$y=f(x)$27scope or AWG$y=f(x)$28ESA end-to-end measurement, where the positive-frequency baseband spectrum is observed and the noise floor is approximately DAC plus analog RF noise; transceiver TX$y=f(x)$29OSA measurement, where $y=f(x)$30 includes TX electrical and optical noise; and card-to-card measurement, where $y=f(x)$31 includes TX noise, RX front-end, and ADC noise, with no DSP. The processing chain is correspondingly minimal: load the notch mask, capture $y=f(x)$32, read out $y=f(x)$33 inside $y=f(x)$34, subtract $y=f(x)$35 in $y=f(x)$36 to recover $y=f(x)$37, and compute SNDR from the preceding equations (Vaquero-Caballero et al., 2021).

A significant methodological issue is crosstalk. With a single notch, residual IQ crosstalk or phase imbalance causes incomplete destructive interference, so that the received power in the notch is spuriously nonzero and the inferred SNDR is finite even when the ideal transmitted spectrum is nulled. The dual-notch construction mitigates this by placing notches at $y=f(x)$38; linear leakage from positive to negative frequency must itself pass through the opposite notch and is therefore also nulled. To first order in crosstalk, $y=f(x)$39, with only higher-order noise and DSP errors remaining. The paper’s numerical example reports that a synthetic crosstalk profile produces SNDR spikes above $y=f(x)$40 inside each single notch, whereas the dual-notch result tends to infinity (Vaquero-Caballero et al., 2021).

The same perturbative framework is also used for impairment detection and skew estimation. A compensation-phase filter

$y=f(x)$41

is applied in the DSP domain, and the skew estimate is obtained by minimizing

$y=f(x)$42

where $y=f(x)$43. The paper states that $y=f(x)$44 is a smooth, convex function of the trial skew and reports experimental repeatability of approximately $y=f(x)$45 (Vaquero-Caballero et al., 2021).

5. Wideband SNDR in massive-MIMO OFDM with clipped power amplifiers

In the massive-MIMO downlink formulation of Taheri et al., wideband SNDR is derived for an OFDM transmitter with nonlinear power amplifiers. The system model comprises a base station with $y=f(x)$46 antennas, each radiating an $y=f(x)$47-point OFDM signal with $y=f(x)$48 active subcarriers, and $y=f(x)$49 single-antenna users served on every subcarrier via Zero Forcing precoding. The small-scale fading coefficients are i.i.d. Rayleigh,

$y=f(x)$50

with large-scale gain $y=f(x)$51. The transmitted precoded symbols satisfy

$y=f(x)$52

and the time-domain OFDM samples are

$y=f(x)$53

User-$y=f(x)$54 power is defined over all antennas and subcarriers as

$y=f(x)$55

Each amplifier is modeled as a memoryless soft limiter with unity small-signal gain and clipping threshold $y=f(x)$56 (Marwaha et al., 8 Sep 2025).

Because each $y=f(x)$57 is the sum of many independent QAM symbols, for large $y=f(x)$58 it is modeled as $y=f(x)$59, enabling a Bussgang decomposition

$y=f(x)$60

where

$y=f(x)$61

is the effective linear gain and $y=f(x)$62 is uncorrelated distortion. Under the soft-limiter model,

$y=f(x)$63

The mean per-antenna distortion power follows from power conservation: $y=f(x)$64 These expressions make the distortion term an explicit function of total allocated power (Marwaha et al., 8 Sep 2025).

After FFT and CP removal, user $y=f(x)$65 on subcarrier $y=f(x)$66 observes

$y=f(x)$67

Under Zero Forcing and $y=f(x)$68, inter-user leakage is neglected. The distortion from $y=f(x)$69 antennas adds as white noise when the channel taps are i.i.d. Rayleigh. Denoting by $y=f(x)$70 the fraction of in-band distortion, the total received distortion power for user $y=f(x)$71 is

$y=f(x)$72

With AWGN variance $y=f(x)$73, the desired signal power is $y=f(x)$74, so the wideband SNDR, termed SINDR in the paper, is

$y=f(x)$75

An equivalent form writes

$y=f(x)$76

This is the closed-form wideband SNDR used for optimization (Marwaha et al., 8 Sep 2025).

The physical interpretation is that clipping by the soft limiter can be re-expressed as a scaled copy of the unclipped waveform plus an error term uncorrelated with it, and that error term behaves like white distortion after aggregation through independent Rayleigh channels. As total power increases, the desired term initially benefits, but clipping reduces $y=f(x)$77 and increases the distortion term. The paper therefore reports an SNDR-efficient operating point without explicit transmit power constraints: the derivative of the sum-rate with respect to total power has a unique root, producing an interior optimum in $y=f(x)$78 and hence in $y=f(x)$79 (Marwaha et al., 8 Sep 2025).

This SNDR is then embedded in the sum-rate objective

$y=f(x)$80

Because each $y=f(x)$81 depends on all powers through both numerator and denominator, the problem is non-convex. The paper introduces

$y=f(x)$82

and alternates between a scalar Distortion-Aware total Power Allocation (DAPA) step that finds the unique $y=f(x)$83 nulling the derivative of the sum-rate, and a Fixed Power Distribution Algorithm (FPDA), described as water-filling-like and convex for fixed $y=f(x)$84 (Marwaha et al., 8 Sep 2025).

6. Recurring technical themes and interpretive cautions

Across these formulations, wideband SNDR is consistently a decomposition-based metric, but the origin of the distortion term depends strongly on the physical layer under study. In memoryless quantization, the distortion term is the Bussgang residual $y=f(x)$85, uncorrelated with the signal and noise components (Burr et al., 2024). In compressive acquisition, the dominant degradations are folded signal noise and quantization noise accumulated through recovery, leading to a reconstruction-domain SNDR expression rather than an output-waveform one (Davenport et al., 2011). In transceiver characterization by notch perturbation, the measured quantity is a spectral profile in which the noise-floor density is estimated directly in intentionally nulled frequency regions (Vaquero-Caballero et al., 2021). In PA-limited OFDM massive MIMO, the distortion term is clipping-induced and propagates through the channel as an effective in-band white impairment (Marwaha et al., 8 Sep 2025).

Several common misconceptions are directly contradicted by these results. First, wideband SNDR is not merely a scalar narrowband figure of merit: it can be defined per frequency through PSDs and then integrated, and in perturbation-based measurements it appears naturally as a stitched spectral curve $y=f(x)$86 across the full bandwidth of interest (Burr et al., 2024, Vaquero-Caballero et al., 2021). Second, more aggressive subsampling does not automatically improve reconstruction fidelity: in compressive wideband acquisition there is an unavoidable $y=f(x)$87 SNR penalty per octave of subsampling from noise folding, even though slower ADC operation may improve dynamic range enough to produce a net SNDR benefit (Davenport et al., 2011). Third, higher transmit power is not guaranteed to improve wideband SNDR in nonlinear front-ends: in the soft-limited massive-MIMO OFDM model, clipping distortion creates an interior optimum, so performance can decline once the input back-off becomes too small (Marwaha et al., 8 Sep 2025).

Taken together, these studies show that wideband SNDR is best understood not as a single closed-form metric with universal parameters, but as a family of rigorously defined ratios tied to a decomposition of signal, noise, and distortion appropriate to the acquisition, quantization, measurement, or transmission mechanism at hand. This suggests that comparing SNDR values across systems is meaningful only when the underlying decomposition, bandwidth convention, and distortion model are made explicit.