S/PDIF Analyzer Metrics and Formulas

Comprehensive technical documentation of all metrics, formulas, and physical models

implemented in the application.

Reference standard: IEC 60958-3 (consumer interface) / IEC 60958-4 (professional AES3 interface) - Biphase Mark Coding (BMC), coaxial signal 0.5 V peak-to-peak on nominal 75 ohm impedance.

Standards and references used:

• IEC 60958-3:2021 - Digital audio interface, Part 3: consumer interface (S/PDIF coaxial), 4th edition
• IEC 60958-1:2021 - Digital audio interface, Part 1: general (frame structure, BMC)
• AES3-2009 (equiv. IEC 60958-4) - Professional AES/EBU interface
• IEC 61000-4-3 - Electromagnetic compatibility, radiated field immunity tests
• MIL-STD-188-124B - Grounding, bonding and shielding (military reference for shielding effectiveness values)
• Pozar, D.M., *Microwave Engineering*, 4th ed., Wiley, 2011 - Transmission line theory, reflections
• Ott, H.W., *Electromagnetic Compatibility Engineering*, Wiley, 2009 - Shielding, EMI coupling, modeling

Table of Contents

• 1.1 Bandwidth Limitation
• 1.1b Skin Effect
• 1.2 Attenuation
• 1.3 Impedance Mismatch Reflections
• 1.4 EMI Noise and Electromagnetic Environment
• 1.5 ISI Jitter (Intersymbol Interference)
• 1.6 Custom Cable

• 2.1 Cell Error Rate (CER)
• 2.2 RMS and Peak-to-Peak Jitter
• 2.3 Voltages (high, low, P-P)
• 2.4 RMS Noise
• 2.5 Parity Errors
• 2.6 Eye Diagram
• 2.7 Waveform Comparison (overlay)
• 2.8 Automatic Interpretations

1. Cable Model

The cable model simulates the physical degradation of an S/PDIF coaxial signal during

propagation. Five phenomena are modeled sequentially in the cable_sim() function,

each corresponding to a real physical effect.

Input Parameters per Cable

Available Cables and Sources

AES/EBU Note: The Belden 1800F is a 110 ohm twisted-pair cable for the professional AES3 interface (IEC 60958-4). The raw attenuation per meter is higher than for 75 ohm coaxial (twisted pair vs. coax), but the AES/EBU signal level is 10 to 20 times higher (2-7 V P-P vs. 0.5 V P-P), which more than compensates. The simulator applies two corrections for AES/EBU cables:

• Effective attenuation: $A_{eff} = \max(0, A_{raw} - 20)$ dB. The 20 dB bonus corresponds to the level ratio 20 log10(5/0.5) between AES/EBU and S/PDIF.
• SNR: $SNR = \max(20, SE + 20 - P_{EMI} - 8 \log_{10}(1 + L/2))$. The additional 20 dB reflects the inherent noise margin of the higher-level signal.
• Reference impedance: $\Gamma = \frac{Z_{cable} - 110}{Z_{cable} + 110}$ (vs. 75 ohm for S/PDIF). A properly terminated 110 ohm cable generates no reflections.

Sources for Belden 1800F data: Belden catalog, Blue Jeans Cable. Conversions: dB/100m = dB/100ft x 3.281. Published attenuation at 5.645 MHz = 2.89 dB/100ft (9.48 dB/100m); interpolated to 5 and 10 MHz points.

Legend:

• No mark: value taken directly from the manufacturer's datasheet
• \*: estimated value (extrapolated, interpolated, or based on similar cables)
• \*\*: value interpolated from the 10 MHz measurement (typical ratio att_5MHz/att_10MHz ≈ 0.7)

Conversions used: Belden datasheets give attenuation in dB/100ft. Conversion: dB/100m = dB/100ft × 3.281.

Sources for shielding values:

Manufacturer datasheets generally do not provide shielding effectiveness in dB directly.

The values used are estimated from the shielding construction according to standard orders

of magnitude (ref. Ott 2009, ch.3; MIL-STD-188-124B):

Unverified values: Bandwidths at 1 m, shielding effectiveness in dB, and some propagation velocities are estimates based on cable construction (shield type, conductor cross-section). These values do not appear in the consulted datasheets.

Each preset cable is defined by the following manufacturer parameters:

The S/PDIF cell frequency is calculated as:

$f_{cell} = \frac{F_s \times 128}{10^6}$ (in MHz)

For $F_s = 44100$ Hz, this gives $f_{cell} \approx 5.645$ MHz.

The factor 128 comes from the IEC 60958 frame structure: each stereo frame consists of

2 sub-frames of 32 bits = 64 time slots, and each BMC bit produces 2 cells, therefore

64 cells per sub-frame x 2 sub-frames = 128 cells per audio frame.

1.1 Bandwidth Limitation

Physical phenomenon: The skin effect and dielectric losses in the cable act as a

low-pass filter. The high-frequency components of the square BMC signal are attenuated,

rounding the rising and falling edges.

Implementation: first-order RC low-pass filter (IIR) via scipy.signal.lfilter.

Effective cutoff frequency formula:

$BW_{eff}(L) = \max\left(5,\ \frac{BW_{1m}}{\sqrt{1 + L/8}}\right)$ (in MHz)

where:

• $BW_{1m}$ is the -3 dB bandwidth for 1 m of cable (manufacturer parameter)
• $L$ is the cable length in meters
• The 5 MHz floor prevents unrealistic filtering for very long cables

Physical justification for the $1/\sqrt{1+L/8}$ model:

The attenuation of a coaxial cable due to the skin effect follows a law $\alpha(f) = \alpha_0 \sqrt{f}$

(cf. Pozar §2.7, "Lossy Transmission Lines"). The -3 dB cutoff frequency corresponds to

the point where the total attenuation reaches 3 dB. Since attenuation is proportional to

$\sqrt{f} \times L$ (cumulative losses along the length), the cutoff frequency decreases as

$L$ increases. The simplified model uses $\sqrt{1 + L/L_0}$ with $L_0 = 8$ m as the reference

length, giving:

• At $L = 0$: $BW_{eff} = BW_{1m}$ (no degradation)
• At $L = 8$ m: $BW_{eff} = BW_{1m} / \sqrt{2}$ (additional 3 dB attenuation)
• At $L = 24$ m: $BW_{eff} = BW_{1m} / 2$ (halved)

The constant $L_0 = 8$ m is calibrated to match the typical attenuation curves of

RG-59/RG-6 coaxial cables in the 1-50 MHz range.

IIR filter coefficient:

$\alpha = \frac{2\pi f_c \cdot \Delta t}{2\pi f_c \cdot \Delta t + 1}$

where:

• $f_c = BW_{eff} \times 10^6$ is the cutoff frequency in Hz
• $\Delta t = t[1] - t[0]$ is the sampling step of the oversampled signal

The filter is applied by the recurrence relation:

$y[n] = \alpha \cdot x[n] + (1 - \alpha) \cdot y[n-1]$

Physical interpretation:

• A reference cable (Belden 1694A, 1.5 m) at $BW_{eff} \approx 233$ MHz: edges remain

nearly square.

• A generic RCA cable of 10 m can drop to $BW_{eff} \approx 17$ MHz: edges become

very rounded, and the decoder struggles to determine transition instants.

Typical ranges:

1.1b Skin Effect

Physical phenomenon: At high frequency, current no longer flows through the entire

cross-section of the conductor but concentrates at the surface, in a layer of thickness $\delta$

(the penetration depth). This increases the effective resistance of the conductor and

therefore the attenuation, proportionally to $\sqrt{f}$.

Penetration depth (Pozar, eq. 2.86):

$\delta = \frac{1}{\sqrt{\pi f \mu \sigma}}$

where:

• $f$ is the frequency (Hz)
• $\mu = 4\pi \times 10^{-7}$ H/m (permeability of copper)
• $\sigma = 5.8 \times 10^{7}$ S/m (conductivity of copper)

For copper at 5.6 MHz: $\delta \approx 28\ \mu m$. At 56 MHz (10th harmonic):

$\delta \approx 8.8\ \mu m$. Higher harmonics "see" a much more resistive conductor.

Implementation in the simulator:

The skin effect is an optional filter (checkbox). When enabled, it applies a frequency-domain

attenuation in the Fourier domain:

$V_{out}(f) = V_{in}(f) \times 10^{-A_{skin}(f) / 20}$

where the per-frequency attenuation follows the square-root law:

$A_{skin}(f) = A_{cell} \times \sqrt{\frac{f}{f_{cell}}}$ (in dB)

with:

• $A_{cell}$ = cable attenuation at the cell frequency (calculated in section 1.2)
• $f_{cell} \approx 5.6$ MHz for 44.1 kHz

Practical effect on the S/PDIF signal:

The BMC signal is a square wave whose spectrum contains odd harmonics

(3rd, 5th, 7th, ...). Without skin effect, all harmonics are attenuated by the same

amount. With skin effect:

High harmonics, which are responsible for the sharpness of rising edges, are

disproportionately attenuated. This rounds the signal edges and reduces

eye opening, beyond what the simple low-pass filter models.

When to enable it? The skin effect is most significant for:

• Long cables (> 5 m) where the baseline attenuation is already notable
• Small-diameter cables (AWG 24-26, e.g., Mogami 2964) where the conductor surface area is reduced
• Fine analyses where a more accurate model of rise times is desired

For short cables (< 3 m) in standard-gauge copper (AWG 18-20), the effect is

negligible (< 0.01 dB difference between harmonics).

1.2 Attenuation

Physical phenomenon: The signal loses amplitude as it propagates through the cable,

due to resistive losses in the conductor and dielectric losses in the insulator.

Attenuation increases with frequency and length.

Formula:

Step 1 - Linear interpolation of attenuation at the cell frequency, between the manufacturer

measurements at 5 MHz and 10 MHz:

$A_{100m} = A_{5MHz} + \frac{f_{cell} - 5}{5} \times (A_{10MHz} - A_{5MHz})$ (in dB/100m)

Step 2 - Scaling to the actual cable length:

$A_{dB} = \max\left(0,\ A_{100m} \times \frac{L}{100}\right)$

Step 3 - Application to the signal (symmetric attenuation around the midpoint):

$v_{out}[n] = v_{mid} + (v_{in}[n] - v_{mid}) \times 10^{-A_{dB}/20}$

where:

• $v_{mid} = \frac{\max(v) + \min(v)}{2}$ is the midpoint of the signal excursion
• The factor $10^{-A_{dB}/20}$ converts dB to a linear voltage factor
• Attenuation is applied symmetrically around $v_{mid}$ to preserve the average DC level of the signal

Variables:

• $A_{5MHz}$, $A_{10MHz}$: attenuation measured by the manufacturer (dB/100m) at reference frequencies
• $f_{cell}$: S/PDIF cell frequency in MHz
• $L$: cable length in meters

Justification of linear interpolation:

The actual attenuation of a coaxial cable follows a $\sqrt{f}$ law (Neumann-Ross model:

$\alpha(f) = \alpha_c \sqrt{f} + \alpha_d \cdot f$, where $\alpha_c$ represents copper losses and $\alpha_d$

dielectric losses). Over the narrow 5-10 MHz interval, linear interpolation is a

sufficient approximation (error < 5% compared to the $\sqrt{f}$ model). Manufacturers

generally provide attenuation at several discrete frequencies (1, 5, 10, 50, 100 MHz),

which justifies using the two points closest to $f_{cell} \approx 5.6$ MHz.

Physical interpretation:

• Attenuation reduces the peak-to-peak amplitude of the signal, decreasing the noise margin.
• A nominal S/PDIF signal is 0.5 V P-P. With 6 dB of attenuation, it drops to 0.25 V P-P,

half the amplitude. At 12 dB, only 0.125 V P-P remains.

• Most S/PDIF receivers stop working correctly below ~0.2 V P-P (approximately 8 dB of attenuation).

Typical ranges:

1.3 Impedance Mismatch Reflections

Physical phenomenon: When the cable impedance does not match the load impedance

(75 ohm for S/PDIF), part of the signal is reflected at the terminations. These

reflections arrive delayed (round-trip propagation time) and superimpose on the useful

signal, creating echoes and distortions.

Reflection coefficient (Pozar §2.3, eq. 2.35):

$\Gamma = \frac{Z_{cable} - Z_{load}}{Z_{cable} + Z_{load}}$

where:

• $Z_{cable}$ is the characteristic impedance of the cable (ohm), given by $Z_0 = \sqrt{L'/C'}$

where $L'$ and $C'$ are the distributed inductance and capacitance (H/m and F/m)

• $Z_{load} = 75\ \Omega$ is the S/PDIF reference impedance (IEC 60958-3 §4.2)
• $\Gamma$ is a dimensionless number between -1 and +1
• $\Gamma = 0$ means perfect matching (no reflection)
• $\Gamma < 0$ when $Z_{cable} < Z_{load}$ (typical case for RCA cables: $Z \approx 40-50\ \Omega$)
• $\Gamma > 0$ when $Z_{cable} > Z_{load}$ (rare in practice)

Round-trip delay:

$\tau_{RT} = \frac{2 \times L}{v_{prop}}$

where:

• $L$ is the cable length in meters
• $v_{prop} = \frac{V_{\%}}{100} \times c$ is the propagation speed in the cable
• $c = 3 \times 10^8$ m/s is the speed of light
• $V_{\%}$ is the velocity factor of the cable (in %)

The delay is converted to a number of samples:

$d = \text{round}\left(\frac{\tau_{RT}}{\Delta t}\right)$

Multiple reflections model:

The simulator models up to 5 bounces (round trips). At each bounce $n$:

$v_{out}[k + n \cdot d\ :] \mathrel{+}= A_n \times v_{in}[: \text{len}(v) - n \cdot d]$

The amplitude of each bounce is:

• Bounce 1: $A_1 = \Gamma$
• Bounce $n$: $A_n = A_{n-1} \times \Gamma^2$

This gives the geometric series:

$A_n = \Gamma^{2n-1}$

The $\Gamma^2$ attenuation per round trip is explained by the fact that the signal is

reflected once at each end (two reflections per round trip).

The simulation stops if:

• $n > 5$ (maximum 5 bounces)
• $|A_n| < 10^{-5}$ (negligible reflection)
• $n \cdot d \geq \text{len}(v)$ (delay exceeds the signal length)

Practical examples:

Physical interpretation:

• A well-matched 75 ohm cable produces $\Gamma = 0$: no reflections.
• A non-standard RCA cable (40-50 ohm) produces significant reflections that manifest

as time-delayed "echoes". This closes the eye diagram, increases jitter and can cause decoding errors.

• Reflections are particularly harmful with long cables because the round-trip delay

increases, and echoes superimpose at different decision instants.

Typical ranges:

1.4 EMI Noise and Electromagnetic Environment

Physical phenomenon: The cable shield protects the signal from electromagnetic

interference (EMI) from the environment. Insufficient shielding or a long cable (which

acts as an antenna) allows noise to superimpose on the useful signal.

The actual shielding effectiveness depends on two factors:

Electromagnetic Environment (EMI)

The analyzer offers three typical environments, each modeling a different level of

interference through an EMI penalty ($P_{EMI}$) subtracted from the cable's

intrinsic shielding:

Why a penalty? The cable shield attenuates interference by a factor $S_{dB}$ under

ideal conditions (laboratory measurement per IEC 62153-4-3, triaxial method). But in a

real environment, the incident EMI field is more intense than in the lab. The penalty

$P_{EMI}$ models this difference: a cable with 80 dB of shielding in an industrial

environment ($P_{EMI} = 25$ dB) behaves as if it only had 55 dB of effective shielding.

In other words, the cable still filters by the same amount, but there is more noise to filter.

Orders of magnitude of EMI fields (ref. IEC 61000-4-3, Ott 2009 ch.6):

The penalty values (0, 10, 25 dB) correspond approximately to the field ratio between

the real environment and laboratory conditions:

$P_{EMI} \approx 20 \log_{10}(E_{env} / E_{lab})$.

For a lab field of ~0.3 V/m, a living room at 1 V/m gives ~10 dB and a stage at 6 V/m

gives ~26 dB.

Effective SNR formula:

$SNR_{eff} = \max\left(20,\ S_{dB} - P_{EMI} \times \min\left(1,\ \frac{L}{5}\right) - 8 \times \log_{10}(1 + L/2)\right)$ (in dB)

where:

• $S_{dB}$ is the cable shielding effectiveness (manufacturer parameter, in dB)
• $P_{EMI}$ is the electromagnetic environment penalty (in dB): 0 (studio), 10 (hi-fi), 25 (industrial)
• $\min(1, L/5)$ is the antenna coupling factor: a short cable captures very little EMI.

The penalty increases linearly with length and reaches its full value from 5 m onward.

• $L$ is the cable length in meters
• The term $8 \times \log_{10}(1 + L/2)$ models the degradation of shielding with length

(the cable increasingly acts as a receiving antenna)

• The 20 dB floor prevents unrealistic noise (below 18 dB, noise alone would produce

massive errors, which is not realistic for a cable of a few meters)

Practical examples:

Physical interpretation:

• In a pro studio, a good cable (Belden, Canare) with 80-90 dB of shielding offers an

SNR > 75 dB: noise is completely negligible. Even a generic cable works

correctly at short lengths.

• In home hi-fi, the 10 dB penalty erodes the margin. A cable with 50 dB of shielding

drops to 40 dB effective - acceptable but without reserve. An unshielded RCA cable (25 dB)

hits the floor and picks up audible noise.

• In industrial/stage, only high-shielding cables (> 80 dB) maintain a

comfortable SNR. A generic cable is unusable beyond a few meters.

Noise application:

Noise is modeled as additive white Gaussian noise:

$v_{out}[n] = v_{in}[n] + \mathcal{N}\left(0,\ \sigma_{noise}\right)$

where the noise standard deviation is calculated from the SNR:

$\sigma_{noise} = \sqrt{\frac{P_{signal}}{10^{SNR_{eff}/10}}}$

with:

$P_{signal} = \frac{1}{N}\sum_{n=0}^{N-1} v[n]^2$

That is, the mean square power of the signal.

Variables:

• $P_{signal}$: mean signal power (V^2)
• $\sigma_{noise}$: standard deviation of added noise (V)
• $\mathcal{N}(0, \sigma)$: zero-mean normal distribution

Typical ranges:

1.5 ISI Jitter (Intersymbol Interference)

Physical phenomenon: When a digital signal passes through a long cable with limited

bandwidth, successive transitions interfere with each other. The energy of one symbol "spills"

into adjacent symbols, shifting transition instants away from the ideal grid.

This phenomenon, called ISI (Intersymbol Interference), is the dominant mechanism of jitter

degradation over long distances and the primary cause of the cliff effect in digital links.

Implemented model:

ISI jitter is modeled as a temporal perturbation applied to the signal after the

filtering, attenuation, reflections, and noise steps. Two components are combined:

$J_{src} = 2\ \text{ns RMS}$

Typical value for an S/PDIF or AES/EBU transmitter (ref. AES-12id-2020).

$J_{ISI} = \frac{K \times \sqrt{L}}{BW_{eff}}$ (in seconds RMS)

where:

• $L$ is the cable length in meters
• $BW_{eff}$ is the effective bandwidth in MHz (section 1.1)
• The $\sqrt{L}$ dependence reflects the statistical accumulation of perturbations along the cable
• $K$ is the ISI coefficient, depending on the protocol:
• S/PDIF coaxial: $K = 20 \times 10^{-9}$ (calibrated against the literature, see below)
• AES/EBU balanced: $K = 7 \times 10^{-9}$ (reduction by a factor of ~3, i.e., ~10 dB).

The differential AES/EBU link reduces effective ISI thanks to: (1) common-mode rejection,

(2) better distributed impedance matching, (3) higher dV/dt slope at the decision threshold

(signal 10x larger). The factor 3 is calibrated so that the Canare DA206

(max spec 360m) works at 300m and the DA202 (max spec 180m) works at 180m.

$J_{total} = \sqrt{J_{src}^2 + J_{ISI}^2}$ (in seconds RMS)

Implementation:

Jitter is applied as a temporal perturbation of the signal in the sampled domain:

with the cable rise times:

$N_{rise} = \max\left(3,\ \left\lfloor\frac{0.35}{BW_{eff} \times 10^6 \times \Delta t}\right\rfloor\right)$

$v_{out}[n] = v_{in}(n + \Delta i[n])$

Calibration against the literature:

The factor $20 \times 10^{-9}$ was calibrated to reproduce jitter values published

in the technical literature:

Cliff effect:

ISI jitter is responsible for the characteristic cliff effect of digital links:

the signal is perfect up to a critical distance, then collapses abruptly.

This behavior emerges naturally from the model because:

• At short distance, $J_{ISI} \ll T_{cell}/2$: transitions remain within the correct

time window, CER = 0%

• At the critical distance, $J_{ISI} \approx T_{cell}/2 \approx 88$ ns: transitions

start to spill into the adjacent cell, CER rises sharply

• Beyond that, the signal is unrecoverable

For the Belden 1694A (75 ohm, BW=250 MHz), the cliff effect occurs between 200 and 250 m,

which is consistent with manufacturer specifications and the AES3 standard.

Typical ranges:

References:

• Dunn, J., "Jitter and Digital Audio Performance Measurements", AES Preprint 3361, 1992
• AES-12id-2020, "AES information document for digital audio measurements - Jitter performance specifications"
• AES3-2009 (IEC 60958-4), Annex B: "Cable characteristics and maximum cable lengths"

1.6 Custom Cable

The interface allows defining a fully custom cable by entering the physical parameters

directly. This makes it possible to simulate a cable whose specifications are known but

that is not in the preset list, or to test the individual influence of each parameter.

Customizable parameters:

The calculation uses the same formulas as for preset cables (sections 1.1 to 1.5).

The EMI environment parameter applies in the same way.

Use cases:

• Testing a cable for which the manufacturer's datasheet is available
• Isolating the influence of a single parameter (e.g., varying only the impedance from 50 to 100 ohm

to visualize the impact of reflections)

• Simulating an exotic cable (converted plastic fiber, repurposed Ethernet cable, etc.)

2. Analysis Metrics

After cable simulation (or import of an oscilloscope capture), the analyzer calculates

a set of quantitative metrics in the full_analysis() function.

2.1 Cell Error Rate (CER)

Definition: The Cell Error Rate (CER) measures the proportion of BMC time cells

that were corrupted during transmission. It is the primary quality metric of the

S/PDIF link.

Cross-correlation alignment method:

Before comparing the reference cells and the decoded cells of the degraded signal,

they must be aligned in time. The analyzer uses cross-correlation:

discrimination:

$r'[n] = 2 \times r[n] - 1$, $c'[n] = 2 \times c[n] - 1$

cells of the reference ($r'$):

$\text{corr}[k] = \sum_{n=0}^{255} r'[n+k] \cdot c'[n]$

$\text{offset} = \arg\max_k\ \text{corr}[k]$

CER formula:

$CER = \frac{E}{N}$

where:

• $E = \sum_{n=0}^{N-1} \mathbb{1}[r_{aligned}[n] \neq c[n]]$ is the number of erroneous cells
• $N = \min(\text{len}(r_{aligned}),\ \text{len}(c))$ is the total number of cells compared
• $\mathbb{1}[\cdot]$ is the indicator function (1 if true, 0 if false)

The CER is displayed as a percentage in the interface: $CER_{\%} = CER \times 100$.

Physical interpretation:

• S/PDIF is a protocol without error correction (unlike professional AES/EBU which has

detection). Any cell error can potentially corrupt an audio sample.

• A single error in a high-weight bit (MSB) of the 16-bit sample produces a very

unpleasant audible "click".

• For a stream at $F_s = 44100$ Hz, there are $44100 \times 128 = 5\,644\,800$ cells per

second. A CER of 0.01% represents ~565 erroneous cells per second.

Typical ranges:

2.2 RMS and Peak-to-Peak Jitter

Definition: Jitter measures the temporal deviation of signal transitions from an

ideal time grid. It reflects the uncertainty in the switching instants and directly

affects the quality of digital-to-analog conversion at the end of the chain.

Calculation method:

Step 1 - Zero-crossing detection by linear interpolation:

For each detected transition (sign change around the threshold $V_{thr}$), the exact

instant is linearly interpolated between two consecutive samples:

$t_{cross} = t[i] + \frac{-\left(v[i] - V_{thr}\right)}{v[i+1] - v[i]} \times \Delta t$

where:

• $V_{thr} = \frac{\max(v) + \min(v)}{2}$ is the decision threshold (midpoint)
• $t[i]$ is the instant of the sample just before the zero-crossing
• $v[i]$, $v[i+1]$ are the voltages on either side of the threshold
• $\Delta t = t[i+1] - t[i]$ is the sampling step

Step 2 - Calculation of intervals between consecutive transitions:

$\Delta T[k] = t_{cross}[k+1] - t_{cross}[k]$

Step 3 - Calculation of deviation from the ideal grid:

Each interval should be an integer multiple of the cell period $T_{cell}$:

$\delta[k] = \Delta T[k] - \text{round}\left(\frac{\Delta T[k]}{T_{cell}}\right) \times T_{cell}$

The $\text{round}$ function determines the nearest ideal cell count (minimum 1), then

$\delta[k]$ is the residual deviation converted to nanoseconds:

$\delta_{ns}[k] = \delta[k] \times 10^9$

RMS jitter formula:

$J_{RMS} = \sqrt{\frac{1}{K}\sum_{k=0}^{K-1} \delta_{ns}[k]^2}$ (in ns)

This is the root mean square of all deviations. RMS jitter is the most commonly used

metric because it gives the statistical weight of the entire distribution.

Peak-to-peak jitter formula:

$J_{PP} = \max(\delta_{ns}) - \min(\delta_{ns})$ (in ns)

This is the total range of deviations, including extreme values.

Physical interpretation:

• Jitter affects the timing accuracy of D/A conversion on the output. In a DAC,

the jitter of the reconstruction clock translates directly into phase modulation noise

(ref. AES-12id-2020, "AES information document for digital audio measurements -

Jitter performance specifications").

• For a signal at 44.1 kHz / 16 bits, the cell period is $T_{cell} = 1 / (44100 \times 128) \approx 177$ ns.

An RMS jitter of 1 ns represents an uncertainty of 0.56% of the cell period.

• PP jitter captures extreme events (worst cases) that can trigger occasional

errors even when RMS jitter is low.

• Model limitation: the analyzer measures interface jitter (TIE - Time Interval

Error), not the DAC output jitter. Real S/PDIF receivers include a PLL

that filters part of the high-frequency jitter. The jitter measured here is therefore an

upper bound on the jitter actually transmitted to the converter.

Jitter audibility and the role of the PLL:

Interface jitter (measured on the cable) is not directly equivalent to the DAC clock jitter.

The receiver PLL (Phase-Locked Loop) acts as a low-pass filter for jitter, strongly

attenuating components at audio frequencies:

PLL filtering model used by the simulator

The simulator assumes that ISI jitter has a white spectral density over [0 ; F_cell], with F_cell = 5.644 MHz (BMC cell rate at 44.1 kHz). The PLL is modelled as a Butterworth low-pass filter of order n. The fraction of jitter reaching the converter is computed via the equivalent noise bandwidth of the filter:

$$B_n = f_c \cdot \frac{\pi}{2n} \cdot \frac{1}{\sin\!\left(\frac{\pi}{2n}\right)}$$

$$J_{DAC} = \sqrt{\left(J_{ISI} \cdot \sqrt{\frac{B_n}{F_{cell}}}\right)^2 + J_{floor}^2}$$

where $f_c$ is the PLL corner frequency, $n$ its order, and $J_{floor}$ the intrinsic floor of the local clock (receiver's own jitter, independent of the cable).

PLL presets:

The ratio $\sqrt{B_n / F_{cell}}$ quantifies attenuation of ISI jitter. For CS8412, ratio = 0.070: 7% of cable jitter reaches the DAC. For ASRC and Word Clock, ratio < 0.001: cable jitter is negligible and the result is governed solely by $J_{floor}$.

Word Clock -- typical setup: a dedicated master clock generator (Mutec MC-3+, Antelope, Aardsync II) drives the DAC clock via BNC at the sample rate (44.1 or 48 kHz). S/PDIF interface jitter is irrelevant since the D/A clock comes from the generator, not the cable. The 5 ps floor corresponds to a typical pro generator (Mutec MC-3+USB: ~3 ps, Antelope Isochrone: ~1 ps, Aardsync II: ~15 ps). A budget generator will typically measure 20-50 ps.

The signal-to-noise ratio degradation caused by jitter follows the formula (Dunn 1997):

$SNR_{jitter} = -20 \log_{10}(2\pi \cdot f_{signal} \cdot J_{RMS})$ (in dB)

where $f_{signal}$ is the audio signal frequency (Hz) and $J_{RMS}$ the RMS jitter (in seconds).

Examples:

Empirical audibility thresholds -- from controlled listening tests:

• PLL-filtered jitter < 10 ns RMS: inaudible -- minimum measured threshold = 10 ns RMS with 17 kHz tone

(Benjamin & Gannon, AES paper 4826, 1998: https://aes2.org/publications/elibrary-page/?id=8354)

• PLL-filtered jitter 10-20 ns RMS: audibility threshold only with 17 kHz pure tone at full level;

inaudible in practice with real music (Benjamin & Gannon 1998)

• PLL-filtered jitter > 20 ns RMS: potentially audible with music under controlled conditions

(Benjamin & Gannon 1998: 20 ns RMS). In practice, Ashihara et al. measured a threshold of 300+ ns RMS with

music for trained professional listeners (Ashihara et al., *Acoustical Science and Technology*,

2005: https://doi.org/10.1250/ast.26.50)

Typical jitter of modern DACs:

References:

• Adams, R.W. (1994). "Clock Jitter, D/A Converters, and Sample Rate Conversion." *The Audio Critic*, No. 21, Spring 1994. https://www.biline.ca/audio_critic/mags/The_Audio_Critic_21_r.pdf
• Dunn, J. (1994). "Jitter and Digital Audio Performance Measurements." *JAES*, Vol. 42, No. 5. https://aes2.org/publications/elibrary-page/?id=6111
• Benjamin, E. & Gannon, B. (1998). "Theoretical and Audible Effects of Jitter on Digital Audio Quality." *AES paper 4826*, 105th AES Conv., San Francisco. https://aes2.org/publications/elibrary-page/?id=8354 -- Minimum measured threshold: 10 ns RMS (17 kHz tone), 20 ns RMS (music).
• Ashihara, K., Kiryu, S. et al. (2005). "Detection threshold for distortions due to jitter on digital audio." *Acoustical Science and Technology*, 26(1), pp. 50–54. https://doi.org/10.1250/ast.26.50 -- Detection threshold: 300+ ns RMS with music (trained professional listeners).
• AES-12id-2020, "AES information document for digital audio measurements — Jitter performance specifications." https://aes.org/publications/standards-store/
• Stuart, J.R. (2004). "Coding for High-Resolution Audio Systems." *JAES*, Vol. 52, No. 3, pp. 117–144. https://aes2.org/publications/elibrary-page/?id=12986
• Siau, J. & Burdick, A.H. (Benchmark Media Systems). "Jitter and Its Effects." https://benchmarkmedia.com/blogs/application_notes/12142221-jitter-and-its-effects

Distribution: The analyzer also produces a 50-bin histogram of the distribution

of $\delta_{ns}[k]$. A healthy signal shows a narrow Gaussian distribution centered

on zero. A degraded signal shows broadening, asymmetry, or secondary peaks.

Typical ranges:

2.3 Voltages (high, low, P-P)

Definition: These metrics measure the voltage levels of the degraded analog signal,

enabling evaluation of signal attenuation and symmetry.

Method: The signal is separated into two populations using the decision threshold:

$V_{thr} = \frac{\max(v) + \min(v)}{2}$

• High samples: $\{v[n]\ |\ v[n] > V_{thr}\}$
• Low samples: $\{v[n]\ |\ v[n] \leq V_{thr}\}$

Formulas:

Mean high voltage:

$V_{high} = \frac{1}{|H|} \sum_{v[n] \in H} v[n]$

where $H = \{v[n]\ |\ v[n] > V_{thr}\}$ is the set of high-level samples.

Mean low voltage:

$V_{low} = \frac{1}{|L|} \sum_{v[n] \in L} v[n]$

where $L = \{v[n]\ |\ v[n] \leq V_{thr}\}$ is the set of low-level samples.

Peak-to-peak amplitude:

$V_{PP} = \max(v) - \min(v)$

Physical interpretation:

• The nominal coaxial S/PDIF signal is 0.5 V P-P (IEC 60958 standard), i.e., $V_{high} \approx 0.5$ V

and $V_{low} \approx 0$ V in the model (the signal oscillates between 0 and $V_{PP} = 0.5$ V).

• Cable attenuation reduces $V_{PP}$ while compressing the high and low levels toward

$V_{thr}$. A $V_{PP}$ below 0.2 V is generally problematic.

• An asymmetry between $(V_{high} - V_{thr})$ and $(V_{thr} - V_{low})$ indicates

nonlinear distortion or a DC offset.

Typical ranges (for a reference signal at 0.5 V P-P):

2.4 RMS Noise

Definition: RMS noise measures the dispersion of voltage samples around their

mean level (high or low). It reflects the amount of noise superimposed on the signal,

whether from EMI interference, cable thermal noise, or reflections.

Formulas:

RMS noise at the high level:

$\sigma_{high} = \sqrt{\frac{1}{|H|} \sum_{v[n] \in H} \left(v[n] - V_{high}\right)^2}$

RMS noise at the low level:

$\sigma_{low} = \sqrt{\frac{1}{|L|} \sum_{v[n] \in L} \left(v[n] - V_{low}\right)^2}$

where $V_{high}$ and $V_{low}$ are the mean voltages defined in the previous section.

Physical interpretation:

• RMS noise determines the "clarity" of the signal's logic levels. If the noise is too

large relative to the gap between $V_{high}$ and $V_{low}$, the decoder can no longer

reliably distinguish the two levels.

• An approximate signal SNR can be defined as:

$SNR_{signal} \approx 20 \times \log_{10}\left(\frac{V_{PP}}{2 \times \max(\sigma_{high}, \sigma_{low})}\right)$

• In general, $\sigma_{high}$ and $\sigma_{low}$ should be of the same order of magnitude.

A large asymmetry indicates a problem specific to one level (for example, reflection

bounces preferentially affecting rising transitions).

Typical ranges:

2.5 Parity Errors

Definition: Each S/PDIF sub-frame (IEC 60958-1 §6.2.5) contains a parity bit

(bit 31, i.e., the last bit of the 32-cell sub-frame). This bit is calculated so

that the parity of all bits 4 to 31 (audio + status + parity) is even. It is the only

error detection mechanism provided by the standard - there is no error correction (FEC).

Verification formula:

$P_{check} = \left(\sum_{i=0}^{26} b[i]\right) \bmod 2$

The encoded parity bit is $b[27]$. If $P_{check} \neq b[27]$, a parity error is counted.

Error counter:

$N_{perr} = \sum_{sf} \mathbb{1}\left[b_{sf}[27] \neq \left(\sum_{i=0}^{26} b_{sf}[i]\right) \bmod 2\right]$

where the sum is over all decoded sub-frames.

Physical interpretation:

• S/PDIF parity can only detect errors affecting an odd number of bits within

the same sub-frame. Errors on an even number of bits go undetected.

• In practice, if parity errors are detected, this means the signal is

significantly degraded, since at least one bit error per affected sub-frame is required.

• The number of parity errors is generally lower than the actual number of bit

errors, because parity does not detect all errors.

Typical ranges:

2.6 Eye Diagram

Definition: The eye diagram is a 2D representation that superimposes all segments

of the signal over a duration of 2 cell periods (2 UI - Unit Intervals). It is the most

powerful visual diagnostic tool for signal integrity.

Construction method:

Step 1 - Calculation of the number of samples per cell:

$SPC = \text{round}\left(\frac{T_{cell}}{\Delta t}\right)$

where $T_{cell}$ is the cell period and $\Delta t$ the sampling step.

Step 2 - Windowing: the observation window is $W = 2 \times SPC$ samples

(2 cell periods = 2 UI).

Step 3 - Segment extraction: the signal is divided into segments of width $W$, offset

by a step of $SPC$ (1 cell). Each segment is a "pass" through the eye diagram:

$\text{seg}_k = v[k \cdot SPC\ :\ k \cdot SPC + W]$

for $k = 0, 1, 2, ...$, as long as the segment fits within the signal.

Step 4 - The horizontal axis is normalized between 0 and 2 UI:

$x = \text{linspace}(0,\ 2,\ W)$ repeated for each segment.

Step 5 - Construction of a 2D histogram (200 x 120 bins):

$H[i, j] = \text{count}(x \in [x_i, x_{i+1}],\ y \in [y_j, y_{j+1}])$

with:

• X axis: 200 bins from 0 to 2 UI
• Y axis: 120 bins from $V_{min} - 0.02$ to $V_{max} + 0.02$ V

The 2D histogram is visualized as a heatmap (color scale "Hot").

Physical interpretation:

The eye "opening" indicates the margin available to the decoder:

• Vertical opening: difference between the high level and the low level at the center of

the eye. The larger it is, the greater the noise margin.

• Horizontal opening: temporal width of the zone where levels are stable.

The larger it is, the greater the jitter margin.

Diagram appearance	Probable cause
Wide open eye	Clean signal, good integrity
Vertically narrowed eye	Excessive attenuation, high noise
Horizontally narrowed eye	High jitter, insufficient bandwidth
Closed eye	Unusable signal, too many cumulative degradations
Multiple traces/echoes	Impedance mismatch reflections
Thickening of traces	EMI noise superimposed on the signal

Rendering parameters:

2.7 Waveform Comparison (overlay)

Definition: The waveform comparison chart superimposes the signals from both cables

with the S/PDIF reference on three synchronized panels, allowing visualization of

differences between cables at different scales.

3-panel structure:

Automatic zoom zone detection:

The algorithm identifies the most "interesting" region of the signal:

$D[n] = |v_A[n] - v_{ref}[n]| + |v_B[n] - v_{ref}[n]|$

$D_{smooth}[n] = \frac{1}{W} \sum_{k=0}^{W-1} D[n+k]$

$n_{center} = \arg\max_n D_{smooth}[n]$

Why 3 panels? The global view (panel 1) is downsampled for performance reasons,

which can mask fine differences between signals. The zoom panel shows these differences

at full resolution over the most relevant zone. The difference panel visually amplifies

gaps by isolating them from the carrier signal.

Synchronization: The three panels share the same X axis (time). Zooming on one

panel automatically zooms the other two, enabling correlation of observations between views.

2.8 Automatic Interpretations

The analyzer generates a dynamic textual interpretation below each chart, based

on the calculated metrics. These interpretations help understand what the charts show

without prior expertise in signal integrity.

Eye diagram - interpretation based on CER, eye opening, and noise:

The vertical eye opening is estimated as:

$O_{eye} = (V_{high} - V_{low}) - 6 \times \max(\sigma_{high}, \sigma_{low})$

The factor 6 (3 sigma on each side) corresponds to the visual closure threshold.

Degradation causes are identified automatically: reflections ($|\Gamma| > 0.05$),

EMI noise ($\sigma > 10$ mV), attenuation ($V_{PP} < 0.35$ V).

Overlay - interpretation based on amplitude loss:

where $\Delta V_{PP} = |0.5 - V_{PP}|$.

Jitter - interpretation based on RMS jitter:

Each interpretation includes a link to the corresponding documentation section

for deeper understanding.

3. Global Verdict

The analyzer assigns a synthetic verdict to each cable by combining CER and

RMS jitter. The thresholds are defined as follows:

Conditions are evaluated from top to bottom; the first matching verdict is retained.

Note: This verdict is a useful simplification for a quick assessment. For a

thorough analysis, all individual metrics should be examined, along with the eye diagram.

Appendix: Global Constants

Appendix: Conversion from Cells to Analog Signal

Before cable simulation, binary BMC cells are converted to an analog waveform

by the cells_to_analog() function:

by $V_{PP} = 0.5$ V.

• Kernel standard deviation: $\sigma_k = \max(1,\ 0.05 \times OVS) = 1.6$ samples
• Kernel width: $K = \max(3,\ \lceil 6\sigma_k \rceil | 1)$ (rounded up to odd)
• The kernel is a normalized Gaussian: $g[n] = \frac{e^{-n^2/2}}{\sum g}$

evaluated over $[-3\sigma, +3\sigma]$

with slightly rounded edges.

CSV Capture Format

The analyzer can import oscilloscope captures in CSV format. Below is the specification of the expected format.

File Structure

# Comment (optional, lines starting with #)
time_s,voltage_V
1.000000e-07,2.50000000e-02
2.000000e-07,4.80000000e-01
3.000000e-07,4.75000000e-01
...

Format Rules

Automatic Time Unit Detection

The analyzer automatically detects the time unit based on the range of values:

Recommended Duration and Sampling Rate

For a reliable S/PDIF signal analysis:

Justification: The S/PDIF signal at 44.1 kHz has a cell frequency of ~5.6 MHz. According to the Nyquist theorem, sampling at at least 2x this frequency (11.2 MSa/s) is required, but in practice 50 MSa/s or more allows good zero-crossing interpolation for jitter measurement.

Example File

An example CSV file can be downloaded from the analyzer interface (button "Download example CSV"). It contains a reference S/PDIF signal (1 kHz sine wave, 44100 Hz, 16 bits) that has passed through a 3 m Belden 1505A cable.

WAV File Import

The analyzer also accepts captures in WAV format (8/16 bits, mono or stereo). The signal is automatically normalized between 0 and 0.5 V to match the S/PDIF scale. This format is useful for USB oscilloscopes that export directly to WAV.

Appendix: Reference Simulation Results

The results below serve as validation of the model. They are generated with a

1 kHz sinusoidal signal, 44100 Hz, 16 bits, 8 frames (1024 cells), fixed random seed (42).

All Cables at 1.5 m (Home Hi-Fi)

Observations:

• All cables at 1.5 m produce 0% CER, including the worst case (generic RCA).
• RMS jitter increases from 0.001 ns (Belden 1694A) to 1.7 ns (generic RCA) - a factor of 1700.
• The Mogami 2964 shows higher jitter than other 75 ohm cables (0.063 vs. 0.001-0.003 ns) due to

its reduced bandwidth (80 MHz vs. 200+ MHz), which rounds the rising edges.

• The generic RCA has a $V_{PP}$ above 0.5 V (0.575 V) because the added noise (SNR=20 dB) increases

the extremes of the voltage distribution.

Effect of Length - Generic RCA Cable (Hi-Fi)

Errors appear from ~15-20 m, when the effective bandwidth

($BW_{eff} \approx 13$ MHz) gets too close to the cell frequency (5.6 MHz), leaving

insufficient margin for the decoder to distinguish cells after adding noise.

Effect of Environment - Belden 1694A, 10 m

Even in an industrial environment, a 10 m Belden 1694A maintains negligible jitter

(0.022 ns) and 0% CER. Its 90 dB shielding easily absorbs the industrial penalty.

Appendix: S/PDIF and AES/EBU Connectors

The simulator does not model connectors. This appendix presents data published by

manufacturers (Amphenol, Neutrik, Kings Electronics) and standards (IEC 61169-8).

> Note: the values below come from manufacturer datasheets and IEC standards. No

> independent measurements were made within the scope of this project.

1. RCA (Phono)

A connector not specified for RF: designed for analog audio in the 1940s, adopted by

convention for consumer S/PDIF coaxial. No impedance or VSWR specification is published.

Effect on signal: each RCA connector pair introduces a localized impedance

discontinuity. On short cables (< 1 m), this reflection arrives during the edge transition

and decays before the next symbol. Beyond 3–5 m, echoes can accumulate and add to the

cable's own reflections.

2. BNC 75 ohm

RF-specified connector conforming to IEC 61169-8. Designed for digital video/audio

systems (controlled 75 ohm impedance).

Effective Γ: < 0.025 at 5.6 MHz (VSWR 1.05), a reflection 13× smaller than a typical

RCA connector. For all practical cable lengths, a BNC connector's impact on the S/PDIF

signal is physically negligible.

3. XLR (AES/EBU)

3-pin balanced connector conforming to IEC 61076-2-103 (Neutrik, Amphenol).

> For AES/EBU, impedance is controlled by the cable (110 ohm balanced). The XLR geometry

> is not RF-optimized, but the connector's electrical length (< 30 mm ≈ 0.6 ns) is

> negligible at 5.6 MHz (λ/100 at 10 MHz).

4. Toslink (optical)

5. Practical impact at 5.6 MHz

6. Why connectors are not modeled

Modeling an RCA connector would require knowing its effective impedance, which varies with

cable geometry, plug diameter, and mating quality — information not available without

vector network analyzer (VNA) measurements. For BNC and XLR connectors, the impact is

so small at 5.6 MHz that modeling would add no useful information.

Appendix: Model Limitations

The simulator is a pedagogical tool. It does not replace a real measurement.

Main simplifications:

Appendix: Validation Against the Literature

The model was compared to published data to verify its consistency with real measurements.

Jitter vs. Distance (Belden 1694A, 75 ohm)

Maximum Distance per Cable

Comparison Sources

• Dunn, J. (1992). "Jitter: Specification and Assessment in Digital Audio Equipment." AES Preprint 3361, 93rd AES Convention, San Francisco. https://aes2.org/publications/elibrary-page/?id=6772
• AES3-1-2009 (r2024), Annex B: recommended maximum distances
• Canare: DA202, DA206 datasheets (published maximum distances)
• Belden: SDI/AES specifications for the 1694A and 1800F

Appendix: Audiophile Myths vs. Digital Signal Physics

This section puts common beliefs about digital cables into perspective and evaluates what

transmission line physics can confirm or refute.

The S/PDIF Signal is Binary, Not Analog

The S/PDIF protocol transmits bits, not a continuous voltage. The receiver compares the

signal to a threshold (200 mV P-P per IEC 60958-3): the bit is either correctly decoded or

in error. There is no progressive degradation of digital audio content due to a

cable - there is either correct transmission or errors.

Direct consequence: two cables both transmitting CER = 0 deliver exactly the

same bits to the DAC. There is no possible audible difference between them.

Critical Distance by Sample Rate

The length above which impedance matching becomes important depends on fs

(source: transmission line calculations, consistent with IEC 60958-3 and field practice):

Above these lengths, cable quality (impedance, shielding) begins to matter.

Below them, all proper 75 ohm cables are physically equivalent.

The Role of the Receiver PLL (DAC)

Even if a cable introduces jitter at the transmitter output, the S/PDIF receiver has

a phase-locked loop (PLL) that re-synchronizes the signal to a local clock.

• The PLL attenuates interface jitter by 40 to 80 dB at audio frequencies (ref. Dunn 1997)
• A 10 ns RMS input jitter becomes < 1 ps after a high-performance PLL
• The audio SNR due to a residual jitter of 1 ns RMS at 20 kHz:

$SNR = -20 \log_{10}(2\pi \times 20000 \times 10^{-9}) \approx 78\ \text{dB}$

which is better than the dynamic range of a 13-bit recording

When the Digital Cable Really Matters

The physical effects identified in this simulator have a real impact under specific conditions:

What Physics Rules Out

For cables < 5 m with matched impedance (75 ohm):

• Attenuation is less than 0.15 dB for all reference cables (Belden, Canare, etc.)
• ISI jitter remains < 0.5 ns RMS, which is > 60 dB below the audibility threshold
• CER is systematically zero
• No 75 ohm cable of adequate quality can produce a different sound from another

The audiophile belief in "high-end digital cables" has been experimentally

refuted by numerous blind tests (ABX tests). It confuses the properties of analog cables

(which affect the signal) with digital cables (which affect binary integrity).

References:

• Dunn, J. (1994). "Jitter and Digital Audio Performance Measurements." *JAES*, Vol. 42, No. 5. https://aes2.org/publications/elibrary-page/?id=6111
• AES-12id-2020, "Jitter performance specifications." https://aes.org/publications/standards-store/
• Siau, J. & Burdick, A.H. (Benchmark Media Systems). "Jitter and Its Effects." https://benchmarkmedia.com/blogs/application_notes/12142221-jitter-and-its-effects

Complete Bibliography

Standards and Specifications

Reference Books

AES Articles and Preprints

Application Notes

Manufacturer Datasheets