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Delta-Sigma Modulator: OSR, Order, Noise Shaping and SNR

Calculate delta-sigma oversampling ratio, in-band quantization noise and ideal SNR. Compare noise-shaping order and estimate the sample rate for a target SNR.

Wired White

Published 30. October 2026 · version 2 of 6. November 2026

#ADC #delta-sigma modulation #noise shaping #oversampling #quantization #SNR

A delta-sigma modulator trades sampling speed and noise shaping for lower quantization noise inside a useful signal band. This calculator helps engineers and students compare oversampling ratio, modulator order and quantizer resolution before choosing a clock rate or running a detailed simulation. It applies to an ideal low-pass, discrete-time noise-transfer function with all shaping zeros at DC.

Choose whether to enter the modulator sample rate or the oversampling ratio. Enter the one-sided signal bandwidth from DC to the band edge. The tool uses OSR = sample rate divided by twice that bandwidth. The sample rate belongs to the modulator before decimation; it is not the final ADC output data rate, OFDM payload rate or optical line baud rate.

Next, set the noise-shaping order, quantizer bits, output voltage span and sine-wave peak amplitude. Order zero provides an oversampling-only comparison. The output span means the separation of the outermost equally spaced reconstruction levels. For a one-bit stream with levels −1 V and +1 V, enter a 2 V span: the step is 2 V. This convention differs from the nominal ADC input range used in some textbook bit-count formulas. The calculation shows the step explicitly so you can check the normalization.

The main results give ideal in-band SNR, RMS quantization noise, the gain from noise shaping, and SNR-equivalent sine bits at the entered signal amplitude. These equivalent bits exclude distortion and are not a measured converter ENOB specification. The order table compares noise and SNR at a common OSR. The OSR chart compares the integrated sine response, its high-OSR approximation and unshaped oversampling. The frequency chart shows noise-transfer gain; its horizontal axis is frequency divided by the modulator clock. Compare the displayed normalized band edge with this axis.

Enter a target SNR to estimate the required continuous OSR and corresponding clock rate. The estimate uses the high-OSR approximation and rounds neither the clock nor the ratio to available hardware settings. It is conservative within this mathematical model because replacing sine by its argument overestimates noise. Select a practical clock and allow transition bandwidth for the actual reconstruction or decimation filter.

The related WiredWhite fronthaul article (https://wiredwhite.com/snr-improved-delta-sigma-transmitter-qam/) uses two quantization branches, optimized transfer functions and PAM-4 transport. This calculator adopts its OSR definition, but does not model that cancellation architecture or reproduce its measured link SNR.

The model assumes white quantization error, unity signal transfer and an ideal brick-wall output filter. It excludes idle tones, overload dynamics, stability, optimized zeros, band-pass modulators, thermal noise, clock jitter, DAC mismatch, finite filter rejection and optical-link impairments. Higher order can improve the theoretical noise floor while making a real loop harder to stabilize. Check amplitude limits and stability with the intended loop implementation; a result here does not establish either.

How it is calculated

The OSR convention follows the WiredWhite article, Section II.B, and Jarman, Intersil AN9504 (May 1995), p. 2: $R=f_s/(2B)$ .

For $2^b$ equally spaced reconstruction levels including both span endpoints, $\Delta=V_{span}/(2^b-1)$ . With sine peak $A$ , $V_{rms}=A/\sqrt{2}$ and unshaped quantization variance is $\Delta^2/12$ (AN9504, Eqs. 1–2).

For the ideal low-pass $NTF(z)=(1-z^{-1})^L$ , its power gain at normalized frequency $\nu=f/f_s$ is $[2\sin(\pi\nu)]^{2L}$ . The in-band noise factor is

K=2\int_0^{1/(2R)}[2\sin(\pi\nu)]^{2L}\,d\nu.

The implementation substitutes $u=f/B$ and uses composite Simpson integration with 128 subintervals. Noise is $\sigma_n=\Delta\sqrt{K/12}$ and $SNR=20\log_{10}(V_{rms}/\sigma_n)$ .

Replacing sine by its argument gives $K_a=\pi^{2L}/[(2L+1)R^{2L+1}]$ , the generalized high-OSR expression in AN9504 Eq. 11. Its SNR improvement per clock doubling is $10(2L+1)\log_{10}2$ dB. The target OSR algebraically inverts this expression and is bounded below by 1. Since $\sin x\leq x$ here, this target is conservative for the chosen NTF.

SNR-equivalent bits are $(SNR-1.7609)/6.0206$ at the entered amplitude, without amplitude correction or distortion.

Assumptions

Zero-mean sinusoidal input with unity in-band signal transfer. The signal frequency is inside 0…B.
White, input-independent quantization error with variance Δ²/12. Idle tones and correlated error are excluded.
Uniform output levels include both endpoints of the specified voltage span. A one-bit span of 2 V means levels ±1 V and Δ = 2 V. The often-quoted 6.02b + 1.76 formula instead uses a nominal range of 2ᵇΔ; it must not be applied to this output span without adjustment.
An ideal low-pass output filter retains DC…B and removes all noise above B. No decimation factor, transition band or implementation is selected.
Orders 0–5 and OSR 1–4096 are supported for evaluating this NTF. The target-rate estimate may exceed that OSR range.

Where it applies

This is an ideal quantization-noise budget, not a nonlinear modulator simulation. “Integrated” means the sine-shaped noise-transfer response is numerically integrated within the white-noise model; it does not mean exact hardware SNR. For third and higher orders, especially single-bit loops, check maximum stable amplitude, coefficients and internal states separately. Even lower-order loops can show tones or overload.

At low OSR, the high-OSR formula can differ substantially from the integrated response; use the integrated result and the displayed difference. Shaping can increase in-band noise when the passband includes frequencies with NTF gain above unity. Total noise after shaping need not be conserved.

The target calculation excludes all other noise and distortion. Its conservatism applies only to this NTF and ideal filtering. SNR-equivalent bits are not noise-free resolution, output word length, measured ENOB or a QAM/EVM/BER guarantee. The WiredWhite article’s two-branch noise cancellation and band-pass examples require a different model.

References

Waqas Javaid, WiredWhite (22 July 2025), An SNR-improved Transmitter of Delta-sigma Modulation Supported Ultra-High-Order QAM Signal for Fronthaul/WiFi Applications, Section II.B. https://wiredwhite.com/snr-improved-delta-sigma-transmitter-qam/
David Jarman, Intersil AN9504, A Brief Introduction to Sigma Delta Conversion (May 1995), pp. 1–3, Eqs. 1–11. Open copy: https://www.engr.colostate.edu/ECE423/docs/intersil_delta_sigma_tutorial.pdf
Walt Kester, Analog Devices MT-022, ADC Architectures III: Sigma-Delta ADC Basics, Rev. A (October 2008), pp. 4 and 10. https://www.analog.com/media/en/training-seminars/tutorials/MT-022.pdf

Inputs

Input	Symbol	Unit	Default
Specify sampling by	`rate_basis`		Sample rate and bandwidth
One-sided signal bandwidth Low-pass band from DC to B. Use the occupied one-sided baseband bandwidth, not the OFDM data rate or the optical baud rate.	`BW`	Hz	20000
Modulator sample rate The clock of the delta-sigma modulator, before filtering or decimation. Enter MHz; 2.56 MHz is 2,560,000 samples per second.	`f_clock`	MHz	2.56
Oversampling ratio OSR = sample rate / (2 × one-sided bandwidth). Fractional ratios are allowed.	`R_set`		64
Noise-shaping order NTF(z) = (1 − z⁻¹)ᴸ. Choose 0 for oversampling without shaping. Orders 3–5 are ideal-model comparisons; loop stability is not simulated.	`L_mod`		2
Quantizer bits There are 2ᵇ equally spaced output levels, including both ends of the output span. One bit means two levels.	`b_q`		1
Quantizer output span (peak-to-peak) Separation of the lowest and highest output levels, not a nominal ADC input range. For levels −1 V and +1 V, enter 2 V; the quantization step is then 2 V.	`V_span`	V	2
Sine-wave peak amplitude A zero-mean sinusoid at the signal input, with unity in-band signal transfer. Peak amplitude is not RMS or peak-to-peak.	`A_sig`	V	0.5
Target quantization-limited SNR Used to estimate a conservative continuous OSR from the high-OSR approximation. No thermal-noise, jitter or distortion budget is included.	`SNR_goal`	dB	90

Results

With the default inputs above, the calc gives:

Result	Value
Oversampling ratio	64
Ideal quantization-limited SNR	73.15 dB
In-band RMS quantization noise	77.757 uV
SNR gain from shaping	59.35 dB
SNR-equivalent sine bits	11.86
Modulator sample rate	2,560,000 Hz
Conservative OSR for target SNR	139.034
Sample rate for target SNR	5,561,342 Hz
SNR correction to high-OSR approximation	0.0012 dB
Signal band edge, f/f_s	0.007813

Checked against sources

Each case below was calculated by this calc and agrees with the value from its source (within the tolerance where one is shown).

AN9504: 12-bit, 2.5 V quantizer noise

Source: Jarman, Intersil AN9504 (May 1995), p. 2: published RMS noise is 176 μV for a 12-bit ADC with 2.5 V full scale. The 0.5 μV tolerance covers the rounded reference and the 2ᵇ versus (2ᵇ−1) span convention (0.024%). This compares quantization noise before shaping.

Inputs: Quantizer bits = 12; Quantizer output span (peak-to-peak) = 2.5 V; Noise-shaping order = 0; Specify sampling by = OSR and bandwidth; Oversampling ratio = 1; the others as above.

Result	Source	This calc
White quantization noise before shaping	1.76 × 10⁻⁴ ± 5 × 10⁻⁷ V	1.762 × 10⁻⁴ V

AN9504: four codes across a 3 V output span

Source: Jarman, Intersil AN9504 (May 1995), p. 1, code example: the four output values are 0, 1, 2 and 3 V, so the adjacent step is 1 V. This checks the endpoints convention, not an alternative nominal ADC range.

Inputs: Quantizer bits = 2; Quantizer output span (peak-to-peak) = 3 V; the others as above.

Result	Source	This calc
Quantization step between adjacent output levels	1 V	1 V
Number of equally spaced quantizer levels	4	4

AN9504: first-order 9 dB per OSR doubling

Source: Jarman, Intersil AN9504 (May 1995), p. 2: first-order high-OSR noise decreases by 9 dB for a factor-of-two clock increase. The source rounds 9.0309 dB to 9 dB; this is an asymptotic slope, not a low-OSR exact gain.

Inputs: Noise-shaping order = 1; the others as above.

Result	Source	This calc
High-OSR SNR improvement per doubling of sample rate	9 ± 0.04 dB	9.031 dB

AN9504: second-order 15 dB per OSR doubling

Source: Jarman, Intersil AN9504 (May 1995), p. 2: second-order high-OSR noise decreases by 15 dB for a factor-of-two clock increase. The source rounds 15.0515 dB to 15 dB; the check allows only that rounding.

Inputs: the default inputs above.

Result	Source	This calc
High-OSR SNR improvement per doubling of sample rate	15 ± 0.06 dB	15.051 dB

WiredWhite: 10 GSa/s and 625 MHz give OSR 8

Source: Waqas Javaid, WiredWhite source article (22 July 2025), Section II.B: fΔΣ = 10 GSa/s, OSR = 8 and BW = 625 MHz. This validates its bandwidth convention only; its cancellation-transmitter SNR is outside this model.

Inputs: Modulator sample rate = 10,000 MHz; One-sided signal bandwidth = 625,000,000 Hz; the others as above.

Result	Source	This calc
Oversampling ratio	8	8
Modulator sample rate	1 × 10¹⁰ Hz	1 × 10¹⁰ Hz

WiredWhite: 10 GSa/s and 500 MHz give OSR 10

Source: Waqas Javaid, WiredWhite source article (22 July 2025), Section II.B: fΔΣ = 10 GSa/s, OSR = 10 and BW = 500 MHz. This checks the inverse sample-rate calculation against the published parameter set.

Inputs: Specify sampling by = OSR and bandwidth; Oversampling ratio = 10; One-sided signal bandwidth = 500,000,000 Hz; the others as above.

Result	Source	This calc
Oversampling ratio	10	10
Modulator sample rate	1 × 10¹⁰ Hz	1 × 10¹⁰ Hz

MT-022: roughly 80 dB at OSR 64, second order

Source: Kester, Analog Devices MT-022 Rev. A (October 2008), p. 10, Fig. 9 and accompanying text give about 80 dB at OSR 64 for an ideal second-order system. With 1-bit levels ±1 V and sine peak 1 V, this NTF model gives 79.17 dB. The 1 dB tolerance is for the source's approximate graph/text value; this theoretical full-amplitude case does not establish loop stability.

Inputs: Specify sampling by = OSR and bandwidth; Sine-wave peak amplitude = 1 V; the others as above.