Mixed-Signal IC Design, Delta-Sigma Data Converters & DSP Decimation
Delta-Sigma ADC Oversampling Ratio (OSR) & ENOB Gain Formula
Calculate Mixed-Signal ADC Oversampling Ratio OSR = f_s / (2*BW), Plain Nyquist Oversampling Extra Bits (+0.5 bit/octave), First-Order Delta-Sigma Noise-Shaping SNR Improvement (dB), and Total Effective Resolution Bits (+1.5 bits/octave).
Contrast +0.5 bit/octave plain oversampling against +1.5 bits/octave ΔΣ noise shaping!
Sampling a baseband signal of bandwidth BW (kHz) at modulator clock f_s (MHz) above the Nyquist rate 2*BW defines the Oversampling Ratio OSR = (f_s * 1000) / (2 * BW)! With a plain Nyquist ADC and digital low-pass decimation filter, white quantization noise spreads flat across [0, f_s/2], adding only 10*LOG10(OSR) dB = +3.01 dB (+0.5 ENOB bit) per octave doubling of OSR: Extra_Bits_White = 0.5 * (LN(OSR) / LN(2))! Wrapping a First-Order Delta-Sigma (ΔΣ) feedback integrator around an N-bit quantizer high-pass-shapes quantization noise as (1 - z^-1), suppressing in-band noise power by π² / (3 * OSR³)—yielding ΔSNR_ΔΣ1 = 10 * LOG10((3 * OSR³) / π²) dB (+9.03 dB = +1.5 ENOB bits per octave)!
Verify on an Audio/Sensor Modulator—Modulator_Clock_fs = 6.144 MHz (B2), Signal_Passband_BW = 24.0 kHz (Nyquist = 48 kHz, B3 -> OSR = 6,144 / 48 = 128 = 2^7 in E2), and MultiBit_Quantizer_N = 4 bits (B4): Plain White Oversampling would add only 0.5 * 7 = 3.50 bits (7.50 total bits, E3), whereas First-Order ΔΣ Noise Shaping boosts SNR by +58.04 dB (E4), lifting Total Ideal Resolution to 4 + 58.0445 / 6.02 = 13.64 bits (E5)!
| ΔΣ Noise-Shaping Metric (f_s = 6.144 MHz, BW = 24 kHz -> OSR = 128, N = 4 Bits) | Formula | Verified Output |
|---|---|---|
| Oversampling Ratio (OSR) | (f_s_MHz * 1000) / (2 * BW_kHz) | 128 (64 @ 3.072 MHz Clock) |
| Plain White-Noise Extra Bits | 0.5 * (LN(OSR) / LN(2)) | 3.50 bits (3.00 bits @ OSR=64) |
| 1st-Order ΔΣ SNR Gain (dB) | 10 * LOG10((3 * OSR^3) / PI()^2) | 58.04 dB (49.01 dB @ OSR=64) |
| 1st-Order ΔΣ Total ENOB (bits) | N_bits + ΔSNR_ΔΣ1_dB / 6.02 | 13.64 bits (12.14 bits @ OSR=64) |
First-Order Delta-Sigma Total Effective Bits Formula
Validate f_s > 0, BW > 0, OSR >= 4 (noise-shaping asymptotic regime), and integer/base quantizer bits N in [1, 16]!
=LET(fs_m, B2, bw_k, B3, nb_q, B4, os_r, (fs_m * 1000) / (2 * bw_k), IF(OR(fs_m <= 0, bw_k <= 0, os_r < 4, nb_q < 1, nb_q > 16), "Invalid Delta-Sigma OSR Input", LET(gn_d, 10 * LOG10((3 * (os_r ^ 3)) / (PI() ^ 2)), ROUND(nb_q + gn_d / 6.02, 2))))Half-Clock Power-Save Octave Check (Set Clock f_s = 3.072 MHz -> OSR = 64)
Halve Modulator_Clock B2 to 3.072 MHz (cutting OSR by 1 octave from 128 to 64 in E2): ΔΣ SNR Gain E4 drops by 9.03 dB to 49.01 dB and Total ENOB E5 steps down by exactly 1.50 bits from 13.64 to 12.14 bits!
=IF(ISTEXT(E5), E5, ROUND(0.5 * (LN(E2) / LN(2)), 2))Try the example
Use the original values or change them to test your understanding.