ADC Errors and Signal-to-Quantization-Noise Ratio

Quantization vs. Sampling: “Opposite” Operations

Aspect	Sampling	Quantization
Axis discretized	Time ($x$-axis)	Amplitude ($y$-axis)
Continuous →	Discrete-time	Discrete-valued
Information loss	Aliasing (if undersampled)	Quantization noise
Control parameter	Sampling frequency $F_s$	Number of bits $B$
Reversibility	Perfect if Nyquist met	Never perfectly reversible

Key insight: Sampling can be lossless (Nyquist); quantization is always lossy. The lost information becomes quantization noise.

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ADC Stage Errors

An Analog-to-Digital Converter introduces errors at multiple stages:

Analog Input → Sample & Hold → Quantizer → Encoder → Digital Output
                 │               │          │
            Aperture      Quantization  Encoding
            Jitter        Noise        Errors

Error Source	Description	Mitigation
Quantization error	Fundamental, unavoidable	Increase bits $B$
Aperture jitter	Uncertainty in sampling instant	Better clock, lower jitter
Integral non-linearity (INL)	Deviation from ideal transfer curve	Calibration, better ADC
Differential non-linearity (DNL)	Uneven step sizes	Laser trimming, calibration
Offset/Gain error	Systematic shift/scale error	Calibration

The quantization stage is the only one that fundamentally cannot be eliminated—only reduced by using more bits.

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Signal-to-Quantization-Noise Ratio (SQNR)

For a full-scale sinusoidal input with $B$ bits:

\[\text{SQNR} \approx 6.02B + 1.76 \text{ dB}\]

Bits	SQNR (dB)	Typical Application
8	49.9	Basic audio, sensors
12	74.0	Instrumentation, audio
16	98.1	CD audio, precision DAQ
24	146.2	Professional audio, scientific

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Non-Uniform Quantization (Advanced)

For signals with non-uniform amplitude distributions (e.g., speech—mostly small amplitudes, occasional large peaks), non-uniform quantization allocates more levels where the signal spends more time.

• $\mu$-law (North America/Japan): $\mu = 255$
• A-law (Europe): $A = 87.6$

These provide ~24 dB improvement for speech at 8 bits vs. uniform quantization.

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