Questions: Quantization Error and Noise Analysis

Question 1

A professional audio studio upgrades its ADC from 16-bit to 24-bit resolution. By approximately how much does this improve the signal-to-quantization-noise ratio?

• A. 6 dB — one additional bit of improvement
• B. 24 dB — the number of bits times 1 dB
• C. 48 dB — eight additional bits times approximately 6 dB per bit
• D. 98 dB — the absolute SQNR of a 16-bit system

Answer: (C)

The SQNR formula is approximately 6.02N + 1.76 dB. For 16 bits: ~98 dB. For 24 bits: ~146 dB. The difference is (24 − 16) × 6 dB = 48 dB. The 6 dB per bit rule comes directly from the step size: each additional bit halves Δ, reducing noise power Δ²/12 by a factor of 4, which is −6 dB. This is the rule audio engineers reach for first when specifying ADC resolution requirements. CD audio (16-bit) gives ~98 dB dynamic range; professional recording (24-bit) gives ~146 dB, allowing headroom for mixing without audible noise floor.

Question 2

An engineer quantizes a 100 Hz sinusoid to 8-bit resolution and examines the frequency spectrum of the quantization error. What is most likely observed?

• A. A flat, spectrally white noise floor uniformly distributed across all frequencies from 0 to Nyquist
• B. Tonal components at harmonics of 100 Hz, because the periodic signal creates correlated (non-white) quantization error
• C. A single spike at 100 Hz because the quantization error is in phase with the signal
• D. Energy concentrated only near the Nyquist frequency where quantization operates

Answer: (B)

The white-noise model for quantization error assumes the signal sweeps through quantization levels in an uncorrelated, pseudo-random way. For a pure sinusoid, the error is periodic — the same amplitude values repeat every cycle, producing the same quantization errors in the same pattern. This correlation manifests as tonal artifacts (harmonics of the signal frequency) in the error spectrum rather than flat white noise. The white-noise model holds well for broadband signals that use the full quantizer range; it breaks down for signals that are periodic relative to the step size, nearly constant, or severely underutilizing the quantizer range.

Question 3

Each additional bit of ADC resolution approximately halves the quantization step size, which reduces quantization noise power by a factor of four and improves SQNR by about 6 dB.

• T. True
• F. False

Answer: True

True. With N bits and full-scale range FS, step size Δ = FS/2^N. Adding one bit doubles 2^N, halving Δ. Noise power σ²_q = Δ²/12 decreases by a factor of 4 (since (Δ/2)²/12 = Δ²/48 = σ²_q/4). A factor-of-4 reduction in noise power is −6 dB in power terms (10 log₁₀(4) ≈ 6 dB). This is the '6 dB per bit' rule, one of the most important rules of thumb in digital signal processing. It holds for sinusoidal signals using the full quantizer range; signals that underutilize the quantizer give less than 6 dB per bit improvement.

Question 4

Dithering improves the signal-to-quantization-noise ratio by randomizing quantization error, eliminating tonal artifacts while also reducing total noise power.

• T. True
• F. False

Answer: False

False. Dithering adds a small random noise signal before quantization, which does convert structured tonal artifacts into spectrally flat (white) noise. But this comes at the cost of slightly increased total noise power — you have added noise deliberately. The ideal SQNR decreases; dithering is never a free lunch in terms of total noise. However, the perceptual trade is almost always worthwhile in audio: a flat noise floor is far less objectionable than harmonic distortion. Dithering trades a small, measurable SQNR penalty for a much larger perceptual quality improvement by converting tones (which our auditory system is highly sensitive to) into white noise (which is much less perceptually salient at the same power level).

Question 5

Under what conditions is treating quantization error as white noise a good approximation, and what happens to this approximation when the input signal is a pure low-frequency sinusoid at low amplitude?

Model answer: The white-noise approximation is valid when the input signal is broadband, varies significantly from sample to sample, and uses most of the quantizer's full range. Under these conditions, the quantization errors at successive samples are approximately uncorrelated — each sample 'lands' in a different part of the step interval randomly — producing a flat error spectrum. For a pure low-frequency sinusoid at low amplitude, both conditions break: the signal is narrowband (periodic at one frequency) and may use only a few quantization levels. The error becomes periodic with the same period as the signal, producing correlated samples and tonal spectral artifacts (harmonics of the signal frequency) rather than white noise. The practical remedy is dithering — adding small broadband noise before quantization to decorrelate the error at the cost of slightly increased total noise power.

This is why professional audio engineers always apply dither when reducing bit depth (e.g., going from 24-bit recording to 16-bit distribution). Without dither, quiet passages in music — where the signal uses only the bottom few bits — develop audible harmonic distortion. With dither, those same passages have a clean noise floor. The white-noise approximation is an engineering model with known limits, not a physical law.