Quantization Error: Derivation and Numerical Example

Quantization Error: The Inevitable Distortion

The difference between input and quantized output is the quantization error $e[n]$:

\[e[n] = x_q[n] - x[n]\]

For a uniform mid-tread quantizer with step size $\Delta$, the error is bounded:

\[-\frac{\Delta}{2} \le e[n] \le \frac{\Delta}{2}\]

Derivation of Quantization Error Formula

Assume the error is uniformly distributed over $[-\Delta/2, \Delta/2]$:

• Mean error: $\mu_e = 0$
• Variance (power): $\sigma_e^2 = \frac{\Delta^2}{12}$
• RMS error: $\sigma_e = \frac{\Delta}{\sqrt{12}}$

This is the quantization noise power.

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Worked Numerical Example

Problem: A signal ranges from -5V to +5V. An 8-bit quantizer is used. Find:

1. Step size $\Delta$
2. Quantized output for input $x = 2.3$V

Solution:

• Full-scale range: $10$ V
• Levels: $L = 2^8 = 256$
• Step size: $\Delta = \frac{10}{256} = 0.0390625$ V $\approx 39.1$ mV
• Quantized output: $x_q = \text{round}\left(\frac{2.3}{\Delta}\right) \cdot \Delta = \text{round}(58.88) \cdot 0.0390625 = 59 \cdot 0.0390625 = 2.3047$ V

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