Advanced: A Linear Model for Quantization

1.11 Advanced: A Linear Model for Quantization

As discussed, the quantization process is non-linear and the quantization error values depend on the quantizer input. But the following linear model for the quantization error has proved to be a good approximation in several applications.

In this model, the error $E_{q} = X - X_{q}$ is assumed to be a uniformly distributed random variable with support $[- Δ ∕ 2, Δ ∕ 2]$ and zero-mean. This is sometimes called quantization noise. Two assumptions are important:

f 1.: The signal has enough variation to span all levels of the quantizer stair. The model fails, for example, if the input signal is a constant (DC) value.
f 2.: There is no correlation between the error $E_{q}$ and the signal to be quantized $X$ .

The main assumption is that, for a quantizer with a step of $Δ$ , the quantization noise $E_{q}$ is modeled by assuming it is a random variable $E_{q}$ with uniform PDF and support $[- Δ ∕ 2, Δ ∕ 2] .$ Because it is zero-mean, the power of $𝔼 [E_{q}^{2}]$ is solely the variance of the uniform PDF given by Eq. (1.20), which in this case is

σ^{2} = Δ^{2} ∕ 12,

(1.61)

as indicated by Eq. (1.20).

Assuming that

Δ = \frac{{\hat{X}}_{max} - {\hat{X}}_{min}}{2^{b}},

the power $P_{n}$ of the quantization noise is

P_{n} = σ^{2} = \frac{Δ^{2}}{12} = \frac{{({\hat{X}}_{max} - {\hat{X}}_{min})}^{2}}{2^{2 b} 12},

(1.62)

It is a good idea to practice using the model by calculating the quantization SNR for sinusoids and cosines, uniformly and normally distributed random signals. The quantization SNR is obtained by using the quantization noise power in the denominator of the SNR expression (the numerator is the signal power, as usual). The following example illustrates the result for quantizing a Gaussian input signal.

Example 1.54. Quantization SNR of a Gaussian signal and the 6 dB per bit rule of thumb. If the signal to be quantized $x (t)$ has samples distributed according to a Gaussian $N (3, 4)$ with mean $μ = 3$ V and variance $σ^{2} = 4$ W, a reasonable alternative (see Eq. (1.43)) is to adopt ${\hat{X}}_{min} = μ - 3 σ$ and ${\hat{X}}_{max} = μ + 3 σ$ . In this case and using Eq. (1.43),

Δ = \frac{{\hat{X}}_{max} - {\hat{X}}_{min}}{M - 1} = \frac{3 + 6 - (3 - 6)}{2^{b} - 1} = \frac{12}{2^{b} - 1},

where $b$ is the number of bits of the quantizer. Using the linear model of quantization:

SNR = \frac{P_{s}}{P_{n}} = \frac{3^{2} + 4}{Δ^{2} ∕ 12} = \frac{13 \times 12}{1 2^{2} ∕ {(2^{b} - 1)}^{2}} = \frac{13 {(2^{b} - 1)}^{2}}{12} \approx 1.083 {(2^{b} - 1)}^{2},

where $P_{s}$ and $P_{n}$ are the signal and noise power, respectively. The ${SNR}_{dB}$ is

\begin{array}{l} 10 {log}_{10} SNR & = 10 [{log}_{10} 1.0833 + 2 {log}_{10} (2^{b} - 1)] \\ \approx 0.3463 + 20 b {log}_{10} 2 \\ \approx 0.3463 + 6.021 b, \end{array}

where the last step assumed that $2^{b} » 1$ .

The result ${SNR}_{dB} \approx 6 b + cte.$ is a well-known “rule of thumb”. The constant (cte.) may vary, but the ${SNR}_{dB}$ typically increases by 6 dB for each extra bit in the quantizer. It is common to use this approximation to suggest, for example, that an ADC of 12 bits has approximately quantization ${SNR}_{dB} = 6 \times 12 = 72$ dB, while an ADC of 16 bits has ${SNR}_{dB} = 6 \times 16 = 96$ dB. $□$