Finite-precision error analysis

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Exact analysis of quantization errors is difficult because quantization is highly nonlinear. Approximating quantization errors as independent, additive white Gaussian noise processes makes analysis tractable and generally leads to fairly accurate results. Dithering can be used to make these approximations more accurate.

Fundamental assumptions in finite-precision error analysis

Quantization is a highly nonlinear process and is very difficult to analyze precisely. Approximations and assumptions are madeto make analysis tractable.

Assumption #1

The roundoff or truncation errors at any point in a system at each time are random , stationary , and statistically independent (white and independent of all other quantizers in a system).

That is, the error autocorrelation function is ${r}_{e}(k)=({e}_{n}{e}_{n+k})={}_{q}^{2}(k)$ . Intuitively, and confirmed experimentally in some (but notall!) cases, one expects the quantization error to have a uniform distribution over the interval $\left[-\left(\frac{}{2}\right) , \frac{}{2}\right)$ for rounding, or $\left(- , 0\right]$ for truncation.

In this case, rounding has zero mean and variance $(Q({x}_{n})-{x}_{n})=0$ ${}_{Q}^{2}=({e}_{n}^{2})=\frac{{}_{B}^{2}}{12}$ and truncation has the statistics $(Q({x}_{n})-{x}_{n})=-\left(\frac{}{2}\right)$ ${}_{Q}^{2}=\frac{{}_{B}^{2}}{12}$

Please note that the independence assumption may be very bad (for example, when quantizing a sinusoid with an integerperiod $N$ ). There is another quantizing scheme called dithering , in which the values are randomly assigned to nearby quantizationlevels. This can be (and often is) implemented by adding a small (one- or two-bit) random input to the signal before atruncation or rounding quantizer.

This is used extensively in practice. Altough the overallerror is somewhat higher, it is spread evenly over all frequencies, rather than being concentrated in spectrallines. This is very important when quantizing sinusoidal or other periodic signals, for example.

Assumption #2

Pretend that the quantization error is really additive Gaussian noise with the same mean and variance as the uniform quantizer. That is, model

This model is a linear system, which our standard theory can handle easily. We model the noise asGaussian because it remains Gaussian after passing through filters, so analysis in a system context is tractable.

• Correlation function

$({r}_{x}(k), ({x}_{n}{x}_{n+k}))$
• Power spectral density

$({S}_{x}(w), \mathrm{DTFT}({r}_{x}(n)))$
• Note ${r}_{x}(0)={}_{x}^{2}=\frac{1}{2\pi }\int_{-\pi }^{\pi } {S}_{x}(w)\,d w$
• $({r}_{\mathrm{xy}}(k), ({x}^{*}(n)y(n+k)))$
• Cross-spectral density

${S}_{\mathrm{xy}}(w)=\mathrm{DTFT}({r}_{\mathrm{xy}}(n))$
• For $y=(h, x)$ : ${S}_{\mathrm{yx}}(w)=H(w){S}_{x}(w)$ ${S}_{\mathrm{yy}}(w)=\left|H(w)\right|^{2}{S}_{x}(w)$
• Note that the output noise level after filtering a noise sequence is ${}_{y}^{2}={r}_{\mathrm{yy}}(0)=\frac{1}{\pi }\int_{-\pi }^{\pi } \left|H(w)\right|^{2}{S}_{x}(w)\,d w$ so postfiltering quantization noise alters the noise power spectrum and may change its variance!
• For ${x}_{1}$ , ${x}_{2}$ statistically independent ${r}_{{x}_{1}+{x}_{2}}(k)={r}_{{x}_{1}}(k)+{r}_{{x}_{2}}(k)$ ${S}_{{x}_{1}+{x}_{2}}(w)={S}_{{x}_{1}}(w)+{S}_{{x}_{2}}(w)$
• For independent random variables ${}_{{x}_{1}+{x}_{2}}^{2}={}_{{x}_{1}}^{2}+{}_{{x}_{2}}^{2}$

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