Quantisation

Figure 3: $\Delta$ is the quantisation step.

A/D converters have a certain resolution. For example, the MAX1271 has a resolution of 12 bits which means that it divides the input range into 4096 equal steps (see Fig 3).

  $$\Delta = \mbox{quantisation step} = \frac{x_{\mbox{max}} - x_{\mbox{min}}}{L - 1}$$ (17)

where $x_{\mbox{max}} - x_{\mbox{min}}$ is the dynamic range in volt (for example $4.096V$) and $L$ is the number of quantisation steps (for example, 4096). $\Delta$ is the quantisation step which defines minimal voltage change which is needed to see a change in the output of the quantiser. The operation of the quantiser can be written down as:

  $$x_{q}(n) = Q[x(n)]$$ (18)

Figure 4: Illustration of the quantisation error. It is zero at $t=0$ and increases to the edges of the sampling interval. Illustrated is the worst case scenario. This repeats in the next sampling interval and so forth.

Fig. 4 shows error produced by the quantiser in the worst case scenario. From that it can be seen that the maximum quantisation error is half the quantisation step:

  $$- \frac{\Delta}{2} \le e(n) \le \frac{\Delta}{2}$$ (19)

The smaller the quantisation step $\Delta$ the lower the error!

What is the mean square error $P_{q}$?

  $$P_{q} = \frac {1}{\tau} \int_{0}^{\tau} e_{q}^{2}(t) dt$$ (20)

$$P_{q} = \frac{1}{\tau} \int_0^\tau \left(\frac{\Delta}{2\tau}\right)^2 t^2 dt$$ (21)

$$= \frac{\Delta^2}{4 \tau^3} \int_0^\tau t^2 dt$$ (22)

$$P_{q} = \frac{\Delta^2}{12\tau^3} \tau^3 =\frac{\Delta^2}{12}$$ (23)

this then results in the almost trival result that the size of the quantisation step $\Delta$ scales linarly with the average error $\sqrt{P_{q}}$. So if we have two times more quantisation steps then the error will half!

What is the relative error to a sine wave?

  $$P_x = \frac{1}{T_p} \int_{0}^{T_p}(A \cos \Omega t)^2 dt = \frac{A^2}{2}$$ (24)

Ratio to signal power to noise:

$$\mbox {SQNR} = \frac {P_x}{P_q} = \frac{A^2}{2} . \frac{12}{\Delta^2}$$ (25)

$$= \frac{6A^2}{\Delta^2}$$ (26)

This equation needs to be interpreted with care because increasing the amplitude of the input signal might lead to saturation if the input range of the A/D converter is exceeded.

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