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6 Quantization and quantization error

A possible solution to solve the bandwidth problem is to reduce the amount of information of the sampled signal i.e. its entropy. Independently from the way in which this is performed, the final compression strategy will be lossy, and the final reconstructed (uncompressed) signal will be corrupted with respect to the original one, degrading in some regard the experimental performances. In this regard, any sort of lossy compression may be seen as a kind of signal rebinning with a coarser resolution (quantization step) in $\Delta T/T$.

There are at least six aspects in PLANCK-LFI operations which may be affected by a coarser quantization:

 Cl and periodical signals reconstruction;
foreground separation;
point like sources detection;
variable sources characterization;
tests for normality of CMB fluctuations.

Since the non linear nature of the quantization process, all of them are hard to be analytically evaluated and for this reason a specific simulation task is in progress for the PLANCK-LFI collaboration (White & Seyfert 1999; Maris et al. 2000). However a heuristic evaluation for the point (1) by analytical means is feasible.

Quantization operates a convolution of the normal distribution of the input signal with the quantization operator $(x:\Delta) =~$sign $(x) \Delta*{\rm floor}(\vert x/\Delta\vert)$. If the quantization error: $(x - (x:\Delta))$ is uniformly distributed its expectation is $\Delta/2$ and its rms is $\Delta/\sqrt{12}$ (Kollár 1994). Quantization over a large amount of samples may be regarded as an extra source of noise which will enhance the variance per sample. If the quantization error is statistically independent from the input quantized signal and if it may be added in quadrature to the white noise variance $\mbox{$\sigma_{{\rm WN}}$ }$, the total variance per sample will be $\approx \mbox{$\sigma_{{\rm WN}}$ }^2 \left( 1 +
\frac{\Delta^2/\mbox{$\sigma_{{\rm WN}}$ }^2}{12}\right)$. So for $\Delta \lesssim
\mbox{$\sigma_{{\rm WN}}$ }$ the expected quantization rms is $\lesssim 4\%$. From error propagation the relative error on the Cl is (Maino 1999):

\frac{\delta C_l}{C_l} =
\sqrt{\frac{4 \pi}{A}}
\left[ 1 + \frac{\sigma^2 \theta^2}{B_l^2 C_l} \right]
\end{displaymath} (11)

so that the quantization contribution to the overall error will be small and dominated by the cosmic variance for a large set of l. However the application of such encouraging result must be considered carefully in a true experimental framework. Apart from the assumptions, it has to be demonstrated indeed that a large quantization error like this will not harm significantly the aforementioned aspects, moreover the impact of signal quantization will depend on how and in which point of the detection chain it will be performed.

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