Digitizer target threshold levels

In Gianni's ALMA Memo 583, footnote 2 indicates that the target level for 2-bit TDM mode should be 1.4 dB higher than FDM. The explanation for this is given below in this email of September 12, 2013.

Hi Todd,

I believe you are correct about the reason for the 1.70 value. In fact, note that on p. 12 of Memo 583, in the paragraph immediately following Eq. 23, Gianni mentions a "nominal level of \sigma=1.706 times the threshold spacing". Here he is referring to the same constant as in Footnote 2, I believe, but with four significant digits rather than three. To six significant digits, the optimal voltage threshold for 8-(equispaced) level sampling is 0.586019\sigma (ALMA memo 591). The reciprocal of 0.586019 is 1.70643. [The 1.706 value is also used on p. 13 of the memo.]

For 4-level quantization, the optimal voltage threshold is 0.995687\sigma. [In my previous recent e-mail I mistakenly wrote 0.9995687] Twice the reciprocal of that value is 2.00866, which rounds to 2.01. Note that where Gianni writes "2.01 times the quantization step in 2 bit TDM mode", the quantization step he is evidently referring to is the 3-bit quantizer step before discarding the lsb. So that's why you want twice the reciprocal value for the effective 2-bit quantization step.

This morning I had a look again at the data sheet describing the ALMA 3-bit, 4Gsps ADC. Table 3 on p. 7 of this document (from Alain Baudry, see attachment) shows the Gray code output of the ALMA sampler as a function of input voltage. In my previous email, I neglected to take cognizance of the fact that the encoding scheme for this output is a 3-bit cyclic Gray code, rather than two's complement format. But the result is equivalent: discarding the least-significant bit from the 3-bit output yields the appropriate 2-bit Gray code. Here's a modified table:

          3-bit                     2-bit
        Gray code   discard lsb   Gray code
+7  <-     100           ->          10 \
                                                       ->   +3
+5  <-     101           ->          10 /

+3  <-     111           ->          11 \
                                                       ->   +1
+1  <-     110           ->          11 /

-1  <-     010           ->          01 \
                                                      ->   -1
-3  <-     011           ->          01 /

-5  <-     001           ->          00 \
                                                      ->   -3
-7  <-     000           ->          00 /

Here I've corrected the earlier paragraph of mine:

"Regarding footnote 2 of Gianni's memo, I probably should re-think this tomorrow, but note that when you discard the lsb from the 3-bit 2's complement levels, you effectively get double the voltage threshold spacings. So the threshold spacings become 2*0.586019\sigma, and the ratio of that to 0.995687\sigma is 1.17712. That is a voltage ratio, and 20 Log_{10}(1.17712) = 1.41638 dB, whose rounded value is equal to Gianni's 1.4 dB."

- Fred

-- ToddHunter - 2013-09-12
Topic attachments
I Attachment Action Size Date Who Comment
Datasheet_Converter.pdfpdf Datasheet_Converter.pdf manage 641 K 2013-09-13 - 11:35 ToddHunter 3-bit digitizer datasheet
Topic revision: r4 - 2013-09-15, ToddHunter
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