OK, here's an expanded version of the simple illustrations I posted on
the bug report.
I have used 2 simple systems - each consists of an ADC the output of
which is lsb extended to 5-bits, a digital mutiplier, DAC and finally
an analogue gain to compensate for the digital multiplier. In all
cases, if the system is transparent, the output should be the same as
the input and there would therefore be zero error.
Key to headings
-----------------
A input
B input as % of fs
C expand to 5-bits
D integerised div
E dac op as % of fs
F analogue gain
G ERROR
System 1
-----------
3-bit ADC, digital multiplier of 4, 5-bit DAC, analogue gain x4
A B C D E F G
7 87.50% 28 7 21.88% 87.50% 0.00%
6 75.00% 24 6 18.75% 75.00% 0.00%
5 62.50% 20 5 15.63% 62.50% 0.00%
4 50.00% 16 4 12.50% 50.00% 0.00%
3 37.50% 12 3 9.38% 37.50% 0.00%
2 25.00% 8 2 6.25% 25.00% 0.00%
1 12.50% 4 1 3.13% 12.50% 0.00%
0 0.00% 0 0 0.00% 0.00% 0.00%
Peak Error 0.00%
Input S/N 18.06 System S/N 18.06
System 2
---------
3-bit ADC, digital multiplier of 5, 5-bit DAC, analogue gain x5
A B C D E F G
7 87.50% 28 5 15.63% 78.13% 9.38%
6 75.00% 24 4 12.50% 62.50% 12.50%
5 62.50% 20 4 12.50% 62.50% 0.00%
4 50.00% 16 3 9.38% 46.88% 3.13%
3 37.50% 12 2 6.25% 31.25% 6.25%
2 25.00% 8 1 3.13% 15.63% 9.38%
1 12.50% 4 0 0.00% 0.00% 12.50%
0 0.00% 0 0 0.00% 0.00% 0.00%
Peak Error 12.50%
Input S/N 18.06 System S/N 12.04
As you can see, the first system, because it uses a modulo 4 multiplier
and makes use of the 'extra' 2 lsbs in the system, produces no errors
and the ouptut S/N ratio is the same as the sampled S/N ratio (3bits =
18dB)
System 2 however, doesn't use a modulo 4 multiplier and therefore
produces rounding errors. The errors are, peak to peak, 1/8th of the
full scale - ie the same as the original sampling error. Output S/N
ratio is therefore halved to 12dB.
Obviously, you can extend the concept from 3-bits lsb extended to
5-bits, to 16/24-bits - you'll just have more numbers. The point is
that the rounding error is significant at the lsb of the sampled signal
NOT the lsb of the system.
I hope that helps - I don't know how better to explain it at the
moment. If you want to hear the difference and you can't, you'll just
have to get an a plane and come over here - I'll demonstrate it to you!
Looser buys the beers ;-)
--
Patrick Dixon
www.at-tunes.co.uk
------------------------------------------------------------------------
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View this thread: http://forums.slimdevices.com/showthread.php?t=17269
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