Cyclic ADC - Differential non linearity error calculation

[mvj]

Dear All,

I was referred to a paper on Cyclic ADC desing. Can somebody help me understand the deduction of input voltages during the major transitions. Please find the paper attached and I am referring to page 959 - I could not figure out how Vx(n-1) is eliminated while determining Vx.

Thanks a lot in advance!!

Best Regards,
M.

View attachment Cyclic_ADC - Copy.pdf

 

[dgnani]

If you are referring to the general formula for Vx(k):
He is simply using induction, you start with a sequence defined by the relation between successive elements and try to generalize to a description of the sequence based only on the index of the sequence

Usually you need to guess the general expression (the one based only on the index) by studying the behavior of the first few elements of the sequence, then you can prove (by induction) that the generalization is correct:
- it is correct for the first element?
- if it is correct for one element is it correct for the following one?

In this case you would start by writing
Vx(0)=Vx
Vx(1)=2(1+eps)Vx(0)-b0 Vref=2(1+eps)Vx-b0 Vref
Vx(2)=2(1+eps)Vx(1)-b1 Vref=2(1+eps)( 2(1+eps)Vx-b0 Vref ) -b1 Vref=
=Vx(2(1+eps))^2 - Vref (b1+2(1+eps)b0)
Vx(3)=...

already at the second element you can estimate the general dependence of the sequence on its index, then you can apply induction to confirm, which is trivial
_________________
If you are referring to the formulas for computing the input signal corresponding to the codes near the MSB transitions:
he is setting
Vx(n-1) ~0
using the first order approximation
(1+eps)^n ~ 1+n eps
and the definition for LSB' as
Vref/(2(1+eps))^(n-1)
since he considering an (n-1)-bit conversion