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A question about Second-order Sigma-Delta modulator

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she_long

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I'm reading Fernando Medeiro's 'Top Down Design of High performance Sigma-Delta Modulators', in chapter 2.3.1 'Second-order Sigma-Delta modulator'. It confused me about the 2nd SD modulator's transfer function. The pic is the same as the book.



This is my result use the gain g2'=2g1g2, and g1'=g1:
z^-2 *X(z)+(1-z^-1)^2*E(z)/(g1g1)=z^-2*Y(z)+2*z^-1*(1-z^-1)Y(z)+(1-z^-1)^2*Y(z)/(g1g2);
It's different from the function of the book. what's matter with my function?
 

Dear friend

g1*g2*(z^-2)*X(z) + E(z)*(1-z^-1)^2 = Y(z)*[(1-g1*g2)*(z^-2) - 2*(1-g1*g2)*(z^-1) +1]

So, I think it is supposed that g1*g2 = 1 -> Y(z)=(z^-2)*X(z) + E(z)*(1-z^-1)^2

Best
 

Yes, if g1*g2 = 1 there comes the function.
I'm wondering how to define the Gains of the modulator. Someone use the way that make (suppose the equation is B.Y(z)=...) B=1 to get the Gains (g1 g1 ...), it's simple for 2nd-order SD modulator, but more complicated for high order modulator. And the Stability is concerned with the Gains. Chapter 2.3.2.1 'Stability considerations' --the stability of 2nd-order modulator of Fig 2.7 is guaranteed if g2'/(g1g2)>1.25. Can anybody explain me the relation of Stability and Gains clearly for 2nd-order and high order ?
Thx.
 

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