Dynamic Range Scaling for Delta Sigma Modulation

Question

I've been trying to follow along with this example in the book "Understanding Delta Sigma Data Converters" and I am completely confused as to how the authors came about the appropriate values for the factors alpha, beta, and gamma.

It seems like the value of beta was found from dividing x2_max by x1_max and gamma was found from dividing x3_max by x2_max. I am not sure about how alpha was calculated but it appears it was found from dividing 12 by x1_max. I am completely unsure as to how the magnitude was strictly limited to 10, 12, and 14 for x1, x2, and x3. I would think that what should happen is alpha would be found by 10/2.605, beta would be found by 12/2.905, and gamma would be found by 14/43.32. Could anyone explain the logic behind the choice of alpha, beta, and gamma and how it achieves the desired maximum values?

Answer 1

I believe it is a typo in the text as the scaling as given results in all nodes being close to 12, not 10, 12 and 14 as described.

The DSM was simulated which resulted in maximum values at the nodes given by $\hat{x_1}$, $\hat{x_2}$, and $\hat{x_3}$ as 2.605, 2.905, and 43.32. As we scale each stage, that scaling would effect the subsequent stages accordingly.

If the first state $\hat{x_1}$ is then scaled by 4.6, this would result in it's maximum value to be close to 12:

$$2.605 \times 4.6 = 11.983$$

That said, note that $\hat{x_2}$ will now also be 4.6 times larger or

$$2.905 \times 4.6 = 13.363$$

Therefore multiplying $\hat{x_2}$ by 0.89 results in a value close to 12:

$$13.363 \times .89 = 11.893$$

Finally, $\hat{x_3}$ will now be $4.6 \times 0.89 = 4.094$ larger or:

$$43.32 \times 4.094 = 177.352$$

Therefore multiplying $\hat{x_3}$ by $\gamma = 0.067$ results in a value close to 12:

$$177.352 \times 0.067 = 11.8825$$