Question

Which of the following expressions define the dead band for a single-pole filter?

(a) |v(n-1)| ≥ \(\frac{(1/2).2^{-b}}{1+|a|}\)

(b) |v(n-1)| ≥ \(\frac{(1/2).2^{-b}}{1-|a|}\)

(c) |v(n-1)| ≤ \(\frac{(1/2).2^{-b}}{1-|a|}\)

(d) None of the mentioned

I got this question by my school principal while I was bunking the class.

The doubt is from Round Off Effects in Digital Filters in chapter Digital Filters Design of Digital Signal Processing

Answer 1

Correct answer is (c) |v(n-1)| ≤ \(\frac{(1/2).2^{-b}}{1-|a|}\)

The best explanation: Since the quantization product av(n-1) is obtained by rounding, it follows that the quantization error is bounded as

|Qr[av(n-1)]-av(n-1)| ≤ \((\frac{1}{2}).2^{-b}\)

Where ‘b’ is the number of bits (exclusive of sign) used in the representation of the pole ‘a’ and v(n). Consequently, we get

|v(n-1)|-|av(n-1)| ≤ \((\frac{1}{2}).2^{-b}\)

and hence

|v(n-1)| ≤ \(\frac{(\frac{1}{2}).2^{-b}}{1-|a|}\) .