Question

If an LTI system is described by the difference equation y(n)=ay(n-1)+bx(n), 0 < a < 1, then what is the parameter ‘b’ so that the maximum value of |H(ω)| is unity?

(a) a

(b) 1-a

(c) 1+a

(d) none of the mentioned

The question was asked in an online interview.

Question is taken from Frequency Domain Characteristics of LTI System topic in division Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Correct answer is (b) 1-a

To elaborate: We know that,

|H(ω)|=\(\frac{|b|}{\sqrt{1-2acosω+a^2}}\)

Since the parameter ‘a’ is positive, the denominator of |H(ω)| becomes minimum at ω=0. So, |H(ω)| attains its maximum value at ω=0. At this frequency we have,

\(\frac{|b|}{1-a}\) = 1 => b=±(1-a).