Designing an all$-$pass filter : An all$-$pass filter has constant magnitude response at all fre$-$

quencies, but the phase response will vary with frequency. Show, by directly calculating

the magnitude response of the numerator and denominator separately, that the magnitude

response of the system with transfer function

H$($s$)=$ s $+$ a$$

s $$ a

is $1.$ Here $()$ denote the complex conjugate of $().$ This implies that if we design a system

that has a zero at s $=$a$$ whenever there is a pole at s $=$ a$,$ we are guaranteed to get constant

magnitude response! Design a second$-$order all$-$pass filter with a pole at s $=2000$ $+$j$2000$$.$

$2$