Complex Arithmetic a let a be real Show that midjomega ajom

Complex Arithmetic (a) let a be real. Show that midjomega - a/jomega + amid = 1 for all omega in R (b) Plot the log-magnitude and phase H(jomega) = a/jomega + b

Solution

To prove :| (jw -a)/( jw +a) | = 1

| jw -a | = sqrt( w^2 +a^2)

|jw +a | = sqrt( w^2 +a^2)

Now :| (jw -a)/( jw +a) | = | jw -a | / |jw +a |

=  sqrt( w^2 +a^2)/   sqrt( w^2 +a^2)

= 1

Hence Proved