Consider a first-order system with transfer function

where K > 0 is a gain, z₁ is a zero, and p₁ is a pole.

(a) If we want unity dc gain, i.e., |H(j0)| = 1, what should be the value of K.

(b) What types of filters (lowpass, bandpass, band-eliminating, all-pass, and high-pass) can you obtain with this first-order system?

(c) Let H₁(s) = 1/(s + 1), corresponding to a low-pass filter, determine the dc gain of this filter and use it to find the half-power frequency of the filter.

(d) Determine the magnitude response of a filter with transfer function H₂(s) = (s ˆ’ 1)/(s + 1). Plot the poles and zeros of this filter. What type of filter is it?

(e) What type of filter has the transfer function H₃(s) = s/(s + 1) ?

s+z1 |H(s) = K s+P1