1.1 For a transfer function with one pole and one zero, where (approximately) in the complex zplane wolud you place the pole and zero z_{1} to obtain a high pass filter characteristic with zero response at f=0 and maximum response at f=f_{s}/2 ?
1.2 Draw a sketch to show the pole and zero locations.
1.3 Draw a sketch of the approximate frequency (magnitude) response.
1.4 Derive the diference equation to implement this high pass filter in terms of and z_{1}.
1.5 Draw a diagram of an impulse response of the filter.
Solution.
1.1
For a filter with one pole and one zero,
H[z]  =  (1) 
H(f)  =  (2)  
=  (3)  
=  (4)  
=  (5) 
where
=  (6)  
=  (7) 
Assume b_{0}=1. From the table on the notes page 49, we can see that the pole should be located on the negative real axis, whereas the zero z_{1} on the positive real axis. If we leave the pole on the negative real axis within the unit circle and move the zero z_{1} to 1, H(f) becomes zero at f=0. For these z_{1} and , H(f) becomes maximum at f=f_{s}/2.
1.2
The pole and zero locations on the zplane are shown in Fig.1(a).
1.3
Magnitude response of the H(f) is shown in Fig.1(b).
1.4
From H(z) in Eq.(1),
By applying inverse ztransform,
1.5 The block diagram of the filter is shown in Fig.2.
