Analysis and Synthesis of Delay Line Periodic Filters

A periodic filter has a frequency characteristic which is periodic. Such filters can be constructed using delay lines where the delay of each line is the reciprocal of the basic frequency period. The network function of the periodic filter is characterized by the presence of the factors of the form
e^{np\tau}
, where
n
is a positive or negative integer,
\tau
is the delay of each delay line, and
p
is the complex frequency variable. Analysis and synthesis are simplified by use of the
z
transform which has been used with much success in the study of sampled data systems. A transformation of the filter network function is made by substituting
z
for
e^{p\tau}
. This substitution transforms the imaginary axis of the
p
plane into the central unit circle in the
z
plane. The properties of the periodic filter are now characterized by the poles and zeros of the
z
-plane transform. A rational method is presented for synthesizing any
z
-plane transform expressed as a rational fraction. Finally, the
z
-transform concept is used to analyze the behavior of periodic filters with pulsed inputs.