Analog Filter Approximation Functions - Practical Analog and Digital Filter Design - page 45

2.5.2 Elliptic Order

The order of the elliptic approximation function required to meet the
specifications for a filter is given in (2.63):

2

CEI ( rt ) ⋅ CEI ( 1 − kn )

n E =

(2.63)

2

CEI ( 1 − rt ) ⋅ CEI ( kn )

where CEI refers to the complete elliptic integral, and the ratio rt and the kernel kn
are defined as

rt = ω

pass /ω

(2.64)

stop

− 0 . a − 0 . a

(2.65)

kn = ( 10

pass

− 1 /( 10

stop

− 1

Example 2.8 Elliptic Order Calculation

Problem: Determine the order of an elliptic filter required to satisfy the
following specifications:

a pass = −1 dB, a stop = −34 dB, ω pass = 1 rad/sec, and ω stop = 2 rad/sec
Solution: In order to determine the order of the elliptic approximation, we
first determine that rt = 0.5 and kn = 0.0101548. Then, using any appropriate math
package, we can determine that

1 . 686 ⋅ 5 . 976

n = = 2 . 97

E

2 . 157 ⋅ 1 . 571

which indicates that a third-order filter will be required. Notice that the standard
and inverse Chebyshev approximations require a fourth-order function to provide
a stop = −33 dB and a Butterworth approximation would require a seventh-order
function to meet this specification.

2.5.3 Elliptic Pole-Zero Locations

The pole and zero locations for the elliptic approximation function are also
dependent on the elliptic integral and the elliptic functions defined in the previous
section. We'll start by defining a variable v o , which is used in the calculation of
the pole and zero locations.