Analog Filter Approximation Functions - Practical Analog and Digital Filter Design - page 40

would be at 0.5 rad/sec.) Therefore the entire frequency response is a factor of 2
too low. The attenuation at ω = 0.5 rad/sec is ~ 1 dB and the attenuation at ω = 1
rad/sec is ~22 dB. The process we use to correct the problem is actually an
unnormalization procedure that is covered in Chapter 3. This unnormalization will
usually occur as part of the total filter design process, but we can make the
adjustment manually in this particular case. The correct transfer function can be
determined by substituting S /2 for S and then simplifying as indicated in the
second transfer function above. This process is mentioned here so we understand
the WFilter coefficients, which are shown in Figure 2.16.

Inv. Chebyshev 3rd-Order Normal. Lowpass

Selectivity: Lowpass
Approximation: Inv. Chebyshev
Implementation: Analog
Passband gain (dB): -1.0
Stopband gain (dB): -22.0
Passband freq (Hz): 0.1591549431
Stopband freq (Hz): 0.3183098862

Filter Length/Order: 03
Overall Filter Gain: 3.09341803036E-01

Numerator Coefficients
QD [S^2 + S + 1 ]
== =========================================
01 0.0 0.00000000000E+00 1.54556432589E+00
02 1.0 0.00000000000E+00 5.33333333334E+00

Denominator Coefficients
QD [S^2 + S + 1 ]
== =========================================
01 0.0 1.00000000000E+00 1.54556432589E+00
02 1.0 1.06745667061E+00 1.64982294953E+00

Figure 2.16 Inverse Chebyshev normalized third-order coefficients from WFilter.

Example 2.7 Inverse Chebyshev Fourth-Order Normalized Transfer
Function

Problem: Determine the order, pole and zero locations, and transfer function
coefficients for an inverse Chebyshev filter to satisfy the following specifications:

a pass = −1 dB, a stop = −33 dB, ω pass = 1 rad/sec, and ω stop = 2 rad/sec
Solution: First, we determine the fundamental constants needed from (2.35),
(2.36), and (2.39):

ε i = 0.022393 n = 3.92 (4th order)

D i = 1.123072 cosh( D i ) = 1.699781 sinh( D i ) = 1.374502