3.3 Chebyshev filter properties  (Page 3/5)

It is possible to develop a theory for Chebyshev passband approximation and arbitrary zero location similar to the Taylor'sseries result in Equation 5 from Butterworth Filter Properties .

Chebyshev filter design procedures

The Chebyshev filter has a passband optimized to minimize the maximum error over the complete passband frequency range, and astopband controlled by the frequency response being maximally flat at $\omega =\infty$ . The passband ripple and the filter order are the two parameters to be determined by the specifications.

The form for the specifications that is most consistent with the Chebyshev filter formulation is a maximum allowed error in thepassband and a desired degree of “flatness" at $\omega =\infty$ . The slope of the response near the transition from pass to stopband at $\omega =1$ becomes steeper as both the order increases and the allowed passband error ripple increases. The dropoff is more rapid than for theButterworth filter [link] .

As stated earlier, the design parameters must be clearly understood to obtain a desired result. The passband ripple is defined tobe the difference between the maximum and the minimum of $\left|F\right|$ over the passband frequencies of $0<\omega <1$ . There can be confusion over this point as two definitions appear in the literature. Most digital [link] , [link] , [link] and analog [link] filter design books use the definition just stated. Approximation literature, especially concerningFIR filters, use half this value which is a measure of the maximum error, $\left|\right|F|-|F_d\left|\right|$ , where $|F_d|$ is the center line in the passband around which $\left|F\right|$ oscillates. The following formulas relate the passband ripple $\delta$ , the passband ripple $a$ in positive dB, and the transfer function parameter $ϵ$ .

In some cases, stopband performance is not given in terms of degree of flatness at $\omega =\infty$ , but in terms of a maximum allowed magnitude $G$ in the stopband above a certain frequency ${\omega }_s$ , i.e., $G>\left|F\right|>0$ for $1<{\omega }_s<\omega <\infty$ . For a given $ϵ$ , this will determine the order as the smallest positive integer satisfying

The design of a Chebyshev filter involves the following steps:

• The maximum-allowed passband variation must be inthe form of $\delta$ or $a$ . From this, the parameter $ϵ$ is calculated using [link] .
• The order $N$ is determined by the desired flatness at $\omega =\infty$ or a maximum-allowed response for frequencies above ${\omega }_s$ using [link] .
• ${\nu }_0$ is calculated from $ϵ$ and $n$ using [link] , and the scale factors $sinh\left({\nu }_0\right)$ and $cosh\left({\nu }_0\right)$ are then determined.
• The pole locations are calculated from [link] or [link] . This can be done by scaling the poles of a Butterworth prototype filter.
• These pole locations are combined in [link] and [link] to give the final filter transfer function.

This process is easily programmed for computer aided design as illustrated in Program 8 in the appendix.

If the design procedure uses [link] to determine the order and the right-hand side of the equation is not exactly an integer,it is possible to improve on the specifications. Direct use of the order with $ϵ$ from [link] gives a stopband gain at ${\omega }_s$ that is less than $G$ , or the same design can be viewed as giving the maximum-allowed gain $G$ at a lower frequency than ${\omega }_s$ . An alternate approach is to solve [link] for a new value of $ϵ$ , then cause [link] to be an equation with the specified ${\omega }_s$ and $G$ . This gives a filter that exactly meets the stopband specifications and gives a smaller passbandripple than originally requested. A similar set of alternatives exists for the elliptic-function filter.