0.2 Filter sizing

The formula for estimation of the fir filter length

For many lowpass filter designs the peak passband excursion ${\delta }_1$ exceeds the peak stopband excursion ${\delta }_2$ by a factor of ten or more. This ratio, earlier denoted as the weight $W$ , was just evaluted in the previous section to have thevalue 28.8 for a typical set of specifications. In this example the stopband attenuation specification drives the requiredfilter order. In this case, and with a few additional assumptions which will be enumerated later, thenumber of coefficients in the impulse response of a high-order FIR linear phase filter, denoted $N$ , can be accurately estimated using the formula:

where the design parameter $\alpha$ is given by the equation:

As before, SBR is the minimum stopband attenuation compared to the nominal passband power transmission level, measured in decibels.

Note that the required filter order $N$ as estimated by [link] and [link] does not depend on the passband ripple PBR or on the exact values of the cutoff and stopbandfrequencies. Thus, when the conditions allowing the underlying assumptions to be met are true, estimating the required filter order $N$ becomes very easy.

[link] provides the values of the design parameter $\alpha$ from [link] for various degrees of stopband suppression. Given also is the range of the passband ripple for which the values of $\alpha$ apply. The column marked maximum passband ripple reflects the the assumption that the passband deviation ${\delta }_1$ is small compared to unity; specifically, the stated value of 1.74 dB correspondsto ${\delta }_1=0.1$ . The rightmost column, denoted minimum passband ripple , is the limit imposed by the assumption that ${\delta }_1>10·{\delta }_2$ . Of course FIR linear phase equal ripple filters can be designed with passband ripple extending beyond the stated range.However, as the PBR specification approaches either of these endpoints the validity of [link] will degrade. The predicted filter length will err on the low side for small PBR values and be overly pessimistic forPBR $>$ 1.74 dB. In such cases, an iteration on design might be necessary to obtain the desired filter characteristics.