Note 37. Designing FIR Filters: Parks-McClellan Algorithm

The Parks-McClellan algorithm is by far the most widely used technique for designing FIR filters. For many years, this algorithm was called the Remez algorithm or the Remez exchange. However, the recent trend has been to name the approach in honor of Thomas Parks and John McClellan, the two individuals who first publicized the utility of using the Remez exchange for designing FIR filters.

An FIR approximation to some ideal desired response typically exhibits error ripples around the ideal response, as shown in Figure 37.1(a). The Parks-McClellan (PM) algorithm is based on the fact that, for a given filter length, the worst-case error is minimized when all of the error extrema are equal in magnitude, as shown in Figure 37.1(b). Hence, a filter resulting from the PM algorithm is often referred to as an equiripple filter.

Figure 37.1. Comparison of magnitude responses for two 36-tap FIR filter designs: (a) suboptimal design showing excessive ripple, (b) optimized equiripple design

The filter produced by the PM algorithm exhibits the following characteristics.

• The passband has ripples that deviate from unity by ±δ_P.

• The stopband has ripples that deviate from zero by ±δ_S.

• The passband edge frequency, ω_P, and stopband edge frequency, ω_S, match the specified values.

• The maximum approximation error is minimized.