Convergence of Remez Exchange

• In theory, $\delta$ is guaranteed to increase monotonically each iteration, ultimately converging to its optimal value. This value is reached when all the extremal frequencies $\omega_k^\ast$ are found.

• In practice, numerical round-off error may cause $\delta$ not to increase monotonically. When this is detected, the algorithm halts and reports a failure to converge.

• Convergence failure is common in practice for FIR filters having more than 300 or so taps and stringent design specifications (such as very narrow pass-bands)

• Further details on Remez exchange are given in [Rabiner and Gold 1975, p. 136]

• As a result of the non-iterative internal solution on each iteration, firpm cannot be used when additional constraints are added, such as those to be discussed in the following sections. In such cases, a more general LP solver such as linprog or cvx must be used.

• Recent advances in convex optimization enable faster solution of much larger problems (see EE364 A,B)