### Chebyshev FIR Design via Linear Programming

We return now to the -norm minimization problem of §4.10.2:

and discuss its formulation as a

*linear programming problem*, very similar to the optimal window formulations in §3.13. We can rewrite (4.46) as

(5.47) |

where denotes the th row of the matrix . This can be expressed as

s.t. | (5.48) |

Introducing a new variable

(5.49) |

then we can write

(5.50) |

and our optimization problem can be written in more standard form:

s.t. | (5.51) |

Thus, we are minimizing a

*linear objective*, subject to a set of

*linear inequality constraints*. This is known as a

*linear programming*problem, as discussed previously in §3.13.1, and it may be solved using the matlab

`linprog`function. As in the case of optimal window design,

`linprog`is not normally as efficient as the Remez multiple exchange algorithm (

`firpm`), but it is more general, allowing for linear equality and inequality

*constraints*to be imposed.

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More General Real FIR Filters

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Least-Squares Linear-Phase FIR Filter Design