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We propose algorithms for the design of FIR compaction filters, which find applications in FIR subband coders. The techniques produce compaction gains that are very close to that of optimal compaction filters, for any fixed filter order and input autocorrelation. The main theme of the paper is the design of multistage FIR compaction filters based on an iterated linear programming approach. The theory behind this is presented followed by design examples and comparisons. Also, a noniterative algorithm that is much faster than other iterative optimization techniques (e.g. linear programming) is mentioned.