We present an algorithm for speech compression which uses the wavelet packet transform, vector quantization, entropy coding and postfiltering of the decoded speech. We address the following issue: obtaining the best speech quality for a given bit rate with minimal algorithmic delay (applying it on the possible shortest segment). The wavelet packet transform provides good compression since it is based on a very close relation between the transform and the actual physical processes in the human ear. The experimental results demonstrate that we can compress speech by factor of 6 - 10 and still have reasonable intelligibility and perceivability of the output speech using an algorithmic delay of 8 msec (64 speech samples). In addition, the proposed algorithm fits well DSP architecture and can be easily ported into any current 40MIPS DSP. By comparing the proposed algorithm in this paper with new CELP-oriented algorithm one can conclude that the former has less delay with higher compression ratio. The postfiltering was found to improve the quality of the decoded speech. We see that by using fixed size segments with 64 samples with wrap-around in the segments border does not degrade the performance in comparison to FIR-implementation without wrap-around. In addition, it is useful to implement different filter in each level of the decomposition.