Chapter 7
CONCLUSION
The results of this research indicate that, for some applications, a multirate approach to the computation of a linear convolution can result in significant reduction in the required number of computations . For the two-channel multirate case, the computational complexity of a large direct time-domain convolution can be reduced by nearly one half at the cost of aliasing distortion in frequencies around one fourth the sampling frequency. For some audio applications, this distortion may be acceptable. A limited A/B listening test indicates that the distortion is generally detectable, but musically insignificant for some types of non-percussive instrumental music.
From a more empirical standpoint, only Vetterli’s (1988) running convolution scheme will outperform well known frequency-domain block convolution algorithms. Computations per sample may be reduced by up to 25%. The designer must take care to recognize that the computational reduction applies to the number of operations per sample per sampling period. If the implementation platform cannot take advantage of this, e.g. if it is not a multi-tasking system, then it cannot take full advantage of the multirate algorithms.
The multirate convolution implementations do exhibit some disadvantages. Besides an increased implementation complexity and possible aliasing distortion, they
introduce additional delay in the system due to the addition of the analysis and synthesis filter banks. Also, the impulse response must undergo significant processing before convolution can begin.
On the other hand, a multirate convolution implementation can produce a convolution result with a lower noise variance and provides a framework for subband specific processing. Expanding the multirate algorithms to more than two channels also holds promise for further improving efficiency. From the discussion regarding Equation 122, future research in the area may yield more efficient multirate filter banks that will benefit frequency domain structures, especially if adaptive or linear prediction algorithms can be designed to remove embedded aliasing in corrupted samples.
While the results of this research may not be directly applicable to existing fast-convolution implementations, they demonstrate a promising direction for future research in both multirate and convolution processing.