(122)Chapter 6
FURTHER RESEARCH
This chapter discusses some improvements to the multirate convolution problem and possible areas for additional research.
An Open Problem in Multirate Convolution without Cross-Convolved Channels
Based on Equation 72, it is evident that in order to completely cancel aliasing with this system, the following condition must hold.
(122)
If the last term is non-zero, the odd output samples must be multiplied by (a + b)/(a - b).
This is due to the following convolution identity. For two causal sequences, h(n) and x(n), with z-transforms H(z) and X(z), define Y1(z) = X(z)× H(z), Y2(z) = X(-z)H(-z), Y3(z) = X(-z)× H(z), and Y4(z) = X(z)× H(-z). From the frequency shifting property of the
z-transform, y1(n) = x(n) * h(n), y2(n) = (-1)n× x(n) * (-1)n× h(n), y3(n) = (-1)n× x(n) * h(n), and y4(n) = x(n) * (-1)n× h(n). From the convolution summation in Equation 4, y1(n) = y2(n) and y3(n) = y4(n) for n even and y1(n) = -y2(n) and y3(n) = -y4(n) for n odd.
The question remains whether Equation 122 is possible. Consider the following conditions:
(123)
In other words, A0(z) and B0(z) contain only even powers of z and A1(z) and B1(z) contain only odd powers of z. Then Equation 72 becomes
(124)
By inspection of Equation 104, it should be clear that any constraints similar to those in Equation 103 that cancel aliasing from X(-z)H(z) and X(z)H(-z) require that the aliasing from X(-z)H(-z) remain.
Additional research might investigate filter combinations for which Equation 122 holds approximately true. For example, if
(125)
Then Equation 122 becomes
(126)
Taking into account the convolution identity described above (or just adding rows in the matrix), this means that aliasing is cancelled for even samples, which are scaled by (12 + 4) = 16 and delayed by z-4. Similarly (by subtracting last row from the first and the third from the second), odd samples retain significant aliasing from the surrounding samples. Specifically, the sample of interest is scaled by (12 - 4) = 8 and has a z-4 delay plus (4 + 4) = 8 times the samples directly adjacent plus (2 + 2) = 4 times the samples two units ahead and behind. With this much information, perhaps a linear predictive method or modification of existing methods can remove significant alias components in the corrupted samples. That is, even samples are uncorrupted. So when an odd sample is output (the z-4 of the matrix), the aliasing from the adjacent even samples (the z-3 and z-5 terms) can be subtracted. The only problem is determining how to remove aliasing from the outer terms (z-2 and z-6).
Another approach might involve finding a simple method to significantly alter the magnitude of a generally allpass (unit delay) characteristic via only a sign change from z to -z. Also, if aliasing is confined to every other sample, then it may be possible to remove the aliasing components based on previously computed correct samples.
Additional Qualitative Tests
Although the data provided by the listening tests performed provides some worthwhile insight into the qualitative performance of the multirate convolution approximation, tests with a greater variety of signals, impulse responses, and filters administered to a larger test pool would provide a more substantive basis for determining when the algorithm becomes a feasible option for a given implementation. It would also be worthwhile to investigate systems running at higher sampling rates. Then the aliased components would be shifted to higher (and possibly inaudible) frequency ranges.
Pre- or Post-Processing
It has been shown that the convolution of a sinusoid with a unit impulse results in a notched frequency response about p/2. If the aliasing from the entire convolution system can be similarly characterized for more generalized signals, then the aliasing may be reduced by a properly defined equalization filter applied at either the inputs or the output of the system. Of course, the computational complexity of the equalization must not negate any computational gain offered by the multirate implementation.
Expansion to M-Channels
It would be advantageous to implement the multirate algorithm in more than two channels. Clearly, aliasing will occur in any analysis/synthesis bank transition band, but the computational gains may override this concern in some applications. Also, since the impulse response is known in advance, it may be possible to design a non-uniform M-channel filter bank such that the information in the transition bands is minimized (see Vaidyanathan 1993a, p. 482-485).
Subband Skipping
Another possible improvement to a multirate convolution algorithm may involve a more active analysis of subband content. Subbands with little energy or a reasonably flat spectrum need not be convolved. One way to accomplish this might be to use adaptive or linear prediction filters in the analysis bank. It may also be worthwhile to investigate the musical usefulness of a partial convolution, where arbitrary subbands are convolved while others are not. This implies that the multirate convolution scheme may be more useful as a platform for frequency specific subband processing.
Subband Processing
Since the analysis bank breaks the signal into multiple subbands, signal processing can be performed on frequency specific areas of the input or impulse response. A spectral compression algorithm, for example, might compress only the low frequencies. Then the clarity provided by the high frequencies is not sacrificed.