I can sometimes describe a way of using certain tools – such as in this case, one of the Discrete Cosine Transforms – which is correct in principle, but which has an underlying flaw, that needs to be corrected, from my first approximation of how it can be applied.
One of the things which I had said was possible was, to take a series of frequency-domain ‘equalizer settings’, which be at one per unit of frequency, not, at so many per octave, compute whichever DCT was relevant, such that the result had the lowest frequency as its first element, and then to apply that result as a convolution, in order finally to apply the computed equalizer to a signal.
One of the facts which I’m only realizing recently is, that if the DCT is computed in a one-sided way, the results are ‘completely non-ideal’, because it gives no control over what the phase-shifts will be, at any frequency! Similarly, such a one-sided convolution can also not be applied as the sinc function, because the amount of sine-wave output, in response to a cosine-wave input, will approach infinity, when the frequency is actually at the cutoff frequency.
What I have found instead is, that if such a cosine transform is mirrored around a centre-point, the amount of sine response, to an input cosine-wave, will cancel out and become zero, thus giving phase-shifts of zero.
But a result which some people might like is, to be able to apply controlled phase-shifts, differently for each frequency, such that those people specify a cosine as well as a sine component, for an assumed input cosine-wave.
The way to accomplish that is, to add-in the corresponding (normalized) sine-transform, of the series of phase-shifted response values, and to observe that the sine-transform will actually be zero at the centre-point. Then, the thing to do is, to apply the results negatively on the other side of the centre-point, which were to be applied positively on one side.
I have carried out a certain experiment with the Computer Algebra System named “wxMaxima”, in order first to observe what happens if a set of equal, discrete frequency-coefficients belonging to a series is summed. And then, I plotted the result of the definite integral, of the sine function, over a short interval. Just as with the sinc function, The integral of the cosine function was (sin(x) – sin(0)) / x, the definite integral of the sine function will be (1 – cos(x)) / x, and, Because the derivative of cos(x) is zero at (x = 0), the limit equation based on the divide by zero, will actually approach zero, and be well-behaved.
(Update 1/31/2021, 13h35: )
There is an underlying truth about Integral Equations in general, which people who studied Calculus 2 generally know, but, I have no right just to assume that any reader of my blog did so. There exist certain standard Integrals, which behave in the reverse way of how the standard Derivatives behave, just because ‘Integrals’ are ‘Antiderivatives’…
When one solves the Derivatives of certain trig functions repeatedly, one obtains the sequence:
sin(x) -> cos(x) -> -sin(x) -> -cos(x) -> sin(x)
Solving the Indefinite Integrals of the same trig functions yields the result:
sin(x) -> -cos(x) -> -sin(x) -> cos(x) -> sin(x)
Hence, the Indefinite Integral of sin(x) is in fact -cos(x), and:
( -(-cos(0)) = +1 )
(End of Update, 1/31/2021, 13h35.)
(Updated 2/04/2021, 17h10…)
Continue reading Exploring the Discrete Sine Transform…