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A linear time-invariant system reacts on a Dirac pulse with its own pulse response. This pulse response fully describes the characteristics of the system under test.
Now the pulse response can be investigated. So e.g. a Fourier transform will give us information about the frequency and phase response. We can consider the pulse response as a filter and feed it with proper signals. A step input signal thus shows us the step response. There are many ways of investigation. Each one shows some detail about the characteristics.
Now we can e.g. derive the frequency response by the Fourier transform and think about a pulse which creates this response in the shortest time and with minimal phase changes. The answer is - not surprising - the minimumphase pulse response. It has the important feature that it is causal. It describes the reaction of the system to an input signal. The system will not react before the input event happens. A loudspeaker will not play music before it gets fed with it. So physical systems usually react causal and minimumphase.
But there is another feature in the game: delay. A sound wave e.g. takes its time to arrive. And thus many things can happen in between and add or deform the pulse response. That's why e.g. the pulse response of a multiway speaker is no longer minimumphase. The drivers have their individual timings and thus the added characteristics does not show the shortest time to create the overall frequency response.
Now we can describe any given NMP pulse response IR as a combination of two underlying pulse responses MP and EP. These sub-pulses MP and EP are connected by convolution and so we can write
NMP = MP * EP (* = convolution)
MP is now the minimumphase pulse response and EP is the so-called excessphase pulse response. We have already talked about minimumphase. The excessphase EP is nothing else than an allpass. It's frequency response is perfectly flat, so every sinewave will pass the allpass filter without change of amplitude. But the allpass allows to change the timing or phase. It simply delays frequencies.
The convolution of MP and EP leads to an indefinite number of resulting impulse responses. We find usually three terms in discussion: minimumphase, linearphase, maximumphase.
All of them are still a result of MP*EP. In case of minimumphase both MP and EP are minimumphase. In case of linearphase EP is cancelling the phase changes of MP. There is no phase change to achieve a desired frequency response. In case of maximumphase EP creates the maximum phase delay. In general MP*EP can be described as mixedphase.
There are some important points regarding MP and EP:
- the inverse of MP is again minimumphase. The frequency response can be seen as mirrored along the frequency axis. It is possible to derive MP from its frequency response.
- the "inverse" of EP is the time-reversal of EP
We can apply these actions when we manage to split IR into MP and EP properly.
And if we like to correct MP and EP we "just" have to apply these two inversion operations. It sounds simple but of course the real world will limit this. Think about the frequency response of a tweeter and invert it. You can see you need to boost a lot in the bass area to get a flat tweeter response. But the tweeter will certainly not survive it.
Think about a comb filter where two sinewaves of same frequencies but 180° phase shift will perfectly cancel to zero. There is no way to boost zero.
These examples are described by the term ill-conditioned. And we can and do find many ill conditions when we try to measure and correct something. The only way is to make cutbacks and reduce the requirements for an ideal correction. The art of correction lies in the treatment of ill conditions.
Just my humble opinion.
Now the pulse response can be investigated. So e.g. a Fourier transform will give us information about the frequency and phase response. We can consider the pulse response as a filter and feed it with proper signals. A step input signal thus shows us the step response. There are many ways of investigation. Each one shows some detail about the characteristics.
Now we can e.g. derive the frequency response by the Fourier transform and think about a pulse which creates this response in the shortest time and with minimal phase changes. The answer is - not surprising - the minimumphase pulse response. It has the important feature that it is causal. It describes the reaction of the system to an input signal. The system will not react before the input event happens. A loudspeaker will not play music before it gets fed with it. So physical systems usually react causal and minimumphase.
But there is another feature in the game: delay. A sound wave e.g. takes its time to arrive. And thus many things can happen in between and add or deform the pulse response. That's why e.g. the pulse response of a multiway speaker is no longer minimumphase. The drivers have their individual timings and thus the added characteristics does not show the shortest time to create the overall frequency response.
Now we can describe any given NMP pulse response IR as a combination of two underlying pulse responses MP and EP. These sub-pulses MP and EP are connected by convolution and so we can write
NMP = MP * EP (* = convolution)
MP is now the minimumphase pulse response and EP is the so-called excessphase pulse response. We have already talked about minimumphase. The excessphase EP is nothing else than an allpass. It's frequency response is perfectly flat, so every sinewave will pass the allpass filter without change of amplitude. But the allpass allows to change the timing or phase. It simply delays frequencies.
The convolution of MP and EP leads to an indefinite number of resulting impulse responses. We find usually three terms in discussion: minimumphase, linearphase, maximumphase.
All of them are still a result of MP*EP. In case of minimumphase both MP and EP are minimumphase. In case of linearphase EP is cancelling the phase changes of MP. There is no phase change to achieve a desired frequency response. In case of maximumphase EP creates the maximum phase delay. In general MP*EP can be described as mixedphase.
There are some important points regarding MP and EP:
- the inverse of MP is again minimumphase. The frequency response can be seen as mirrored along the frequency axis. It is possible to derive MP from its frequency response.
- the "inverse" of EP is the time-reversal of EP
We can apply these actions when we manage to split IR into MP and EP properly.
And if we like to correct MP and EP we "just" have to apply these two inversion operations. It sounds simple but of course the real world will limit this. Think about the frequency response of a tweeter and invert it. You can see you need to boost a lot in the bass area to get a flat tweeter response. But the tweeter will certainly not survive it.
Think about a comb filter where two sinewaves of same frequencies but 180° phase shift will perfectly cancel to zero. There is no way to boost zero.
These examples are described by the term ill-conditioned. And we can and do find many ill conditions when we try to measure and correct something. The only way is to make cutbacks and reduce the requirements for an ideal correction. The art of correction lies in the treatment of ill conditions.
Just my humble opinion.
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And there are many ways to skin a cat.
