The Audio Pages

Introduction
Time Delay
Crossover Filters
Distortion	(30 Jul)
Phase Audibility	(30 Jul)
Conclusion
Update	(15 Jun)
References

One only needs to look at a few web sites to realise that there is actually very little useful information on phase in audio systems in general, and loudspeakers in particular.  There are a great many conflicting claims and counterclaims, but little real data.  There is naturally a great deal of rubbish, mostly describing why "Brand X" loudspeaker (for example) is demonstrably superior to every other speaker on the planet (which is why no-one has ever heard of them).  Expect to see claims that "this speaker is the only design that will accurately reproduce a square wave" or something similar.  As we shall see, this is realistically possible, but has (or should have) a huge "who cares" factor that will be discussed in greater detail a little later.

This article is not for the faint hearted, as it discusses amplitude, phase and delay, and the complex interactions between them.  There is only so much that can be accomplished by diagrams and graphs, and many of the concepts do not lend themselves to easy analysis.  I have tried to keep the information in a logical form, but unfortunately, all of the things discussed occur simultaneously.  This is not always easy to visualise, and is even harder to write.

The many diagrams and graphs were produced using SIMetrix, an excellent simulator available from Newbury Technology in the UK (http://www.newburytech.co.uk).  It is available as a free demo system, and is the best simulator I have used so far.  All images are reduced for speed - to see the full version, just click on the displayed image.

Since I am going to be using a 6 dB/ octave filter for many of the examples below, Figure 1 shows the response of a conventional 1st order (6 dB) filter.  This is normalised to 1k ohm and 1uF, which gives a crossover frequency of 159 Hz.  Although most discussions will revolve around higher frequencies, this is of no consequence.  The graph is designed to show the rolloff slopes of the high and low pass sections - not the absolute performance at any specific frequency.

Note that all graphs on this page are shown as reduced size. For the full sized version, click on the image - this will open a new window for the full sized graph.

The line labelled "1-Input" is the input applied to the filter, and is also equal to the electrically summed outputs from the two sections.  Both are flat, and are at 1 Volt relative level.  Note that the simulator claims slightly different -3 dB frequencies for the two signals - this is not a simulator aberration, but the result of the simulator calculating to the absolute limits of accuracy.  The crossover frequency is in fact 159 Hz as calculated, and at that frequency, the level is exactly 0.707 volt.  If expressed accurately, -3 dB is in fact 0.7079, and not 0.707 as is commonly used.  This is a small error, and may safely be ignored.

All filters come with some pretty rigid rules - these are determined by the laws of physics, and are not open to discussion, although some of the snake oil vendors will still try.  Filters are described in "orders" - 1st, 2nd etc.  Each order has an ultimate rolloff (i.e. achieved at some point distant from the cutoff frequency) that increases by 6 dB steps for each successive order, so 6, 12, 18, 24 dB/ octave is a common way to describe the filter's response.  They are further divided into "even" and "odd" order (even and odd numbers - it doesn't matter much, but is commonly used anyway).

A brief numerical description of each filter type is shown below, along with its rolloff characteristics and power level above the "cutoff frequency", typically defined as that frequency where the response is reduced by 3 dB.  This is not always used as the crossover frequency - Linkwitz-Riley aligned crossovers use the -6 dB point instead, and achieve a flat response as a result (not applicable to 1st order filters).
 

Table 1 - Filter Characteristics

Order	Slope	Voltage	Power	Theoretical
None	Flat	1 Volt	1 Watt	1 Watt
1st	6 dB/ octave	439 mV	193 mW	250 mW
2nd	12 dB/ octave	371 mV	138 mW	64 mW
3rd	18 dB/ octave	195 mV	38 mW	16 mW
4th	24 dB/ octave	122 mV	15 mW	4 mW

Voltage in the above table is the voltage one octave above the -3dB frequency (assuming an input of 1 Volt and a low pass filter), and power at the same frequency, referred to 1 Watt.  For example, 138 mW is about 1/7th Watt.  The performance of a high pass filter is exactly the same as shown.  The "Theoretical" value quoted is the power that should appear in theory - you may even see it quoted by manufacturers who have neglected to actually perform the maths, and have simply used the filter rolloff to arrive at a convenient looking number.

The above is not exhaustive, but it covers the filters most commonly used in audio.  For all filters above 1st, the table is based on the Bessel (minimum settling time) alignment, which is also typical of Linkwitz-Riley designs.

Firstly, there are many ways that the phase of a wave can be shifted, with the most common being time delay.  At its most extreme, there is a delay of days to decades between the material being recorded and you listening to it - and no, this is not meant as a marginally humourous comment - this is a genuine time delay.  The important thing is that all of the signal is delayed by the same amount, and it doesn't matter if this delay is measured in milliseconds or millennia, the sound will emerge intact and completely recognisable.

The situation is very different if some of the sound is delayed, while the rest is not.  The listening experience would not be enhanced if the high frequencies were to be reproduced half an hour later than the bass or vice versa.  This is quite obvious, but let's reduce the time to something more realistic.  What if the treble were to be delayed by 20 milliseconds?  The effect would be awful - this is a time difference we can easily pick, and we use these cues to determine the original sound from reflected sound for localisation.

We can continue reducing the time delay, and the effect will become less and less discernible as the time is reduced.  Finally, we get to a point where the delay represents less than a wavelength (in air), and (perhaps surprisingly), the differences are still audible.  Consider a 1 kHz sine wave, reproduced from two sources, but with one delayed by 500 us - just 1/2 millisecond.  As one source creates a compression, the other creates a rarefaction - the waves are 180 degrees apart, and will attempt to cancel each other.  Early reflections and a multitude of other effects will ensure that we still hear the sound (at least at that frequency), but there will be a noticeable drop in level.

Now, there are some who will claim that reversing the phase of one source will bring everything back to where it was, so there is no harm done, and the net result is the same as if the two sources were not delayed at all.  While this will obviously work at 1 kHz, at other frequencies this is not the case.

Now, let's look at some of the physics involved here.  How would a 500 us delay be introduced in the first place?  In reality, this is not uncommon, but we shall reduce the time delay to something more realistic before continuing.  Any two loudspeakers that reproduce the same signal at the same time will exhibit this phenomenon, but for our purposes on a smaller scale.

If we look at a midrange driver and a tweeter, in the common vertical alignment in an enclosure, we have a time delay.  The "acoustic centre" of the tweeter will most likely be a small distance closer to the listener than that of the midrange driver, and for the sake of this discussion, let us assume a difference of 50 mm, because it is a realistic and typical offset for common loudspeakers.

Before continuing, it is important that the concept of "wavelength" is properly understood.  Sound travels at about 345 m/s in dry air at sea level.  This changes with temperature, humidity and altitude, but we shall not concern ourselves with this, and there is little we can do about it most of the time.  A sound at 345 Hz has a wavelength of 1 metre, at 34.5 Hz the wavelength is 10 metres, and at 3450 Hz, it is 100 mm.  This is quite linear, and works for all frequencies.  Another useful thing to know is the period (the actual time required to reproduce one cycle at the selected frequency).

wavelength = velocity / frequency
period = 1 / frequency

If we return to the midrange and tweeter mentioned above, their acoustic centres are offset by 50 mm - this is exactly 1/2 wavelength if the crossover frequency is 3450 Hz.  We can account for the 1/2 wavelength by reversing the wires to the tweeter, so it is 180 degrees out of phase with the midrange.  The two drivers are now aligned in phase, so in theory, they are time aligned.  Unfortunately, this is not the case.  Although the signal is in alignment at the crossover frequency, it will not be aligned any more when the frequency changes.

What is really needed is to delay the signal going to the tweeter by 145 us (1/2 of the period of a 3450 Hz waveform), or align the acoustic centres of the two drivers in the vertical plane.  Such "time alignment" is commonly achieved by angling the baffle so that at the listening position, the signals are properly in phase and time.  Stepped baffles have also been used, but often create more problems with diffraction than are solved by the time alignment.

In short, time alignment is a good goal, but does not necessarily guarantee that the sound will be any better than a "conventional" flat baffle, with the phase of the drivers appropriately switched to ensure that the signal is in phase at the crossover frequency.  It must be understood that with any flat baffle, an octave each side of the crossover frequency will see the phase out of alignment again, so it is essential that a high order crossover is used to prevent unwanted cancellations and reinforcements at different frequencies.

With a flat baffle and a "time displacement", above or below the crossover frequency the signals are in and out of phase - the exact amount can be calculated, and this can be very important in the greater scheme of things.
 

Table 2 - Acoustic centre displacement 50 mm 
(145 us time delay) 1 driver reverse phase

Octave	Frequency	Wavelength	Phase Angle
-1	1725 Hz	200 mm	90 degrees
-1/2	2439 Hz	141 mm	45 degrees
0	3450 Hz	100 mm	0 degrees
+1/2	4878 Hz	70 mm	90 degrees
+1	6900 Hz	50 mm	180 degrees

Expect a dip at an octave above the crossover frequency, since the two signals (from the midrange and tweeter) are 180 degrees out of phase at this frequency - not because of the crossover, but because of the time delay of 145 us.  The only way to ensure that this dip is inaudible is to use a steep filter!  If a 6 dB/octave filter were to be used, the signal level is only down to 0.447 of the total (7 dB).  On the other side, at 1 octave above crossover frequency, the tweeter will only have 0.894 of the full signal (0.97 dB down).  These voltage relationships can be seen in Figure 1, above.

Hang on - this is a 6 dB/ octave filter, and it's 7 dB down an octave from crossover frequency.  How can that be?

Remember that we are already 3 dB down at the crossover frequency, but because a 1st order crossover has a very low Q (or in other words is highly damped), the rolloff is not as steep initially as expected.  It should be down by 9 dB an octave away, but this will never happen.

Ignoring the acoustic centres of midrange and tweeter for a moment, Figure 2 shows the waveform response of the filter at crossover frequency, together with the input signal.  The RMS voltages of each are also shown.  The waveform at 1 octave above crossover frequency is not shown - the absolute phase will be different, but relative phase (between outputs) remains at 90 degrees for all frequencies.  This is the electrical response only - the acoustical response will be different if the drivers are not time aligned!

Now (and this is where it gets tricky), what happens if we sum the electrical signals reproduced by the simple 1st order crossover?  Assume an input of 1 volt for convenience.  Adding 894 mV and 447 mV electrically (at any frequency) will give an output of 1.34 volts - this is clearly not correct, since the input is only 1 volt to begin with.

Analysis of a 6 dB/ octave crossover shows that the high and low pass signals are in fact 90 degrees out of phase at all frequencies ("Yes but ... isn't the 1st order crossover supposed to be phase coherent?").  Yes and no.  It is phase coherent in that all signals at all frequencies are 90 degrees out of phase.  I know that you have seen web sites that say that there is no phase shift through a 1st order crossover, but this is simply untrue!  At crossover, the high pass section is leading - the signal appears to emerge from the filter 45 degrees before the input, not possible it would seem.  This sort of behaviour is standard with all filters with a "steady state" signal - you don't have to really understand it, so I suggest that you just live with it.

The low pass filter has a lagging response, so the signal emerges 45 degrees after the input.  This is easier to comprehend, but may still seem a little strange (which I suppose it is for a filter that many claim has no delays).

So, if we make the essential correction, and shift the relative phase of either signal by 90 degrees, we can recalculate the summing of the two signals.  Predictably, 894 mV + 447 mV with a 90 degree phase shift now gives a summed response of 1V - this is as we would expect, and is shown in Figure 3.

You can see the phase relationship between the 3 signals quite clearly.  I doubt that this will be terribly meaningful for the most part, but it is essential to the understanding of the relationships - time and phase are inextricably entwined with each other, and cannot be separated.

The electrical and acoustical relationships only coincide if the acoustic centres of the speakers are in exact alignment.  As soon as there is a misalignment (introducing a time delay), everything changes.  To see the effect, imagine the original setup, with the acoustic centres misaligned by 50 mm.  The tweeter's output will now be heard 145 us before that of the midrange.  For the purpose of explanation, we shall ignore the 90 degree phase shift introduced by the crossover, and indeed, this is only present in the 1st order design.  In fact, for many of the following explanations I will use signals of equal amplitude, and will ignore the crossover altogether.  This provides for a worst case - reality will be somewhat tamer.

If we use two signals of equal amplitude, when summed we get a signal of double that of each signal - after all, the concept of 1 + 1 = 2 is not uncommon (except in corporate financial circles :-)  If the level is any different, then there is phase shift (or delay) that causes the error.

Figure 4 shows what happens when the 3450 Hz signal is produced from both speakers simultaneously, but with a 145 us time delay (representing the 50 mm offset).  The red line is the combined signal - there is no signal!  This is electrical summing, which is much more critical than acoustical summing, so in reality we will still hear something, but nowhere near what we should.  This is commonly referred to as a "suckout" by reviewers, and there will be a pronounced dip in frequency response.  Now, we know that this is easily fixed by reversing the phase of one driver, and everything will be back where it should be - but (and this is the clincher here) - only at one frequency!  At all other frequencies there will be interference effects, and the lower the filter order, the worse it becomes.

Rather than take vast amounts of bandwidth to display as whole series of similar waveforms, I have tabulated the resultant signal level below, for 2 signals of equal amplitude but with one delayed by 145 us.  These are the same frequencies we looked at earlier.  In all cases, the result should be 2 volts ...
 

Table 3 - Summed Signals

Octave	Frequency	Amplitude
-1	1725 Hz	1.414 V
-1/2	2439 Hz	0.887 V
0	3450 Hz	0 V
+1/2	4878 Hz	1.195 V
+1	6900 Hz	1.959 V

Now, bear in mind that the above table is actually meaningless (it looks impressive though).  All of the information must be presented in a simultaneous manner for any of it to make real sense.  To expand on this a little further, have a look at a frequency scan of two drivers reproducing the same signal, but with one delayed by 145 us.  This produces a comb filter effect.  Now, in real life, the signals will not be at the same amplitude, so the effect is reduced.  The signals are also summed acoustically, reducing the effect even further, but the crucial point here is that the crossover and acoustical summing reduce - not eliminate - the problem.  But this is still not real! (It is marginally useful though, just so you can see where all this is going.)

We can see the notch predicted in earlier examples at the crossover frequency of 3450 Hz, but we also see another at 10.26 kHz, and another at 17.4 kHz.  The final notch shown is unlikely to be audible for most of us at 24 kHz.  If the delay is increased, the effect becomes worse.  It is also worth noting that even with the relatively small delay used for this example, the combined signal is down 3 dB at 1737 Hz.  Remember that this is worst case, with no crossover network.

The combined effect of the delay and crossover can be expected to be a little less daunting, so the trusty simulator has been stretched a little here, and Figure 6 shows what happens when both the delay and the crossover are used, with the phase of one driver reversed as required to prevent the cancellation at crossover frequency.  Oh dear!  There might not be a major problem at the crossover frequency, but an 8 dB dip at 2 kHz is less than desirable.  The 2.8 dB peak at 5.1 kHz is no bonus either.  Less daunting?  When all the material is presented, then the whole picture is available.

Note that this was missed in the table above, since I only looked at the 1/2 octave boundaries and with equal amplitudes.  Little omissions can leave major gaps in ones actual knowledge!  A small (cunningly disguised) trick of calculation or description can leave one thinking that a designer has achieved something special, so always make sure you have all of the information.

The effect is not as severe (note the depth of the notches - read the voltage levels!), but in quite a few respects it is actually worse than the "fake" graph of the previous example!

Just to make sure, I reduced the time delay to 10 us then 1 us, to verify that nothing was awry with my simulations.  As expected, the response was almost flat, and with no delay at all, the response was completely flat.

So, the next question has to be ... What difference does it make if the filter order is increased?  Figure 7 shows the response with a 2nd order filter, using a Bessel (Linkwitz Riley) alignment.  The ripples have been reduced considerably, but are still quite obvious.  Figure 8 shows the response with a 24 dB/ octave L-R crossover.  In both cases, the signal to the tweeter is inverted to account for the 145 us time delay, which as we know reverses the effective phase of the driver.

As can be seen, the ripple is reduced as filter order is increased.  Remember that all filters shown will sum electrically and acoustically flat if there is no time delay.  All ripple is a direct result of the time misalignment.  To put this into perspective, the room and furnishings (including the speaker box itself) will have a much greater effect on the response than the 12 or 24 dB/ octave filters introduce - however, there is no good reason to muck up the response before the room has a chance.

Using DSPs (Digital Signal Processors), it is possible to delay the signal to speakers to compensate for the physical offset.  At present, this is still frightfully expensive, but we can expect digital crossovers with adjustable time alignment delays to become commonplace in a few years.  They exist now, but few of us can afford the luxury, and many will be unwilling to insert yet another set of analogue-digital-analogue converters into their system

I have always liked 1st order filters.  Most loudspeaker drivers do not like 1st order filters.  The ideal system would use no filters at all.  With the partial exception of electrostatic loudspeakers (ESLs), the ideal speaker does not exist.  Why "partial" exception?  ESLs are bi-directional, and as a result of a relatively small baffle, do not reproduce low frequencies well.  ESLs are also hardly a point source - the radiating panel is quite large, and this makes for a small "sweet spot" for listening because of the off-axis response of any large radiating surface.

As always, we must make compromises, and the ideal would be to have a single point source driver that could reproduce all frequencies equally well, and with no distortion.  The smaller the driver, the better it will reproduce high frequencies without lobing, most easily described as listening angle dependent response peaks and dips.  Low frequencies require that a lot of air be moved, so the small driver will do a very poor job - larger drivers are needed.  This is the reason that most high fidelity speakers use at least two, and commonly three different loudspeakers to cover the audible range.

This is where the filters come into play - they are an essential part of the compromise, and separate the signal into ranges that can be accommodated by the individual drivers.  The 1st order (6 dB/ octave) filter has the lowest phase shift and the best transient response of all the possibilities.  It also has the slowest rolloff, so undesirable effects will be heard from the loudspeakers as they are excited by the signals outside their optimum operating frequency range.

Contrary to what you may read elsewhere, all crossover networks (filters) bar none introduce phase shift.  This is actually the least important characteristic of a filter, and provided that the low and high frequency waveforms remain in phase with each other, their absolute phase is not important.  Such a filter is described as phase coherent, and this is extremely important to the sound quality obtained.

Since filters introduce a phase shift, they also introduce a time delay.  This is not the fixed delay referred to above, but varies with frequency.  Perhaps surprisingly, this frequency dependent delay is not overly important to the overall sound, but it requires considerable care to ensure that audible artefacts are not created as a result of the delay.

The conventional crossover of old was the Butterworth.  Maximally flat frequency response, a Q of 0.707 (damping factor of 1.414), and 3 dB down at the crossover frequency.  It has been shown by many workers in acoustics that this is actually wrong, as a 3 dB peak is experienced at the crossover frequency.  It should be noted that this only occurs with 2nd order (12 dB/ octave) filters and higher - a 1st order filter does not have that problem.

The response of this filter is shown in Figure 9, and the peak at the crossover frequency is clearly visible.  Figure 10 shows the phase response at one octave below crossover frequency - the signals are perfectly in phase (after inversion of one signal - the 12 dB crossover alwaysinverts one signal with respect to the other.

What about a square wave?  This is supposed to be the most telling aspect of a design, which is interesting in itself since a synthesiser is the only instrument that is capable of producing a square wave, and no-one ever uses an unfiltered square wave anyway.  Well, the result is shown in Figure 11, and the combined signal looks nothing like a square wave.  The fact of the matter is that all frequencies that make the square wave are still present in their exact amplitude relationships, but they are shifted in phase.  This is completely inaudible, and that has been proven many, many times.  Human hearing is not sensitive to absolute phase, and responds to relative phase only if it causes a peak or dip in the frequency response.  I suggest that you treat any claim to the contrary with the utmost suspicion, as the writer has a hidden agenda (to sell you his product being the most common).

Now, for reasons that are unclear (to me anyway), to obtain a license to use the term "Time Aligned", the speaker must be demonstrably capable of reproducing a square wave.  Que?  License??  Oh yes - the term is trademarked, and one may not advertise speakers as "Time Aligned" unless the appropriate fee is paid (presumably - I have no idea how much this costs), and the requirements are met.  The biggest problem faced with getting any crossover to pass a square wave is simply phase shift.  1st order filters do it, but few drivers can cope with the low rolloff.  An interesting tradeoff is the so-called "subtractive" crossover.  This uses a single filter (of any slope), and subtracts the output of that from the input signal.  The result is perfect square wave response, and a flat summed response.

Do you see the anomalies?  There is a bump in the high pass response, and although the low pass is 12 dB/ octave, the high pass is only 6 dB/ octave.  Even if the "real" filter is 24 dB/ octave, the subtracted one is still 6 dB/ octave.  Figure 13 shows the combined waveform and the high and low pass waveforms (input is a square wave).

One driver will have an easy enough time, as it will be prevented from entering into the region where it becomes "hostile", with unpleasant lobing effects and possible cone breakup.  The tweeter has no such luck!  The design frequency is not as expected either (the diagrams shown used the same filter that gave a crossover frequency of 2.8 kHz in Figure 9).  When we obtain a crossover frequency of 3450 Hz, the signal to the tweeter is down by only 11 dB at 1 kHz (typical of the resonance frequency of many high end tweeters).

The primary issues that confront the crossover designer are the constraints of the drivers themselves.  As soon as the diameter of the radiating surface (the cone) of a driver becomes "significant" with respect to wavelength, you will have problems with lobing.  This causes poor off-axis response, and makes the overall sound power output something of a gamble.  A safe enough rule of thumb is that no speaker should be asked to reproduce any frequency where the cone diameter is greater than one wavelength.  A typical 150 mm (6") mid-bass driver should not be operated above about 2300 Hz, and a 100 mm (4") driver is limited to around 3450 Hz.  In addition, all loudspeakers will have cone breakup at some frequency - this can be "soft", causing no gross unpleasant sounds, or "hard", where the sound is quite objectionable.  Generally, the more rigid the cone material, the worse it will be when it is finally incapable of true pistonic movement.  This is one of the reasons that paper cones are so popular.  I do not propose to cover this particular area in detail - further information is available on the Web (right or wrong, subjective or measured - this is up to you to determine).

It is very important that no appreciable power is supplied to a driver at or above the frequency where the cone breaks up or where the cone diameter exceeds one wavelength.  The result is almost always a sonic disaster at the high frequency end.  A relatively steep crossover is the only way to ensure that this colouration is kept below audibility.

Likewise, no speaker should be operated through its resonant frequency (pity about the bass driver!).  For typical tweeters, this is between about 900 to 1500 Hz, and it is imperative that no appreciable power is allowed to get to the tweeter at its resonant frequency - the result is audible, not always insufferably unpleasant, but usually fatiguing and the sound is definitely coloured.  With passive crossovers, the resonant peak also changes the characteristics of the crossover network (see High Quality Passive Crossover Design for more details).

This is surely one of the major quandaries facing any loudspeaker designer.  To use a steep rolloff crossover, with its attendant transient response problems (and yes, these are real), or a simple 1st order design, that will allow the signal through that will excite the speaker at frequencies it will handle poorly.  Despite some of the claims that you may see, there is no evidence that anyone has actually made a speaker that can handle more than about 6 octaves, and most will not come close to managing that.  I would normally expect that a driver (any driver) will handle about 4 octaves reasonably well.  The table below is an example of using 4 octaves per driver.  In a 3 way system, it almost makes it across the full audio band.
 

Table 4 - Octave Division

Oct.	Lo	Mid	Hi
1	32	300	2,400
2	75	600	4,800
3	150	1,200	9,600
4	300	2,400	19,200

Naturally, if the number of drivers is reduced, the bandwidth they must cover is much greater - ever wondered why some (many?) large 2 way systems just don't seem to cut it?  One of the biggest problems (and rarely spoken of) is intermodulation.  If a cone is moving back and forth reproducing a low frequency, as well as "jiggling" back and forth simultaneously reproducing a higher frequency, what will happen to the high frequency?

This is not an electrical system, this is pure mechanics and high school physics.  Remember the Doppler effect?  As a car (for example) comes towards you, the sound is higher in pitch as the sound waves are "squashed" together by the forward motion of the vehicle.  As it passes directly past you, the pitch falls to normal, and becomes lower as the car retreats  from your observation point.  Everyone has heard this effect, but not everyone has equated it with loudspeakers.  Admittedly, the magnitude is much reduced from the above example, but it is nonetheless real, and can be very audible.

The Doppler effect is caused by compression or rarefaction of the wavefront, depending upon whether the object is approaching or retreating from your position.  A loudspeaker cone does exactly the same thing!  The high frequency tones are frequency modulated by the cone movement caused by the low frequency tones.

Now we have identified a new distortion (well, "new" is hardly the right word, but you know what I mean) ... or do we?

My thinking on this issue was recently challenged (thanks Chuck :-) and we did a bit of e-mail exchange, and some demonstartion calculations, and I ran some tests on a speaker in my workshop.  The simple fact of the matter is that doppler distortion is a furphy, not quite a lie, but a calculated twisting of the truth ....

... so, while it can be "proven", further analysis shows the real truth of the matter - the "bad guy" is intermodulation, and the good Prof. Doppler can rest easy, since the effects we hear are not related to his discovery of the Doppler effect at all.  Further reading on this matter will be forthcoming, but for now, assume that Doppler "distortion" does not exist (despite the snake oil liberally spread over the topic by various individuals who shall remain nameless, but have also annointed cables with their alleged "magic" - charlatanism at its very finest  :-)
[Update added 30 Jul 2002]

The real problem is intermodulation, and this is one of the major arguments for using ported enclosures, since it reduces cone excursions at the lowest frequencies, and therefore reduces the tendency of the voice coil to partially leave the magnetic field, and introduce amplitude modulation distortion of the higher frequencies (i.e. intermodulation).  The difficult load this presents to the power amp, and the phase irregularities of ported enclosures are well known, and I will not dwell on them here.  Other alternatives exist ...

• transmission lines - usually very good performance, but rather bulky and hard to build
• horn loaded enclosures - not to everyone's taste, in size or sound
• multiple speakers to share the load - Ok at low frequencies, but causes problems when the distance between drivers exceeds 1 wavelength
• remove all the low bass from the main speakers and use a subwoofer - often an excellent choice, but not to everyone's taste (and subwoofer positioning can be almost impossible in some listening spaces)

The major effect we hear is simple loudspeaker intermodulation distortion.  A loudspeaker driver uses a motor, consisting of a voice coil, which is "immersed" in an intense magnetic field.  The radiating element (usually a cone or dome) is coupled to the motor, and supported by a surround of corrugated material, rubber (usually synthetic) or foam.  Additional support is provided by the spider, which is attached to (or near) the voice coil former - this is essential to prevent the cone from shifting, and causing the voice coil to rub on the magnetic pole pieces (poling).

The surround, spider and the motor itself are linear over a limited range.  The maximum excursion of a driver (Xmax) describes the maximum physical movement allowed, but usually does not guarantee that this full range of movement will be linear.  If it is not linear, the speaker will distort - subtle with some, gross with others.  How do you know what a driver will be like at its limits?  You can ...

• believe the manufacturer - not always a good idea
• test it yourself - be prepared for the odd damaged driver if you go too far
• ask others for their experiences - expect many conflicting responses
• make sure that your design will stay well within limits, perhaps 25% of the maximum - pay more for the driver
• use a servo system - feedback from the cone to the amp will linearise the movement, but it only works at low frequencies

The ideal is naturally to limit the excursion to the absolute minimum, but this is not always possible, especially with bass drivers.  In this case, it is far better to relegate the bass to its own speaker altogether - a subwoofer is not just for home theatre - it can work absolute magic on normal musical programme material as well, including music that does not appear to have a great deal of low bass.

The audibility of absolute phase is nil. 

I must explain this further, as this is a somewhat contentious issue.  It can be proven in ABX tests that there are some signals where the difference between a non-inverted and inverted signal is audible.  Certain waveforms and instruments are highly asymmetrical, and if listened to in isolation will sound different if the phase is reversed.  The difference is not subtle, either - it can be very pronounced.  This is much more likely to be a result of loudspeaker driver behaviour than anything else, and the "correct" phase is anyone's guess - should it be inverted or not?  We don't know the answer, since we will be unsure of what the instrument sounded like "live" - it is possible that neither the inverted or non-inverted recorded signal will sound like the original, so the point is moot.

The key issue here is that if we listen to a saxophone (a good example of an asymmetrical waveform) with the phase normal then reversed, all we hear is a difference - there is not necessarily a "right" or "wrong" phase, since it depends on the way the instrument was miked in the first place.  If the period between listenings is extended to a few minutes, the chance of us hearing the difference will be minimal, and we still won't know which is "right" and which is "wrong" - all that this proves is that there is a difference, and it only becomes audible with some instruments.

This is probably the only case where an ABX test proves something that is not relevant in the general sense - so yes, absolute phase can be audible, but it is (generally) irrevelant. [Update added 30 Jul 2002]

The net result is that our ears do not care if there is a slight misalignment between the fundamental and harmonics of any instrument known.  This is likely to cause howls of protest from people who won't actually bother to read this article in its entirety (if at all), but it has been demonstrated a great many times, and by various techniques.

A simple all pass filter will shift the phase of an audio signal by 180 degrees over a frequency range determined by the component selection, and it is completely inaudible - provided the source is music, and provided the phase sweep is performed slowly enough for our ears and brain to make the necessary adjustments.  In fact, I have demonstrated this as the filter is adjusted (very slowly), and the sound quality remains the same.  Nearly every (Ok, not nearly - every) recording ever made has been recorded using a microphone, had some equalisation applied, and/ or has had some additional treatment in the recording process.  All of these introduce some degree of phase shift, but does it ruin a good recording?  No.  As the signal emerges from the vast majority of crossover networks, there are huge shifts of phase, as has been described above.  A square wave subjected to phase shift still has all of its harmonics present, they are just slightly misplaced in time.

The sort of delay we will experience is dependent on the frequency, but it doesn't matter.  Vented speaker boxes do "awful" things to phase, as do many highly regarded "feedback free" single ended triode (SET) amps.  Any equaliser, be it a constant Q graphic, parametric, or just a simple tone control, will introduce phase shift as well as equalisation.  The phase of a waveform changes as you move about - but your best friend sounds like your best friend regardless of your relative positions in a room, even though there are massive changes in phase as we walk around.

If we believe the "absolute phase" lunatics, this would not be the case, so your wife may sound like your wife in one part of the room, but sound like the milkman in another.  We all know that this doesn't happen - the tonal structure of a sound does not rely on the phase integrity of the received sound, only the relative amplitudes of the fundamental and harmonics.  So a speaker that has perfectly flat frequency response but is not 100% phase coherent will sound the same as one that is also flat, but totally phase coherent.  This does not include colourations caused by the cabinet or drivers - of course these are important.  Assume the same enclosure, same drivers, but a phase shift applied to one, and not the other.

In isolation, they will sound the same.  Put them together, and you will hear strange reinforcements and cancellations as you move about.  This is relative phase, and is very audible indeed.  What we need to concern ourselves with is relative phase only.  Two amplifiers with different phase responses used as a stereo pair will sound terrible if the shift is sufficient.  Use two of the same amplifier, and there is no problem.

Absolute phase is inaudible within reason - a 3,600 degree phase shift represents a time delay that is significant, but a 360 degree phase shift will not be heard.  Inverting a signal (e.g. reversing the connections to a loudspeaker driver) creates a 180 degree phase inversion, but this is not the same thing as a 180 degree phase shift!  This is a point missed by many.

Relative phase is audible, depending on the amount, the frequency and the context.  Two speakers side by side with 90 degrees phase shift between them will sound dreadful - and the sound will change as you move about.  The relative phase of two musical instruments playing in harmony makesthe sound you hear - take away the phase shifts, and it will sound flat and lifeless.

An example of a pair of very typical all pass filters is shown in Figure 14.  These are connected differently so I could show the different behaviour (not actually different, the phase of one is simply reversed from the other).

The resulting output and phase response of the filters is shown in Figures 15 and 16 respectively.  I only included the phase response graph for one version - the other is simply the reverse of that shown.

Note that this particular filter is called "all pass" - it passes all frequencies equally.  Not much of a filter by normal standards, but a useful tool nonetheless.  Interestingly, if the input and output of an all pass filter are summed, the result is an ordinary filter.  High and low pass responses are available.  Not that there is a great deal of point, since this is vastly more complex than a 6 dB/ octave filter built conventionally.  I just thought I'd mention it - someone might be interested :-)

For what it's worth, I originally started this article not to praise, but to debunk the theory that time alignment is the only way a speaker should ever be designed.  Having done the research, run tests, and written the article, I confess that I must agree with many (perhaps even most) of the points made by the time alignment proponents.  Mind you, there is still a lot that you will hear and read that is either gross exaggeration or a downright lie, and it can be very difficult to tell the difference unless you know exactly what the real story is.

My overall opinion, based on the research for this article (primarily tests and simulations), is that time alignment is a very good thing, and perhaps all speakers should be designed this way.  On the negative side, the offset required to achieve time alignment can lead to diffraction effects that may damage the sound quality far more than the misalignment.  A sloped baffle means that you are always listening off axis from the drivers - not by a great deal perhaps, but off axis nonetheless.  This conundrum can be resolved, and it has been by several manufacturers, each in their own way.

Use of 1st order crossovers means that the vertical axis of the speaker is very narrow - the speaker will sound entirely different when you stand up!  This means that the signal propagated into the room is uneven, so the natural reverberation of the listening area is not excited evenly at all frequencies.  Higher order crossovers are better in this respect, but cause their own problems.  Relatively poor transient response is always claimed, but in reality, a great many high end manufacturers are using 24 dB /octave filters, especially with electronic crossovers, and achieve extraordinary results.  My own system loudspeakers are triamped using my version of the Linkwitz-Riley 24 dB crossover, and they sound very good indeed.  They are not time aligned, but based on the results of my work on this article, I would expect that when (not if) I rebuild the boxes (or just make a new system altogether) they can sound even better.

Reproduction of a square wave is something of a myth.  I have received a very passable square wave response from a pair of small hi-fi boxes I use in my workshop.  All I have to do is select a good position for the measuring microphone.  How many sites have you visited in your quest for "the ultimate loudspeaker", where they claim (or show) the square wave response?  How many admitted that the positioning of the measurement mic has a very great bearing on whether a square wave is reproduced or not?  From what I have seen, no-one has ever claimed that a square wave is received perfectly, regardless of mic position, nor have they disclosed the actual measurement setup that was used - is this at the listening position in a "typical" room, or 300 mm in front of the speaker in an anechoic chamber?  We shall never know.

Indeed, the room itself is still the greatest offender - even a coffee table that is in the acoustic path of the loudspeaker will have a profound effect on the overall response.  Very few rooms are acoustically dead enough (IMO), and I have seen a great many photos of people's systems set up on polished marble (or whatever) floors in relatively bare rooms, with almost no acoustic deadening materials to be seen.

Human hearing is very adept at picking the original sound from the reverberant field, provided the early reflections are not so early that they influence the direct sound.  Given the highly reverberant listening rooms of some people, I have difficulty understanding how that can even tell what anything really sounds like - yet they will happily espouse their theories on what makes the sound better, ignoring the fact that their room will destroy the sound of any loudspeaker.

Finally, the quality of much of the recorded material available is absolutely woeful.  Equalised to within an inch of its life (so it will sound "good" on crappy systems), compressed, "aurally excited" (ptooey!), and generally mangled beyond all recognition.  To be sure, quality recordings are available, but are they available from your favourite artist(s)?  Usually not, so you either have to change your musical tastes to experience a decent recording, or put up with the rubbish that is often the only version of the artist/ song available.  I have so many CDs and vinyl recordings that I find unlistenable on a decent system that it's not funny - for one CD, I have to switch off my subwoofer, or all my windows will fall out!

This article started as a short explanation, intended to dispel some more snake oil, and has become the missive you see due to the vast amount of information I collected as I ran the tests and simulations.  I Hope that it has been of value to you - having read this far, I suppose it must have been. Expect an update shortly, after I have had a chance to figure out a way to determine the acoustic centre of typical drivers - perhaps manufacturers could supply this information as a part of their specifications (hint, hint).

Although the majority of this work is the result of tests and simulations I have carried out, there are a few other sources as well.  Many are part of the ESP site, and I shall not bother referencing my own work.  The only other real reference used is shown below.

ASA 130th Meeting - St. Louis, MO - 1995 Nov 27 .. Dec 01  1pEA4. Time Align(registered) loudspeaker crossovers.
Edward M. Long, E. M. Long Assoc., 4107 Oakmore Rd., Oakland, CA 94602

http://www.auditory.org/asamtgs/asa95stl/1pEA/1pEA4.html

Copyright Notice. This article, including but not limited to all text and diagrams, is the intellectual property of Rod Elliott, and is Copyright  2002. Reproduction or re-publication by any means whatsoever, whether electronic, mechanical or electro- mechanical, is strictly prohibited under International Copyright laws. The author (Rod Elliott) grants the reader the right to use this information for personal use only, and further allows that one (1) copy may be made for reference.  Commercial use is prohibited without express written authorisation from Rod Elliott.