Interpretation and measurement of acoustic phase response
Massimo Costa
ALBEDO Loudspeakers
Via C. Calisse, 78 - 00053 Civitavecchia
e-mail: info@albedoaudio.com
Giuseppe Pucacco
Department of Physics – University of Roma “Tor Vergata”
Via della Ricerca Scientifica, 1 - 00133 Roma
e-mail: pucacco@roma2.infn.it
1 Introduction
This note is inspired to an article published some years ago on Fedeltà del Suono [1]. The motivation
to write something about acoustic phase and its implications on listening was originated from the
reading of an interesting “Application Note” from Brüel & Kjær [2]: a perfect example of how it
is possible to treat in an exemplary, clear way, a not so very intuitive subject like this one. Many
technicians and amateurs in this field read and studied this type of publications, and often have
taken them as a starting point to improve their research or divulgation activity. It is therefore
puzzling that this work, which was presented on occasion of the 48th AES Convention held in
California in 1974, could remain largely ignored for a long time. One of the dark sides of high
fidelity is that some of its important technical aspects are present and well known in the relevant
literature and nevertheless are systematically ignored by designers and sound enthusiasts.
2 Frequency response
Let us consider the transfer function of an electro-acoustic linear system, that is, an ideal systems
that does not distort the input signal. In this case we know that this function is reflected in the
following expression1
H(s) = A(s)eiφ(s) , (1)
1
In this note we only show two equations: the one here above marked (1) should not intimidate the non-expert
more than necessary since we only use it to introduce the main quantities under consideration and the second one,
marked (2) in the following, in its great simplicity, will be the one more properly related and relevant for this article.
1
where s = iω = i2πf is the complex pulsation. It is important to note how this equation is made of
a part, A(s), which is the modulus or amplitude response, and of φ(s), which is the phase response.
The behavior of a speaker (or of a system of speakers) follows this rule and therefore will always
be characterized by these two responses. The first function, A(s), is well known as “the” frequency
response, while φ(s) is generally disregarded or not properly taken into account: a serious error as
we shall soon see.
In reality this attitude was, until a short while ago, a tacit acceptance of insurmountable
limitations. In fact, in order to have reliable measurements of acoustic phase it was necessary to
have significant budgets to be able to invest in test equipment, and this could only be attained in
large companies. The very fact that, according to the results, such large companies would not elect
to pursue this type of research is a matter for other considerations. Fortunately today, after the
massive spread of IT technology, things have changed radically to the point that many integrated
audio boards for digital measurement are also capable of providing reliable phase measurements.
We want to underline that by this we do not mean that any phase measurement generated by
an audio card is correct. We are saying that, once properly utilized, the audio card can provide
some indication for the correct measurement of acoustic phase. But why would we want to give
such special importance to the acoustic phase of a speaker system? Let us see how to clarify some
fundamental concepts.
3 Phase response and impulse response
We know that an impulse can be considered as the combination of an infinite number of “over-
lapped” sinusoidal components. The phase response of a system tells us the temporal relation
which relates these various frequencies; in other words, it tells us which is the phase distortion of
the reproduced signal with respect to the input signal, at different frequencies. On the other hand
we know that a phase angular value can be seen, for all purposes, as a temporal delay. Now, a very
bad phase reproduction means that the various sinusoidal components which form an impulse will
not be reproduced contemporarily but rather with delays which are different per each frequency,
which means in other words that the original impulse form will be rendered severely distorted.
From now on, therefore, ne need to keep very clear in the back of our mind that a good phase
response is the same thing as the correct reproduction of the impulse. This point is very important
as it also explains why it is absolutely incorrect to design a loudspeaker system only examining the
frequency response. Such system will only be able to accurately reproduce sinusoidal signals, but
will fail in the reproduction of typical transient signals in a musical message.
Once we have clarified that acoustic phase response is very important, two problems arise
immediately: how can it be measured and evaluated, and which are the technical choices which
allow to optimize it An entire book would not be sufficient for an answer to the second question,
and furthermore, every designer has his own ideas on the subject so that there is no universal
recipe to apply (without mentioning the fact that some other designers simply decide to continue
to ignore the problem).
On the other hand, as far as the first question is concerned, let us see if we can clarify our ideas
a little bit. The concept of the identity between impulse and phase responses is well represented
in Fig.1. For ease of graphic representation let us consider a square wave, in place of an impulse.
This square wave is a kind of “sustained” impulse and can be assumed to be made of the “union”
between a fundamental sine wave and its (infinite) odd harmonics, which in this case, still for
2
Figure 1: Under the original square wave, the charts shows its first three harmonic components
and their superposition. Further below, the same harmonics with a 90-degree phase delay are also
superposed, with a substantially different result with respect to the original square wave.
3
Figure 2: Phase response with delays independent from frequency.
reasons of graphic representation, we limited to two. As we can see, in the absence of phase delay,
the various components are superposed into a waveform which is virtually identical to the original
one. This is not the case for a phase delay of, for instance, 90 degrees. The sum of the fundamental
sine wave with its two first harmonics, all phase delayed of 90 degrees, yields a final waveform which
is quite different from the original one. The reason is also graphically intuitive: a constant phase
delay of 90 degrees implies different time delays per each wavelength being considered. Since 90
degrees are actually one quarter of a period (which obviously is 360 degrees), it results that a phase
delay of 90 degrees, for instance at 1000 hertz corresponds to (1/1000)/4=0.00025 seconds, or 0.25
milliseconds; at 10.000 hertz the same phase delay corresponds to (1/10.000)/4=0.000025 seconds
or 0.025 milliseconds, and so on for the various frequencies in question. From these considerations it
derives that, if we want to represent an ideal behavior of the phase angle as a function of frequency,
we should try to obtain a horizontal line, absolutely flat; in other words, under varying frequencies
the phase delay with respect to the input signal should always be zero. At this point however it
is important to observe that we have seen how, at different frequencies, equal phase delays (for
instance 90 degrees) are not reflected in equal time delays. If we continue with this reasoning we
must also observe that equal time delays do not correspond, in turn, to constant phase rotations.
Let us now examine Fig.2. A system with no phase delay at varying frequencies would be repre-
sented with a straight line, coinciding with the frequency axis, just to make it clear, the horizontal
one. If, on the other hand, we introduce some time delay, applied equally to all frequencies, the
straight line gets slanted towards the low part of the chart, with an angle which depends on the
delay in question, and becomes for example lines a or b. This line has equation
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2πD
φ(f ) = − f, (2)
c
where c is the speed of sound, and D is the distance from the source to the microphone. Attention
should be paid, however, as we are talking of the same physical system to which we just added some
delay. It is appropriate to remember that “flight” maintains the phase relation, as the sound speed
across air is independent of frequency. But then, why does the straight line get slanted? Based on
what we have seen in Fig.1, a certain time delay will cause, for instance, a phase shift of 90 degrees
for a given frequency, a phase shift twice as high for its second harmonic, three times higher for its
third harmonic, and so on. In other words, since the phase shift for a constant value of time delay
varies linearly with frequency, the straight line remains in fact straight, but get slanted downward
(it would become inclined upward if we had to deal with phase gains).
In Fig.3 we can see the curve of an ideal acoustic phase generated by a known system of
measurement, Clio from Audiomatica [5]. The red curve, which in fact is a straight line, lays on
the horizontal line of 0 degrees and it is important since it represents our ideal behavior: no phase
shift with varying the frequencies. The green curve is the same as the previous one but with the
addition of a delay of 0.01 millisecond and the yellow one is still the same, this time with a 0.1
millisecond delay. But, didn’t we just say that the original straight line, in case we added a delay,
had to become slanted downward? In fact, this is true only in a plot where frequency is represented
linearly. In most common graphics the frequency scale is represented logarithmically and therefore
it is obvious that the straight line is no longer a line but becomes a curve. But let us understand
that this is only a problem of graphic representation. If we went to read, patiently, the values of
phase changes as a function of frequency, we would find out that they are the “correct” ones. The
reason being that in a logarithmic plot a straight line remains as such only if it lays on the 0-degree
abscissas axis; otherwise it becomes a curve. Is this important? For sure, because if we are not
capable to distinguish a correct behavior but only a delay affected one, how can we evaluate all the
other behaviors which are not perfect (and in real life they will never be)?
Another interesting case is the one indicated in the top part of Fig.4. The straight line, still on
a linear frequency scale, is inclined but does not pass through the origin. A question then: in such
case, does the system have a good phase response? Well, not at all. If we apply the reasoning we
just made related to the delay and “take away” a proper amount of delay to this measurement, the
straight line of the top example becomes the one in the example below, both in Fig.4. Except for
the case of φ equal to π or one of its entire multiples (which also means to rotate everything by 180
degrees or its multiples), in which case nothing happens, in all other cases we would have a phase
distortion which is constant with frequency and this (Fig.3 considered a value of φ = 90◦ ) does not
allow a good response to impulse. Furthermore, in the case of a logarithmic scale, in which zero
corresponds to “minus infinite” on the abscissas axis, it results rather difficult to determine if the
straight line passes or not through the origin of the axis! We have been speaking several times of
delay, but somebody could correctly remark “But why don’t we get rid of this delay, so we simplify
our life?” As if it were easy...
4 Talking about measurements
The users of measuring equipment adopting the MLS technique, the one most commonly used today
to obtain acoustic phase response (in addition to a wide range of other things [3, 4]), are aware
that, in order to obtain correct anechoic curves, they must apply a window to the time sequence
5
Figure 3: Response with zero phase delay (red curve) and with uniform delay of 0.01 milliseconds
(green curve) and 0.1 milliseconds (yellow curve) in logarithmic representation.
Figure 4: Top: response with a linear phase shift. Bottom: same response, uniformely delayed.
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of the impulse (Fig.5). On one side this window must exclude the “flight time”, that is the time
taken by sound to go through the space between speaker and microphone and, on the other one, the
first “disturbing” reflection. Leaving aside whatever happens after the main impulse, that is the
problem of reflections, which for the moment is of no interest for us, let us go and see what happens
before the impulse. Now, the first thing that could come to mind is to start the analysis from the
first non-vanishing sample of the impulse or, in other words, to eliminate any form of delay. Well,
this which theoretically would be the most correct thing to do, can result inaccurate with some
test systems. Let us look into the reasons right away: the measured delay is not only due to the
sound “flight”, that is, to the time interval corresponding to the distance between microphone and
speaker, even if this is by far the largest contribution. Actually, there are other smaller but notable
components that affect the total value. One of them, in this respect, is the intrinsic delay of the
test card which depends on the chosen hardware architecture. Attention should be paid to this
problem, therefore, which can present itself in different forms depending on the instrument chain
being used.
Figure 5: “Windowed” impulse response.
Other elements which contribute to “mask” the true pattern of the phase response are the phase
distortions due to microphone, to its pre–amplifier used for the test and to the amplifier which feeds
the speaker. As far as the phase distortions introduced by the front-end mike-preamplifier, these
are generally well contained since the bandwidth is generally very wide (or at least so we do
hope). The same thing cannot be said any more for the amplifiers which are sometime used in
measurements, that can present a limited bandwidth. In this case the deviations at the extremes
of the band are notable, even if the phase distortion at low frequencies is more pronounced than
the one associated with the high extreme. There are two ways to neutralize this phenomenon: find
out an amplifier with an extremely wide bandwidth, or create a calibration file which includes its
own phase non-linearity.
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Figure 6: Response with no phase shift (red curve) and with uniform 20 microsecond (green curve)
and 40 microsecond (yellow curve) delay.
There is another problem in the measurement of acoustic phase, a significant one, which is tied
to the temporal resolution of the measuring system, that is, to the sampling frequency adopted. In
the case of Clio, this frequency is 48,200 hertz and this means that, in the process of sampling, we
can measure intervals that are no smaller than approximately 0.02 milliseconds, or 20 microseconds.
“Isn’t this sufficient?” somebody may say. Well, it depends, because in 20 microseconds sound
goes through air a little less than 7 millimeters. In Fig.6 we can see our ideal straight line with the
application of delays of 20 and 40 microseconds, respectively. This means two things: first, that
we cannot evaluate the exact distance between microphone and the acoustic center of the driver
being measured without a minimum error of 7 mm, and second, that the consequent error in the
measurement of the phase, or possibly better said, in its graphical representation (as a function of
the introduced delay), extends into an area included between the horizontal line which indicated
0 degrees and the curve associated with the 20 microsecond delay, or, which is the same thing,
between the curve of 20 and 40 microseconds.
It is important to recall the concept that a measurement of an acoustic phase affected by delay is
no less true than the one where the delay has been completely eliminated: it is only more difficult to
read or, to an extreme, not usable at all. On the other hand, the only way to reduce this uncertainty
is to increase the sampling frequency, but then increase by far the cost of the measuring equipment.
So, coming back to the original problem, in order to have a correct acoustic phase where to have
to place the starting point of the window to apply to the impulse? Still a little patience.
Here the concepts of “minimum phase” and “Hilbert phase” come to partial help. Starting with
the modulus of the frequency response, we can trace back the phase pattern using an operation
called “Hilbert transformation” [6]. However, this operation only makes sense if the system under
consideration is a “minimum phase” system, that is when the points of minimum and maximum
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of the modulus correspond to the points of inflection in the phase curve and when a flat modulus
actually corresponds to a vanishing phase shift. Let us avoid entering into any further details, but
suffice to say that a single speaker, in its linear zone, can be reasonably assimilated to a minimum
phase system, while the same thing is not true, for instance, with a system of multiple speakers
under crossover network. The “Hilbert” phase is thus, indeed, a “theoretical” phase which we
derive from a modulus curve, but is also a phase that is independent of any delay and therefore
that will be almost coinciding, in the linear range of the speaker under measure, with the correctly
measured phase, that is, excluding any delay which is external to the system. In practice therefore
it may be useful to generate the minimum phase and, comparing this with the one measured in the
speaker linear range (Fig.7, blue curve), then vary the starting point of our analysis window until
we obtain a curve as close as possible with the minimum phase (Fig.7, red curve), still keeping
in consideration as valid all problems of temporal resolution we were mentioning before. In other
words, it would be possible to have the minimum phase in a position such that it cannot be close
to the one measured, because a sample added or subtracted may introduce an excessive change
in the shape of the curve. In this case it is useful to try and move the microphone used for the
measurement forward or backward by a few millimeters and in any case no more that the 7 mm
that correspond to a sampling step, otherwise we are back to our starting blocks. In the case we
just showed, we added to the phase measured using the start of the window immediately before the
impulse (blue curve) an appropriate phase gain (a “negative” delay, red curve) to make it similar
to the minimum phase.
Figure 7: Phase response with start of the window immediately after the impulse (blue curve) and
with a suitable phase gain (red curve).
As we just said, taking Hilbert phase as reference is allowed only for systems that can be
considered with “minimum phase”. It is therefore useful whenever we want to generate curves for
non-filtered drivers to import in simulation programs, where an incorrect evaluation of acoustic
9
phase induces fatal errors in the simulation of drivers–network behavior at the crossing point. On
the other hand this technique is to be avoided in case it is necessary to evaluate the acoustic phase of
a complete speaker system since, as we have seen, the eventual minimum phase generated, without
taking into account possible reciprocal delays between different drivers, would have an “optimistic”
behavior, in any case not representative of the real measured phase, independently from the total
delay considered.
5 Conclusions
The overall performance of a loudspeaker system depend on a large number of parameters, to which
any different designer assigns his own priorities, and only few of them can be easily associated with
the listening perception. In other words, we came across speakers that once measured did not have
an exceptional phase response, but still showed an optimal general performance. On the other
hand, it is also true that we never heard good speed and coherence in speakers that had a very bad
phase response. Without pretending that it is a panacea against all evils, it is however evident that
the control of acoustic phase is an indispensable criterion to achieve the maximum performance for
sound reproduction in a loudspeaker system.
In any case, there is the sad consideration of noticing that, in almost 35 years from when
Henning Møller of B&K wrote this article [2], only a handful of designers actually treasured it.
One more mystery, in fact, of High Fidelity.
References
[1] M. Costa & G. Pucacco: La Fase Acustica, come misurarla e come interpretarla, Fedeltà del
Suono, 39, 75–82 (1995).
[2] H. Møller: Loudspeaker phase measurements transient response and audible quality, Brüel
& Kjær Application Notes, 48th Convention of the Audio Engineering Society, California
(1974).
[3] J. D’Appolito: Testing Loudspeakers, Audio Amateur Press (1998).
[4] V. Dickason: Loudspeaker Design Cookbook, 7th edition, Audio Amateur Press (2006).
[5] M. Bigi & M. Jacchia: Clio User Manual, Audiomatica (2008).
[6] E. Gatti, P. F. Manfredi & A. Rimini: Elements of Linear Networks, Editrice Ambrosiana
(1966).
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