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by Richard C. Heyser
THE FREQUENCY RESPONSE of a speaker consists of both an amplitude of sound pressure level and a phase of that sound pressure. Frequency response is measured by applying a sine wave of voltage to the speaker terminals and measuring the resultant pressure. This will be of the form.
P = A sin (2π ft + Φ). The number A is the amplitude of the pressure and tells us "how much" sound pressure is generated at the frequency, F. The angle (2π ft = Φ) is the phase of the sound pressure and contains the information telling us "when" the sound arrives at our microphone position. The number 0 is the initial phase of the angle and is a measure of the polarity of the sound pressure. Since phase is an angle, it is measured in degrees.
While the standard exists for measurement of amplitude, there is as yet no agreed-upon standard for phase measurement, because it is such a new measurement for loudspeakers.
In order to provide an accurate measurement of phase which can be defined on an absolute basis, AUDIO uses an interim standard which we have defined specifically for these tests and which has been derived from both speaker and microphone convention. If a positive voltage is applied to the positive (or red) terminal, the cone should move outward from the magnet assembly. (Unfortunately, while the great majority of speaker system manufacturers use this standard, there are a few notable exceptions). If this speaker were used as a direct radiator in the conventional manner, this means that a sound pressure increase will occur in front of the speaker with application of a properly-phased voltage. Pressure-calibrated micro phones are polarized so that a pressure increase generates a positive voltage at the positive terminal. AUDIO has joined these two conventions to determine a phase standard. An absolute reference of zero degrees of phase is defined when a positive going voltage applied to the positive marked speaker terminal produces a pressure increase exactly in phase with that voltage when the time delay of the air path between speaker and microphone is removed from the measurement. The definition of what constitutes a phase lag or lead directly follows the defined physical basis of the measurement. A retardation of phase with no change in amplitude constitutes time delay.
One industry benefit of absolute phase is that it is possible to know that a pressure increase in the recording microphone of a session can be properly processed by simple bookkeeping to assure that the ultimate listener will also experience a pressure increase during the proper time sequence. For those who feel that a pistol shot (an overpressure followed by an under pressure) must sound the same whether it is a pop or suck, considerations of absolute phase are meaningless. There is some evidence, however, that the difference is not inaudible. AUDIO has taken the step of introducing phase measurements and standardizing them in order to fill an industry gap. We hope others soon follow.
More significant to the readers of the speaker reviews is the fact that you have a basis for comparing different speakers for their compatibility in quad and stereo reproduction.
We all know that an out-of-phase woofer can seriously impair low frequency response. Consider the effect of an in-phase woofer but out-of-phase midrange or tweeter.
All that is required to cause an audible difference in stereo is to reverse the midrange of one channel relative to the other. The principal problem will be an instrumental wandering as one changes seating position and as a function of pitch. As long as both channels of a stereo installation are phased the same, instrumental wander will seldom be a problem, but change one channel and problems arise. If you are considering expanding an existing stereo installation into quad, the phase measurements we provide can be of great value in deciding what to purchase that will be compatible with what you now have.
From a diagnostic point of view, the phase response can quickly reveal sonic difficulties. A wandering minstrel in stereo in quite fine, unless he were supposed to be firmly placed for artistic balance when the session was recorded. Human sound perception does a remarkable job of recreating a stereo or quad ensemble of sounds from rigidly fixed speakers. In general, we come to expect that a serious audition should present the sonic illusion of artistic presence and configuration. Twenty-four foot wide pianos were swell gimmicks in the early days of stereo, but their charm has somewhat diminished with audience maturity.
Flanging and phase effects are good techniques for artistic embellishment of a performance, particularly of what has come to be known as electronic music. One might suspect that comparable manipulation of the phase of a signal purported to represent a natural sound may not yield a completely realistic reproduction. A rapid change of phase spectrum with seating location will yield such unnatural affects assuming that the change is other than that represented by distance only.
Speakers having highly angle-and position-dependent phase spectra generally demand sitting on a carefully determined chalk mark for good stereo imagery of direct sound.
The classic 12-dB-per-octave crossover network demands a phase reversal between drivers in order that the pressure amplitude spectrum not have a dip at the crossover frequency.
It can be shown rather easily that the only independent passive crossover networks capable of complementary phase and amplitude response are the 6-dB per-octave RC networks. Subtractive networks in which the signal to one channel is subtracted from the input to yield the complementary channel can be made ideal for any crossover slope. The phase and amplitude frequency response readily shows how well the speaker designer met his goal.
The fly in the ointment is that the speakers are not in the same physical and acoustic location. You might be able to touch the cone of the woofer but its effective acoustic position may be a number of inches inside the enclosure. Many manufacturers place the tweeter well in front where little diffraction of sound can take place. The time difference between crossed-over drivers can then be enough to upset the ideal crossover conditions. This causes both amplitude and phase disruptions. Is this bad? In some cases it is and in others it is not, but you know what to listen for when supplied with the complete frequency response.
Historically the great majority of audio networks (amplifiers, tone controls, etc.) have been of minimum phase type. The logarithmic simplicity of being able to add dBs to calculate network cascading, quickly forced the engineer to use the amplitude response for all his measurements.
Because he didn't have to worry about phase response--it came along in well behaved fashion--he got used to a situation where "flattening" the frequency response gave better square waves and such. It wasn't until quite recently, when we began seriously making speaker phase measurements, that most engineers began understanding why it didn't follow that flattening a loudspeaker amplitude response did not always produce the best transient response.
That will only happen with a minimum phase transfer function, and many times a loudspeaker is non minimum phase.
You will occasionally see situations where phase transitions occur which are in excess of 360 degrees. These measurements are accurate and meaningful. In mathematical terms: nature does not function modulo 2π. The presence of such a characteristic is indicative of a generally non-minimum phase behavior. A network with such properties has a narrow band "ringing" sonic characteristic. It is not necessary to exhibit the sonic equivalent of resonance.
This is a common characteristic you often hear on long distance telephone communication due to extreme all pass equalization.
General non-minimum phase acoustic performance is evident from a joint inspection of the amplitude and phase spectrum. A theoretical derivation of the geometric features to be expected for amplitude and phase has recently been presented in technical literature by this author. In the plots presented by AUDIO, the overall characteristic of a minimum phase speaker are as follows: A peak in amplitude should occur at a maximum downward slope of phase with linear increasing frequency. A dip in amplitude should occur at a maximum upward slope in phase. A peak in phase should occur at a maximum upward slope in amplitude, and a dip in phase should occur at a maximum downward slope in amplitude. It may sound complicated but it is easy to verify a minimum phase network once you get the hang of it. An easy way to remember the behavior is to imagine the peak and slope relationships between a sine wave and a cosine wave plotted as a linear function of frequency. If the amplitude plot is a sine wave, then the phase plot will be a cosine wave if the device is minimum phase.
The interest in minimum phaseness centers around the fact that the majority of equalizer circuits you can use are also minimum phase in transfer characteristic. This means that equalization of a minimum phase speaker by conventional amplitude equalizers will result in more accurate sound. Conventional amplitude equalizers of a non-minimum phase response irregularly may improve the sound or actually make it less accurate depending on the irregularity. The concern lies with accurate sound.
The actual acoustic crossover frequency may be readily determined from the phase response. This is immediately obvious on a linear frequency basis. Occasionally an embarrassing discrepancy may be noted between the acoustic crossover and that frequency specified by the manufacturer. If the actual acoustic crossover frequency is very far below the stated crossover, then beware of a higher frequency unit being driven near its lower acoustic limit.
"Stretching" a midrange or tweeter can lead to relatively high driver distortion.
The acoustic position of drivers can be obtained from the phase measurement made on a linear frequency basis. Occasionally AUDIO finds it necessary to make separate phase measurements on each driver because they are so far apart in space that one phase response would be meaningless.
A time delay corresponding to an offset of just under one foot will have a phase spectrum with a uniform phase slope passing through 360 degrees for every kiloHertz increase in frequency.
Even if you do not subscribe to the philosophy that all the sound should recombine as though from one source, you should note the behavior of the transition between drivers. The transition in phase should be uniform without severe discontinuities.
The frequency response may, in certain cases, have a special form of redundancy such that the amplitude response and the phase response are mathematically connected. When this occurs, we can use simple methods to "flatten" the amplitude response and the phase response will also be automatically corrected. The transient performance of the loudspeaker with this property can thus be simultaneously corrected for most accurate timbre by the use of conventional frequency equalization. A device which has this type of redundancy is said to be "minimum phase." The term derives from the early days of network theory when it was observed that it was possible to design a great many different networks that all had the same amplitude response but only one of them had a minimum accumulation of phase lag (negative phase shift) as you swept up through the frequency range. All others had a greater phase accumulation and were called "non-minimum phase." Technically, a minimum phase behavior has all of its zeros in the left half frequency plane Mathematically, this places all logarithmic singularities where an analysis based on steady-state frequency can be made unequivocally, and there is only one possible amplitude response for each possible phase response. If you know one of them, you know the other. The mathematical relation between amplitude and phase is then that of Hilbert transform. In audio networks most of our concern rests only with the description of frequency performance as an amplitude part and a phase part, since these are logarithmic expressions of the transfer function. We, in fact, use amplitude and phase BECAUSE they are logarithmic. Cascading networks (which multiplies frequency responses) may thus be treated by ADDING the amplitude responses in dB to get the amplitude of the resultant, and ADDING the phase responses in degrees (or possibly radians) to get the resultant phase. It makes audio analysis much easier.
(Audio magazine, Dec. 1974; Richard C. Heyser)
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