Time & Group Delay in Loudspeaker Systems

Time Delay in Loudspeaker Systems The time delays experienced by different frequency components in passing through a loudspeaker system are governed by certain fundamental laws of physics. The consequences of these laws are definite constraints between the amplitude and the phase responses as functions of frequency. The phase and time delay characteristics of a loudspeaker system can be controlled only inasmuch as the amplitude response can be controlled, unless nonminimum phase all-pass networks are incorporated. The capabilities of the PAM technology with regard to the laws of physics and the phase and group delays in particular are reviewed. Measurements for an example loudspeaker system are included in this paper. Introduction

Mathematically speaking…

Time and phase concepts The behaviour of a linear network The system function of a linear network or system is usually defined or system can be described as the ratio of the Laplace transform of a response or output variable mathematically. The ability to to the Laplace transform of an excitation or input variable. make such a definition then  Alternatively, the Fourier transform can be used in place of the enables the use of all the powerful Laplace transform [4, pp. 2–5, 86–88] if the analytic properties of the tools of mathematics to assist in system function are not being examined and only the sinusoidal understanding and predicting steady-state response is required. behaviour of the system. The Laplace transform of a real function of the real time The work of Laplace and Fourier variable t is provided powerful tools to enable the analysis of general systems including loudspeaker systems – the Fourier and Laplace transforms. These tools enabled analysts to view the behaviour of a which is a complex function of the complex frequency variable system quite consistently in either the time domain or the frequency domain. For loudspeakers in particular this represented a very useful facility. The imaginary axis of the s-plane contains the real angular frequencies (radians/s). For example, when appropriately modelled, the frequency response For a loudspeaker system, the output variable is usually the sound of a minimum phase system pressure at a standard distance from the driver while the input analysed over the full frequency range of interest can be used to variable is the voltage applied to the driver of a passive system construct the impulse response of or to the amplifier of an active system. The system function is then the system in the time domain. defined as  Alternatively, given given the reciprocal nature of the analysis tool, the impulse response of a minimum phase network can be used to determine the full frequency response in the frequency domain. For sinusoidal steady-state analysis, and the system function can be expressed in cartesian form with real and imaginary In addition, the analysis approach parts or in polar form with an amplitude and phase angle enables the use of complex variables (argument): to effectively represent both the amplitude and the phase frequency response of the network or s ystem. The The con conce ce t of of am am litu litude de res res onse onse is

generally well understood, but some The complex natural logarithm of of the work relating to phase response is little understood and warrants some clarification. What is apparent from examination of the mathematics is that there is a rigorous definable relationship  between amplitude amplitude and phase for many classes of network – but not all.

is

where is the attenuation in nepers and angle in radians.

is the phase

Basic phase and amplitude relationships The simplest case for defining phase relationships is the study of relative timing for single frequency sinusoids. Study of this case provides a useful basis for understanding of the underlying concepts. But a word of caution; care must be exercised when forming conclusions for extended cases of multifrequencies. Exceptions are described in the following section on non-minimum-phase networks. networks. Considering a single sinusoid as a function of time, the phase of a point can be referenced to the start of the sinusoid by simply considering the period as comprising either 2p radians or more commonly 360 degrees. Thus, for example, a 100 Hz sinusoid has a period of 10 milliseconds. After a time from the start of a cycle of say 5 milliseconds, the phase of the point relative to the start of the cycle is forward 180 degrees. Note that the specification of the phase requires the defining of a reference point for measurement of phase, and a statement of the relativity. Thus there is a relationship between time and phase for a sinusoid. For the reverse relationship, a constant time delay can be expressed as a phase relationship for the single frequency sinusoid. For the above example, 5 milliseconds of delay corresponds to a phase lag (relative to the undelayed sinusoid) of 180 degrees for a 100 Hz sinusoid. If another frequency of sinusoid was chosen, a different phase lag would result. For example, for a 50 Hz sinusoid, 5 milliseconds of delay corresponds to a phase lag of 90 degrees. Generalising this relationship, a constant time delay corresponds to a linear change in phase angle with frequency, and therefore a network having a linear phase variation with frequency would be a constant time delay network. The following graphs show the phase lag for a constant time delay of 1, 2 and 5 milliseconds.

What is clear from this example is first that the rate of change or slope of slope  of the phase lag with frequency relates to the time delay existing, and second that for time-delay purposes the absolute value of the phase is arbitrary. Given that loudspeakers are usually measured at a distance from the source, considerable phase slopes will result and will be an indication of the total system response modified by the time delay for final propagation to the observing point. In acoustics, it is usual to use logarithmic frequency plots but it should be noted that the phase lag for a constant time delay is not represented by a straight line on such log plots.

-p h a s e c o n s t r a i n t s   Non-minimum phase networks A m p l i t u d e -p

The analytic nature of the system function implies that the most general The term minimum phase is phase is used in the mathematical sense  form is given by the quotient of two polynomials in s with real to describe the behaviour of a coefficients: system where the output does not go to zero at any point in the measured frequency band (strictly for any complex frequency in the right half of the s-plane). The most simple example of an acoustic network of the type where this can occur is when there are two paths of propagation enabling signals to arrive completely out of phase and cancel at a single or even

The complex roots

of the numerator polynomial are called

"zeros" while the complex roots of the denominator polynomial are called "poles" of the system function.

multiple frequencies. This is easy to create in practice  A minimum-phase system function has no zeros in the right half-plane of and occurs for example with the s-plane. A general system function can always be factored into the most measurements where  product of a minimum-phase minimum-phase system function and an all-pass system ground reflections occur. Here there is a reflection which is time-delayed because of the  function, of the form , which has unit magnitude magnitude along the extra propagation path length. The condition is also achieved imaginary axis [3, p. 285]. with ported enclosures and dipole radiators for example, where there are two separate sources of acoustic output. The resulting non-minimum phase network is able to be modelled and analysed mathematically, but presents a case where the rules governing the link between attenuation and phase are not valid. Care should be exercised when analysing these types of networks. The term non-minimum phase arises from the fact that the network does not follow a well behaved continuity of phase over the band of interest. Minimum phase networks on the other hand exhibit well behaved attenuation and phase characteristics where in particular there is a formal relationship between the attenuation and phase response of the network. As a consequence, with minimum phase networks, once the amplitude response is defined the phase response is fully defined. Equally, defining the phase response means that the amplitude response is then fixed.

Non-minimum phase all-pass networks Networks exhibiting nonminimum phase characteristics can certainly provide continuous output across a band of interest.

The real and imaginary parts of a general system function are related as  Hilbert transform pairs:

 Provided the system function is of minimum-phase type, the attenuation attenuation and phase angles are also related as Hilbert transform pairs. These theorems allow the unique determination of one member of the pair from the other. A modified theorem shows how to construct a minimum-phase  system function when the attenuation and phase angle are specified on complementary parts of the imaginary axis [4].

The detailed modelling and analysis of these networks does require more careful consideration than the more mathematically tractable minimum-phase networks where relationships between amplitude and phase are constrained. Time, phase and group delay  As discussed previously, the rate of change of phase with frequency equates to a measure Definitions o f time delay  of time delay. The system function can be written as Phase delay, or more accurately the time delay associated with change of phase with frequency, f requency, is generally a function of frequency. It is defined as the difference between the zerofrequency reference phase and in which the phase delay is defined as the measured value of the phase normalised to the frequency of measurement. The choice of zero-frequency phase is of little consequence for Usually the zero-frequency phase angle loudspeaker systems. Group delay is again a measure of time delay and is defined as the rate of change or slope of negative phase with frequency and is also a function of frequency. It has relevance only over a narrow band of frequencies.

and the zero-frequency phase

delay are of no consequence in a loudspeaker system as they result in no waveform distortion in the time domain. Deviations from phase linearity (constant time delay) occur when transient waveform distortion [6].

differs from

and cause

The group delay is defined as

On a plot of negative phase against frequency on linear axes, the group delay at a given and is relevant in determining the time delay of the envelope of narrow-band frequency is the slope of the waveforms like tone-bursts [4, p. 135]. Now: tangent line whereas the phase delay is the slope of the chord line drawn from the phase value at zero frequency to the phase value at the given frequency. Two conditions are of passing interest. Therefore First, at frequencies where the rate of change of the phase delay is zero, the group delay equals the phase delay. This can occur at any point in the pass

band. Hence, provided , Second, as the frequency approaches zero (or for the , lowest frequency asymptote), the  Also, when frequency or dc) insertion delay. group delay again equals the phase delay. This delay value is called the insertion delay (or the zero-frequency or dc delay).

whenever

.

which is called the (zero-

System response and the phase and group delay. There are some fundamentally important rules applicable to the phase and group delay of a system. First, all sections in the filtering chain contribute to the phase and group delay. The delays for filter sections are generally additive and so both low and high pass filter sections contribute to the delay. For loudspeakers, this includes all components in the chain; the recording chain, the reproduction amplifiers, filters, loudspeakers, enclosures and the acoustic environment of the listener.

There are several important theorems concerning phase and group delay. For T There example, it is easily shown that each pole and zero of the system function contributes individually to the phase delay and to the group delay [3, pp. 230– 232]. Furthermore, the amplitude-phase constraints on system functions translate into constraints upon the phase and group delays.

P ec ec u l i a r i t ie ie s o f h i g h - p a s s a n d b a n d - p a s s s y s t e m f u n c t i o n s  

For high-pass and band-pass systems, such as loudspeaker systems, the amplitude response drops to zero as the frequency approaches zero. Hence the zero-frequency insertion delay is impossible to measure directly and may even be difficult to estimate at all. Typically the group delay reaches a Second, for high-pass and band- maximum at the lower cutoff frequency and the zero-frequency phase angle pass systems, the amplitude can be estimated such that the phase delay at the cutoff frequency has a response drops to zero as the value similar to the group delay at the cutoff frequency. Each zero of the system frequency is lowered and so it is impossible to measure the so function at the origin contributes a zero-frequency phase angle of radians called zero frequency delay (90 degrees) and an attenuation slope of 2.303 nepers/decade (20 dB/decade). (phase and group delay are equal at this limit.). All The conventional frequency transformations for mapping a prototype low-pass loudspeaker systems are either system to a high-pass or band-pass system do not preserve phase function high-pass or band-pass in linearity or constant time delay [3, p.535]. Phase linearity is improved, however, nature, with sealed systems if a frequency transformation is used that has arithmetical symmetry rather than being of lowest order. geometrical symmetry [10, 11]. Third, the low frequency cutoff of the loudspeaker is predictable from the loudspeaker design. Closed box designs have a second order rolloff. This rolloff exhibits a low frequency slope of 40 dB per decade and a low frequency phase shift of 180 degrees. Vented boxes and dipoles exhibit a low frequency slope of 80 dB er deca decade de and and a low low fre uenc uenc

In loudspeaker systems the type of loudspeaker enclosure determines the attenuation slope and phase angle below cutoff. Closed boxes have secondorder rolloff (two zeros at the origin giving asymptotes of 40 dB/decade and 180 degrees) while vented boxes have fourth-order rolloff (four zeros at the origin giving asymptotes of 80 dB/decade and 360 degrees).

limit phase shift of 360 degrees. High-pass filters in the amplifier chain will add to these figures as will dividing networks [12]. The time delay of a loudspeaker system can be adjusted to some extent by controlling the alignment, that is, the amplitude response as a function of frequency, but only within the bounds dictated by the amplitude-phase constraints. In practice, the adjustment is achieved more readily at the upper end of the pass-band.

PAM technology PAM (parametric acoustic modelling) technology has been described in an AES preprint [13]. Basically PAM describes a technology for low frequency enclosure design where the driver displacement can be controlled over an extended range of frequencies whilst at the same time controlling the amplitude and/or phase response of the enclosure over these frequencies. Maximum efficiency is maintained in the band by utilising controlled acoustic impedance structures with low loss. What can be said about PAM PAM provides four significant advantages over existing enclosure designs. Higher efficiency  The enclosure design provides for more sound pressure level over the desired band than conventional vented or sealed designs for a given driver displacement. Loss is not deliberately added over the pass-band. Lower distortion The enclosure design provides for lower distortion over the band by reducing cone displacement, and by providing an additional low-pass filter structure. Controlled amplitude response More flexible control of the amplitude response is possible than with other enclosure types. The amplitude response can be tailored to suit different acoustic environments. Controlled group delay response  A limited control of the group delay is possible possible in association with the control of the amplitude response. The group delay can be tailored but not independently of the amplitude response. What cannot be said about PAM 

PAM is not magic.

PAM does not defy the laws of physics. The amplitude and the phase response are irrevocably linked and one will affect the other. But the enclosure can be designed for controlled amplitude response or group delay or both. Proof of the pudding – measured response The EMW320 is a passive subwoofer loudspeaker system utilising the PAM technology for low frequencies up to around 200 Hz (suitable for crossing over at 100 Hz or below). Some design effort was invested in providing group delay alignment consistent with a smooth amplitude response. The measured phase angles were adjusted to pass through zero degrees at about 80 Hz. Allowance was made for the propagation time from the vent opening to the microphone one metre away in an on-axis position. The first graph plots the magnitude and phase of the SPL while the second graph shows the phase delay and the group delay for the loudspeaker system. At extremely low frequencies the measurement noise dominates. Also, being a derivative, the group delay is very sensitive to small fluctuations in the phase function.

From the delay graph it can be seen that the phase delay rises steadily from about 10 ms at 100 Hz to a peak of 23 ms at 28 Hz. The group delay is only about 3 ms on average at the top of the pass-band then rises through 16 ms at 30 Hz to a maximum of about 35 ms below 20 Hz. As expected the group delay equals the phase delay at 28 Hz because the gradient of the phase delay is zero there. Beyond the top of the useable pass-band the phase and group delays converge to about 2.5 ms. This is mainly explained by the fact that the driver of this particular loudspeaker system is actually well inside the enclosure about half a metre from the vent opening.

For interest a third curve has been added to the SPL graph. It shows the phase lag associated with a delay of 20 milliseconds being the measured phase delay of the loudspeaker system averaged over the lower end of the pass-band (say 18 Hz to 50 Hz). References [1] H. W. Bode, Network Analysis and Feedback Amplifier Design (Van Nostrand, Princeton, N. J., 1945) (14th printing with supplementary material added by R. E. Krieger, New York, 1975, from the 13th printing by Van Nostrand, 1959). [2] E. A. Guillemin, Theory of Linear Physical Systems (Wiley, New York, 1963). [3] L. Weinberg, Network Analysis and Synthesis (McGraw-Hill, New York, 1962). [4] A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962). [5] R. W. Daniels, Approximation Methods for Electronic Filter Design (McGraw-Hill, New York, 1974). [6] M. J. Di Toro, "Phase and Amplitude Distortion in Linear Networks," Proc. IRE, vol. 36, pp. 24–36 (1948 Jan.). [7] M. H. Hebb, C. W. Horton and F. B. Jones, "On the Design of Networks for Constant Time Delay," J. Appl. Phys., vol. 20, pp. 616–620 (1949 J une). [8] G. G. Gouriet, "Two Theorems Concerning Group Delay with Practical Application to Delay Correction," Proc. IEE, pt. C, vol. 105, pp. 240–244 (1958 Mar.) (or Monograph 275R, 1957 Dec.). [9] E. J. Post, "Note on Phase-Amplitude Relations" Proc. IRE (Corresp.), vol. 51, p. 627 (1963 Mar.). [10] P. R. Geffe, "On the Approximation Problem for Band-Pass Delay Lines," Proc. IRE (Corresp.), vol. 50, pp. 1986–1987 (1962 Sept.). [11] H. Blinchikoff, "A Note on Wide-Band Group Delay," IEEE Trans Circuit Theory (Corresp.), vol. CT-18, pp. 577–578 (1971 Sept.). [12] A. N. Thiele, "Optimum Passive Loudspeaker Dividing Networks," Proc. IREE Australia, vol. 36, pp. 220–224 (1975 July).

[13] G. J. Huon and G. K. Cambrell, "New Low-Frequency Enclosure Configuration," AES preprint #4038, 5th Australian Regional Convention, Sydney, Sydney, (1995 April 26-28).