Abstract
This chapter describes the physical modeling and output-based measurement of loudspeakers, essential hardware components in sound-field control. A gray box model represents linear, time-variant, nonlinear, and non-deterministic signal distortions. Each distortion component requires a particular measurement technique that includes test stimulus generation, sound pressure measurement at selected points in 3D space, and signal analysis for generating meaningful metrics. Near-field scanning measures all signal components at a large signal-to-noise ratio with minor errors caused by loudspeaker positioning, air temperature, room reflections, and ambient noise. Holographic postprocessing based on a spherical wave expansion separates the direct sound from room reflections to assess the linear output and signal distortion. New metrics are presented that simplify the interpretation of the loudspeaker properties at single points, sound zones, and over the entire sound-field.
Keywords
- loudspeaker directivity
- near-field scanning
- signal distortion
- nonlinear loudspeaker modeling
- sound-field control
- spatial sound application
1. Introduction
Loudspeakers play an essential role in spatial sound applications, such as conventional multi-channel sound reproduction, beam steering [1], wave-field reconstruction [2], higher-order ambisonics [3], immersive audio [4], and multi-zone contrast control [5]. Those techniques require many loudspeakers arranged in linear, planar, circular, and spherical arrays [6] to satisfy the spatial sampling theorem at higher frequencies and provide desired directivity, sufficient sound power output, and audio quality. Cost, size, weight, and energy consumption are critical factors limiting the practical application.
Sound-field control techniques can use model-based or data-based methods to calculate the individual driving signals for the loudspeakers. Both approaches prefer an idealized loudspeaker model, usually assuming a linear, time-invariant transfer behavior and omnidirectional radiation while ignoring undesired properties (e.g., distortion) and physical limitations of the loudspeaker.
Loudspeakers are not always omnidirectional, especially at high frequencies. Various theories [7, 8, 9] consider and exploit the loudspeaker directivity in sound-field control. There are exciting opportunities for loudspeaker arrays exploiting a higher-order spherical wave model used in reverberant rooms [10].
Standard characteristics describe the loudspeaker directivity in the far-field [11]. Still, this information is less relevant in applications for home, automotive, or public address systems where either the radiating surface is large (e.g., arrays, flat panel) or the distance to the listener is small. Choi et al. [12] showed that active control could cope with those conditions if the near-field properties of the loudspeaker are considered.
Xiaohui et al. [13] showed that loudspeaker nonlinearities degrade the performance of spatial sound control, as nonlinear distortions limit the acoustic contrast between “bright” and “dark” sound zones. Cobianchi et al. [14] proposed a method for measuring the directivity of the nonlinear distortion in the far-field by using sinusoidal and multi-tone stimuli. Such tests performed in the near and far-field generate a significant test effort and a high amount of data that can be difficult to interpret.
Olsen and Møller [15] showed that typical ambient temperature variations in automotive applications change the loudspeaker properties in ways that compromise the sound zone performance significantly. Production variability, heating of the voice coil, fatigue, and aging of the suspension and other soft parts (cone) can change the loudspeaker properties over time and degrade the performance in a non-adaptive control solution.
This chapter presents models and measurement techniques to assess the loudspeaker transfer behavior from the input to the sound pressure at any point in the sound-field. The objective is to generate comprehensive information for selecting loudspeakers for spatial sound applications, simulating the performance, including room interaction, and maintaining sound quality over product life.
Such measurements are intended to provide meaningful characteristics that describe the sound pressure at a local point, over a listening zone, or in all directions, simplifying loudspeaker diagnostics.
2. General loudspeaker modeling
A single loudspeaker system used in spatial audio applications can be modeled by a multiple-input-multiple-output system (MIMO), as shown in Figure 1.
The loudspeaker input signals
are generated by sound-field control or other DSP algorithms fDSP applied to audio signals
while assuming a negligible coupling between the loudspeaker channels in the electrical, mechanical, or acoustical domain. This assumption is valid for transducers radiating sound independently into the free-field but not for multiple transducers mounted in one enclosure and working on the same air volume.
The function fSP,
The following chapter describes a single loudspeaker channel’s modeling, measurement, and quality assessment while omitting the subscript
Figure 2 shows a gray box model representing the nonlinear, time-variant function fSP(
The time-variant transfer function
Nonlinear subsystem NI and ND generate harmonics and intermodulation distortions at higher amplitudes. The first nonlinear system NI in the feedback loop in Figure 2 represents the dominant nonlinearities [22] in the transduction and the mechanical suspension such as force factor, voice coil inductance, and stiffness of a moving coil speaker [23]. A network with lumped parameters models the nonlinear dynamics by generating equivalent input distortion
The second nonlinear subsystem ND(
The nonlinear distortions
Imperfections in the design, manufacturing problems, overload, and other malfunction (“rub&buzz”) generate irregular dynamics perceived as abnormal distortion
3. Acoustical loudspeaker measurements
The free model parameters and other signal-dependent characteristics introduced in the gray box model presented in Section 2 can be identified by acoustic measurements.
The sound pressure can be modeled as a superposition of desired and undesired signal components in the time domain as
and in the frequency domain as a corresponding Fourier spectrum:
The component
New output-based measurement techniques compliant with IEC 60268–21 [11] provide accurate data with sufficient spatial resolution in a non-anechoic environment with minimum test effort (time, equipment).
The following sections will discuss those signal components in greater detail.
3.1 Loudspeaker positioning
The positioning of the loudspeaker in the 3D space is clearly defined by IEC 60268–21 [11] using a spherical coordinate system using the polar angle
3.2 Test environment
To ensure the reproducibility of the test result, it is common practice to measure loudspeakers under free-field conditions using a full-space (4π) or half-space (2π) environment. A half-space anechoic room with a solid ground floor is convenient for moving large and heavy loudspeaker systems and measuring loudspeakers mounted in or placed at a short distance from walls. The IEC standard [11] defines various methods of testing and postprocessing to generate simulated free-field conditions in a non-anechoic environment.
3.3 Far-field measurement
The traditional way to assess the loudspeaker directivity is the measurement of the spatial transfer function
using the wavenumber
The choice of measured directions determines the angular resolution of the directional gain [11], the accuracy of coverage angle [11], and other derived far-field characteristics. 2-degree angular resolution, needed for some professional loudspeakers, requires about 16,000 measurement points. Rotating a large and heavy loudspeaker over all combinations of the two angles requires robust and accurate robotics with speed ramps to accelerate and deaccelerate the mass. A microphone array speeds up the test by simultaneously measuring the sound pressure at multiple points without moving the loudspeaker.
Common far-field measurements usually provide no information about the accuracy of the measured data. They cannot indicate errors related to the positioning of loudspeakers or microphones, insufficient sampling of complex directivity patterns, or acoustical disturbances due to wind, air temperature, static sound pressure, or ambient noise [15].
Minor positioning errors and normal variation of the speed of sound, which is usually not critical for the amplitude response, can cause significant errors in the phase response and degrade the performance of 3D sound applications. For example, a deviation of the room temperature by 2 Kelvin during the test changes the speed of sound by 1.2 m/s and the acoustic propagation time by 50 μs at a measurement distance
3.4 Near-field measurement
The IEC standard 60268–21 [11] recommends measurements in the near-field, which overcome the restrictions and problems faced in the far-field. However, the 1/
Figure 3 shows a scanning system used for measuring the sound pressure generated by a loudspeaker placed at a fixed position on a post. The microphone moves in three axes in cylindrical coordinates (
The scanning points are distributed on two concentric layers, as shown in Figure 3, to measure the local derivative of the sound pressure like a sound intensity probe. That is the basis for separating the outgoing wave comprising direct sound radiated by the loudspeaker (e.g., diaphragm) from the incoming wave generated by reflections on the positioning arm of the robotics, ground floor, and room walls. The close distance to the sound source increases the direct sound, which increases the signal-to-noise ratio (SNR) by more than 20 dB and significantly reduces the phase error caused by varying air properties in far-field measurements.
4. Spatial transfer function
The spatial transfer function
The spherical coordinates allow a separation of angular dependency using the spherical harmonics
Figure 4 illustrates the expansion for a woofer operated in a sealed enclosure at 200 Hz. The measured directivity pattern is presented as a target on the lower left-hand side and compared with the wave model for rising maximum order
The Hankel function
Figure 5 shows the power Π
4.1 Parameters of the linear model
The optimum coefficients
Normalizing the mean squared error in Eq. (7) with the total output power gives a valuable criterion
Figure 6 shows the normalized fitting error
This example shows that the loudspeaker properties determine the maximum order
For acoustic, esthetic, or technical reasons, most loudspeakers have a natural symmetry in the diaphragm’s shape, the cone placement on the front side of the cabinet, and the enclosure’s geometry. Symmetry factors [24] calculated from identified coefficients
4.2 Simulated free-field condition
The measurement of the spatial transfer function requires free-field conditions or at least simulated free-field conditions as defined in IEC standard 60268–21 [11].
The absorption of the lined walls in “anechoic” rooms is usually imperfect at low frequencies where the wavelength of the standing waves exceeds the thickness of the lining. Gating the sound pressure signal and windowing of the impulse response provides good results at higher frequencies but degrade the frequency resolution at low frequencies.
The wave separation technique based on near-field scanning on two surfaces [25] can be used to separate the direct sound from the room reflections at low and middle frequencies and complements the windowing technique at higher frequencies. The measured transfer function
considering outgoing wave
The coefficients
4.3 Interpretation of the spatial transfer function
The interpretation of the spatial transfer function
using a fixed RMS value
The phase response at point
provides essential information for combining multiple loudspeaker channels in systems and arrays and applying DSP processing to control the sound-field. The total phase response
comprising the latency
The (real) sound power ΠL(
This sound power ΠL(
5. Time-variant distortion
The gray box model from Figure 2 describes the time-variant distortion spectrum
Using the spatial transfer
of two spatial transfer functions
This model is able to predict the amplitude compression at any point
and the phase deviation:
The voice coil heating in professional stage loudspeakers can cause significant amplitude compression (up to 6 dB) in the output signal. Fatigue and climate changes can also shift the resonance frequencies of modal cone vibrations, causing more than 90-degree phase deviation. Those variations can impair the intended superposition of multiple loudspeakers’ output in spatial sound applications.
6. Nonlinear distortions
The regular nonlinear distortions found in the sound pressure output
A typical audio signal (e.g., music) has a dense excitation spectrum, as shown in Figure 11, which makes separating the nonlinear distortion
As shown in Figure 11, a sparse multi-tone complex is a stimulus able to represent typical program material such as music and speech by having similar properties such as spectral distribution and crest factor. This stimulus has pseudo-random properties generated by a standardized algorithm [11] to ensure reproducible and comparable test results. The excitation tones are not dense but sufficiently activate harmonics, intermodulation, and other nonlinear distortion components, which can easily be detected and separated from the fundamental response in the spectrum.
The prevalent measurement technique uses a single tone stimulus with a constant or varying excitation frequency
The measurement technique presented in the following section can also be applied to a burst signal, two-tone signal, white or pink noise, and other input signals.
6.1 Nonlinear distortion in 3D space
A comprehensive measurement of the nonlinear distortion in the 3D space requires near-field scanning providing the distortion spectrum
Applying the spherical wave expansion to the measured distortion spectrum
The coefficients in vector
However, there is a significant difference between the nonlinear coefficients
The sound power spectrum calculated as
is a valuable global metric to assess the nonlinear distortion radiated by the loudspeaker in all directions.
6.2 Equivalent input distortion
The standard IEC 60268–21 calculates the equivalent input distortion (EID) for a single point measurement
using the time-variant transfer functions
The lower middle panel in Figure 12 shows the total harmonic distortion as an absolute SPL frequency response LTH,N(
Those artifacts in the equivalent input distortion (EID) can be removed by minimizing the mean squared error between the estimated and the measured nonlinear distortion spectrum at the scanning points
This fitting provides the voltage level response
Figure 13 shows the equivalent input distortion spectrum
The EID spectrum
The sound power spectrum
The transfer functions
This fact simplifies the distortion measurement and motivates the definition of relative distortion metrics discussed in Section 6.4. Furthermore, nonlinear control techniques [17] that cancel the EID at the loudspeaker input by synthesized compensation signal can reduce the sound pressure distortion
6.3 Distributed nonlinear distortion
The distributed nonlinear distortion
Eq. (26) uses the basic functions
The residual error in Eq. (27) can be used to find the maximum order
The coefficients
The distributed distortion can be ignored if the sound power ΠD(
6.4 Relative distortion metrics
This section introduces metrics that simplify the interpretation of the distortion components. These equations use a symbol # as a placeholder for N, I, or D representing the total, equivalent input, or distributed distortion.
Comparing the spectral components at frequency
The SNDR is usually negative and describes the SPL difference between the distortion and the linear component at the same spectral frequency
It is a proper physical metric for broad-band stimuli such as typical audio signals, noise, and other artificial test stimuli. It also applies to sparse multi-tone stimuli with a resolution smaller than one-third octave by using
However, SNDR) is less useful for sinusoidal stimuli generating only a single tone with constant or varying excitation frequency (e.g., chirp) because the harmonics have a significant spectral distance to the fundamental.
An alternative approach considers the total energy ratio between the nonlinear distortion
This metric can be applied to all kinds of stimuli but is very popular for the total harmonic distortion THD measured with a single tone and plotted versus the excitation frequency
Referring the nonlinear sound power spectrum Π#(
For a multi-tone stimulus representing typical program material (IEC 60268–21), the SPDR becomes an essential, single-value characteristic for the assessment of the audio quality in a global sense.
The spectral equivalent input distortion ratio (SEIDR) defined in decibel as
compares the spectral components of distortion
7. Abnormal distortion
Loudspeaker defects such as voice coil rubbing, mechanical vibrations of loose parts, air turbulences, and other irregular nonlinear dynamics that are neither intended nor considered in the design can generate particular distortion that can significantly degrade the audio quality. A loudspeaker generating abnormal distortion, usually called “rub & buzz” should not be shipped to a customer!
Modern measurement techniques exploit unique features of abnormal distortion. Time-analysis applied to a distorted single-tone stimulus reveals a complex fine structure comprising spikes, transients, and noise-like patterns [29]. Contrary to the harmonic and intermodulation distortion discussed in Section 6, the abnormal distortions cover the entire audio band. However, they have a low RMS value, are usually close to the noise floor, and thus require a near-field measurement. Spherical wave expansion or averaging over multiple periods removes the random features of the abnormal distortion.
The IEC standard 60268–21 [11] recommends a chirp stimulus at varying excitation frequency
The crest factor
This fact initiated the measurement of the impulsive distortion (ID) defined in IEC 60268–21 as a peak level in decibel as
Using a peak found over a period length
is the basis for calculating the maximum impulsive distortion ratio (IDR) defined according to IEC 60268–21 [11] as
using a reference sound pressure level LREF measured at the standard evaluation point (on axis,
Those metrics compared with meaningful limits for passing or failure are essential for the quality control of loudspeakers in manufacturing and maintenance.
8. External noise
The SNR in decibel is defined as
using reference SPL
9. Metrics for sound zones
Audio quality assessment, loudspeaker diagnostics, and active sound-field control require metrics that assess the properties of the sound-field at a specific listening point described by a probability
10. Maximum SPL output
The maximum sound pressure output (max SPL) rated according to IEC standard 60268–21 [11] plays a primary role in adjusting the amplitude of the test stimulus in output-based testing. The max SPL can be used to calibrate any input channel (digital, analog) in passive and active systems and provides a maximum input RMS value
11. Conclusions
Acoustical measurement in the near-field of the loudspeaker can provide much of the relevant information required for designing and assessing spatial sound control applications. The spatial transfer function
The time-variant transfer function
The multi-tone complex is a valuable artificial stimulus that can simplify the interpretation of the amplitude compression and the nonlinear distortion. The sinusoidal chirp is required to measure the impulsive distortion ratio, a sensitive characteristic for detecting loudspeaker defects and abnormal behavior degrading the audio quality.
An anechoic room is usually not required for performing the essential loudspeaker measurements at superior accuracy.
The methods for measuring loudspeaker characteristics presented in this chapter are compliant with modern international loudspeaker standards. They are the basis for simplifying the numerical simulation of sound-field control and selecting optimal hardware components offering a maximum performance-cost ratio.
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