**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 [7–9] consider and exploit the loudspeaker directivity in sound-

#### *Advances in Fundamental and Applied Research on Spatial Audio*

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 nonadaptive 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**.

*Modeling and Testing of Loudspeakers Used in Sound-Field Control DOI: http://dx.doi.org/10.5772/intechopen.102029*

The loudspeaker input signals

$$\mathbf{u}\_{i} = \mathbf{f}\_{\text{DSP}}(w\_{1}, w\_{2}, \dots, w\_{N\_{\text{DSP}}}) \text{ } i = \mathbf{1}, \dots, N\_{u} \tag{1}$$

are generated by sound-field control or other DSP algorithms fDSP applied to audio signals *w*m. The input signal *ui* can be an analog voltage at the loudspeaker terminals or a digital data stream using other electrical, optical, or wireless transmission means. For each input signal *u*i, the loudspeaker system uses at least one electro-acoustical transducer (woofer, tweeter, full-band driver) that generates a sound pressure *pi*(**r**) at an evaluation point **r** under free-field condition. In modern loudspeaker systems, the transduction block fSP,*i*(*ui*,**r**) also performs amplification, equalization, active speaker protection against mechanical and thermal overload [16], and adaptive nonlinear control to cancel undesired signal distortion [17]. The total sound pressure output *p*T(**r**) is a linear superposition of the contributions *pi*(**r**) from all transduction blocks described as

$$p\_T(\mathbf{r}) = \sum\_{i=1}^{N\_u} p\_i(\mathbf{r}) = \sum\_{i=1}^{N\_u} \mathbf{f}\_{\text{SP},i}(u\_i, \mathbf{r}) \tag{2}$$

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,*i*(*ui*, **r**) describes the nonlinear and time-variant relationship between input *ui* and output signal *pi*(**r**).

The following chapter describes a single loudspeaker channel's modeling, measurement, and quality assessment while omitting the subscript *i* in the input voltage *u* (u*<sup>j</sup>* = 0 for *j* 6¼ *i*) and the sound pressure output *p*(**r**).

**Figure 2** shows a gray box model representing the nonlinear, time-variant function fSP(*u*, **r**) under free-field conditions. At small input signal amplitudes, the linear spatial transfer function *H*L(f,**r**) describes the loudspeaker behavior, assuming that other signal distortions are negligible. Still, additional noise *n*(**r**) generated by electronics or external sources can corrupt the sound pressure output.

The time-variant transfer function *H*V(*f*,*t*) represents reversible and nonreversible changes in the loudspeaker properties caused by the stimulus, climate [15], heating [18], aging, fatigue [19], and other external influences. The function *H*V(*f*,*t*) is independent of the evaluation point **r** because the dominant time-variant processes are in the electrical and mechanical domains. For example, the voice coil resistance [18], the natural frequencies, and loss factors of the modal vibrations [20] affect the sound-field in the same way. Variations of the mode shape, box geometry, and other boundaries can change the loudspeaker directivity but are

**Figure 2.**

*Gray box model of a single loudspeaker channel describing the relationship between the input signal u and sound pressure output p(r) at an evaluation point r in the free-field.*

neglected in the modeling. The *H*V(*f*,*t*) variation can be monitored by endurance, environmental or accelerated-life testing defined in various loudspeaker standards [11, 21].

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 *u*<sup>I</sup> added to the input signal *u* and transferred via the linear transfer path to any point **r** in the sound-field [11].

The second nonlinear subsystem ND(**r**) in **Figure 2** represents nonlinearities in the cone, diaphragm, surround, horn, port, and other acoustic elements and generates distributed distortion *p*D(**r**). The distributed distortion pD(**r**) depends on the point **r** and cannot be represented by equivalent input distortion.

The nonlinear distortions *u*<sup>I</sup> and *p*D(**r**) are considered in loudspeaker design because they affect the maximum output, audio quality, size, cost, and reliability. Finally, the distortions accepted as regular properties give the best performancecost ratio for the end-user.

Imperfections in the design, manufacturing problems, overload, and other malfunction ("rub&buzz") generate irregular dynamics perceived as abnormal distortion *p*ID(**r**) that is partly not deterministic and not predictable.

### **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

$$p(t, \mathbf{r}) = p\_{\rm L}(t, \mathbf{r}) + p\_{\rm V}(t, \mathbf{r}) + p\_{\rm N}(t, \mathbf{r}) + p\_{\rm ID}(t, \mathbf{r}) + n(t, \mathbf{r})\tag{3}$$

and in the frequency domain as a corresponding Fourier spectrum:

$$\begin{split} P(f, \mathbf{r}) &= F\{p(t, \mathbf{r})\} \\ &= P\_{\mathcal{L}}(f, \mathbf{r}) + P\_{\mathcal{V}}(f, \mathbf{r}) + P\_{\mathcal{N}}(f, \mathbf{r}) + P\_{\text{ID}}(f, \mathbf{r}) + N(f, \mathbf{r}) \end{split} \tag{4}$$

The component *p*<sup>L</sup> represents the desired linear output separated from signal distortion components *p*V, *p*N, *p*ID, and *n* corresponding to the time-variant properties, regular loudspeaker nonlinearities, and abnormal distortion generated by irregular vibration and measurement noise*,* respectively.

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 *θ,* azimuthal angle *ϕ,* and distance *r*. The origin **O** is placed at a convenient reference point **r**ref,

usually on the radiator's surface, grill, or enclosure, close to the supposed acoustical center. A reference axis **n**ref is orthogonal to the radiator's surface, and the orientation vector **o**ref usually points upwards in a vertical direction.
