*<sup>L</sup>*SPð Þ¼ *<sup>f</sup>*, **<sup>r</sup>** 20lg j j *<sup>H</sup>*Lð Þ *<sup>f</sup>*, **<sup>r</sup>** *<sup>u</sup>*<sup>~</sup> *pref* !dB (10)

#### **Figure 8.**

*Generating simulated free-field conditions at low frequencies by separating direct sound (solid line) from the room reflections (dashed line) in the measured SPL frequency response (thin line).*

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

using a fixed RMS value *u*~ of the input signal *u*(*t*) and the reference sound pressure *pref* = 20μPa. The SPL frequency response displayed in 2D or 3D plots (polar, balloon, contour) shows the directional dependency versus angles *θ* and *ϕ* in the far-field *r* > *r*far as shown in **Figure 9** and the local dependence versus Cartesians coordinates *x,y,z* in the near and far-field in **Figure 10**.

The phase response at point **r** calculated as

$$\begin{split} \rho(f, \mathbf{r}) &= \arg(H\_{\mathcal{L}}(f, \mathbf{r})) \\ &= \rho\_{\mathcal{M}}(f, \mathbf{r}) + \rho\_{\mathcal{A}}(f, \mathbf{r}) - 2\pi f \tau(\mathbf{r}) \end{split} \tag{11}$$

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 *φ*(*f*,**r**) can be decomposed into three parts: The minimal phase *φ*M(*f*, **r**) corresponds to the amplitude response |*H*L(*f,* **r**)| via the Hilbert Transform. The all-pass phase *φ*M(*f*,**r**) reveals the polarity and other loudspeaker properties. A critical part is a total time delay

**Figure 9.**

*Visualization styles for the far-field directional SPL response LSP(f,r,θ, ϕ) in spherical coordinates.*

#### **Figure 10.**

*Visualization of the SPL of the direct sound-field LSP(f,x,y,z) generated by a loudspeaker at 2 kHz outside the scanning surface.*

*Advances in Fundamental and Applied Research on Spatial Audio*

$$\tau(\mathbf{r}) = \tau\_{\text{DSP}} + \frac{|\mathbf{r} - \mathbf{r}\_{\epsilon}|}{c\_0(T\_A(\mathbf{r}), P\_0)} \tag{12}$$

comprising the latency *τ*DSP [11] in DSP processing and the acoustical delay depending on the distance |**r**-**r**e| and the local speed of sound c0, which is a function of the temperature field *T*A(**r**) and the static sound pressure *P*0.

The (real) sound power ΠL(*f*) radiated by the loudspeaker into the far-field can be calculated by multiplying the wave coefficients **C**L(*f*) with its Hermitian transpose:

$$\Pi\_L(f) = \frac{\mathbf{C}\_L(f)\mathbf{C}\_L^H(f)}{2\rho\_0 c k^2} \left| U(f) \right|^2 \tag{13}$$

This sound power ΠL(*f*) is a valuable metric for describing the global acoustic output of the loudspeaker by a single value. Still, it is also a convenient basis to estimate the mean sound pressure of the diffuse sound generated in a non-anechoic room if the reverberation time is known [11].

### **5. Time-variant distortion**

The gray box model from **Figure 2** describes the time-variant distortion spectrum *P*v(*f*,**r**|*t*) at any point **r** in the sound-field as

$$P\_{\mathbf{V}}(f, \mathbf{r}|t) = (H\_{\mathbf{V}}(f|t) - \mathbf{1})H\_{L}(f, \mathbf{r})U(f) \tag{14}$$

Using the spatial transfer *H*L(*f*,**r**), and the input spectrum *U*(*f*), and the timevariant transfer function *H*(*f*|*t*), which can be identified as the ratio

$$H\_{\mathbf{V}}(f|t) \approx \frac{H(f, \mathbf{r}|t)}{H(f, \mathbf{r}|t\_0)}\tag{15}$$

of two spatial transfer functions *H*(*f*,**r**|*t*0) and *H*(*f*,**r**|t) measured on the same loudspeaker unit under identical measurement conditions (environment, evaluation point **r**) at a reference time *t*<sup>0</sup> and a later evaluation time *t*. The reference measurement at *t*<sup>0</sup> assesses the loudspeaker under climatized standard conditions using a small stimulus generating negligible heating and nonlinear distortion. The subsequent measurement at time *t* can be performed with any stimulus providing sufficient excitation of the loudspeaker. This measurement requires no scanning process, and the calculated time-variant transfer function *H*v(*f*|*t*) is independent of the choice of the evaluation point **r**. Placing the microphone in the near-field ensures a good SNR.

This model is able to predict the amplitude compression at any point **r** in the sound-field defined in agreement with IEC standard 60268–21 [11] in decibel as

$$C\_{AC}(f, \mathbf{r}|t) = -20 \lg(|H\_V(\, f|t)|) dB \tag{16}$$

and the phase deviation:

$$
\Delta\rho(f,\mathbf{r}|t) = \arg(H\_\mathcal{V}(f|t))\tag{17}
$$

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.
