**3.3 Far-field measurement**

The traditional way to assess the loudspeaker directivity is the measurement of the spatial transfer function *H*L(*f*,*r*D,*θ,ϕ*) between the input *u* and the sound pressure output *p*(*r*D,*θ,ϕ*) under far-field condition [11]. The distance *r*<sup>D</sup> between the loudspeaker and microphone should be much larger than the size of the speaker and acoustic wavelength. The 1/*r* law valid in the far-field allows extrapolating the complex transfer function to other distances *r* as

$$H\_{\mathcal{L}}(f, r, \theta, \phi) = H\_{\mathcal{L}}(f, r\_{\mathcal{D}}, \theta, \phi) \frac{r\_{\mathcal{D}}}{r} e^{-jk(r - r\_{\mathcal{D}})} \tag{5}$$

using the wavenumber *k=2πf/c0* and the speed of sound *c*0. Large loudspeakers such as loudspeaker arrays, soundbars, flat-panel speakers, and horn loudspeakers require a large measurement distance *r*<sup>D</sup> and a sizeable anechoic room with good air conditioning to keep the variance of the temperature field sufficiently small.

The choice of measured directions determines the angular resolution of the directional gain [11], the accuracy of coverage angle [11], and other derived farfield 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 *r* = 5 m, which is required to ensure far-field condition for large loudspeakers. This time delay corresponds to a positioning error of 17 mm and generates a phase error of 36 degrees at 2 kHz, increasing linearly with frequency and reaching 180 degrees at 10 kHz.

**Figure 3.** *Nearfield measurement by placing the loudspeaker at a fixed position and moving a microphone with robotics over the scanning grid close to the speaker surface.*

#### **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/*r* law in Eq. (5) is not applicable, and a holographic measurement technique that scans the sound pressure and fits a spherical wave model to measured data is required.

**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 (*r,φ,z*) to multiple test points **r**<sup>k</sup> ∈ Sr distributed on a double layer grid Sr close to the speaker's surface [24]. Moving a lighter microphone instead of rotating the heavier loudspeaker simplifies the robotics, allows faster speed ramps, and reduces the positioning error. Those opportunities make it possible to generate redundancy in the collected data and check the measurement's accuracy.

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 *H*L(*f*,**r**) describes the linear relationship between input spectrum *U*(*f*) and sound pressure spectrum *P*L(*f*,**r**) generated by the loudspeaker at any point **r** under the free-field condition as a spherical wave expansion in Eq. (6) using general solutions **B**out(*f*, **r**) of the Helmholtz equation weighted by complex coefficients in vector **C**L(*f*) [25]:

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

$$\begin{split} H\_{\mathcal{L}}(f, \mathbf{r}) &= \frac{P\_{\mathcal{L}}(f, r, \theta, \phi)}{U(f)} = \mathbf{C}\_{\mathcal{L}}(f) \mathbf{B}\_{\text{OUT}}(f, \mathbf{r}) \\ &= \sum\_{n=0}^{N} \sum\_{m=-n}^{n} c\_{n,m}^{\mathcal{L}}(f) h\_{n}^{(2)}(kr) Y\_{n}^{m}(\theta, \phi) \end{split} \tag{6}$$

The spherical coordinates allow a separation of angular dependency using the spherical harmonics *Y<sup>m</sup> <sup>n</sup>* ð Þ *θ*, *ϕ* from the radial dependency using the Hankel function of the second kind *h*<sup>n</sup> (2)(*kr*). The spherical harmonics have orthonormal properties representing a monopole (*n* = 0), dipoles (*n* = 1), quadrupoles (*n* = 2), and more complex sources with increasing order *n*.

**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 *N*. The expansion can be truncated at *N* = 3 because 16 coefficients weighting the spherical harmonics provide sufficient accuracy. Higher-order terms can be ignored at 200 Hz because they are 50 dB below the total sound power. The contribution of the higher-order terms rises with frequency and is required to explain the directivity pattern at 1 kHz, as shown in the upper diagram on the right-hand side.

The Hankel function *h*<sup>n</sup> (2)(*kr*) in Eq. (6) models the decay of the sound pressure with rising distancer *r* from expansion point **r**<sup>e</sup> of the spherical wave expansion. In the near-field for *r* < *r*far, the 1/*r* law is not valid anymore because sound pressure and particle velocity are not in phase, generating an increase in the apparent power at lower distances [24]. In the far-field *r*> > *r*far, the sound pressure decreases inversely with the rising distance *r* giving 6 dB less output for doubling the distance. Thus, the apparent sound power radiated from the loudspeaker is constant and corresponds to the real power.

**Figure 5** shows the power Π*n*(*r*) contributed by spherical waves of order *n* to the total apparent power Πa(*r*). Only the order *n =* 0 (monopole) generates a constant power output for all distances while the steepness of the power curve Π*n*(*r*) in the near-field increases with the order *n* of the waves.

**Figure 4.**

*Modeling the total sound power frequency response (upper right) and directivity pattern at 200 Hz (below) of a loudspeaker by spherical wave model (upper left).*

#### **Figure 5.**

*Total apparent sound power Πa(r) (thick line) generated by a loudspeaker versus radial distance r and the contribution Πn(r) of the spherical waves of order n (thin lines).*

#### **4.1 Parameters of the linear model**

The optimum coefficients **C**L(*f*) in the spherical wave model in Eq. (6) can be calculated by minimizing the mean squared error between the response *H*<sup>0</sup> <sup>L</sup> (*f*,**r***k*) measured at scanning points **r***<sup>k</sup>* ∈ Sr and the modeled responses as

$$\mathbf{C}\_{\rm L}(f) = \arg\text{MIN}\sum\_{\mathbf{C}}^{K\_{\rm r}} \left| H\_{\rm L}^{\prime}(f, \mathbf{r}\_{k}) - \mathbf{C}(f)\mathbf{B}\_{\rm OUT}(f, \mathbf{r}\_{k}) \right|^{2} \tag{7}$$

Normalizing the mean squared error in Eq. (7) with the total output power gives a valuable criterion *e* for checking the measurement's spherical wave expansion accuracy [24].

**Figure 6** shows the normalized fitting error *e* in the wave expansion with rising total order *N*. A single monopole expansion (*N* = 0) already gives an error reduction of 10 dB at 100 Hz. Considering the monopole and the three dipoles (N = 1) can reduce the error to minus 20 dB at 100 Hz, which means the model can explain 99% of the output power. A wave expansion of order *N* = 5 requiring at least 36 measurement points describes the sound output of the woofer channel below 1 kHz with sufficient accuracy (*e* < 1%). The increase of the fitting error at higher frequencies indicates that higher-order terms are required in the expansion to model the directivity at higher frequencies.

This example shows that the loudspeaker properties determine the maximum order *N* of the expansion, the number of measurement points *K*<sup>r</sup> required to identify the coefficients **C***i*(*f*), and the total scanning time.

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 **C**L(*f*) during the scanning process reveal the loudspeaker's left/ right or top/bottom single-plane, dual-plane or rotational symmetry. This information can be used to align the loudspeaker position and orientation with spherical harmonics to reduce the number of measurement points required to fit the wave

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

**Figure 6.**

*Normalized fitting error e versus frequency f of the spherical wave expansion truncated at maximum order N (above) and corresponding identified directivity pattern shown as a balloon-plot for the corresponding order N compared with the measured target response (left-hand side below).*

**Figure 7.**

*Exploiting symmetry in the loudspeaker geometry to reduce the number of measurement points required for the spherical wave expansion.*

expansion. As illustrated in **Figure 7**, considering the rotational symmetry can reduce the number of measurement points to 4%, significantly speeding up the scanning process.

#### **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 *H*<sup>0</sup> *<sup>L</sup>* (*f*,**r***k*) with **r***<sup>k</sup>* ∈ Sr corrupted by room reflections can be modeled by a spherical wave expansion [26]

$$\begin{split} H\_{L}^{\prime}(f, \mathbf{r}\_{k}) &= \mathbf{C}(f)\mathbf{B}(f, \mathbf{r}\_{k}) = \mathbf{C}\_{L}(f)\mathbf{B}\_{\text{OUT}}(f, \mathbf{r}\_{k}) + \mathbf{C}\_{\text{SR}}(f)\mathbf{B}\_{\text{SR}}(f, \mathbf{r}\_{k}) \\ &= \sum\_{n=0}^{N} \sum\_{m=-n}^{n} \left( c\_{n,m}^{\text{out}} h\_{n}^{(2)}(kr) + c\_{n,m}^{\text{SR}} J\_{n}(kr) \right) Y\_{n}^{m}(\theta, \varphi) \end{split} \tag{8}$$

considering outgoing wave **B**OUT(f,**r***k*) radiated by the loudspeaker as used in Eq. (6) and reflected waves **B**SR(*f*,**r***k*) represented by Bessel functions of the first kind *J*n(*kr*). The optimal coefficients **C**<sup>L</sup> and **C**SR minimizing the mean squared error between measured and modeled response can be estimated by

$$\begin{split} \mathbf{C}(f) &= \begin{bmatrix} \mathbf{C}\_{\mathrm{L}}(f) & \mathbf{C}\_{\mathrm{SR}}(f) \end{bmatrix} \\ &= \arg\text{MIN} \sum\_{k=1}^{K\_{\mathrm{r}}} \left| H'\_{\mathrm{L}}(f, \mathbf{r}\_{k}) - \mathbf{C}(f) \mathbf{B}(f, \mathbf{r}\_{k}) \right|^{2} \end{split} \tag{9}$$

The coefficients **C**SR(*f*) provide the SPL response of the sound reflections shown as a dashed curve in **Figure 8** that corrupts the measurement and causes a significant error below 1 kHz in the measured SPL response (thin green solid line). The **C**L(*f*) represents the SPL direct sound (thick blue solid line) measured under simulated free-field conditions.

#### **4.3 Interpretation of the spatial transfer function**

The interpretation of the spatial transfer function *HL*(*f*,**r**) can be simplified by calculating the SPL frequency response at point **r** in decibel as
