**6. Nonlinear distortions**

The regular nonlinear distortions found in the sound pressure output *p*N(*t*,**r**) are symptoms of loudspeaker nonlinearities modeled by subsystems NI and ND(**r**) shown in **Figure 2**. The input signal *u* strongly influences the generation process and the spectral and temporal properties of the nonlinear distortion [22].

A typical audio signal (e.g., music) has a dense excitation spectrum, as shown in **Figure 11**, which makes separating the nonlinear distortion *p*<sup>N</sup> in the sound pressure output *p* more difficult. An adaptive linear filter can model the linear and timevariant components *p*<sup>L</sup> + *p*<sup>V</sup> in the output [27]. The difference signal *e*(*t*) between the measured and the modeled signal comprises nonlinear distortion and noise.

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 *f*<sup>e</sup> (e.g., sinusoidal chirp [11]). The harmonic components generated at multiple frequencies *nf*<sup>e</sup> with *n* = 2, 3, 4 can be easily separated from the fundamental part at *f*e. This measurement technique has a long tradition and is simple but has a significant drawback: It does not consider the intermodulation distortion generated by multiple tones and music.

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 *P*N(*f*,**r***k*) at the grid points **r***<sup>k</sup>* ∈ Sr. The small distance between the microphone and loudspeaker ensures sufficient SNR to cope with noise. The measurement performed at high amplitudes can be integrated into the scanning process for spatial transfer function *H*L(*f,***r**) measured at low amplitude (see Section 4).

Applying the spherical wave expansion to the measured distortion spectrum *P*N(*f*,**r***k*) gives the optimal coefficients

**Figure 11.**

*Spectra of reproduced test stimuli used for nonlinear distortion measurement.*

*Advances in Fundamental and Applied Research on Spatial Audio*

$$\mathbf{C}\_{\rm N}(f) = \arg\text{MIN}\sum\_{\mathbf{C}}^{K\_{\rm r}} \left| P\_{\rm N}(f, \mathbf{r}\_{k}) - \mathbf{C}(f)\mathbf{B}\_{\rm out}(f, \mathbf{r}\_{k}) \right|^{2} \tag{18}$$

The coefficients in vector **C**N(f) allow extrapolation of the distortion to any point **r** outside the scanning surface:

$$P\_N(f, \mathbf{r}) = \mathbf{C}\_N(f)\mathbf{B}\_{\text{out}}(f, \mathbf{r})\tag{19}$$

However, there is a significant difference between the nonlinear coefficients **C**N(*f*) and the linear coefficients **C**L(*f*) discussed in Section 4. The linear coefficients **C**L(*f*) are parameters of a linear system. They can be identified with any broad-band stimulus and used to transfer another input signal into the sound-field, including music and speech. The nonlinear coefficients **C**N(*f*) describes the results (distortion) of loudspeaker nonlinearities that depend on the particular stimulus [22].

The sound power spectrum calculated as

$$\Pi\_{\rm N}(f) = \frac{\mathbf{C}\_{N}(f)\mathbf{C}\_{N}^{H}(f)}{2\rho\_{0}ck^{2}}|U(f)H\mathbf{v}(f,|t)|^{2} \tag{20}$$

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 **r**<sup>k</sup> by a simple approximation [28]

$$U\_{\rm I}(f, \mathbf{r}\_k) = \frac{P\_{\rm N}(f, \mathbf{r}\_k)}{H\_{\rm V}(f, |t|) H\_{\rm L}(f, \mathbf{r}\_k)} \tag{21}$$

using the time-variant transfer functions *H*V(*f*|*t*) and spatial transfer function *H*L(*f*|**r**). This inverse filtering transforms the sound pressure distortion *p*N(**r**k) into virtual input signal *u*' (**r**k), as illustrated in **Figure 12**.

**Figure 12.**

*Block diagram illustrates the calculation of equivalent input distortion (EID) by applying inverse filtering (right) or optimal estimation (left) based on three sound pressure measurements in the near-field (middle).*

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

The lower middle panel in **Figure 12** shows the total harmonic distortion as an absolute SPL frequency response LTH,N(*f*e,**r**) measured at three different distances **r**<sup>k</sup> in an office room (in-situ). The near-field measurement at 2 cm provides a relatively smooth curve, while the 30 and 60 cm measurements have a lower SPL and are affected by room reflections. The filtering of the sound pressure signals p (**r**k) with the inverse transfer function *H*(*f*,**r**k) �<sup>1</sup> generates a voltage signal u' (rk) with the total harmonics level *L*TH, I + D(*fe*,**r**k) on the lower right-hand side in **Figure 12**. This filtering removes the peaky curve shape caused by the room reflections, and the three curves become virtually identical between 100 Hz and 1 kHz. However, noise corrupts the measurement at low frequencies, and the distributed distortion *p*<sup>D</sup> causes minor deviations above 800 Hz.

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 **r**<sup>k</sup> with *k* = 1,.., *K*<sup>r</sup> and *K*<sup>r</sup> ≥ 1:

$$U\_{\rm I}(f) = \arg\text{MIN}\sum\_{\rm U\_{\rm EID}}^{K\_{\rm r}} \left| H\_{\rm V}(f, |t|) H\_{\rm L}(f, \mathbf{r}\_{k}) U\_{\rm I}(f) - P\_{\rm N}(f, \mathbf{r}\_{k}) \right|^{2} \tag{22}$$

This fitting provides the voltage level response *L*TH,I(*f*) on the left-hand side in **Figure 12**, representing the EID.

**Figure 13** shows the equivalent input distortion spectrum *U*I(*f*) generated by multi-tone stimuli with a different spectral shaping to represent typical test signals and selected audio material. All the stimuli have the same RMS value. Cello music provides the highest low-frequency components, generating the highest voice coil displacement and harmonic components at 500 Hz. Pink noise and IEC noise [11], representing typical program material, cause harmonic and intermodulation distortion at the same SPL over a wide frequency band. The nonlinear distortion rise to higher frequencies for voice and white noise stimuli.

The EID spectrum *UI*(*f*) at the input of the loudspeaker can also be easily transferred to at any point **r** in the 3D space by applying linear filtering:

**Figure 13.**

*Relative equivalent input distortion LI(f) measured with various broad-band stimuli at the same RMS input voltage.*

*Advances in Fundamental and Applied Research on Spatial Audio*

$$\begin{split} P\_{\mathbf{l}}(f, \mathbf{r}) &= H\_{\mathbf{V}}(f|t) H\_{\mathbf{L}}(f, \mathbf{r}) U\_{\mathbf{l}}(f) \\ &= \mathbf{C}\_{\mathbf{L}}(f) \mathbf{B}\_{\text{out}}(f, \mathbf{r}) H\_{\mathbf{V}}(f|t) U\_{\mathbf{l}}(f) \end{split} \tag{23}$$

The sound power spectrum *Π*I(*f*) of the equivalent input distortion radiated into the far-field can be similarly calculated as the linear power *Π*L(*f*) in Eq. (13) by using the same wave coefficients **C**L(*f*) of the linear wave modeling:

$$\Pi\_{\rm I}(f) = \frac{\mathbf{C}\_{\rm L}(f)\mathbf{C}\_{\rm L}^{\rm H}(f)}{2\rho\_{0}ck^{2}} \left| U\_{\rm I}(f)H\_{\rm V}(f,|t) \right|^{2} \tag{24}$$

The transfer functions *H*L(*f*,**r**)*H*v(*f*|**r**) shape the spectral components of equivalent input distortion and the input stimulus in the same way. Thus, the ratio between distortion and linear signal part is identical in the voltage, sound pressure at any point **r**, and power output:

$$\frac{|U\_{\mathbf{l}}(f)|}{|U(f)|} = \frac{|P\_{\mathbf{l}}(f,\mathbf{r})|}{|P\_{\mathbf{L}}(f,\mathbf{r})|} = \sqrt{\frac{|\Pi\_{\mathbf{l}}(f)|}{|\Pi\_{\mathbf{L}}(f)|}}\tag{25}$$

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 *P*I(*f*,**r**) everywhere in the 3D space.

#### **6.3 Distributed nonlinear distortion**

The distributed nonlinear distortion *p*D(**r**) introduced in Section 2 is the remaining distortion part in the sound-field that EID cannot represent:

$$\begin{split} P\_{\rm D}(f, \mathbf{r}\_k) &= (P\_{\rm N}(f, \mathbf{r}\_k) - H\_{\rm V}(f|t)H\_{\rm L}(f, \mathbf{r}\_k)U\_{\rm l}(f)) \\ &= \mathbf{C}\_{\rm D}(f)\mathbf{B}\_{\rm out}(f, \mathbf{r}\_k)H\_{\rm V}(f, |t)U(f) \end{split} \tag{26}$$

Eq. (26) uses the basic functions **B**OUT(*f*, **r**) from Eq. (6) for the spherical wave expansion but determines the coefficients **C**D(*f*) as:

$$\mathbf{C}\_{\rm D}(f) = \arg\text{MIN}\sum\_{k=1}^{K\_{\rm r}} \left| P\_{\rm D}(f, \mathbf{r}\_k) - \mathbf{C}(f)\mathbf{B}\_{\rm out}(f, \mathbf{r}\_k) \right|^2 \tag{27}$$

The residual error in Eq. (27) can be used to find the maximum order *N* of the wave expansion, as discussed in Section 4. The symmetry properties of the particular loudspeaker are also valuable for minimizing the scanning effort.

The coefficients **C**D(*f*) provide the sound power spectrum ΠD(*f*) of the distributed nonlinear distortion radiated into the far-field as:

$$\Pi\_{\rm D}(f) = \frac{\mathbf{C}\_{\rm D}(f)\mathbf{C}\_{\rm D}^{H}(f)}{2\rho\_{0}c\mathbf{k}^{2}} \left| U(f)H\_{\rm V}(f, |t|) \right|^{2} \tag{28}$$

The distributed distortion can be ignored if the sound power ΠD(*f*) is smaller than one-tenth of the EID sound power ΠI(*f*). Then a single test in the near-field of the loudspeaker is sufficient to measure the dominant EID and predict the total distortion *p*<sup>N</sup> in the 3D space.
