**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 *f* in the nonlinear distortion *P*#(*f*,**r**) with the linear output signal *P*L(*f*,**r**) from Eq. (6) at the same point **r** leads to a spectral nonlinear distortion ratio (SNDR) defined in decibel as:

$$L\_{\boldsymbol{\theta}}(f, \mathbf{r}) = 20 \lg \left( \frac{|P\_{\boldsymbol{\theta}}(f, \mathbf{r})|}{|P\_L(f, \mathbf{r})|} \right) \mathbf{dB} \quad \boldsymbol{\pi} \in \{\mathbf{N}, \mathbf{I}, \mathbf{D}\} \tag{29}$$

The SNDR is usually negative and describes the SPL difference between the distortion and the linear component at the same spectral frequency *f*.

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 *P*#(*fi,***r**) in the nominator of Eq. (29) and the fundamental component *P*L(*fj,* **r**) in the denominator with the smallest frequency difference |*fi- fj*| for each spectral distortion component.

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 *P*# and the linear output signal *P*<sup>L</sup> for a particular stimulus. It leads to the total distortion ratio (TDR) defined in percent as:

$$R\_{\boldsymbol{\theta}}(\mathbf{r}) = \sqrt{\frac{\int |P\_{\boldsymbol{\theta}}(f, \mathbf{r})|^2 d\boldsymbol{f}}{\int |P\_{\boldsymbol{\mathsf{L}}}(f, \mathbf{r})|^2 d\boldsymbol{f}}} \mathbf{100\% \quad \boldsymbol{\#} \in \{\mathbf{N}, \mathbf{I}, \mathbf{D}\} \tag{30}$$

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 *f*e. This metric does reveal the spectral distribution of the nonlinear distortion (second, third, and higher-order harmonics).

Referring the nonlinear sound power spectrum Π#(*f*) to the linear sound power ΠL(f) in Eq. (13) provides a sound power distortion ratio (SPDR):

$$R\_{\Pi, \mathfrak{e}} = \sqrt{\frac{\int \Pi\_{\mathfrak{e}}(f) df}{\int \Pi\_{\mathcal{L}}(f) df} \mathbf{100\%}} \mathbf{100\% \quad \mathfrak{e} \in \{\mathcal{N}, \mathcal{I}, \mathcal{D}\}} \tag{31}$$

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

$$L\_{\rm I}(f) = 20 \lg \left( \frac{|U\_{\rm I}(f)|}{|U(f)|} \right) \text{dB} = \rm L\_{\rm I}(f, \mathbf{r}) \approx \rm L\_{\rm N}(f, \mathbf{r}) \tag{32}$$

compares the spectral components of distortion *U*I(*f*) with the input signal *U*(*f*). The metric *L*I(*f*) is identical with the metric *L*I(*f,* **r**), assessing the EID at any point **r** in the sound-field. It is a valid approximation for the total distortion metric *L*N(*f,* **r**) if the distributed distortion *P*D(*f*,**r**) is negligible.
