## 5 Pre-Processing Techniques
### Non-Uniform Frequency Resolution
#### 5.2.2 Frequency Warping  
Another technique that provides a nonuniform frequency resolution is _\"frequency warping\"_[98]. The original idea of frequency warping is presented in [99], where a nonuniform Fourier transform is introduced. The technique consists of replacing the unit delay \(z^{-1}\) of digital filters with a first-order all-pass filter,  
\[D_{1}(z)=\frac{z^{-1}-\lambda}{1-\lambda z^{-1}}\,, \tag{3}\]  
thus obtaining a bilinear mapping of the unit circle on itself. The warping effect can be adjusted to approximate the spectral representation of the ear [100]. In [101], analytic expressions that approximate the Bark and ERB scale are provided. They allow for a very good approximation of the Bark scale and less accurate approximation of the ERB scale, due to the higher frequency resolution required, particularly at low frequencies. The effect of the frequency warping can be easily reversed by again replacing the unit delay \(z^{-1}\) with the all-pass filter  
\[\widetilde{D}_{1}(z)=\frac{z^{-1}+\lambda}{1+\lambda z^{-1}}. \tag{4}\]  
Figure 5 shows an example of the effect of frequency warping on an RTF for different values of \(\lambda\). The reader should note the expansion of the low frequency range and the compression of the high frequencies obtained with positive values of the warping parameter \(\lambda\).  
Warped FIR and IIR filters can be obtained by replacing the tapped delay line with a chain of first-order all-pass filters, but while the implementation of warped FIR filters is immediate [66], warped IIR filters require appropriate structures to avoid delay-free loops [102]. Warped FIR filters are strictly related to the Laguerre filters [103], the only difference being the fact that in a Laguerre filter there is an additional prefilter placed before the all-pass chain [66]. A logarithmic frequency scale can also be approximated, but in this case the all-pass chain has to be replaced with a filterbank formed by all-pass filters.  
Frequency warping has been exploited in many audio applications, from LPC analysis [100], audio equalization [104; 105], loudspeaker equalization [106; 107; 108], and physical modeling of guitar bodies [109], to head-related transfer function (HRTF) filtering [66; 109]. The reader is referred to [66] for a review of frequency warping techniques and their applications. In the context of RRE, frequency warping has been used by many researchers in minimum-phase equalization to improve the equalizer performance by expanding the resolution at low frequencies and compressing it at high frequencies. A psychoacoustically-motivated frequency scale--in most cases the Bark scale--is used. For example, in the approach of [70; 71], a prototype room response is first frequency warped to an approximate Bark scale. Then, an all-pole model of the room response is obtained in the warped domain using  
Figure 4: Complex smoothing of a measured room transfer function (RTF): (**a**) RTF; (**b**) 1/12 octave-band complex smoothing; (**c**) 1/3 octave-band complex smoothing.  
LPC analysis. Eventually, a minimum-phase equalizer is derived in the time domain by de-warping the inverse of the all-pole model with (4). The main disadvantage of this approach is represented by the high computational cost of the frequency warping operation. In [73; 75], frequency warping was efficiently implemented by nonlinearly sampling a high-resolution fast Fourier transform (FFT) of the prototype room response.