Prior Use of First-Order Conformal Maps as Frequency Warpings

In 1971, Oppenheim, Johnson, and Steiglitz proposed forming an FFT filter bank with non-uniformly spaced bins by taking the FFT of the outputs of a cascade chain of first-order allpass filters [25].

In 1980, ``warped linear prediction'' was proposed by Strube [37] for obtaining better formant models of speech: The frequency axis ``seen'' by LPC is made to approximate a Bark scale using the first-order allpass transformation. It was noted in [37] that setting the allpass coefficient to 0.47 gave a ``very good approximation to the subjective Bark scale based on the critical bands of the ear'' at a 10 kHz sampling rate. It was concluded that low-order LPC was helped significantly by the frequency warping, because the first and second formants of speech become well separated on a Bark scale and therefore better resolved by a low-order predictor. However, higher order LPC fits could actually be made worse, e.g., due to splitting of the first formant as a result of four poles being used in the LPC fit instead of two.

In 1983, the Bark bilinear transformation was also developed independently for audio digital filter design [34]. In that work, the frequency response fit was carried out over an approximate Bark scale provided by the allpass transformation. The allpass coefficient $\rho$ was optimized as a function of sampling rate using the method of bisection under a least-squares norm on the error between the allpass and Bark frequency warpings. The root mean square errors were found to range from 0.0034f_s at f_s=6 kHz to 0.0068f_s at f_s=27 kHz, where f_s denotes the sampling rate. The frequency warp dictated by the optimal allpass transformation ${\cal A}_{\rho }$ determined an interpolated resampling of the desired filter frequency response $H(e^{j\omega })$ which converted its support to an approximate Bark scale $H(e^{j\omega })=H[{\cal A}_{\rho }(e^{ja(\omega )})]$. Any filter design method could then be carried out to give an optimal match $H^*[e^{ja(\omega )}]$ over the warped, sampled frequency response. Many filter design methods were compared and evaluated with respect to their audio quality. Finally, the optimal warped filter $H^*(\zeta )$ was unwarped by applying the inverse allpass transformation ${\cal A}_{-\rho }$ to the warped filter transfer function using polynomial manipulations to obtain $H^*[{\cal A}_{-\rho }(z)]$.

The first-order allpass transformation has been used traditionally in digital filter design to scale the cut-off frequency of digital lowpass and highpass filters, preserving optimality in the Chebyshev sense [26,4]. Higher order allpass transformations have been used to convert lowpass or highpass prototype filters into multiple bandpass/bandstop filters [23]. Allpass transformations of order greater than one appear not to have been used in frequency warping applications, since allpass transformations of order N map the unit circle to N traversals of the unit circle, and a one-to-one mapping of the unit circle to itself is desired.³

More recently, in 1994 [15], an allpass coefficient of 0.62 was used to generate a frequency warping closely approximating the Bark scale for a sampling rate of 22 kHz. Experiments comparing the performance of warped LPC and ``normal'' LPC for speech coding and speech recognition applications showed that warped LPC required less than half the predictor model order for comparable performance.

Very recently, the first-order allpass transformation was used to implement audio-warped filters directly in the warped domain [13,14]. In this application, a digital filter is designed over the warped frequency axis, and in its implementation, each delay element is replaced by a first-order allpass filter which implements the unwarping on the fly. Advantages of this scheme include (a) reducing the necessary filter order by a factor of 5 to 10 (more than compensating for the increased cost of implementing a delay element as a first-order allpass filter), (b) avoiding numerical failures which can occur (even in double-precision floating point) when attempting to unwarp very high-order filters (e.g., much larger than 30), and (c) providing a dynamic warping modulation control which tends to act as a frequency-scaling parameter associated with ``acoustic size'' and is therefore musically useful.

The critical feature of the first-order conformal map in the z plane is that it does not increase filter order; it is the most general order-preserving frequency-warping transformation for rational digital filters. In view of this constraint, it is remarkable indeed that a ``free'' filter transformation such as this can so closely match the Bark frequency scale.