Prior Use of First-Order Conformal Maps as Frequency Warpings

A conformal map is a complex-valued function of a complex variable that is an angle-preserving mapping from one complex plane to another. That is, the mapping preserves the angles between intersecting curves in the complex plane. — Click for http://mathworld.wolfram.com/ConformalMapping.html

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A frequency warp is a non-uniform scaling of frequency. Frequency warps are commonly used in audio signal processing in order to better mimic the non-uniform frequency resolution of human hearing. — Click for https://ccrma.stanford.edu/~jos/bbt/Filter_Design_Example.html

The transfer function is defined for LTI filters as the z transform of the filter output signal, divided by the z transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Transfer_Function_Analysis.html

Bandlimited interpolation is the ideal type of interpolation for properly sampled signals. — Click for https://ccrma.stanford.edu/~jos/resample/

Sampling is the process of converting a continuous-time signal into a discrete-time signal. — Click for http://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

The frequency response is defined for LTI filters as the Fourier transform of the filter output signal divided by the Fourier transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Frequency_Response_I.html

A digital filter can be visualized as a 'black box' that accepts a sequence of numbers and emits a new sequence of numbers. In digital audio signal processing applications, such number sequences usually represent sounds. For example, digital filters are used to implement graphic equalizers and other digital audio effects. — Click for https://ccrma.stanford.edu/~jos/filters/

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In the context of linear time-invariant filter analysis, a zero is a point in the complex s or z plane at which the transfer function goes to zero. A pole is a point in the s or z plane at which the transfer function goes to infinity. The s plane is used to analyze analog filters while the z plane is used for digital filters. — Click for https://ccrma.stanford.edu/~jos/filters/Pole_Zero_Analysis_I.html

The sampling rate of a discrete-time signal is defined as the number of samples per second. Its units are thus in Hertz (Hz). Shannon's Sampling Theorem states that the original continuous-time signal can be recovered exactly from the samples if and only if the sampling rate is higher than twice the highest frequency present in the original signal. Any higher frequencies will alias to frequencies below half the sampling rate. — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

Linear Prediction is an important signal modeling tool which captures correlation information in the form of prediction coefficients. In digital audio signal processing, linear prediction computes a superior spectral envelope that emphasizes spectral peaks. — Click for http://en.wikipedia.org/wiki/Linear_prediction

A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.html

Fast Fourier Transforms (FFT) are fast algorithms for computing the Discrete Fourier Transform (DFT) — Click for https://ccrma.stanford.edu/~jos/mdft/Fast_Fourier_Transform_FFT.html

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.