Characteristics-Based Design of Generalized-Exponent Bandpass Filters

TL;DR

The paper presents a characteristics-based design framework for Generalized Exponent Filters (GEFs), enabling direct design from native filter characteristics such as peak frequency, bandwidth, and group delay rather than traditional frequency-response targets. By deriving analytic mappings from filter characteristics to the underlying pole/exponent parameters (Θ) via sharp-filter approximations, the authors obtain closed-form parameterizations that support simultaneous control over magnitude and phase-based metrics, yielding sharp, low-delay bandpass filters and scalable multi-band filterbanks. The approach delivers high accuracy (typical relative errors < ~1.5%), inherent stability, and computational efficiency, with extensions to related filters (V-type) and multi-band configurations; open-source code is provided for practical adoption. This characteristics-based paradigm offers a direct, interpretable route for static, adaptive, and filterbank designs, enabling precise tailoring of both selectivity and synchronization for diverse seismic, audio, and sensing applications.

Abstract

We develop characteristics-based filter design methods for a class of IIR bandpass filters, which we refer to as Generalized-Exponent Filters (GEFs) and that are represented as second-order filters raised to non-unitary exponents. GEFs have a peak, are effectively linear phase, and are useful for seismic signal phase-picking, cochlear implants, and equalizers. The native frequency-domain specifications for GEFs are not on given frequency responses but rather on filter characteristics such as peak frequency, bandwidth, and group delay. Our characteristics-based method for filter design accommodates direct specification of a trio of frequency-domain characteristics from amongst the peak frequency, convexity, ndB quality factors, equivalent rectangular bandwidth, maximum group delay, and phase accumulation. We achieve this by deriving filter parameterizations with sets of filter characteristics which involves deriving closed-form analytic expressions mapping sets of filter characteristics to the original filter constants by making sharp-filter approximations. This results in parameterizations for GEFs including ones with simultaneous specification of magnitude-based and phase-based characteristics (e.g. bandwidths and group delays). This in turn enables designing sharply tuned filters without significant group delay, and simultaneous control over frequency selectivity and synchronization which is important in designing filterbanks. Our filter design methods with direct control over characteristics may also be utilized beyond static filter design for higher-order variable bandpass filter design and may be useful for characteristics-based adaptive filtering. Our methods are inherently stable, highly accurate in meeting strict specifications on desired characteristics, simple, and computationally efficient. The methods extend to the design of related bandpass and multiband filters.

Figures (7)

• Figure 1: Schematic diagram describing the approach for developing the characteristics-based filter design methods and characteristics-based filter parameterizations: We describe this approach in detail in the main text of section \ref{['s:tunabilityApproach']}. We refer to relevant sections, tables, and figures in the schematic so that the reader may easily refer to these. We start with the transfer function in green which is parameterized by filter constants and end with the transfer function in purple which is parameterized by filter characteristics that are the native specifications used for filter design. We have used the arrow notation to emphasize the vector nature of the filter constants and filter characteristics - though we avoid doing so in the main text for simplicity.
• Figure 2: Validity of sharp-filter approximation and its dependence on filter constants: The plots show the magnitude (left) and phase (right) of GEF/$\mathbf{P}$ (dashed red lines) and its sharp-filter approximation, $\mathbf{P}_{sharp}$ (solid blue lines) (normalized to peak magnitude and referenced to zero phase) as a function of $\beta$ generated using (\ref{['eq:Pmag']}) and (\ref{['eq:Pphase']}). Each row is generated using different values of the filter constants, $A_p, B_u$ with the last row not fulfilling the sharp-filter condition $A_p < 0.2$. For all plots, $b_p = 1$. The following may be observed: (a) the phase of $\mathbf{P}$ and $\mathbf{P}_{sharp}$ are almost equivalent, (b) the magnitudes are similar near the peak and especially so when the sharp-filter condition is fulfilled, and (c) the region of phase corresponding to an appreciable magnitude has a linear slope.
• Figure 3: The filter design method is highly accurate in fulfilling desired GEF characteristics: The error in computed characteristics for $Q_{erb}$ (left) and $N$ (right) are shown as a function of desired values for $Q_{erb}$ and $N$ that are most appropriate for AFs (we have imposed $\beta_{peak} = 1$ throughout). This figure is used to demonstrate the accuracy of the filter design method - as can be deduced from the range of values (in the colorbar), rather than to demonstrate any error pattern.
• Figure 4: The relative error in achieving desired filter characteristics is small: The plot shows $\mathbf{P}_{sharp}(\beta), \mathbf{P}(\beta), \mathbf{V}(\beta)$ designed using specified (desired) values for filter characteristics. The magnitudes and phases (top) illustrate similarity near the peak, thereby supporting our choice of parametizing $\mathbf{P}$ and $\mathbf{V}$ using peak-centric filter characteristics of $\mathbf{P}_{sharp}$. The bottom graph shows that the relative errors between the desired filter characteristics used to derive the filters and those computed from the generated transfer functions. This is done for $\mathbf{P}_{sharp}$ (left, blue bars), $\mathbf{P}$ (middle, red bars), and $\mathbf{V}$ (right, yellow bars). This figure was generated using filter constant values as per the figure title, or equivalently, using a subset of the following desired values for filter characteristics: $\beta_{peak} = 1, N_\beta = 19.1 \textrm{ cyc}, \phi_{accum} = 3 \textrm{ cyc}, Q_{erb} = 25.9, \textrm{ERB}_\beta = 0.039, Q_{10} = 14.6, \textrm{BW}_{10,\beta} = 0.69, S_\beta = 2.08 \times 10^4 \textrm{ dB}, \frac{Q_{erb}}{N} = 1.35 \textrm{ cyc}^{-1}, \frac{Q_{10}}{N} = 0.77 \textrm{ cyc}^{-1}, \frac{Q_{erb}}{Q_{10}} = 1.77$. An accompanying figure with an alternate set of filter characteristic values for parameterization is Fig. \ref{['fig:RespAndErrs6And0052']}.
• Figure 5: The relative error in achieving desired filter characteristics is small: See Fig. \ref{['fig:RespAndErrs6And005']} for a full description. This figure was generated using filter constant values (in the figure titles) corresponding to the following values of desired filter characteristic: $\beta_{peak} = 1, N_\beta = 11.1 \textrm{ cyc}, \phi_{accum} = 3.5 \textrm{ cyc}, Q_{erb} = 14.1, \textrm{ERB}_\beta = 0.071, Q_{10} = 8, \textrm{BW}_{10,\beta} = 0.12, S_\beta = 6.08 \times 10^3 \textrm{ dB}, \frac{Q_{erb}}{N} = 1.27 \textrm{ cyc}^{-1}, \frac{Q_{10}}{N} = 0.72 \textrm{ cyc}^{-1}, \frac{Q_{erb}}{Q_{10}} = 1.76$. Note that the filter is less sharp than that in Fig. \ref{['fig:RespAndErrs6And005']} and so the relative errors are larger (though still very small).