A filter representation of diffraction at infinite and finite wedges

Acoustic diffraction occurs at objects and corners in urban outdoor and indoor environments with perceptually notable or specifically desired frequency-dependent attenuation, e.g., for sound barriers, exhibiting a lowpass characteristic when the direct sound path is occluded. Effects of diffraction are highly relevant in the context of room acoustics simulation (e.g., Torres et al., 2004) and in (interactive) virtual acoustic environments (VAEs) (e.g., Savioja et al., 1999; Funkhouser et al., 2004; Stephenson, 2010; Pisha et al., 2020; Schissler et al., 2021), with dynamic sound sources and receivers. In addition to applications in architectural acoustics, such VAEs have recently gained interest in hearing research and development aiming at ecological validity (e.g., Keidser et al., 2020) or a high degree of perceptual plausibility (e.g., Brinkmann et al., 2019).

For acoustics simulation, geometrical acoustics (GA), assuming ray-like sound propagation, offers advantages with regard to computational complexity (e.g., Wendt et al., 2014) and has become a de-facto standard. Effects of diffraction can be included in GA by constructing sound ray propagation paths involving “bending” at edges such as building or room corners (e.g., Tsingos et al., 2001; Funkhouser et al., 2004).

To account for edge diffraction, several exact and asymptotic solutions in the frequency and time domain exist. Keller (1962) coined the concept of edge diffracted rays in the geometrical theory of diffraction (GTD), considering the incident and reflected diffracted field at straight and curved edges. The uniform theory of diffraction (UTD; Kouyoumjian and Pathak, 1974) has been established as an asymptotic high-frequency solution for diffraction of electromagnetic waves and similar solutions for acoustic diffraction have been derived (e.g., Pierce, 1974). An exact time-domain solution was suggested by Biot and Tolstoy (1957) and extended by Medwin (1981) and Svensson et al. (1999), often referred to as BTMS. A reformulation in the frequency domain was presented in Svensson et al. (2009). Svensson et al. (1999) and Svensson et al. (2009) use line integrals along the physical edge based on the concept of secondary sources located along the edge, particularly suited for handling finite edges.

Several approximations have been suggested, e.g., using empirically based simplifications (Maekawa, 1968), for the required Fresnel terms in UTD (Kawai, 1981), for the secondary source model (e.g., Calamia and Svensson, 2006) in the context of a directive line source model (e.g., Menounou and Nikolaou, 2017), or for half-planes (Ouis, 2019). With considerable computation time spent on diffraction path finding, existing approximations for VAEs typically use filters derived by evaluating diffraction solutions at relatively low spectral resolution and restriction to infinite wedges in the shadow zone (e.g., Schissler et al., 2021) or from a coarse approximation of prototypical object shapes (e.g., Pisha et al., 2020). Parametric [infinite impulse response (IIR)] filter approximations of diffraction have been optimized in Pulkki and Svensson (2019) using machine learning, and have been heuristically derived using geometric parameters for a diffraction lowpass filter in Kirsch and Ewert (2021), also with restrictions to the shadow zone.

While application-specific filter approximations exist in the context of VAEs, no general filter representation exists for edge diffraction. However, such a filter representation might offer advantages in providing easily interpretable parameters like gains and cutoff frequencies as well as in connecting different diffraction solutions, given that the existing asymptotic and exact solutions must be representable in a unified way at high frequencies and near the GA boundaries, where the respective underlying terms converge to a flat frequency response.

Here, a unified filter representation of the singly diffracted sound field by an arbitrary wedge is suggested. Wedge diffraction is described as superposition of (up to four) fractional half-order lowpass filters, representing the diffracted incident and reflected sound field. It is shown that the cutoff frequencies and gains of the same underlying lowpass filter function can be derived from the asymptotic solution for diffraction of Pierce (1974, 2019) and Kouyoumjian and Pathak (1974). The suggested filter representation can be extended by an alternative asymptotic low-frequency filter function to approximate the exact BTMS solution, helping to interconnect the existing solutions. Expressions for the filter transfer functions and the impulse responses are provided. Based on the filter representation for infinite wedges, finite wedges are described by truncation in the time domain or convolution with the according sinc function in the frequency domain.

A rigid infinite wedge is considered, formed by two intersecting planes at an open angle $θw$ as depicted in Fig. 1. A cylindrical coordinate system with the z-axis along the edge is assumed. A point source is assumed to be located at an angle $θs$ from the (closer) wedge plane $Pe,s$ while the receiver is located at an angle $θr$ from the wedge plane $Pe,s$⁠. $θb=θr−θs$ denotes the bending (deflection) angle of the sound path. At $θSB=π+θs$⁠, the shadow boundary (SB) separates the shadow zone where the GA direct sound component disappears from the “illuminated” zone. At $θRB=π−θs$⁠, the reflection boundary (RB) separates the illuminated zone from the reflection zone, where additionally the GA specular reflection exists, originating from an image source in the wedge plane $Pe,s$⁠, located at $θis=−θs$⁠. $φ$ is the angle of the source and receiver in their respective incident and excitant planes $Ps$ and $Pr$ with respect to the edge and the apex point $za$⁠. $ds$⁠, $dr$ are the distances to $za$ and $d=ds+dr$ is the (shortest) path distance from source to receiver over the edge via the apex point $za$⁠.

Based on Pierce (1974, 2019) an asymptotic expression for the diffracted sound field of a point source can be derived as the sum of two diffraction functions,

with wave number $k=2πf/c$ and speed of sound $c$⁠. The two diffraction functions $AD(X)=sgnXfX+ig(X)$ are composed of auxiliary Fresnel functions.¹ In the above, $AD$ for $X+=Xθ+=Xθr+θs$ and $X−=Xθ−=Xθr−θs$ represent the diffracted reflected and diffracted incident component, respectively, with

where $ν=π/θw$ is the exterior edge index and

reflects the distance, wavelength λ, and $φ$ dependency. Furthermore,

reflects the (real-valued) gain dependence on $θ$ and $ν$ which lets the diffracted field disappear at integer $ν$⁠. It should be noted that Pierce (1974, 2019) defined $θr$ and $θs$ from the opposed excitant wedge plane, however, the relevant $cosνθ$ term is invariant to that change.

Inspection of the diffraction function $AD$ (see Fig. 2, top left) reveals a fractional half-order lowpass filter characteristic, with a $1/f$ slope (–3 dB/octave) at high frequencies (see also, e.g., Maekawa, 1968; Kirsch and Ewert, 2021). Cutoff frequencies $fc±θ$ of such filters with the desired asymptotic high-frequency slope of $1/f/fc$ can be derived from Eq. (2) with $f=c/λ$ and by assuming a fixed value of $X±θ=2/π$ (dotted vertical line in Fig. 2) at $fc±θ$⁠, resulting in

with

where $d*=2drds/d$ is the characteristic distance.

Further, using the bending angle $θb=θr−π−θs$⁠, relative to the shadow boundary, and evaluating Eq. (3) at $θ=θr−θs$ and $θ=θr+θs$⁠, the cutoff frequencies for the diffracted incident, $fc−$⁠, and diffracted reflected component, $fc+$⁠, are

It is obvious from Eq. (5) that $fc,−$ exhibits a singularity at the shadow boundary SB (⁠$θb=0$⁠, $θr=π+θs$⁠), where the two cosine terms in the denominator cancel. Here, the transfer function of the diffraction lowpass filter becomes flat with unit gain. Likewise, $fc+$ shows the same behavior at the reflection boundary RB (⁠$θb=−2θs$⁠, $θr=π−θs$⁠). Given that $1/fc$ is later used in the filter expressions (see Sec. 2.5) numerical issues can be avoided. In combination with the according discontinuities in the GA components at those boundaries, an overall continuous sound field arises (see, e.g., Pierce, 1974; Kouyoumjian and Pathak, 1974).

Finally, considering the phase inversions (at SB and RB) in $AD$⁠, substituting $e−iπ/42AD$ results in a real-valued filter expression for the diffracted sound field,

where $H+$ and $H−$ are fractional half-order lowpass filters (see Sec. 2.5) with cutoff frequencies $fc+$ and $fc−$⁠, respectively.

The above asymptotic diffraction solution can be extended with an additional filter function based on Pierce (1974) [their Eq. (5b)] resulting in a modification of Eq. (1),

with

Given that $X±$ rises proportional to $f$ as a consequence of the contained $Γ$ term [Eq. (3)], the last term in Eq. (9) can be represented by fractional half-order lowpass with a cutoff frequency well below the desired (audio) frequency range, providing the required $1/f$ slope. Accordingly, only a single additional lowpass filter $HΔ$ is required for the extended expression, with a fixed cutoff frequency $fΔ≪1$ Hz and a gain representing the sum for the diffracted incident and reflected component,

where $X±,1$ indicates that $X±$ is evaluated at 1 Hz. This results in the extended filter expression

The left column of Fig. 3 shows the above two-filter (dotted) and extended three-filter (dashed) representation in comparison to the original form (solid). Three example geometric configurations with the source at $θs=45°$ and exterior wedge angles $θw=360°, 270°, 185°$ (top, second, and third row) for varied bending angle (color coded) are shown. Source and receiver distances are 1 m. Deviations only arise from the (0.1 dB) errors caused by the filter approximation. The lower left panel in Fig. 3 shows the underlying lowpass terms of the three-filter representation and their sum (bold; replotted from the second row) for two example bending angles $θb=−67.5°, 37.5°$ (red, blue) in comparison the BTMS solution (light colors).

Based on UTD (Kouyoumjian and Pathak, 1974) as alternative asymptotic diffraction expression, an additional filter expression can be derived, using four fractional half-order lowpass filters of the same form as above, with (except for some geometrical arrangements, see Sec. 2.4) different cutoff frequencies and gains.

The UTD expression for the singly diffracted field of a point source by an infinite rigid wedge is

with

Here, $N±$ are the integers which most nearly satisfy $2πN±−vθ=±vπ$⁠, $L=d*/2sin2(φ)$⁠, and $F$ (referred to as transition function in the original publication) involves a Fresnel integral (for more details see Kouyoumjian and Pathak, 1974). The two diffraction functions $VD±$ are each evaluated at $θ=θ+=θr+θs$ and $θ=θ−=θr−θs$ resulting in overall four terms. The former $±$ subscript, representing the evaluation at $θ±$ has been omitted for clarity here and in the following. It is not to be confused with the superscript which refers to the $±$ in the respective terms.

The function $F(kLa)$ can be represented by a fractional half-order highpass filter (see Fig. 2, top right), complementary to $AD$ in the Pierce notation (see also Sec. 2.5), with the cutoff frequency $1/π$ (dotted vertical line). As above, assuming this fixed argument resulting in the fixed attenuation $F(1/π)$⁠, a cutoff frequency $fc$ can be derived. In combination with the $e−iπ/4/k∝ 1/if$ dependency in the pre-factor in Eq. (13), the highpass transition function $F$ is converted in a half-order lowpass and a cutoff-frequency-dependent gain. Using similar algebraic reformulations as above for the Pierce expressions, the (four) cutoff frequencies and the cutoff-frequency-dependent gains result in

In combination with the remaining terms in the pre-factor in Eqs. (13) and (14), the final filter expression with the total (real-valued) gain factors $GD±θ$ becomes

where $HD+$ and $HD−$ each refer to two fractional half-order lowpass filters with cutoff frequencies $fc+θ±$ and $fc−θ±$⁠, $θ+=θr+θs$⁠, and $θ−=θr−θs$⁠, respectively.

The middle column of Fig. 3 shows this filter representation (dashed) in comparison to the original form (solid) for the three example geometric configurations and the underlying lowpass filters in the bottom panel. Differences to the two Pierce solutions (left row) occur in the low frequency region for the square edge (second row) and particularly for the condition with a near flat surface (third row), in line with earlier reported validity ranges in Kawai (1981).

The exact BTMS solution uses four terms based on the same angular differences as in UTD, however, it differs from the asymptotic solutions in that no fixed filter transfer function exists. Nevertheless, BTMS can be approximated by blending between the above-described asymptotic filters at high cutoff frequencies and an alternative asymptotic filter for low cutoff frequencies. This filter, resembling the transfer function of each BTMS term, shows a steeper slope transition around the cutoff frequency (see Fig. 3, bottom right panel). The gain $GD,B$ for the suggested BTMS filter approximation can be derived numerically by evaluating the frequency domain solution (Svensson et al., 2009) at DC. Given that the alternative low-frequency filter (see Sec. 2.5) must match the same asymptotic $1/f$ high frequency behavior as its according UTD filter, the cutoff frequency of the alternative filter can be calculated as $fc,B=fcGD2/GD,B2$⁠, where $fc$ and $GD$ are the cutoff frequency and gain of the according UTD component, as derived above. Inspection of the filter shapes originating from the BTMS solution, suggests that the transition between the two filter shapes can be achieved by blending the two filters with the empirically derived transition factors

with the modified distance $d̂=d/(1+dΛ/(8c))$⁠. The transition factor for the asymptotic high-frequency filter becomes $qa=1$ for $kd≫1$⁠, where the asymptotic solution is valid (and q_B = 0 for the low-frequency BTMS filter).

The right column of Fig. 3 shows the resulting approximation (dashed) in comparison to the original BTMS solution (solid). As for Pierce and UTD, deviations are mainly apparent in regions with strongly attenuated diffraction, where the underlying lowpass filters (partially) cancel. As expected, all underlying solutions (Pierce, UTD, BTMS) are identical at high frequencies and very similar for the entire frequency range in case of the knife edge (upper row). Comparison of Pierce, UTD, and BTMS (replotted as light colors in the left and middle panel of the lower row) shows the different filter shape at low frequencies.

Three special geometrical cases can be considered in which the above diffraction filter expressions further simplify.

The first case occurs when the source (or the receiver, obeying reciprocity of the problem) is located in one of the wedge planes, here assuming $θs=0$⁠. In this case, the source and image source are collocated, the incident and reflected field merge, and the shadow boundary and reflection boundary are coplanar. In the Pierce notation, thus both diffraction filters are identical with the same cutoff frequency as expressed in Eq. (5). Accordingly, the frequency characteristic of the diffracted field resembles that of a single half-order lowpass filter when using the basic filter expression in Eq. (6) or that of a single half-order lowpass filter in combination with a low-frequency lowpass for the extended expression in Eq. (12). In the UTD notation Eq. (16), and accordingly the BTMS approximation, two of the four lowpass filters are identical, thus effectively the diffracted field is composed of two different lowpass filter functions.

The second case is that of a thin half plane (knife edge, see upper row of Fig. 3) with $θw=2π$ and $ν=1/2$⁠, where $sinνπ=1$ and $cosνπ=0$⁠. Here, for the Pierce notation, the additional component in the extended expression disappears and Eq. (12) becomes Eq. (6). Moreover, the gains in Eq. (4) become $Gν±=1$⁠. A simplified cutoff frequency expression for both the diffracted incident and reflected field may be used,

Likewise, for the UTD notation (and accordingly BTMS) the cutoff frequencies of two filters are identical. For UTD the gains add to 0.5, identical to Pierce, where a pre-factor of 0.5 is present. It is easily observed that the same simplified cutoff frequency expression as for Pierce is also gained for UTD from Eq. (15) for $ν=1/2$⁠.

The third case occurs for the combination of the first two cases, i.e., the knife edge with either receiver or sender located in the wedge plane. Again, the incident and reflected field merge and both filters for Pierce as well as all four filters for UTD (and accordingly BTMS) are identical, with all gains adding up to unity gain for Pierce and UTD. For both notations, the common cutoff frequency may conveniently be expressed as a function of the bending angle,

This simplified form has earlier been considered in Kirsch and Ewert (2021) for diffraction in the shadow zone, with an additional factor of $π/3$⁠, providing a slightly better approximation in the design frequency range with a different parametrization of the lowpass filter.

Resembling the frequency characteristic of the diffraction lowpass function $AD$ and the according complementary highpass for $F$ (see solid dark gray traces in the top row of Fig. 2), a parametric approximation of the diffraction transfer function using a modified fractional-order lowpass filter is proposed,

where $fc$ is the cutoff frequency in Hz and $α=0.5$ is the fractional filter order. The blending exponent $b=1.44$ is used to achieve a smooth roll-off around the cutoff frequency matching the diffraction function $AD$⁠. The parameters $Q=0.2$ and $r=1.6$ have been optimized to approximate $AD$ and likewise the according complementary highpass $if/fcHf$ for $F$ with a maximum and identical error of about ±0.1 dB (see red traces in Fig. 2). To approximate the asymptotic low-frequency filter shape required in the BTMS approximation (Sec. 2.3), $b=1.05$ and $Q=0.37$ are used in Eq. (20). A filter transition can be achieved by either blending two filters or by directly blending the filter parameters $b$ and $Q$ in Eq. (20) with the blending factors $qa$ and $qB$ from Eq. (17).

Alternatively, with a larger maximum error of about ±0.45 dB, a simpler modified fractional-order lowpass filter (indicated in blue in Fig. 2) based on Kirsch and Ewert (2021) may be used,

Here, $α=0.5$ and $b=1.53$ are used to approximate $AD$ (and $F$⁠), while for $b=1$ a regular fractional-order lowpass filter is realized which is suited for the low-frequency filter $HΔ$ (Sec. 2.1) and can also be used as alternative asymptotic low-frequency filter in the BTMS approximation.

An alternative approximation of the asymptotic diffraction filter transfer function (with a resulting frequency-domain error smaller than ±0.1 dB) is suggested in the time domain by the impulse response

Here, $H(t)$ denotes the Heaviside step function and $t=0$ reflects the onset corresponding to the shortest path between source and receiver over the apex point $za$ (see Fig. 1). The impulse response initially decays proportionally to $1/t$ and converges against $1/t3$ (see Fig. 4, bottom left), with a transition point depending on $fc$⁠. This basic behavior was already described in Medwin (1981) and Ouis (2002) and a similar form of impulse response is given in Menounou and Nikolaou (2017). The impulse response shows a singularity at $t=0$⁠, reflecting a delta pulse for infinite cutoff frequencies (at the shadow and reflection boundaries). The integral of the impulse response (equal to the DC gain of the filter) is finite and equals one, which can be easily shown by the indefinite integral of $ht$⁠, reflecting the step response,

For the asymptotic low-frequency impulse response of the BTMS approximation a similar expression (frequency-domain error smaller than ±0.1 dB) is suggested,

and converges against a steeper $1/t5$ decay with an identical onset as in Eq. (22) (see Fig. 4, bottom right). The closed form solution for the step response is not provided here in the interest of space.

The overall diffraction impulse response is always a superposition of multiple terms [e.g., four impulse responses according to Eq. (22) for UTD] with respective cutoff frequencies and gains, equivalent to the filter representation [Eq. (16) for UTD] in the frequency domain.²

Any finite wedge transfer function and impulse response may be conveniently obtained from the infinite edge, based on the concept of (truncated) infinite half edges: The infinite edge is thought to be composed of two symmetric infinite half edges on both sides of the apex point (see Fig. 1). For any finite wedge, two cases exist: (i) The apex point is contained within the physical edge and the finite edge is the sum of two finite half edges from the apex point to both sides of the edge extension. If the apex point is exactly at either end of the edge, one of the half edges disappears. (ii) The (virtual) apex point is located outside the physical edge and the finite edge is the difference of the finite half edge from the virtual apex point to the farther edge end and the finite half edge to the closer edge end.

The impulse responses of the underlying finite half edges can be obtained by truncation of the infinite edge impulse response at the travel time difference $τ$ of the path from source to receiver over the edge end point and the path including the (virtual) apex point (assigned to $t=0$⁠). Accordingly, the transfer function is the convolution the infinite edge transfer function and the sinc function obtained by Fourier transform of the rectangular truncation time window. Identically, the finite half edge can be obtained by subtracting the (offset) virtual infinite half edge (starting at the end of the physical edge) with an impulse response starting at $τ$>0 from the infinite half edge. Figure 4 shows the resulting transfer functions (top row) and impulse responses (bottom row) for the three underlying filter notations.

The finite length wedge transfer functions (red, green) exhibit a low frequency plateau which results from the removal of low-frequency content in the respective later part of the infinite edge impulse response, i.e., the offset virtual infinite half edge (blue, magenta). The ripples result from the convolution with the sinc function, respective time domain truncation. In the right panel, the frequency-domain line integral reference solution for BTMS (gray) is exactly matched using the suggested filter expression and truncation. It obvious that the transfer function of the later part of the impulse response (offset virtual infinite half edges; blue, magenta) resembles that of a first-order (6 dB/oct) lowpass, given that the $1/t$ slope only prevails in the initial part of the impulse response. In comparison to BTMS, Pierce and UTD (for this configuration identical) show a higher DC gain with apex point outside the edge (green), caused by the slower decay of the impulse response.

Diffraction for finite scattering objects can be obtained by a combination of several finite wedges; however, it should be noted that higher-order diffraction might be necessary to include.

A filter representation of the singly diffracted sound field at arbitrary infinite and finite wedges was suggested based on two asymptotic solutions and an exact solution from the literature. While the filter representation is exact for the asymptotic solutions of Pierce (1974) and Kouyoumjian and Pathak (1974) (UTD) for which it is shown that their respective diffraction and transition functions $AD$ and $F$ can be represented in a unified way by (the same) single underlying filter function, the BTMS solution is approximated by blending between the same filter function at high frequencies and a second asymptotic filter function valid at low frequencies. In addition to the accessibility of the diffraction filter representation, the current work thus helps to connect the different asymptotic and the exact solution from the literature in an easily interpretable way. Furthermore, the suggested filter representation might be helpful for parametric digital filter implementations of diffraction and further simplifications in the context of virtual acoustics.

The author would like to thank C. Kirsch for fruitful discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG)—Project-ID 352015383–SFB 1330 C5 and DFG SPP Audictive—Project ID 444827755.