# Frequency-Dependent Schroeder Allpass Filters
## 1 Introduction  
Since the introduction of feedforward-feedback comb allpass filters by Schroeder and Logan [1], its popularity has not diminished due to its computational efficiency and versatile applicability in artificial reverberation [2]. Compared to general high-order allpass filters [3], the Schroeder allpass is a sparse filter, which is less flexible, but more computationally efficient. Multiple Schroeder allpass filters can be combined in series or by nesting [4] to create more complex structures while retaining the allpass characteristic. Allpass filters can further be generalized to multiple input and output (MIMO) systems based on unitary networks [5] or feedback delay networks (FDNs) [6]. An ideal high-order allpass filter might possess the following properties: (a) being strictly allpass instead of an approximation; (b) good control of system pole locations while maintaining ease of design; and (c) efficient implementation independent from the total order. However, an inherent disadvantage of Schroeder allpass filters is the dependency between the dry-wet-ration and the decay time. Nonetheless, allpass filters were applied to with a wide range of roles including: (1) increasing the echo density as preprocessing to an artificial reverberator [1, 7, 8]; (2) increasing echo density of in the feedback loop of reverberators [9, 10, 11, 12, 13]; (3) decorrelation for widening the auditory image of a sound source [14, 15, 16, 17]; (4) as reverberator in electro-acoustic reverberation enhancement systems [6, 12, 18, 19]; and (5) dispersive system design [20]. In artificial reverberators and decorrelators, the allpass characteristic is most important when used as short diffusing filters. When used with longer decay times, only the local comb-like spectrum is perceived. For reverberation enhancement systems and feedback delay networks, the allpass characteristic is mathematically important to maintain the system stability.  
In this work, we present an extension to the Schroeder allpass filter [1] by introducing frequency-dependent feedforward and feedback gains while maintaining the allpass characteristic. The problem addressed in this paper was already realized by [PERSON] [8] when he remarked: \"This gives us the entertaining result that either [feedforward] and [feedback gain] must be FIR filters, or [the feedforward gain] must be an unstable filter when viewed in isolation.\" and effectively concludedthat it is unfeasible to employ an IIR filter. However, a few years earlier, [PERSON] [5] showed that it is possible to apply some IIR filters with a more elaborate structure. In this paper, we show that indeed, it is possible to achieve the desired effect on the compact feedforward/-back structure with any FIR or IIR filter. By this, we directly improve upon the design of [PERSON] and Jot [11], which exhibits a frequency-dependent absorption but does not preserve the allpass property. At the same time, we also improve upon [PERSON]'s allpass filter [5] as our design is both less restrictive and computationally more efficient.  
The remaining manuscript is structured as follows. Section 2 introduces some filter design background and reviews related Schroeder allpass filter structures. In Section 3, we present the proposed frequency-dependent Schroeder allpass alongside an example case. We give a full derivation of the pole and zero locations depending on the feedforward and feedback gains. Section 4 presents an application of the proposed filter for decorrelation. The paper is then concluded in Section 5. For reproducibility, the MATLAB code to reproduce all data and figures are published at [[https://github.com/[PERSON]/FrequencyDependentSchroederAllpass](https://github.com/[PERSON]/FrequencyDependentSchroederAllpass)]([https://github.com/[PERSON]/FrequencyDependentSchroederAllpass](https://github.com/[PERSON]/FrequencyDependentSchroederAllpass)).