# Frequency-Dependent Schroeder Allpass Filters
## 4 Application in Decorrelation  
[PERSON] introduced decorrelation filters to widen the spatial image of sound sources [14]. The decorrelators should ideally break up the phase relationships in a signal while leaving the magnitude response unaltered. Thus, allpass filters are the natural choice for decorrelators. For audio applications, the time-smearing caused by the phase alteration shall be limited to avoid degradation of transient sounds. The acceptable frequency-dependent group delay was found to be 60 ms at low frequencies and 10-20 ms in the high frequencies [22]. Alternative decorrelator designs include velvet noise decorrelators [23; 24] and frequency-domain reverberators [25].  
The group delay of a single Schroeder allpass filter has a comb-like shape (see Figure 7d). Multiple of such comb characteristics can be superposed to achieve a less regular structure. This, in turn, is equivalent to a series of Schroeder allpasses with different delays \(m_{i}\) and gains \(g_{i}(z)\), but with the same target decay curve, \(\left|g_{i}(e^{i\omega})\right|^{1/m_{i}}\). The delays \(m_{i}\) of the \(i^{th}\) allpass filter are \(42,60,86,91,120\) samples for \(i=1,\ldots,5\). A second decorrelation filter is designed with delays \(41,93,94,134,144\). In the following, the sample rate is 48 kHz and we study a series combination of five allpass filters. For ease of interpretation, the decorrelator pole magnitudes are given in terms of reverberation time, i.e., the time required for the corresponding exponential to decay by 60 dB. The gains \(g_{i}(z)\) are realized by a first-order shelving filter [26] such that each Schroeder allpass filter has 7 non-zero coefficients. In total, the proposed decorrelator has only 35 non-zero coefficients, which is less than 1% compared to the decorrelator introduced in [22] and similar to the velvet noise decorrelator in [24].  
Figure 7: Numerical example of a frequency-dependent Schroeder allpass filter. (**a**) Poles and zeros of the gain \(g(z)\) (\(\times/\circ\)) and magnitude response \(|g(e^{i\omega})|\) (solid red); (**b**) Poles and zeros of the Schroeder allpass (\(\times/\circ\)) and magnitude response \(|g(e^{i\omega})|^{1/m}\) (solid red); (**c**) Impulse response of the Schroeder allpass; (**d**) Group delay of the Schroeder allpass.  
Figure 8a shows the poles of the proposed decorrelator alongside the target frequency-dependent decay curve. The poles of the individual allpass filters follow the target curve closely with only some deviation for the lowest frequency poles. Figure 8b depicts the corresponding impulse response. In Figure 8c, we present the total group delay of the decorrelator alongside the group delay of the individual allpass filters. The cross-correlation between two decorrelators in each third octave band is shown in Figure 8d. The maximum values are 0.6 for low and mid frequencies and 0.2 for high frequencies, which can be considered sufficiently low for many audio applications. Listening examples of this decorrelator can be found online ([[https://www.sebastianjiroschlecht.com/publication/Frequency-Dependent-Schroeder-Allpass-Filters/](https://www.sebastianjiroschlecht.com/publication/Frequency-Dependent-Schroeder-Allpass-Filters/)]([https://www.sebastianjiroschlecht.com/publication/Frequency-Dependent-Schroeder-Allpass-Filters/](https://www.sebastianjiroschlecht.com/publication/Frequency-Dependent-Schroeder-Allpass-Filters/))).  
Figure 8: Decorrelator based on the proposed frequency-dependent Schroeder allpass.