Microphone and signal simulation techniques 291

Figure 7.19 First-order IIR low-pass filters.

7.5.2 IIR filter algorithm of late reverberation

For the late diffused reverberation of a room, the temporal density of reflections from various directions increases as time is extended because of the increasing number of reflections from the boundary surfaces, as expressed in Equation (1.2.17). The overall energy of reverberations exponentially decays as time is prolonged because of the absorption of boundary surfaces. Late reverberation signals can be simulated with various perceived-based reverberation algorithms based on the statistical acoustic parameters of reverberation sound fields. Reverberation signals with a certain temporal density of reflections are enough to enhance auditory perceptions because of the limited resolution of human hearing. Schroeder (1962) suggested that a temporal density of not less than 1000/s is enough. On the basis of the results of psychoacoustic experiments, Kuttruff (2009) suggested a limit of 2000/s. Some other studies have also suggested a temporal density of ≥4000/s (Rubak and Johansen, 1998).

A plain reverberation algorithm simulates the successive reflections and decay in a room. It is implemented by combining the outputs of an infinite number of delay lines with lengths of m, 2m, 3m… samples and gains of g, g2, g3 …. (g < 1) relative to the original signal ex(n)

The impulse response hREV(n) in Equation (7.5.23) consists of an infinite series of unit impulses weighted with an infinite power series of g. The algorithm given by Equation (7.5.23) is called the plain reverberation algorithm, which is implemented by the IIR filter structure shown in Figure 7.20. The structure comprises a feedback loop with an m-sample

292 Spatial Sound

Figure 7.20 IIR filter structure for the plain reverberation algorithm.

delay line and g. The input and output of Figure 7.20 satisfy Equation (7.5.21), and this conclusion is easy to prove.

Equation (7.5.23) shows that each simulated reflection is delayed by m samples and attenuated g times compared with the preceding reflection. The magnitude of the qth reflection is gq times that of direct sound. As stated in Section 1.2.4, the reverberation time of the plain reverberation algorithm is evaluated by assuming 20 log10gq = −60 (dB):

For a given sampling frequency and delay line length m, T60 increases with the feedback g. The plain reverberation algorithm is simple with adjustable reverberation time and accommodates the exponential energy decay of natural reverberation. However, it suffers from the

following drawbacks:

1.Reverberation time is frequency independent, preventing it from simulating frequencydependent reverberation time caused by surface and air absorptions, which usually decrease as frequency increases in real rooms.

2.The equal time interval between two successive reflections tends to create fluttering.

Moreover, the simulated reflection density fs/m is usually small and invariable with time, which contradicts the phenomenon of increasing reflection density with time in real rooms.

3.The system function HREV(z) in Equation (7.5.24) includes m poles located at an equal interval in a circle with the radius g1/m in the Z-plane:

These poles cause the response magnitude |HREV(ω)| to vary with frequency, resulting in comb filtering characteristics similar to those shown in Figure 7.17(b). These characteristics cause subjective coloration in timbre. The peaks in |HREV(ω)| correspond to the poles of HREV(z). The digital angular frequencies of peaks are evaluated by substituting z = exp(jω) in Equation (7.5.26) as

A delay less than m = 44 in the plain reverberation algorithm is needed to obtain the temporal density of reflections larger than 1000/s at a sampling frequency of 44.1 kHz.

Microphone and signal simulation techniques 295

Figure 7.23 Schroeder reverberation algorithm.

where τi = mi / fs is the delay of the ith plain reverberation unit, mi is the corresponding delay in the measured sample, and fs is the sampling frequency. The series connection of all-pass reverberation units is intended to increase the reflection density. Other elements in Figure 7.23 are similar to those in the aforementioned discussion. However, the simulated temporal density of reflections in Figure 7.23 is constant over time. The low-pass reverberation units in Figure 7.21 can be used to replace the plain reverberation units in Figure 7.23, thereby enabling the simulation of increasing reflection density with time in real rooms (Gardner, 2002). The parameters of the reverberation algorithm can be chosen in terms of statistical acoustic characteristics of a target hall. In practice, they can be chosen on the basis of various optimal perception methods (Bai and Bai, 2005).

On the basis of the work of Stanuter and Puckette (1982), Jot and Chaigne (1991) proposed a feedback delay network (FDN) structure of reverberation algorithms. As shown in Figure 7.24, the outputs of N parallel delay lines with different lengths are connected to all N inputs by an N × N feedback matrix [A]. The feedback matrix and delay in each delay line may be designed so that the outputs of delay lines are uncorrelated. Generally, the matrix [A] is chosen as a unitarity multiplying with a gain | g | < 1 to ensure the stability of the algorithm. The FDN structure is applicable to creating multichannel uncorrelated reverberations, and the inputs of N parallel delay lines can serve as multichannel signal inputs. Stanuter and Puckette illustrated an example of a four-input–four-output structure. Appropriate filters (such as low-pass filters) can be inserted before/after each delay line to simulate frequency-dependent reverberation time, but they also cause reflection density to increase with time. The FDN is the general form of an IIR reverberation structure. In the FDN structure with N inputs and N outputs, if the feedback matrix [A] is a diagonal matrix, channels are decoupled, and their structure is simplified into the case of N independent plain or low-pass reverberation units

Various reverberation algorithms in the time domain are discussed above. Some reverberation algorithms in the time-frequency domain have also been proposed to simulate the frequency-dependent decay of late reverberation in rooms. Nikolic (2002) used quadrature mirror filters (QMF) or wavelet decomposition to decompose the input signal into 2–16