# Late Reverberation Synthesis Using Filtered Velvet Noise +
## 3 Filtered Velvet Noise Reverberation Algorithm  
The key idea of the FVN reverberation algorithm is to divide the RIR into short non-overlapping segments and to approximate each segment as filtered white noise. Velvet noise is used instead of standard white noise, such as Gaussian noise, since then the convolution with the input signal is fast to compute.  
Figure 3 shows the block diagram of the basic FVN reverberation algorithm. The delay lines of each VNC block serve two purposes: they delay the input signal appropriately for the next stage, as indicated by the right-hand-side output signal \(x(n-L)\) in Figure 2, and they provide the state variables of the sparse multi-tap delay line used to implement the VNC, i.e., a very efficient multiplication-free convolution of input signal with the velvet-noise sequence. The sparse sum of each segment is next filtered by its own spectral coloration filter \(H_{m}(z)\) and attenuated appropriately by the gain term \(G_{m}\), as shown in Figure 3.  
Figure 2: Velvet-noise convolution: Convolving the signal \(x(n)\) with a velvet-noise sequence \(s(n)\) reduces to the multiplication-free process of computing two sparse sums of delayed input signal samples and their difference. Blocks containing \(z^{-T_{d}}\), where \(z\) is the complex variable of the Z transformation, refer to delay lines of \(T_{\text{d}}\) samples. The output tap of each delay-line element is located at the sample point determined by sequence \(s(n)\).  
Uniform segmentation of an RIR should not be used, as the constant frame rate causes a periodic disturbance in the synthetic response. This is reminiscent of the flutter echo effect, which is a common problem in room acoustics. Much effort has been made to reduce this effect in recursive reverberation algorithms that use a pseudo-random noise sequence [13; 14]. Thus, it makes sense to use a non-uniform segmentation scheme in the FVN algorithm, as suggested in [16]. Another motivation to use a non-uniform framing is that the filter for each segment would be sufficiently different. In a typical RIR in which the exponential decay is faster at high frequencies than at low, a constant decrease in bandwidth, such as a 1-kHz narrowing, takes place non-uniformly in time--quickly in the beginning and slower towards the end of the RIR. This also motivates the use of longer segments at the end than at the beginning of the RIR. Figure 4a shows an example of a RIR and its segmentation. The impulse response has been measured in the concert hall in Pori, Finland (this impulse response of the concert hall is available online at [[http://legacy.spa.aalto.fi/projects/poririrs/](http://legacy.spa.aalto.fi/projects/poririrs/)]([http://legacy.spa.aalto.fi/projects/poririrs/](http://legacy.spa.aalto.fi/projects/poririrs/))).  
Figure 4: (**a**) Measured room impulse response (RIR) of the concert hall in Pori, Finland, with the boxes indicating every second segment used for modeling, and (**b**) its spectrogram showing frequency-dependent decay.  
Figure 3: Basic principle of the filtered velvet noise (FVN) algorithm [16]. The delay lines between the filtering branches of length \(L_{m}\) are in practice combined with velvet-noise convolution (VNC) blocks, cf. Figure 2. Blocks \(H_{m}\) and \(G_{m}\), for \(m=1,2,...,M\), represent the spectral coloration filters and gain factors for each segment, respectively.