## 3 Dynamic Replacement of Impulse Responses
The replacement itself will not introduce output artefacts as long as the coefficients are replaced just in time and in the correct order. For completeness, we will examine a third filter of length \(N\) where the coefficients are replaced at two different times, \(n_{s1}<n_{s2}\):  
\[h_{C+D+E}(\tau,t)= \begin{cases}c_{m-n},&m<n_{s1},\\ d_{m-n},&n_{s1}\leq m<n_{s2},\\ e_{m-n},&m\geq n_{s2}.\end{cases}  \tag{18}\]  
If the time interval \(n_{s2}-n_{s1}>N\), then the two transition regions \([n_{s1},n_{s1}+N-1]\) and \([n_{s2},n_{s2}+N-1]\) each behave as in Equation (17). If not, we get a subinterval \([n_{s2},n_{s1}+N-1]\) of the transition region where all three filters contribute to the output:  
\[y(n)=\sum_{l=0}^{n-n_{s2}}e_{l}x(n-l)+\sum_{l=n-n_{s2}+1}^{n-n_{s1}}d_{l}x(n-l )+\sum_{l=n-n_{s1}+1}^{N-1}c_{l}x(n-l)\qquad(n_{s2}\leq n\leq n_{s1}+N-1). \tag{19}\]  
Similar to the above, the coefficients \(e_{l}\), \(d_{l}\), and \(c_{l}\) do not contribute to the same output sample \(y(n)\) for any \(l,n\). Hence, several filter replacements may be carried out simultaneously without artefacts. However, the number of coefficients involved in the convolution sum will be limited by the interval \([n-n_{s2}+1,n-n_{s1}]\) as seen in the middle term above, for a total of \((n_{s2}-n_{s1})\) summands. An alternative view of this filter replacement scheme is that the input signal \(x(n)\) is split in segments \([0,n_{s1}-1]\), \([n_{s1},n_{s2}-1]\), and \([n_{s2},\infty)\) each convolved with just one set of coefficients, \(c_{l}\), \(d_{l}\), and \(e_{l}\), respectively. The output is the sum of contributions from all three filters. Convolution has by nature a ramp characteristic as exemplified in Figure 1 where a simple sinusoidal signal time-limited by a rectangular window is convolved with itself (\(x(n)=h(n)\)). When doing a gradual replacement of filter coefficients, this inherent ramp characteristic ensures a smoothly overlapping transition region between the filters \(h_{C}\) and \(h_{E}\). This technique of stepwise filter replacement outperforms other methods of dynamic filter updates:  
Figure 1: Convolving a time-limited sine sequence with itself. * It avoids the discontinuities and artefacts caused by instantaneous switching between filters. It makes no assumptions on similarity between filters as in the commutation [17] or on input signal properties as for the switched convolution reverberator [18]. * It saves the computational effort necessary to superpose or cross-fade parallel convolution processes [12; 22]. * It provides the minimum possible latency for a filter update and even allows convolution with a filter to start in parallel with the generation/recording of the filter impulse response itself. When filter length increases direct convolution in the time-domain is normally replaced with computationally more efficient methods in the frequency domain. A popular approach is _partitioned convolution_[6; 36] where the filter impulse response and the input signal are broken into partitions and the convolution computed as multiplication of these partitions in the frequency domain (more on implementation in Section 5). The method of stepwise filter replacement still works, but now the _partition_ is the unit of replacement. Replacement must be initiated at a time index \(n_{s}\) equal to an integer multiple of the partition length \(N_{P}\): \(n_{s}=kN_{P}\). A latency equal to the partition length \(N_{P}\) is also introduced.