Chapter 6

Room Simulation

Room simulation artificially reproduces the acoustics of a room. The foundations of room acoustics are found in [Cre78, Kut91]. Room simulation is mainly used for post-processing signals in which a microphone is located in the vicinity of an instrument or a voice. The direct signal, without additional room impression, is mapped to a certain acoustical room, for example a concert hall or a church. In terms of signal processing, the post-processing of an audio signal with room simulation corresponds to the convolution of the audio signal with a room impulse response.

6.1 Basics

6.1.1 Room Acoustics

The room impulse response between two points in a room can be classified as shown in Fig. 6.1. The impulse response consists of the direct signal, early reflections (from walls) and subsequent reverberation. The number of early reflections continuously increases with time and leads to a random signal with exponential decay called subsequent reverberation. The reverberation time (decrease in sound pressure level by 60 dB) can be calculated, using the geometry of the room and the partial areas that absorb sound in the room, from

where T₆₀ is the reverberation time (in s), V the volume of the room (m³), S_n the partial areas (m²) and α_n the absorption coefficient of partial area S_n.

The geometry of the room also determines the eigenfrequencies of a three-dimensional rectangular room:

where n_x, n_y, n_z are integer number of half waves (0, 1, 2,…), l_x, l_y, l_z are dimensions of a rectangular room, and c is the velocity of sound.

Figure 6.1 Room impulse response h(n) and simplified decomposition into direct signal, early reflections and subsequent reverberation (with |h(n)|).

For larger rooms, the eigenfrequencies start from very low frequencies. In contrast, the lowest eigenfrequencies of smaller rooms are shifted toward higher frequencies. The mean frequency between two extrema of the frequency response of a large room is approximately inversely proportional to the reverberation time [Schr87]:

The distance between two eigenfrequencies decreases with increasing number of half waves. Above a critical frequency

the density of eigenfrequencies becomes so large that they overlap each other [Schr87].

6.1.2 Model-based Room Impulse Responses

The methods for analytically determining a room impulse response are based on the ray tracing model [Schr70] or image model [All79]. In the case of the ray tracing model, a point source with radial emission is assumed. The path length of rays and the absorption coefficients of walls, roofs and floors are used to determine the room impulse response (see Fig. 6.2). For the image model, image rooms with secondary image sources are formed which in turn have further image rooms and image sources. The summation of all image sources with corresponding delays and attenuations provides the estimated room impulse response. Both methods are applied in room acoustics to get insight into the acoustical properties when planning concert halls, theaters, etc.

Figure 6.2 Model-based methods for calculating room impulse responses.

6.1.3 Measurement of Room Impulse Responses

The direct measurement of a room impulse response is carried out by impulse excitation. Better measurement results are obtained by correlation measurement of room impulse responses by using pseudo-random sequences as the excitation signal. Pseudo-random sequences can be generated by feedback shift registers [Mac76]. The pseudo-random sequence is periodic with period L = 2^N − 1, where N is the number of states of the shift register. The autocorrelation function (ACF) of such a random sequence is given by

where a is the maximum value of the pseudo-random sequence. The ACF also has a period L. After going through a DA converter, the pseudo-random signal is fed through a loudspeaker into a room (see Fig. 6.3).

Figure 6.3 Measurement of room impulse response with pseudo-random signal x(t).

At the same time, the pseudo-random signal and the room signal captured by a microphone are recorded on a personal computer. The impulse response is obtained with the cyclic cross-correlation

For the measurement of room impulse responses it has to be borne in mind that the periodic length of the pseudo-random sequence must be longer than the length of the room impulse response. Otherwise, aliasing in the periodic cross-correlation r_XY(n) (see Fig. 6.4) occurs. To improve the signal-to-noise ratio of the measurement, the average of several periods of the cross-correlation is calculated.

Figure 6.4 Periodic autocorrelation of pseudo-random sequence and periodic cross-correlation.

Sine sweep measurements [Far00, Mül01, Sta02] are based on a chirp signal x_S(t) of length T_C and an inverse signal x_Sinv(t), where both satisfy the condition

The chirp signal can be applied to the room by a loudspeaker and then a signal inside the room y(t) = x_S(t)*h(n) is recorded. By performing the convolution of the received signal y(t) with the inverse signal x_Sinv(t) one obtains the impulse response from

6.1.4 Simulation of Room Impulse Responses

The methods just described provide means for calculating the impulse response from the geometry of a room and for measuring the impulse response of a real room. The reproduction of such an impulse response is basically possible with the help of the fast convolution method as described in Chapter 5. The ear signals at a listening position inside the room are computed by

where h_L(n) and h_R(n) are the measured impulse responses between the source inside the room, which generates the signal x(n), and a dummy head with two ear microphones. Special implementations of fast convolution with low latency are described in [Soo90, Gar95, Rei95, Ege96, Mül99, Joh00, Mül01, Garc02] and a hybrid approach based on convolution and recursive filters can be found in [Bro01]. Investigations regarding fast convolution with sparse psychoacoustic-based room impulse responses are discussed in [Iid95, Lee03a, Lee03b].

In the following sections we will consider special approaches for early reflections and subsequent reverberation, which allow a parametric adjustment of all relevant parameters of a room impulse response. With this approach an accurate room impulse response is not possible, but with a moderate computational complexity a satisfying solution from an acoustic point of view can be achieved. In Section 6.4 an efficient implementation of the convolutions (6.9) and (6.10) with a multirate signal processing approach [Zöl90, Sch92, Sch93, Sch94] is discussed.

6.2 Early Reflections

Early reflections decisively affect room perception. Spatial impression is produced by early reflections which reach the listener laterally. The significance of lateral reflections in creating spatial impression was investigated by Barron [Bar71, Bar81]. Fundamental investigations of concert halls and their different acoustics are described by Ando [And90].

6.2.1 Ando's Investigations

The results of the investigations by Ando are summarized in the following:

• Preferred delay time of a single reflection: with the ACF of the signal, the delay is determined from |r_xx(Δt₁)| = 0.1 · r_xx(0).
• Preferred direction of a single reflection: ±(55° ± 20°).
• Preferred amplitude of a single reflection: A₁ = ±5 dB.
• Preferred spectrum of a single reflection: no spectral shaping.
• Preferred delay time of a second reflection: Δt₂ = 1.8 · Δt₁.
• Preferred reverberation time: T₆₀ = 23 · Δt₁.

These results show that in terms of perception, a preferred pattern of reflections as well as the reverberation time depend decisively on the audio signal. Hence, for different audio signals like classical music, pop music, speech or musical instruments entirely different requirements for early reflections and reverberation time have to be considered.

6.2.2 Gerzon Algorithm

The commonly used method of simulating early reflections is shown in Figs 6.5 and 6.6. The signal is weighted and fed into a system generating early reflections, followed by an addition to the input signal. The firstm M reflections are implemented by reading samples from a delay line and weighting these samples with a corresponding factor g_i (see Fig. 6.6). The design of a system for simulating early reflections will now be described as proposed by Gerzon [Ger92].

Figure 6.5 Simulation of early reflections.

Figure 6.6 Early reflections.

Craven Hypothesis. The Craven hypothesis [Ger92] states that the human perception of the distance to a sound source is evaluated with the help of the amplitude and delay time ratios of the direct signal and early reflections as given by

where d is the distance of the source, d′ the distance of the image source of the first reflection, g the relative amplitude of the direct signal to first reflection, c the velocity of sound, and T_D the relative delay time of the first reflection to the direct signal.

Without a reflection, human beings are not able to determine the distance d to a sound source. The extended Craven hypothesis includes the absorption coefficient r for determining

For a given reverberation time T₆₀, the absorption coefficient can be calculated by using exp(−rT₆₀) = 1/1000, that is,

With the relationships (6.15) and (6.17), the parameters for an early reflections simulator as shown in Fig. 6.5 can be determined.

Gerzon's Distance Algorithm. For a system simulating early reflections produced by more than one sound source, Gerzon's distance algorithm can be used [Ger92], where several sound sources are placed at different distances as well as in the stereo position into a stereophonic sound field. An application of this technique is mainly used in multichannel mixing consoles.

By shifting a sound source by −δ (decrease of relative delay time) it follows that from the relative delay time of the first reflection T_D − δ/c = [d′ − (d +δ)]/c, and the relative amplitude according to (6.17),

This results in a delay and a gain factor for the direct signal (see Fig. 6.7) as given by

Figure 6.7 Delay and weighting of the direct signal.

By shifting a sound source by +δ (increase in relative delay time) the relative delay time of the first reflection is T_D − δ/c = [d′ − (d − δ)]/c. As a consequence, the delay and the gain factor for the effect signal (see Fig. 6.8) are given by

Using two delay systems in the direct signal as well as in the reflection path, two coupled weighting factors and delay lengths (see Fig. 6.9) can be obtained. For multichannel applications like digital mixing consoles, the scheme in Fig. 6.10 is suggested by Gerzon [Ger92]. Only one system for implementing early reflections is necessary.

Figure 6.8 Delay and weighting of effect signal.

Figure 6.9 Coupled factors and delays.

Stereo Implementation. In many applications, stereo signals have to be processed (see Fig. 6.11). For this purpose, reflections from both sides with positive and negative angles are implemented to avoid stereo displacements. The weighting is done with

For each reflection, a weighting factor and an angle have to be considered.

Generation of Early Reflection with Increasing Time Density. In [Schr61] it is stated that the time density of reflections increases with the square of time:

After time t_C the reflections have a statistical decay behavior. For a pulse width of Δt, individual reflections overlap after

Figure 6.10 Multichannel application.

Figure 6.11 Stereo reflections.

To avoid overlap of reflections, Gerzon [Ger92] suggests increasing the density of reflections with t^p (for example, p = 1, 0.5 leads to t or t^0.5). In the interval (0, 1], with initial value x₀ and a number k between 0.5 and 1 the following procedure is performed:

The numbers y_i in the interval (0, 1] are now transformed to time delays T_i in the interval [T_min, T_min + T_max] by

The increase in the density of reflections is shown by the example in Fig. 6.12.

Figure 6.12 Increase in density for nine reflections.

6.3 Subsequent Reverberation

This section deals with techniques for reproducing subsequent reverberation. The first approaches by Schroeder [Schr61, Schr62] and their extension by Moorer [Moo78] will be described. Further developments by Stautner and Puckette [Sta82], Smith [Smi85], Dattarro [Dat97], and Gardner [Gar98] led to general feedback networks [Ger71, Ger76, Jot91, Jot92, Roc95, Roc96, Roc97, Roc02] which have a random impulse response with exponential decay. An extensive discussion on analysis and synthesis parameters of subsequent reverberation can be found in [Ble01]. Apart from the echo density, an important parameter of subsequent reverberation [Cre03] is the quadratic increase in

with frequency. The following systems exhibit a quadratic increase in echo density and frequency density.

6.3.1 Schroeder Algorithm

The first software implementations of room simulation algorithms were carried out in 1961 by Schroeder. The basis for simulating an impulse response with exponential decay is a recursive comb filter as shown in Fig. 6.13.

The transfer function is given by

Figure 6.13 Recursive comb filter (g = feedback factor, M = delay length).

with

With the correspondence of the Z-transform the impulse response is given by

The complex poles are combined as pairs so that the impulse response can be written as

The impulse response is expressed as a summation of cosine oscillations with frequencies Ω_k. These frequencies correspond to the eigenfrequencies of a room. They decay with an exponential envelope rⁿ, where r is the damping constant (see Fig. 6.15a). The overall impulse response is weighted by 1/Mg. The frequency response of the comb filter is shown in Fig. 6.15c and is given by

It shows maxima at Ω = 2πk/M(k = 0, 1,…, M − 1) of magnitude

and minima at Ω = (2k + 1)π/M(k = 0, 1,…, M − 1) of magnitude

Another basis of the Schroeder algorithm is the all-pass filter shown in Fig. 6.14 with transfer function

From (6.46) it can be seen that the impulse response can also be expressed as a summation of cosine oscillations.

Figure 6.14 All-pass filter (M = delay length).

The impulse responses and frequency responses of a comb filter and an all-pass filter are presented in Fig. 6.15. Both impulse responses show an exponential decay. A sample in the impulse response occurs every M sampling periods. The density of samples in the impulse responses does not increase with time. For the recursive comb filter, spectral shaping due to the maxima at the corresponding poles of the transfer function is observed.

Frequency Density

The frequency density describes the number of eigenfrequencies per hertz and is defined for a comb filter[Jot91] as

A single comb filter gives M resonances in the interval [0, 2π], which are separated by a frequency distance of Δf = f_S/M. In order to increase the frequency density, a parallel circuit (see Fig. 6.16) of P comb filters is used which leads to

The choice of the delay systems [Schr62] is suggested as

and leads to a frequency density

Figure 6.15 (a) Impulse response of a comb filter (M = 10, g = −0.6). (b) Impulse response of an all-pass filter (M = 10, g = −0.6). (c) Frequency response of a comb filter. (d) Frequency response of an all-pass filter.

In [Schr62] a necessary frequency density of D_f = 0.15 eigenfrequencies per hertz is proposed.

Echo Density

The echo density is the number of reflections per second and is defined for a comb filter [Jot91] as

For a parallel circuit of comb filters, the echo density is given by

Figure 6.16 Parallel circuit of comb filters.

With (6.50) and (6.52), the number of parallel comb filters and the mean delay length are given by

For a frequency density D_f = 0.15 and an echo density D_t = 1000 it can be concluded that the number of parallel comb filters is P = 12 and the mean delay length is ms. Since the frequency density is proportional to the reverberation time, the number of parallel comb filters has to be increased accordingly.

A further increase in the echo density is achieved by a cascade circuit of P_A all-pass filters (see Fig. 6.17) with transfer function

These all-pass sections are connected in series with the parallel circuit of comb filters. For a sufficient echo density, 10000 reflections per second are necessary [Gri89].

Avoiding Unnatural Resonances

Since the impulse response of a single comb filter can be described as a sum of M (delay length) decaying sinusoidal oscillations, the short-time FFT of consecutive parts from this impulse response gives the frequency response shown in Fig. 6.18 in the time-frequency domain. Only the maxima are presented. The parallel circuit of comb filters with the condition (6.49) leads to radii of pole distribution as given by ). In order to avoid unnatural resonances, the radii of the pole distribution of a parallel circuit of comb filters must satisfy the condition

Figure 6.17 Cascade circuit of all-pass filters.

This leads to the short-time spectra and the pole distribution as shown in Fig. 6.19. Figure 6.20 shows the impulse response and the echogram (logarithmic presentation of the amplitude of the impulse response) of a parallel circuit of comb filters with equal and unequal pole radii. For unequal pole radius, the different decay times of the eigenfrequencies can be seen.

Figure 6.18 Short-time spectra of a comb filter (M = 8).

Reverberation Time

The reverberation time of a recursive comb filter can be adjusted with the feedback factor g which describes the ratio

of two different nonzero samples of the impulse response separated by M sampling periods. The factor g describes the decay constant per M samples. The decay constant per sampling period can be calculated from the pole radius r = g¹/M and is defined as

Figure 6.19 Short-time spectra of a parallel circuit of comb filters.

Figure 6.20 Impulse response and echogram.

The relationship between feedback factor g and pole radius r can also be expressed using (6.57) and (6.58) and is given by

With the constant radius and the logarithmic parameters R = 20 log₁₀ r and G_p = 20 log₁₀g_p, the attenuation per sampling period is given by

The reverberation time is defined as decay time of the impulse response to −60 dB. With −60/T₆₀ = R/T_S, the reverberation time can be written as

The control of reverberation time can either be carried out with the feedback factor g or the delay parameter M. The increase in the reverberation time with factor g is responsible for a pole radius close to the unit circle and, hence, leads to an amplification of maxima of the frequency response (see (6.43)). This leads to a coloring of the sound impression. The increase in the delay parameter M, on the other hand, leads to an impulse response whose nonzero samples are far apart from each other, so that individual echoes can be heard. The discrepancy between echo density and frequency density for a given reverberation time can be solved by a sufficient number of parallel comb filters.

Frequency-dependent Reverberation Time

The eigenfrequencies of rooms have a rapid decay for high frequencies. A frequency-dependent reverberation time can be implemented with a low-pass filter

in the feedback loop of a comb filter. The modified comb filter in Fig. 6.21 has transfer function

with the stability criterion

Figure 6.21 Modified low-pass comb filter.

The short-time spectra and the pole distribution of a parallel circuit with low-pass comb filters are presented in Fig. 6.22. Low eigenfrequencies decay more slowly than higher ones. The circular pole distribution becomes an elliptical distribution where the low-frequency poles are moved toward the unit circle.

Figure 6.22 Short-time spectra of a parallel circuit of low-pass comb filters.

Stereo Room Simulation

An extension of the Schroeder algorithm was suggested by Moorer [Moo78]. In addition to a parallel circuit of comb filters in series with a cascade of all-pass filters, a pattern of early reflections is generated. Figure 6.23 shows a room simulation system for a stereo signal. The generated room signals e_L(n) and e_R(n) are added to the direct signals x_L(n) and x_R(n). The input of the room simulation is the mono signal x_M(n) = x_L(n) + x_R(n) (sum signal). This mono signal is added to the left and right room signals after going through a delay line DEL1. The total sum of all reflections is fed via another delay line DEL2 to a parallel circuit of comb filters which implements subsequent reverberation. In order to get a high-quality spatial impression, it is necessary to decorrelate the room signals e_L(n) and e_R(n) [Bla74, Bla85]. This can be achieved by taking left and right room signals at different points out of the parallel circuit of comb filters. These room signals are then fed to an all-pass section to increase the echo density.

Besides the described system for stereo room simulation in which the mono signal is processed with a room algorithm, it is also possible to perform complete stereo processing of x_L(n) und x_R(n), or to process a mono signal x_M(n) = x_L(n) + x_R(n) and a side (difference) signal x_S(n) = x_L(n) − x_R(n) individually.

6.3.2 General Feedback Systems

Further developments of the comb filter method by Schroeder tried to improve the acoustic quality of reverberation and especially the increase in echo density [Ger71, Ger76, Sta82, Jot91, Jot92, Roc95, Roc96, Roc97a, RS97b]. With respect to [Jot91], the general feedback system in Fig. 6.24 is considered. For the sake of simplicity only three delay systems are shown. The feedback of output signals is carried out with the help of a matrix A which feeds back each of the three outputs to the three inputs.

Figure 6.23 Stereo room simulation.

Figure 6.24 General feedback system.

In general, for N delay systems we can write

The Z-transform leads to

with

and the diagonal delay matrix

With (6.68) the Z-transform of the output is given by

and the transfer function by

The system is stable if the feedback matrix A can be expressed as a product of a unitary matrix and a diagonal matrix with g_ii < 1 (derivation in [Sta82]). Figure 6.25 shows a general feedback system with input vector X(z), output vector Y(z), a diagonal matrix D(z) consisting of purely delay systems z^−mi, and a feedback matrix A. This feedback matrix consists of an orthogonal matrix U multiplied by the matrix G which results in a weighting of the feedback matrix A.

If an orthogonal matrix U is chosen and the weighting matrix is equal to the unit matrix G = I, the system in Fig. 6.25 implements a white-noise random signal with Gaussian distribution when a pulse excitation is applied to the input. The time density of this signal slowly increases with time. If the diagonal elements of the weighting matrix G are less than one, a random signal with exponential amplitude decay results. With the help of the weighting matrix G, the reverberation time can be adjusted. Such a feedback system performs the convolution of an audio input signal with an impulse response of exponential decay.

The effect of the orthogonal matrix U on the subjective sound perception of subsequent reverberation is of particular interest. A relationship between the distribution of the eigenvalues of the matrix U on the unit circle and the poles of the system transfer function cannot be described analytically, owing to the high order of the feedback system. In [Her94], it is shown experimentally that the distribution of eigenvalues within the right-hand or left-hand complex plane produces a uniform distribution of poles of the system transfer function. Such a feedback matrix leads to an acoustically improved reverberation. The echo density rapidly increases to the maximum value of one sample per sampling period for a uniform distribution of eigenvalues. Besides the feedback matrix, additional digital filtering is necessary for spectrally shaping the subsequent reverberation and for implementing frequency-dependent decay times (see [Jot91]). The following example illustrates the increase of the echo density.

Figure 6.25 Feedback system.

Example: First, a system with only one feedback path per comb filter is considered. The feedback matrix is then given by

Figure 6.26 shows the impulse response and the amplitude frequency response.

Figure 6.26 Impulse response and frequency response of a 4-delay system with a unit matrix as unitary feedback matrix (g = 0.83).

With the feedback matrix

from [Sta82], the impulse response and the corresponding frequency response shown in Fig. 6.27 are obtained. In contrast to Fig. 6.26, an increase in the echo density of the impulse response is observed.

Figure 6.27 Impulse response and frequency response of a 4-delay system with unitary feedback matrix (g = 0.63).

6.3.3 Feedback All-pass Systems

In addition to the general feedback systems, simple delay systems with feedback have been used for room simulators (see Fig. 6.28). Theses simulators are based on a delay line, where single delays are fed back with L feedback coefficients to the input. The sum of the input signal and feedback signal is low-pass filtered or spectrally weighted by a low-frequency shelving filter and is then put to the delay line again. The first N reflections are extracted out of the delay line according to the reflection pattern of the simulated room. They are weighted and added to the output signal. The mixing between the direct signal and the room signal is adjusted by the factor g_MIX. The inner system can be described by a rational transfer function H(z) = Y(z)/X(z). In order to avoid a low frequency density the feedback delay lengths can be made time-variant [Gri89, Gri91].

Increasing the echo density can be achieved by replacing the delays z^−M_i by frequency dependent all-pass systems A(z^−M_i). This extension was first proposed by Gardner in [Gar92, Gar98]. In addition to the replacement of z^−M_i → A(z^−M_i), the all-pass systems can be extended by embedded all-pass systems [Gar92]. Figure 6.29 shows an all-pass system (Fig. 6.29a) where the delay z^−M is replaced by a further all-pass and a unit delay z⁻¹ (Fig. 6.29b). The integration of a unit delay avoids delay-free loops. In Fig. 6.29c the inner all-pass is replaced by a cascade of two all-pass systems and a further delay z^−M₃.

Figure 6.28 Room simulation with delay line and forward and backward coefficients.

The resulting system is again an all-pass system [Gar92, Gar98]. A further modification of the general all-pass system is shown in Fig. 6.29d [Dat97, Vää97, Dah00]. Here, a delay z^−M followed by a low-pass and a weighting coefficient is used. The resulting system is called an absorbent all-pass system. With these embedded all-pass systems the room simulator shown in Fig. 6.28 is extended to a feedback all-pass system which is shown in Fig. 6.30 [Gar92, Gar98]. The feedback is performed by a low-pass filter and a feedback coefficient g, which adjusts the decay behavior. The extension to a stereo room simulator is described in [Dat97, Dah00] and is depicted in Fig. 6.31 [Dah00]. The cascaded all-pass systems A_i(z) in the left and right channel can be a combination of embedded and absorbent all-pass systems. Both output signals of the all-pass chains are fed back to the input and added. In front of both all-pass chains a coupling of both channels with a weighted sum and difference is performed. The setup and parameters of such a system are discussed in [Dah00]. A precise adjustment of reverberation time and control of echo density can be achieved by the feedback coefficients of the all-passes. The frequency density is controlled by the scaling of the delay lengths of the inner all-pass systems.

6.4 Approximation of Room Impulse Responses

In contrast to the systems for simulation of room impulse responses discussed up to this point, a method is now presented that measures and approximates the room impulse response in one step [Zöl90b, Sch92, Sch93] (see Fig. 6.32). Moreover, it leads to a parametric representation of the room impulse response. Since the decay times of room impulse responses decrease for high frequencies, use is made of multirate signal processing.

The analog system that is to be measured and approximated is excited with a binary pseudo-random sequence x(n) via a DA converter. The resulting room signal gives a digital sequence y(n) after AD conversion. The discrete-time sequence y(n) and the pseudo-random sequence x(n) are each decomposed by an analysis filter bank into subband signals y₁,…, y_P and x₁,…, x_P respectively. The sampling rate is reduced in accordance with the bandwidth of the signals. The subband signals y₁,…, y_P are approximated by adjusting the subband systems H₁(z) = A₁(z)/B₁(z),…, H_P(z) = A_P(z)/B_P(z). The outputs of these subband systems give an approximation of the measured subband signals. With this procedure, the impulse response is given in parametric form (subband parameters) and can be directly simulated in the digital domain.

Figure 6.29 Embedded and absorbing all-pass system [Gar92, Gar98, Dat97, Vää97, Dah00].

Figure 6.30 Room simulator with embedded all-pass systems [Gar92, Gar98].

Figure 6.31 Stereo room simulator with absorbent all-pass systems [Dah00].

Figure 6.32 System measuring and approximating room impulse responses.

By suitably adjusting the analysis filter bank [Sch94], the subband impulse responses are obtained directly from the cross-correlation function

The subband impulse responses are approximated by a nonrecursive filter and a recursive comb filter. The cascade of both filters leads to the transfer function

which is set equal to the impulse response in subband i. Multiplying both sides of (6.76) by the denominator 1 − giz^−Ni gives

Truncating the impulse response of each subband to K samples and comparing the coefficients of powers of z on both sides of the equation, the following set of equations is obtained:

The coefficients b₀,…, b_M and g in the above equation are determined in two steps. First, the coefficient g of the comb filter is calculated from the exponentially decaying envelope of the measured subband impulse response. The vector [1, 0,…, g]^T is then used to determine the coefficients [b₀, b₁,…, b_M]^T.

For the calculation of the coefficient g, we start with the impulse response of the comb filter H(z) = 1/(1 − gz^−N) given by

We further make use of the integrated impulse response

defined in [Schr65]. This describes the rest energy of the impulse response at time k. By taking the logarithm of h_e(k), a straight line over time index k is obtained. From the slope of the straight line we use

to determine the coefficient g [Sch94]. For M = N, the coefficients in (6.78) of the numerator polynomial are obtained directly from the impulse response

Hence, the numerator polynomial of (6.76) is a direct reproduction of the first M samples of the impulse response (see Fig. 6.33). The denominator polynomial approximates the further exponentially decaying impulse response. This method is applied to each subband. The implementation complexity can be reduced by a factor of 10 compared with the direct implementation of the broad-band impulse response [Sch94]. However, owing to the group delay caused by the filter bank, this method is not so suitable for real-time applications.

Figure 6.33 Determining model parameters from the measured impulse response.

6.5 Java Applet – Fast Convolution

The applet shown in Fig. 6.34 demonstrates audio effects resulting from a fast convolution algorithm. It is designed for a first insight into the perceptual effects of convolving an impulse response with an audio signal.

The applet generates an impulse response by modulating the amplitude of a random signal. The graphical interface presents the curve of the amplitude modulation, which can be manipulated with three control points. Two control points are used for the initial behavior of the amplitude modulation. The third control point is used for the exponential decay of the impulse response. You can choose between two predefined audio files from our web server (audio1.wav or audio2.wav) or your own local wav file to be processed [Gui05].

6.6 Exercises

1. Room Impulse Responses

1. How can we measure a room impulse response?
2. What kind of test signal is necessary?
3. How does the length of the impulse response affect the length of the test signal?

Figure 6.34 Java applet – fast convolution.

2. First Reflections

For a given sound (voice sound) calculate the delay time of a single first reflection. Write a Matlab program for the following computations.

1. How do we choose this delay time? What coefficient should be used for it?
2. Write an algorithm which performs the convolution of the input mono signal with two impulse responses which simulate a reflection to the left output yL(n) and a second reflection to the right output yR(n). Check the results by listening to the output sound.
3. Improve your algorithm to simulate two reflections which can be positioned at any angle inside the stereo mix.

3. Comb and All-pass Filters

1. Comb Filters: Based on the Schroeder algorithm, draw a signal flow graph for a comb filter consisting of a single delay line of M samples with a feedback loop containing an attenuation factor g.
   • (a) Derive the transfer function of the comb filter.
   • (b) Now the attenuation factor g is in the feed-forward path and in the feedback loop no attenuation is applied. Why can we consider the impulse response of this model to be similar to the previous one?
   • (c) In both cases how should we choose the gain factor? What will happen if we do otherwise?
   • (d) Calculate the reverberation time of the comb filter for f_S = 44.1 kHz, M = 8 and g specified previously.
   • (e) Write down what you know about the filter coefficients, plot the pole/zero locations and the frequency response of the filter
2. All-pass Filters: Realize an all-pass structure as suggested by Schroeder.
   • (a) Why can we expect a better result with an all-pass filter than with comb filter? Write a Matlab function for a comb and all-pass filter with M = 8, 16.
   • (b) Derive the transfer function and show the pole/zero locations, the impulse response, the magnitude and phase responses.
   • (c) Perform the filtering of an audio signal with the two filters and estimate the delay length M which leads to a perception of a room impression.

4. Feedback Delay Networks

Write a Matlab program which realizes a feedback delay network.

1. What is the reason for a unitary feedback matrix?
2. What is the advantage of using a unitary circulant feedback matrix?
3. How do you control the reverberation time?

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