A re-weighted ZA-LMS (RZA-LMS) algorithm has also been designed for further improving the modeling ability under sparse conditions [5]. The room impulse ...
On the Design of a Sparse Adaptive Room Equalizer Rushi Jariwala, Jyoti Maheshwari, Nithin V. George Department of Electrical Engineering, Indian Institute of Technology Gandhinagar, Gujarat, India Email: {rushi.jariwala, jyoti.maheshwari, nithin}@iitgn.ac.in Abstract—Adaptive room equalization is a technique which compensates for the modification of a sound signal caused by the impulse response of the room in which the sound is played. Room impulse responses as well as the impulse response of the equalizer are generally sparse in nature. However, traditional adaptive room equalizers are not designed to make use of this sparseness. In order to utilize the sparse nature of the scenario, this paper proposes a sparse adaptive room equalizer. The proposed scheme has been shown through a simulation study to provide improved room equalization over conventional methods.
I.
I NTRODUCTION
Equalization of the room impulse response has become a fascinating research area in recent times. Sound travels an acoustic path in a room before reaching the listener. The listener receives a distorted sound which may not be pleasing to the ears or may not be of desired quality. Reflections from the wall and the interference of other sources present in the room affect the impulse response of the room and it leads to appreciable distortion in the sound quality. In order to solve this problem, it is necessary to equalize the effect of the acoustic path to get a good approximation of the desired signal. Several different approaches have been proposed to establish the equaliser based on the impulse response. Basic method of equalisation involves designing the equaliser based on the impulse response obtained from a single location from a room [1], [2], [3]. The traditional least mean square (LMS) algorithms are not effective in modeling sparse systems. A set of sparse LMS algorithms has been recently proposed to utilize the sparseness of the system being identified [4]. A zero-attracting LMS (ZA-LMS) algorithm, which forces some of the near zero coefficients to zero has been discussed in [5] for sparse system identification. A re-weighted ZA-LMS (RZA-LMS) algorithm has also been designed for further improving the modeling ability under sparse conditions [5]. The room impulse responses are generally sparse in nature and the steady state equalizer impulse responses also contains many near zero values which may be approximated as zeros without considerable degradation in equalizer performance. Under the above mentioned approximation, the equalizer may be considered as a sparse system [6] and an attempt is made in this paper to design an effective learning rule in such a scenario. A new method for automatic selection of the regularization parameter has also be developed in this paper. The rest of the paper is organized as follows. The proposed scheme is developed in Section II. The new equalization mechanism has been tested using a simulation study in Section III and the concluding remarks are made in Section IV.
Fig. 1.
Schematic diagram of an adaptive room equalization system.
II.
P ROPOSED S CHEME
Figure 1 shows the basic schematic diagram of an adaptive room equalization scenario. The primary objective of the adaptive equalizer is to ensure that the signal z(n) sensed at the listener location is as close as possible to the delayed version of the input signal x(n). Let w(n) = [w1 (n), w2 (n), . . . , wi (n), . . . , wN (n)]T be the impulse response of the adaptive equalizer (of size N ), the input of which is x(n). The response of the equalizer is given by y(n) = wT (n) ∗ x(n),
(1) T
where x(n) = [x(n), x(n − 1), . . . , x(n − N + 1)] is the tap delayed input signal vector. The equalizer output y(n) acts as the input to the loudspeaker and a microphone is kept near the listener to measure the characteristics of the sound signal received at the location of the listener. The electro-acoustic path between the input of the loudspeaker and the output of the microphone may be modeled as the room impulse response h(n) = [h1 (n), h2 (n), . . . , hM (n)]T of length M . Using the room impulse response, the signal measured at the microphone may be written as g(n) = hT (n) ∗ y(n),
(2)
where y(n) = [y(n), y(n − 1), . . . , y(n − M + 1)]T is the tap delayed equalizer response vector. In a conventional room equalizer, the weights w(n) are updated using a gradient descent approach, which minimizes the cost function ξ(n) = E e2 (n) ≈ e2 (n) (3) where E[·] is the expectation operator and e(n) = x(n − D) − g(n)
(4)
N 2.
The weights are updated in conventional room with D = equalizer using the filtered-x least mean square (FxLMS) algorithm as w(n + 1) = w(n) + µe(n)xf (n),
(5)
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Fig. 3.
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Experiment 1: Room impulse response (K = 4, η = 0.3). 30 FxLMS FxRZA−LMS
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Fig. 2. Schematic diagram of the room impulse response measurement setup.
where µ is the learning rate and xf (n) is x(n) filtered through a model of the room impulse response. As discussed in the introduction section, the steady state equalizer impulse response may be approximated as a sparse system, the weights of which may be updated with an objective to minimize N X e2 (n) |wi (n)| ξ1 (n) = +γ log 1 + , (6) 2 0 i=1 where γ is the penalty factor and 0 = 1 with as the shrinkage magnitude. The weights of the equalizer are updated using w(n + 1) = w(n) − µ with
∂ξ1 (n) . ∂w(n)
∂ξ1 (n) sgn [w(n)] = −e(n)xf (n) + ρ , ∂w(n) 1 + |w(n)|
(7)
(8)
where ρ = γµ is the regularization parameter. The new weight update rule is given by w(n + 1) = w(n) + µe(n)xf (n) − ρ
sgn [w(n)] , 1 + |w(n)|
(9)
which is hereafter referred to as the filtered-x RZA-LMS (FxRZA-LMS) algorithm. We have considered ρ = βe2 (n), where β is a small constant. The contribution of the sparse penalty in the cost function will reduce as the error decreases which will ensure that all the weights are not reduced to zero and that the performance is not affected for less sparse systems. III.
S IMULATION S TUDY
The capability of the proposed FxRZA-LMS based room equalization mechanism is tested in this section. We have considered a room of size 20 m ×19 m ×21 m, with the primary sound source located at position A as shown in Figure 2. The room also contains K virtual secondary sources, which interfere with the sound generated by the primary source. In Figure 2 we have depicted four virtual sources V1 to V4 . The room boundary is assumed to have a reflection coefficient of η and a microphone is placed at position B to measure the room
MSE (dB)
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Fig. 4. Experiment 1: Comparison of convergence characteristics obtained as an average of 100 independent trials.
impulse response using the method by McGovern [7], [8]. A set of three equalization scenarios have been considered in this paper, for the same room with different number of virtual sources or reflection coefficients. The input signal x(n) used in all the three experiments is a random signal, with magnitude uniformly distributed in the range [−0.5, 0.5] and the length of the equalizer used is three times the length of the room impulse response. Number of samples by which the desired signal is delayed is equivalent to half the length of the equalizer. The performance of the proposed scheme has been compared with a traditional LMS algorithm based equalizer using the mean square error (MSE) as the metric for comparison. Figure 3 shows the room impulse response used in the first experiment. The impulse response is sparse, with 106 nonzero elements out of 229 tap coefficients and the response has been obtained with K = 4, η = 0.3. Figure 4 shows the comparison of convergence characteristics of FxRZA-LMS and FxLMS algorithm based equalizer systems. The improved equalization provided by the proposed sparse equalizer is clear from the convergence characteristics. The mean MSE, computed as an average of last 1000 samples is −32.79 dB for FxLMS and −35.17 dB for FxRZA-LMS based equalizer. In this experiment, the sparse equalizer has a steady state sparsity of 318 687 . The various other simulation parameters used are: µ = 1×10−3 (for FxLMS as well as FxRZA-LMS algorithms), β = 1 × 10−4 , = 1. In the second experiment, the reflection coefficient is
TABLE I. C OMPARISON OF STEADY STATE MSE (dB) OF F X LMS F X RZA-LMS ALGORITHMS . T HE DATA REPORTED ARE OBTAINED AS AN AVERAGE OF LAST 1000 SAMPLES IN EACH CASE .
1
AND
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Algorithm FxLMS FxRZA-LMS
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Experiment 2 -11.90 -19.35
Experiment 3 -26.60 -32.65
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Experiment 1 -32.79 -35.17
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Experiment 2: Room impulse response (K = 4, η = 0.6).
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Fig. 7.
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Experiment 3: Room Impulse Response (K = 8, η = 0.3).
−10
adaptive room equalizer, which utilizes the sparse nature of the equalization scenario has been designed in this paper. The new system has been shown to provide improved room equalization over conventional FxLMS algorithm based scheme.
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ACKNOWLEDGEMENT
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Fig. 6. Experiment 2: Comparison of convergence characteristics obtained as an average of 100 independent trials.
increased to 0.6 and the impulse response obtained is shown in the Figure 5 with K = 4. The impulse response have 222 coefficients with 152 non-zero elements and the convergence characteristics is shown in Figure 6. The enhanced equalization provided by the proposed equalizer is evident form the mean MSEs shown in Table I. The sparse equalizer has a steady state 74 in this experiment and the other simulation sparsity of 666 parameters employed are: µ = 2 × 10−4 (for FxLMS as well as FxRZA-LMS algorithms), β = 3 × 10−6 , = 1. Another experiment, with an impulse response of 424 coefficients with 125 non-zero elements (Figure 7) has been conducted to further check the performance of the proposed scheme. The impulse response has been computed considering eight secondary sources and a reflection coefficient of 0.3. The proposed method has been shown to provide improved room equalization in comparison with FxLMS algorithm. The comparison of convergence characteristics is made in Figure 8 and the MSEs have been compared in Table I. The steady 536 state sparsity of the proposed equalizer is 1272 in this case and the simulation parameters used are: µ = 6 × 10−4 (for FxLMS as well as FxRZA-LMS algorithms), β = 4 × 10−5 , = 1.
This work was supported by the Summer Research Internship Programme (SRIP) of Indian Institute of Technology Gandhinagar. R EFERENCES [1]
S. J. Elliott and P. A. Nelson, “Multiple-point equalization in a room using adaptive digital filters,” Journal of the Audio Engineering Society, vol. 37, no. 11, pp. 899–907, 1989. [2] S. Cecchi, A. Primavera, F. Piazza et al., “An adaptive multiple position room response equalizer,” in Signal Processing Conference, 2011 19th European. IEEE, 2011, pp. 1274–1278. [3] M. Schneider and W. Kellermann, “Adaptive listening room equalization using a scalable filtering structure in the wave domain,” in Acoustics, Speech and Signal Processing (ICASSP), 2012 IEEE International Conference on. IEEE, 2012, pp. 13–16.
30 FxLMS FxRZA−LMS
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IV.
C ONCLUSIONS −50
Room impulse responses as well as room equalizer impulse responses are generally sparse in nature. Traditional room equalization schemes are not designed to make use of the sparse nature and generally offer slow convergence. An
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Fig. 8. Experiment 3: Comparison of convergence characteristics obtained as an average of 100 independent trials.
[4]
[5]
[6]
[7]
[8]
N. Kalouptsidis, G. Mileounis, B. Babadi, and V. Tarokh, “Adaptive algorithms for sparse system identification,” Signal Processing, vol. 91, no. 8, pp. 1910–1919, 2011. Y. Chen, Y. Gu, and A. O. Hero, “Sparse LMS for system identification,” in IEEE International Conference on Acoustics, Speech and Signal Processing, 2009. IEEE, 2009, pp. 3125–3128. L. Fuster, M. De Diego, M. Ferrer, A. Gonzalez, and G. Pinero, “A biased multichannel adaptive algorithm for room equalization,” in Signal Processing Conference (EUSIPCO), 2012 Proceedings of the 20th European. IEEE, 2012, pp. 1344–1348. S. G. McGovern, “Fast image method for impulse response calculations of box-shaped rooms,” Applied Acoustics, vol. 70, no. 1, pp. 182–189, 2009. S. McGovern, “The image-source reverberation model in an Ndimensional space,” in 14th International Conference on Digital Audio Effects, 2011, pp. 11–18.
