Abstract

Nonlinear system modelling plays an important role in the field of digital audio systems whereas the most of real-world devices shows a nonlinear behaviour. Amongnonlinear models, Hammerstein systems are particular nonlinear systems composed of a static nonlinearity cascaded with a linear filter. In this paper, a novel approachfor the estimation of the static nonlinearity is proposed based on the introduction of an adaptive CatmullRom cubic spline in order to overcome problems related to theadaptation of high-order polynomials necessary for identifying highly nonlinear systems. Experimental results confirm the effectiveness of the approach, making alsocomparisons with existing techniques of the state of the art.

Linear systems can be fully characterized by theirimpulse responses (IRs) in the time domain andtheir transfer functions in the frequency domain.

0 500 1000 1500 2000 2500 3000 3500 4000 4500−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Samples

Figure 1: Example of an IR.

Linear SystemEven if linear systems are in some cases reasonable approx-imations, nonlinear models have to be introduced in order tocorrectly describe the most of real-world phenomena [1].

Figure 2: Nonlinear devices.

Real World

State of the Art

Several techniques have been presented in the literature in order to manage nonlinear systems.

• Dynamic convolution [2, 3], based on the application of signals with different levels to a device under test in order to derive the resulting IRs for each level.

• Volterra series [4] and diagonal Volterra kernels (tipically used for weak nonlinearities) [5].

• Identification by using swept sine signals [6].

• Adaptive filtering of orthogonal nonlinear functions [1] for the estimation of nonlinear systems.

In order to identify nonlinear systems suitable models have to be taken into account. The choice of the model influences the accuracy of the analysis and the computational costof the process. Among the different proposed nonlinear models, the Hammerstein cascade consists of a memory-less nonlinearity in cascade with a linear dynamic system [4] asshown in Figure 3. Although less general than functional expansion models, e.g., Volterra series [4], a Hammerstein cascade allows the representation of hard nonlinearities usinga reduced number of parameters still obtaining an accurate modelling [7, 8, 9].

Figure 3: Block diagram of the Hammerstein system.

Fi(ui) = v(ui)Q(i).

where

Q(i) =[Qi Qi+1 Qi+2 Qi+3

]Tand

v(ui) = T (ui)1

2

−1 3 −3 12 −5 4 −1−1 0 1 00 2 0 0

.

The cubic CR spline is introduced in the proposed approach in replacement ofthe polynomial, taking advantage of its interesting features, i.e., local interpola-tion and regularization characteristics [10]. The spline is described by an arrayof control knot points [Q1, ..., QNs] of length Ns, where each point is defined by itsx and y coordinates Qi = [qx,i qy,i]

T , being (·)T the transpose operator.

x(n)

uΔx

x

f(x)

Δx

QiQi+1 Qi+2

Qi+3

Figure 4: The i-th CR spline with fixed

distance ∆x between two adjacent control

points.

CatmullRom Spline

Proposed Approach

In this paper a new adaptive algorithm for the estimation of the Hammerstein model coefficients is proposed.The memory-less nonlinearity is modelled through an adaptive Catmull-Rom cubic spline instead of an adaptive polynomial whereas the dynamic linear part ismodelled through an adaptive IIR filter as described in [11]. Then, the system coefficients are updated according to a stochastic gradient procedure in order to minimizethe istantaneous squared estimation error.

The linear part of the Hammerstein system is modelled through an IIRfilter [11]. Thus, the input-output relationship results described by thefollowing equation:

d(n) = −N∑i=1

ai(n)d(n− i) +M∑j=0

bj(n)z(n− j),

Hammerstein model adaptation

The coefficients of the nonlinear model are updated according to astochastic gradient procedure in order to minimize the istantaneoussquared estimation error at each iteration, as fully discussed in [11]. Giventhe estimated error at each iteration n:

e(n) = d(n)− d(n),

the model coefficients are then updated in order to minimize the squaredof equation (1) as follows:

θ(n + 1) = θ(n) + µnR−1(n + 1)φ(n)

δ +HT (n)µnφ(n)e(n),

being

θ(n) =[a(1) . . . a(N) b(1) . . . b(M) Q1(n) . . . QNs(n)

]T.

Results

( a ) Identified characteristic in Test 1 ( b ) MSE in Test 1

Figure 5: Identified non linear input-output characteristic and MSE for the proposed approach (red line) and the approach in[11] (green line) in relation to the measured characteristic (blue line).

( a ) Identified characteristic in Test 2 ( b ) MSE in Test 2

Figure 6: Identified non linear input-output characteristic and MSE for the proposed approach (red line) and the approach in[11] (green line) in relation to the measured characteristic (blue line).

( a ) Identified characteristic in Test 3 ( b ) MSE in Test 3

Figure 7: Identified non linear input-output characteristic and MSE for the proposed approach (red line) and the approach in[11] (green line) in relation to the measured characteristic (blue line).

( a ) Synthesis of the signal with the identified nonlinearity. ( b ) MSE in Test 4

Figure 8: Real signal versus signal synthesized using the identified nonlinearity and MSE for the proposed approach (redline) and the approach in [11] (green line) in relation to the measured characteristic (blue line).

Simulated Polynomial Nonlinearity

z(n) = 0.1x(n)− 0.075x2(n) + 0.05x3(n) + 0.2x4(n)− 0.3x5(n)

Linear Component

H(z) =1 + 0.1z−1

1− 0.4z−1 + 0.2z−2.

Simulated Guitar Distortion (Symmetric)

z(n) = 1 + kx(n)

1 + k|x(n)|, with k =

2a

1− awith a = 0.5.

Linear Component

H(z) =1 + 0.1z−1

1− 0.4z−1 + 0.2z−2.

VST Overdrive PlugIn with Asymmetric Nonlinearcharacteristic followed by the linear system

H(z) =1 + 0.1z−1

1− 0.4z−1 + 0.2z−2.

Real-World Device (BOSS DS-2 Guitar DistortionPedal)

Test 1

Test 2

Test 3

Test 4

Linear coefficients identification.

Table 1: Real and estimated coefficients of the dynamic linear system.

Coefficients Value Approach of [11] Proposed approach

a1 −0.4 −0.4036 −0.4020a2 0.2 0.2013 0.1999

b1 0.1 0.0957 0.0981

( a ) Test1

Coefficients Value Approach of [11] Proposed approach

a1 −0.4 −0.4070 −0.4003a2 0.2 0.2040 0.2001

b1 0.1 0.0947 0.0998

( b ) Test2

Coefficients Value Approach of [11] Proposed approach

a1 −0.4 −0.3607 −0.3998a2 0.2 0.1732 0.2010

b1 0.1 0.1423 0.0997

( c ) Test3

ConclusionA novel approach has been proposed in this paper for estimating the Hammerstein coefficients based on the introduction of an adaptive CR cubic spline for the identification of thestatic nonlinearity and an adaptive IIR filter for the identification of the linear part. Experimental results show the effectiveness of the proposed approach, making also comparisonswith another technique known in the literature using a polynomial-based approach for estimating the nonlinear part of the Hammerstein system. Better performance in terms ofMSE behaviour and coefficients approximation can be obtained with the proposed approach, especially when high-orders polynomials are required. Informal listening tests havepreliminary carried out, confirming the objective results. Future work will be oriented to the introduction of a FIR filter for estimating the dynamic linear system in order to avoidstability problems, especially noticed during tests on real-world device.

References[1] L. Romoli, M. Gasparini, S. Cecchi, A. Primavera, and F. Piazza, “Adaptive Identification of Nonlinear Models Using Orthogonal Nonlinear Functions,” in Proc. 48th Audio Engineering Society Conference, Monaco, Germany, Sep. 2012.

[2] M. Kemp, “Analysis and Simulation of Non-Linear Audio Processes using Finite Impulse Responses Derived at Multiple Impulse Amplitudes,” in Proc. 106th Audio Engineering Society Convention, Munich, Germany, May 1999.

[3] A. Farina and E. Armelloni, “Emulation of Not-Linear, Time-Variant Devices by the Convolution Technique,” in Congresso AES Italia, Como, Nov 2005.

[4] T. Ogunfunmi, Adaptive Nonlinear System Identification: The Volterra and Wiener Model Approaches. Springer, 2007.

[5] A. Farina, A. Bellini, and E. Armelloni, “Non-Linear Convolution: a New Approach for the Auralization of Distorting Systems,” in Proc. 110th Audio Engineering Society Convention, Amsterdam, NL, May 2001.

[6] A. Novak, L. Simon, F. Kadlec, and P. Lotton, “Nonlinear System Identification Using Exponential Swept-Sine Signal,” IEEE Trans. Instrum. Meas., vol. 59, no. 8, pp. 2220–2229, Aug. 2010.

[7] D. T. Westwick and R. E. Kearney, “Identification of a Hammerstein model of the stretch reflex EMG using separable least squares,” in Proc. 22nd Annual Int. Conference of the Engineering in Medicine and Biology Society, Chicago, IL, USA, Jul. 2000.

[8] E. J. Dempsey and D. T. Westwick, “Identification of Hammerstein models with cubic spline nonlinearities,” IEEE Trans. Biomed. Eng., vol. 51, no. 2, pp. 237–245, 2004.

[9] X. Wu, N. Zheng, X. Yang, J. Shi, and H. Chen, “A Spline-Based Hammerstein Predistortion for 3G Power Amplifiers with Hard Nonlinearities,” in Proc. 2nd Int. Conference on Future Computer and Communication, Wuhan, China, May 2010.

[10] S. Guarnieri, F. Piazza, and A. Uncini, “Multilayer feedforward networks with adaptive spline activation function,” IEEE Trans. on Neural Networks, vol. 10, no. 3, pp. 672–683, 1999.

[11] J. Jeraj and V. J. Mathews, “A Stable Adaptive Hammerstein Filter Employing Partial Orthogonalization of the Input Signals,” IEEE Trans. Signal Process., vol. 54, no. 4, pp. 1412–1420, 2006.

Paper ID 8913

System Identication based on Hammerstein Models using Cubic Splines

Michele Gasparini1, Andrea Primavera1, Laura Romoli1, Stefania Cecchi1, Francesco Piazza11A3Lab - DII - Universita Politecnica delle Marche

Via Brecce Bianche 1, 60131 Ancona, Italywww.a3lab.dii.univpm.it