NONLINEAR SYSTEM IDENTIFICATION USING DETERMINISTIC MULTILEVEL SEQUENCES Ender M. Ek¸ sio˘ glu Department of Electrical and Electronics Engineering, Istanbul Technical University, Istanbul,Turkey {ender}@ehb.itu.edu.tr
Nonlinear System Identification Using Deterministic Multilevel Sequences MAIN HEADINGS • Purpose • Multivariate Kernel Vector Representation • Identification Of The Kernel Vectors Using Multilevel Input Signals • Identification of the 1-D Kernel Vectors • Identification of the ℓ -D Kernel Vectors • Simulations • Concluding Remarks Istanbul Technical University, 2002 - DSP Conference, Santorini, Greece 1
Nonlinear System Identification Using Deterministic Multilevel Sequences PURPOSE • Nonlinear system identification is important due to the shortcoming of linear models when applied to inherently nonlinear problems which are abundant in real life applications. • The truncated (or “doubly finite”) Volterra series representation constitutes an appealing nonlinear system model, since the output is linearly dependent on the kernel parameters, hence making the identification process mathematically tractable. • This paper proposes a novel partitioning of the Volterra kernels, resulting in simple closed form solutions when deterministic multilevel input sequences are used. Istanbul Technical University, 2002 - DSP Conference, Santorini, Greece 2
Nonlinear System Identification Using Deterministic Multilevel Sequences MULTIVARIATE KERNEL VECTOR REPRESENTATION • The usual Volterra filter representation: y ( n ) = N [ x ( n )] M N N N � � � � · · · b k ( i 1 , i 2 , . . . , i k ) x ( n − i 1 ) x ( n − i 2 ) · · · x ( n − i k ) = k =1 i 1 =0 i 2 = i 1 i k = i k − 1 Here, M is the order and N is the memory length of the Volterra filter and b k ( i 1 , i 2 , . . . , i k ) is the triangular Volterra kernel of degree k . Istanbul Technical University, 2002 - DSP Conference, Santorini, Greece 3
Nonlinear System Identification Using Deterministic Multilevel Sequences • The kernels b k ( i 1 , i 2 , . . . , i k ) are grouped together according to the order k of the nonlinear input term x ( n − i 1 ) x ( n − i 2 ) · · · x ( n − i k ) . We propose a different grouping for the kernels. The output y ( n ) can be rewritten as the sum of the outputs of M different multivariate cross-term nonlinear subsystems, H ( ℓ ) . • The proposed Volterra filter representation: M M � � y ( ℓ ) ( n ) = H ( ℓ ) [ x ( n )] y ( n ) = N [ x ( n )] = ℓ =1 ℓ =1 H ( ℓ ) [ x ( n )] = N h (1) T ( i ) x (1) � h ( n − i ) ℓ = 1 i =0 I ¯ Q ℓ − 1 Q 1 qℓ − 1 h ( ℓ ) T ( q 1 , . . . , q ℓ − 1 ; i ) x ( ℓ ) � · · · � � h ( q 1 , . . . , q ℓ − 1 ; n − i ) 2 � ℓ � M q 1 =1 q ℓ − 1 =1 i =0 Istanbul Technical University, 2002 - DSP Conference, Santorini, Greece 4
Nonlinear System Identification Using Deterministic Multilevel Sequences • In this representation, the symbol H ( ℓ ) [ · ] , which represents ℓ summations, is called as an ℓ -D cross-term Volterra operator and h ( ℓ ) ( q 1 , . . . , q ℓ − 1 ; i ) is called as an ℓ -D kernel vector. • The ℓ -D input vector can be expressed in the following form:: x ( ℓ ) h,ℓ ( q 1 , . . . , q ℓ − 1 ; n ) x ( ℓ ) h,ℓ +1 ( q 1 , . . . , q ℓ − 1 ; n ) x ( ℓ ) h ( q 1 , . . . , q ℓ − 1 ; n ) = . . . x ( ℓ ) h,M ( q 1 , . . . , q ℓ − 1 ; n ) Here, the subinput vectors x ( ℓ ) h,k ( q 1 , . . . , q ℓ − 1 ; n ) consist of all possible inputs of degree k , x ( p 1 , ··· ,p ℓ ) = x p 1 ( n ) x p 2 ( n − q 1 ) · · · x p ℓ ( n − q 1 − · · · − q ℓ − 1 ) ( q 1 , . . . , q ℓ − 1 ; n )ˆ h,k Istanbul Technical University, 2002 - DSP Conference, Santorini, Greece 5
Nonlinear System Identification Using Deterministic Multilevel Sequences • The corresponding ℓ -D kernel vector h ( ℓ ) ( q 1 , . . . , q ℓ − 1 ; i ) can be rewritten in terms of subkernels. There exists an equivalent triangular kernel b k ( i 1 , i 2 , . . . , i k ) for each component of the subkernel vector h ( ℓ ) k ( q 1 , . . . , q ℓ − 1 ; i ) . • We introduced the concept of delay-wise dimensionality and cross-term subsystem to replace the multiplicational dimensionality of the regular Volterra kernels. This novel grouping enables us to devise an exact closed form algorithm for identifying the Volterra kernels using deterministic multilevel sequences. Istanbul Technical University, 2002 - DSP Conference, Santorini, Greece 6
Nonlinear System Identification Using Deterministic Multilevel Sequences IDENTIFICATION OF THE KERNEL VECTORS USING MULTILEVEL INPUT SIGNALS • In this section, we derive an efficient algorithm to identify the kernel vectors h ( ℓ ) ( q 1 , . . . , q ℓ − 1 ; i ) by using multilevel input sequences with ℓ distinct impulses. Istanbul Technical University, 2002 - DSP Conference, Santorini, Greece 7
Nonlinear System Identification Using Deterministic Multilevel Sequences • Identification of the 1-D Kernel Vectors: � M Multilevel single impulses, x (1) ( m 1 ; n ) = a m 1 δ ( n ) , for m 1 = 1 , 2 , . . . , � can be 1 used to obtain the 1-D kernel vectors. Using the cross-term representation, it is trivial to prove that the higher dimensional outputs are zero for these multilevel single impulses, i.e., y ( ℓ ) ( n ) = 0 for ℓ > 1 . Hence, N h (1) T ( i ) u (1) � � � y (1) ( m 1 ; n ) = N x (1) ( m 1 ; n ) h ( m 1 ; n − i ) = i =0 � T u (1) x ( m 1 ; n ) x 2 ( m 1 ; n ) · · · x M ( m 1 ; n ) � h ( m 1 ; n ) = Istanbul Technical University, 2002 - DSP Conference, Santorini, Greece 8
Nonlinear System Identification Using Deterministic Multilevel Sequences � M Now we can write all � ensemble outputs in the matrix form as follows:: 1 e ( n ) = H (1) � � y (1) x (1) = U (1) h (1) ( n ) e ( n ) e Here, x (1) e ( n ) , y (1) e ( n ) and U (1) e ( n ) denote the ensemble input, ensemble output vectors and the ensemble input matrix, respectively. Istanbul Technical University, 2002 - DSP Conference, Santorini, Greece 9
Nonlinear System Identification Using Deterministic Multilevel Sequences Therefore, provided the inverse of the M × M matrix U (1) exists, all the 1-D kernel e vectors can be obtained as: � − 1 � h (1) ( n ) = U (1) y (1) e ( n ) e This result shows that all 1-D kernels with one cross-term can be determined by using only the inverse of an M ensemble matrix times the ensemble output matrix. Note that the linear FIR filter identification via the impulse response is covered by this method as the special case M = 1 . Istanbul Technical University, 2002 - DSP Conference, Santorini, Greece 10
Nonlinear System Identification Using Deterministic Multilevel Sequences • Identification of the ℓ -D Kernel Vectors: The ℓ -D input ensemble vector can be written as: ℓ T ( M ) e ( n − n ( ℓ ) � x ( ℓ ) ℓ,i x (1) e ( q 1 , . . . , q ℓ − 1 ; n ) = i ) i =1 Let v ( m,k ) ( q 1 , . . . , q m − 1 ; n ) denote the ensemble output of the k -D subsystem H ( k ) e from the m -D input ensemble x ( m ) ( q 1 , . . . , q m − 1 ; n ) . e Istanbul Technical University, 2002 - DSP Conference, Santorini, Greece 11
Nonlinear System Identification Using Deterministic Multilevel Sequences • Using these definitions, the response of the nonlinear system to the ensemble input in x ( ℓ ) e ( q 1 , . . . , q ℓ − 1 ; n ) can be written in terms of the outputs of the subsystems. ℓ � y ( ℓ ) v ( ℓ,k ) e ( q 1 , . . . , q ℓ − 1 ; n ) = ( q 1 , . . . , q ℓ − 1 ; n ) e k =1 • The subsystem outputs v ( ℓ,k ) ( q 1 , . . . , q ℓ − 1 ; n ) , k = 1 , 2 , . . . , ℓ − 1 can be obtained e from the previous lower dimensional subsystem outputs. ( k ) ℓ S ( M ) ( q ( ℓ,k ) ; n − n ( ℓ,k ) � v ( ℓ,k ) ( q 1 , . . . , q ℓ − 1 ; n ) = ℓ 1 ,j v ( k,k ) ) e j j j =1 Istanbul Technical University, 2002 - DSP Conference, Santorini, Greece 12
Nonlinear System Identification Using Deterministic Multilevel Sequences • It is possible to determine the ℓ -D Volterra kernel vectors by: � − 1 � h ( ℓ ) ( q 1 , . . . , q ℓ − 1 ; n − ¯ U ( ℓ ) v ( ℓ,ℓ ) q ℓ − 1 ) = ( q 1 , . . . , q ℓ − 1 ; n ) e e Here, ( 1 ) ℓ S ( M ) ( n − n ( ℓ, 1) � v ( ℓ,ℓ ) ( q 1 , . . . , q ℓ − 1 ; n ) = y ( ℓ ) ℓ 1 ,j v (1 , 1) e ( q 1 , . . . , q ℓ − 1 ; n ) − ) e e j j =1 ( k ) ℓ ℓ − 1 S ( M ) ( q ( ℓ,k ) , . . . , q ( ℓ,k ) j,k − 1 ; n − n ( ℓ,k ) � � ℓk,j v ( k,k ) − ) e j,i j j =1 k =2 Istanbul Technical University, 2002 - DSP Conference, Santorini, Greece 13
Nonlinear System Identification Using Deterministic Multilevel Sequences • U ( ℓ ) � M � M is an � � input ensemble matrix composed of terms in the form of × e ℓ ℓ a p 1 i 1 a p 2 i 2 · · · a p ℓ � � . i ℓ • Fig. 1 depicts the identification of the Volterra kernels of orders one through M using the proposed algorithm. Istanbul Technical University, 2002 - DSP Conference, Santorini, Greece 14
Nonlinear System Identification Using Deterministic Multilevel Sequences Figure 1. Proposed Volterra kernel identification method using multilevel deterministic sequences as inputs. Istanbul Technical University, 2002 - DSP Conference, Santorini, Greece 15
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