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convolution encoder using matrix multiplication
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  APPENDIX 9 Matrices and Polynomials The Multiplication of Polynomials Let  α ( z ) =  α 0 + α 1 z + α 2 z 2 + ··· α  p z  p and  y ( z ) =  y 0 + y 1 z + y 2 z 2 + ··· y n z n betwo polynomials of degrees  p  and  n  respectively. Then, their product  γ  ( z ) = α ( z ) y ( z ) is a polynomial of degree  p  +  n  of which the coefficients comprisecombinations of the coefficient of   α ( z ) and  y ( z ).A simple way of performing the multiplication is via a table of which themargins contain the elements of the two polynomials and in which the cellscontain their products. An example of such a table is given below:(1) α 0  α 1 z α 2 y 0  α 0 y 0  α 1 y 0 z α 2 y 0 z 2 y 1 z α 0 y 1 z α 1 y 1 z 2 α 2 y 0 z 3 y 2 z 2 α 0 y 1 z 2 α 1 y 2 z 3 α 2 y 2 z 4 The product is formed by adding all of the elements of the cells. However, if theelements on the SW–NE diagonal are gathered together, then a power of theargument  z  can be factored from their sum and then the associated coefficientis a coefficient of the product polynomial.The following is an example from the table above:(3) γ  0 + γ  1 z + γ  2 z 2 + γ  3 z 3 + γ  4 z 4 = α 0 y 0 + ( α 0 y 1  + α 1 y 0 ) z + ( α 0 y 2  + α 1 y 1  + α 2 y 0 +) z 2 + ( α 1 y 2  + α 2 y 1 ) z 3 + α 2 y 4 z 4 . The coefficients of the product polynomial can also be seen as the products of the convolutions of the sequences  { α 0 ,α 1 ,α 2 ...α  p }  and  { y 0 ,y 1 ,y 2 ...y n } .1  D.S.G. POLLOCK: ECONOMETRICS  The coefficients of the product polynomials can also be generated by asimple multiplication of a matrix by a vector. Thus, from the example, weshould have(3)  γ  0 γ  1 γ  2 γ  2 γ  v  =  y 0  0 0 y 1  y 0  0 y 2  y 1  y 0 0  y 2  y 1 0 0  y 2  α 0 α 1 α 2   =  α 0  0 0 α 1  α 0  0 α 2  α 1  α 0 0  α 2  α 1 0 0  α 2  y 0 y 1 y 2  . To form the elements of the product polynomial  γ  ( z ), powers of   z  may beassociated with elements of the matrices and the vectors of values indicated bythe subscripts.The argument  z  is usually described as an algebraic indeterminate. Itsplace can be taken by any of a wide variety of operators. Examples are providedby the difference operator and the lag operator that are defined in respect of doubly-infinite sequences.It is also possible to replace  z  by matrices. However, the fundamentaltheorem of algebra indicates that all polynomial equations must have solutionsthat lie in the complex plane. Therefore, it is customary, albeit unnecessary,to regard  z  as a complex number. Polynomials with Matrix ArgumentsToeplitz Matrices There are two matrix arguments of polynomials that are of particularinterest in time series analysis. The first is the matrix lag operator. Theoperator of order  T   denoted by(4)  L T   = [ e 1 ,e 2 ,...,e T  − 1 , 0]is formed from the identity matrix I  T   = [ e 0 ,e 1 ,...,e T  − 1 ] by deleting the leadingvector  e 0  and by appending a column of zeros to the end of the array. Ineffect,  L T   is the matrix with units on the first subdiagonal band and with zeroselsewhere. Likewise,  L 2 T   has units on the second subdiagonal band and withzeros elsewhere, whereas  L T  − 1 T   has a single unit in the bottom left (i.e.  S–W  )corner, and  L T  + rT   = 0 for all  r  ≥  0. In addition,  L 0 T   =  I  T   is identified with the2  POLYNOMIALS AND MATRICES  identity matrix of order  T  . The example of   L 4  is given below:(5) L 04  =  1 0 0 00 1 0 00 0 1 00 0 0 1  , L 4  =  0 0 0 01 0 0 00 1 0 00 0 1 0  ,L 24  =  0 0 0 00 0 0 01 0 0 00 1 0 0  , L 34  =  0 0 0 00 0 0 00 0 0 01 0 0 0  . Putting  L T   in place of the argument  z  of a polynomial in non-negative powerscreates a so-called Toeplitz banded lower-triangular matrix of order  T  . Anexample is provided by the quadratic polynomials  α ( z ) =  α 0  + α 1 z + α 2 z 2 and y ( z ) =  y 0  +  y 1 z  +  y 2 z 2 . Then, there is  α ( L 3 ) y ( L 3 ) =  y ( L 3 ) α ( L 3 ), which iswritten explicitly as(6)  α 0  0 0 α 1  α 0  0 α 2  α 1  α 0  y 0  0 0 y 1  y 0  0 y 2  y 1  y 0   =  y 0  0 0 y 1  y 0  0 y 2  y 1  y 0  α 0  0 0 α 1  α 0  0 α 2  α 1  α 0  . The commutativity of the two matrices in multiplication reflects their polyno-mial nature. Such commutativity is available both for lower-triangular Toeplitzand for upper-triangular Toeplitz matrices, which correspond to polynomialsin negative powers of   z .The commutativity is not available for mixtures of upper and lower trian-gular matrices; and, in this respect, the matrix algebra differs from the corre-sponding polynomial algebra. An example is provided by the matrix version of the following polynomial identity:(7) (1  − z )(1  − z − 1 ) = 2 z 0 −  ( z  + z − 1 ) = (1  − z − 1 )(1  − z )Putting  L T   in place of   z  in each of these expressions creates three differentmatrices. This can be illustrated with the case of   L 3 . Then, (1  −  z )(1  −  z − 1 )gives rise to(8)  1 0 0 − 1 1 00  − 1 1  1  − 1 00 1  − 10 0 1   =  1  − 1 0 − 1 2  − 10  − 1 2  , whereas (1  − z − 1 )(1  − z ) gives rise to(9)  1  − 1 00 1  − 10 0 1  1 0 0 − 1 1 00  − 1 1   =  2  − 1 0 − 1 2  − 10  − 1 1  . 3  D.S.G. POLLOCK: ECONOMETRICS  A Toeplitz matrix, in which each band contains only repetitions of the sameelement, is obtained from the remaining expression by replacing  z  by  L 3  in2 z 0 −  ( z  + z − 1 ), wherein  z 0 is replaced by the identity matrix:(10)  2  − 1 0 − 1 2  − 10  − 1 2   =  − 1 1 0 00  − 1 1 00 0  − 1 1  − 1 0 01  − 1 00 1  − 10 0 1  . Example.  It is straightforward to derive the dispersion matrices that are foundwithin the formulae for the finite-sample estimators from the corresponding au-tocovariance generating functions. Let  γ  ( z ) =  { γ  0 + γ  1 ( z + z − 1 )+ γ  2 ( z 2 + z − 2 )+ ···}  denote the autocovariance generating function of a stationary stochasticprocess. Then, the corresponding dispersion matrix for a sample of   T   consec-utive elements drawn from the process is(11) Γ =  γ  0 I  T   + T  − 1  τ  =1 γ  τ  ( L τ T   + F  τ T  ) , where  F  T   =  L  T   is in place of   z − 1 . Since  L T   and  F  T   are nilpotent of degree  T  ,such that  L qT  ,F  qT   = 0 when  q   ≥  T  , the index of summation has an upper limitof   T   −  1. Circulant Matrices In the second of the matrix representations, which is appropriate to afrequency-domain interpretation of filtering, the argument  z  is replaced by thefull-rank circulant matrix(12)  K  T   = [ e 1 ,e 2 ,...,e T  − 1 ,e 0 ] , which is obtained from the identity matrix  I  T   = [ e 0 ,e 1 ,...,e T  − 1 ] by displacingthe leading column to the end of the array. This is an orthonormal matrixof which the transpose is the inverse, such that  K   T  K  T   =  K  T  K   T   =  I  T  . Thepowers of the matrix form a  T  -periodic sequence such that  K  T  + qT   =  K  qT   forall  q  . The periodicity of these powers is analogous to the periodicity of thepowers of the argument  z  = exp {− i2 π/T  } , which is to be found in the Fouriertransform of a sequence of order  T  .4

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Jul 23, 2017
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