## FD01AD

### Fast recursive least-squares filtering

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

      To solve the least-squares filtering problem recursively in time.
      Each subroutine call implements one time update of the solution.
      The algorithm uses a fast QR-decomposition based approach.

Specification

          SUBROUTINE FD01AD( JP, L, LAMBDA, XIN, YIN, EFOR, XF, EPSBCK,
         $                   CTETA, STETA, YQ, EPOS, EOUT, SALPH, IWARN,
         $                   INFO )
    C     .. Scalar Arguments ..
          CHARACTER         JP
          INTEGER           INFO, IWARN, L
          DOUBLE PRECISION  EFOR, EOUT, EPOS, LAMBDA, XIN, YIN
    C     .. Array Arguments ..
          DOUBLE PRECISION  CTETA(*), EPSBCK(*), SALPH(*), STETA(*), XF(*),
         $                  YQ(*)

Arguments

Mode Parameters

      JP      CHARACTER*1
              Indicates whether the user wishes to apply both prediction
              and filtering parts, as follows:
              = 'B':  Both prediction and filtering parts are to be
                      applied;
              = 'P':  Only the prediction section is to be applied.

Input/Output Parameters

      L       (input) INTEGER
              The length of the impulse response of the equivalent
              transversal filter model.  L >= 1.

      LAMBDA  (input) DOUBLE PRECISION
              Square root of the forgetting factor.
              For tracking capabilities and exponentially stable error
              propagation, LAMBDA < 1.0 (strict inequality) should
              be used.  0.0 < LAMBDA <= 1.0.

      XIN     (input) DOUBLE PRECISION
              The input sample at instant n.
              (The situation just before and just after the call of
              the routine are denoted by instant (n-1) and instant n,
              respectively.)

      YIN     (input) DOUBLE PRECISION
              If JP = 'B', then YIN must contain the reference sample
              at instant n.
              Otherwise, YIN is not referenced.

      EFOR    (input/output) DOUBLE PRECISION
              On entry, this parameter must contain the square root of
              exponentially weighted forward prediction error energy
              at instant (n-1).  EFOR >= 0.0.
              On exit, this parameter contains the square root of the
              exponentially weighted forward prediction error energy
              at instant n.

      XF      (input/output) DOUBLE PRECISION array, dimension (L)
              On entry, this array must contain the transformed forward
              prediction variables at instant (n-1).
              On exit, this array contains the transformed forward
              prediction variables at instant n.

      EPSBCK  (input/output) DOUBLE PRECISION array, dimension (L+1)
              On entry, the leading L elements of this array must
              contain the normalized a posteriori backward prediction
              error residuals of orders zero through L-1, respectively,
              at instant (n-1), and EPSBCK(L+1) must contain the
              square-root of the so-called "conversion factor" at
              instant (n-1).
              On exit, this array contains the normalized a posteriori
              backward prediction error residuals, plus the square root
              of the conversion factor at instant n.

      CTETA   (input/output) DOUBLE PRECISION array, dimension (L)
              On entry, this array must contain the cosines of the
              rotation angles used in time updates, at instant (n-1).
              On exit, this array contains the cosines of the rotation
              angles at instant n.

      STETA   (input/output) DOUBLE PRECISION array, dimension (L)
              On entry, this array must contain the sines of the
              rotation angles used in time updates, at instant (n-1).
              On exit, this array contains the sines of the rotation
              angles at instant n.

      YQ      (input/output) DOUBLE PRECISION array, dimension (L)
              On entry, if JP = 'B', then this array must contain the
              orthogonally transformed reference vector at instant
              (n-1). These elements are also the tap multipliers of an
              equivalent normalized lattice least-squares filter.
              Otherwise, YQ is not referenced and can be supplied as
              a dummy array (i.e., declare this array to be YQ(1) in
              the calling program).
              On exit, if JP = 'B', then this array contains the
              orthogonally transformed reference vector at instant n.

      EPOS    (output) DOUBLE PRECISION
              The a posteriori forward prediction error residual.

      EOUT    (output) DOUBLE PRECISION
              If JP = 'B', then EOUT contains the a posteriori output
              error residual from the least-squares filter at instant n.

      SALPH   (output) DOUBLE PRECISION array, dimension (L)
              The element SALPH(i), i=1,...,L, contains the opposite of
              the i-(th) reflection coefficient for the least-squares
              normalized lattice predictor (whose value is -SALPH(i)).

Warning Indicator

      IWARN   INTEGER
              = 0:  no warning;
              = 1:  an element to be annihilated by a rotation is less
                    than the machine precision (see LAPACK Library
                    routine DLAMCH).

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value.

Method

      The output error EOUT at instant n, denoted by EOUT(n), is the
      reference sample minus a linear combination of L successive input
      samples:

                            L-1
         EOUT(n) = YIN(n) - SUM h_i * XIN(n-i),
                            i=0

      where YIN(n) and XIN(n) are the scalar samples at instant n.
      A least-squares filter uses those h_0,...,h_{L-1} which minimize
      an exponentially weighted sum of successive output errors squared:

          n
         SUM [LAMBDA**(2(n-k)) * EOUT(k)**2].
         k=1

      Each subroutine call performs a time update of the least-squares
      filter using a fast least-squares algorithm derived from a
      QR decomposition, as described in references [1] and [2] (the
      notation from [2] is followed in the naming of the arrays).
      The algorithm does not compute the parameters h_0,...,h_{L-1} from
      the above formula, but instead furnishes the parameters of an
      equivalent normalized least-squares lattice filter, which are
      available from the arrays SALPH (reflection coefficients) and YQ
      (tap multipliers), as well as the exponentially weighted input
      signal energy

          n                                              L
         SUM [LAMBDA**(2(n-k)) * XIN(k)**2] = EFOR**2 + SUM XF(i)**2.
         k=1                                            i=1

      For more details on reflection coefficients and tap multipliers,
      references [2] and [4] are recommended.

References

      [1]  Proudler, I. K., McWhirter, J. G., and Shepherd, T. J.
           Fast QRD based algorithms for least-squares linear
           prediction.
           Proceedings IMA Conf. Mathematics in Signal Processing
           Warwick, UK, December 1988.

      [2]  Regalia, P. A., and Bellanger, M. G.
           On the duality between QR methods and lattice methods in
           least-squares adaptive filtering.
           IEEE Trans. Signal Processing, SP-39, pp. 879-891,
           April 1991.

      [3]  Regalia, P. A.
           Numerical stability properties of a QR-based fast
           least-squares algorithm.
           IEEE Trans. Signal Processing, SP-41, June 1993.

      [4]  Lev-Ari, H., Kailath, T., and Cioffi, J.
           Least-squares adaptive lattice and transversal filters:
           A unified geometric theory.
           IEEE Trans. Information Theory, IT-30, pp. 222-236,
           March 1984.

Numerical Aspects

      The algorithm requires O(L) operations for each subroutine call.
      It is backward consistent for all input sequences XIN, and
      backward stable for persistently exciting input sequences,
      assuming LAMBDA < 1.0 (see [3]).
      If the condition of the signal is very poor (IWARN = 1), then the
      results are not guaranteed to be reliable.

Further Comments

      1.  For tracking capabilities and exponentially stable error
          propagation, LAMBDA < 1.0 should be used.  LAMBDA is typically
          chosen slightly less than 1.0 so that "past" data are
          exponentially forgotten.
      2.  Prior to the first subroutine call, the variables must be
          initialized. The following initial values are recommended:

          XF(i) = 0.0,        i=1,...,L
          EPSBCK(i) = 0.0     i=1,...,L
          EPSBCK(L+1) = 1.0
          CTETA(i) = 1.0      i=1,...,L
          STETA(i) = 0.0      i=1,...,L
          YQ(i) = 0.0         i=1,...,L

          EFOR = 0.0          (exact start)
          EFOR = "small positive constant" (soft start).

          Soft starts are numerically more reliable, but result in a
          biased least-squares solution during the first few iterations.
          This bias decays exponentially fast provided LAMBDA < 1.0.
          If sigma is the standard deviation of the input sequence
          XIN, then initializing EFOR = sigma*1.0E-02 usually works
          well.

Example

Program Text

    *     FD01AD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT, NOUT1
          PARAMETER        ( NIN = 5, NOUT = 6, NOUT1 = 7 )
          DOUBLE PRECISION ZERO, ONE
          PARAMETER        ( ZERO = 0.0D0, ONE = 1.0D0 )
          INTEGER          IMAX, LMAX
          PARAMETER        ( IMAX = 500, LMAX = 10 )
          DOUBLE PRECISION LAMBDA
          PARAMETER        ( LAMBDA = 0.99D0 )
    *     .. Local Scalars ..
          CHARACTER        JP
          INTEGER          I, INFO, IWARN, L
          DOUBLE PRECISION DELTA, EFOR, EOUT, EPOS, XIN, YIN
    *     .. Local Arrays ..
          DOUBLE PRECISION CTETA(LMAX), EPSBCK(LMAX+1), SALPH(LMAX),
         $                 STETA(LMAX), XF(LMAX), YQ(LMAX)
    *     .. External Functions ..
          DOUBLE PRECISION XFCN, YFCN
          EXTERNAL         XFCN, YFCN
    *     NOTE: XFCN() generates at each iteration the next sample of the
    *           input sequence. YFCN() generates at each iteration the next
    *           sample of the reference sequence. These functions are user
    *           defined (obtained from data acquisition devices, for
    *           example).
    *     .. External Subroutines ..
          EXTERNAL         FD01AD
    *
    *     .. File for the output error sequence ..
          OPEN ( UNIT = NOUT1, FILE = 'ERR.OUT', STATUS = 'REPLACE' )
    *     ..  Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) L, DELTA, JP
          IF ( L.LE.0 .OR. L.GT.LMAX ) THEN
             WRITE ( NOUT, FMT = 99992 ) L
          ELSE
             IF ( DELTA.LT.ZERO ) THEN
                WRITE ( NOUT, FMT = 99991 )
             ELSE
    *
                DO 10 I = 1, L
                   CTETA(I)  = ONE
                   STETA(I)  = ZERO
                   EPSBCK(I) = ZERO
                   XF(I) = ZERO
                   YQ(I) = ZERO
       10       CONTINUE
                EPSBCK(L+1) = ONE
                EFOR = DELTA
    *           .. Run least squares filter.
                DO 20 I = 1, IMAX
                   XIN = XFCN(I)
                   YIN = YFCN(I)
                   CALL FD01AD( JP, L, LAMBDA, XIN, YIN, EFOR, XF, EPSBCK,
         $                      CTETA, STETA, YQ, EPOS, EOUT, SALPH, IWARN,
         $                      INFO)
                   WRITE(NOUT1,*) EOUT
       20       CONTINUE
                CLOSE(NOUT1)
    *           NOTE:  File 'ERR.OUT' now contains the output error
    *                  sequence.
    *
                IF ( INFO.NE.0 ) THEN
                   WRITE ( NOUT, FMT = 99998 ) INFO
                ELSE
                   WRITE ( NOUT, FMT = 99997 )
                   DO 30 I = 1, L
                      WRITE ( NOUT, FMT = 99996 ) I, XF(I), YQ(I), EPSBCK(I)
       30          CONTINUE
                   WRITE ( NOUT, FMT = 99995 ) L+1, EPSBCK(L+1)
                   WRITE ( NOUT, FMT = 99994 ) EFOR
                   IF ( IWARN.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99993 ) IWARN
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' FD01AD EXAMPLE PROGRAM RESULTS', /1X)
    99998 FORMAT (' INFO on exit from FD01AD = ', I2)
    99997 FORMAT ('  i', 7X, 'XF(i)', 7X, 'YQ(i)', 6X, 'EPSBCK(i)', /1X)
    99996 FORMAT ( I3, 2X, 3(2X, F10.6))
    99995 FORMAT ( I3, 28X, F10.6, /1X)
    99994 FORMAT (' EFOR = ', D10.3)
    99993 FORMAT (' IWARN on exit from FD01AD = ', I2)
    99992 FORMAT (/' L is out of range.',/' L = ',I5)
    99991 FORMAT (/' The exponentially weighted forward prediction error',
         $         '  energy must be non-negative.' )
    *
          END
    *
    *     .. Example functions ..
    *
          DOUBLE PRECISION FUNCTION XFCN( I )
    *     .. Intrinsic Functions ..
          INTRINSIC        DBLE, SIN
    *     .. Local Scalar ..
          INTEGER          I
    *     .. Executable Statements ..
          XFCN = SIN( 0.3D0*DBLE( I ) )
    * *** Last line of XFCN ***
          END
    *
          DOUBLE PRECISION FUNCTION YFCN( I )
    *     .. Intrinsic Functions ..
          INTRINSIC        DBLE, SIN
    *     .. Local Scalar ..
          INTEGER          I
    *     .. Executable Statements ..
          YFCN = 0.5D0 * SIN( 0.3D0*DBLE( I ) ) +
         $       2.0D0 * SIN( 0.3D0*DBLE( I-1 ) )
    * *** Last line of YFCN ***
          END

Program Data

     FD01AD EXAMPLE PROGRAM DATA
       2    1.0D-2     B

Program Results

     FD01AD EXAMPLE PROGRAM RESULTS

      i       XF(i)       YQ(i)      EPSBCK(i)

      1      4.880088   12.307615   -0.140367
      2     -1.456881    2.914057   -0.140367
      3                              0.980099

     EFOR =  0.197D-02

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