### 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  and  (the
notation from  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  and  are recommended.

```
References
```    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.

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

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

  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 ).
If the condition of the signal is very poor (IWARN = 1), then the
results are not guaranteed to be reliable.

```
```  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
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. 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 ..
*
*     .. 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
```