Finite Precision
zyxwvu
nalysis
of the Conventional
QR
Decomposition
RLS
Algorithm
z
M.
zyxwv
.
Siqueira
and
P.
zyx
zy
iniz
Programa de Engenharia ElCtrica COPPE/EE/Federal University of Rio de Janeiro Caixa Postal 68504
zyxwvu
Rio de Janeiro

RJ
21945

Brazil
zyxwvu
Abstract
This paper presents relations for predicting the mean squared values in the devi ations
of
the outputs in the Conventional
QRD
zyxw
LS
algorithm
[l]
due to quantization effects. A set of formulas are presented, along with a pro cedure to employ them to calculate the expect ed value of squared norm of the tap deviation and the mean squared value of the error signal deviation. Simulation results are presented to show the accuracy of the obtained results.
I.
INTRODUCTION
The Conventional QRDRLS algorithm has been known for
a
long time
as
a
numerically stable algorithm for solving the Recursive Least Squares Problem for adaptive filtering. Bounds for its internal variables have been calculated
[2],
but so
far
no detailed analysis of the errors propagation in this algorithm has been presented. This paper shows
a
set of relations, rather than closed formulas, for calculating the mean squared values
of
deviations in the 6lter outputs with the help of
a
simple procedure. This paper uses relations for mean squared values
of
the internal variables of the Conventional QRDRLS algorithm, which can be found on
131.
The QRDRLS algorithm consists of the following steps
[2],
when implemented with inhite precision arithmetic:
Matiwral
Fonnulatzon:
For
zyxwvutsrqp
=
0
.
N
o
zyxwvutsrq
X k)
=
Q”(k). Q’ )Xp(k)
=
Q(1)X”k)
Ewer.
calculation:
IJse
ne
of
the equations
c(k)
=
d(k)
WH(k)X(k)
zyxw
(k)
=
Eq k)C05:
Onr(k). Co.O,(k)
11.
MEAN
SQUARED VALUES
OF
DEVIATIONS
N THE
INTERNAL ARIABLES
In this section, the kedpoint quantization errors are defmed, and recursive difference equations that describe the recursive error propagation are derived and solved. We assume that the input signal has been scaled to avoid overflow
[3],
so
that no overflow is considered in the analy sis. Two’s complement arithmetic is used, and consequently no errors are considered when additions and subtractions are performed. We also assume that the instantaneous quan tizations are performed by rounding, for
any
type of arith metic. The quantization error has zero mean and variance
ux
=
2Pb
12, where
b
is the number
of
bits excluding the sign. We
also
assume that multiplication, division and square root operations introduce, respectively, quantization errors described by where
a
and
b
are scalars. The accumulated quantization error in each quantity is defined
as
he difference between its value in infinite precision
(3)
0 7803 2428 5195
4.00
1995
IEEE
990
implementation and its value in finite precision,
zyxwvut
s
shown below
zyxwvutsrq
1
zyxwvu
j O
zyxwvutsrq
A.
zyxwvutsrqpo
ean Squared Value
of
Aai,l(k)
C.
Mean Squared Value
of
Ac;(k)
The quantity
aij(k)
is the intermediate value that the jth element of the first row of
XP(k)
assumes, during the triangularization process, by the time the ith Givens Rota, tion is applied. These quantities can be expressed
as
The quantity
ci(k)
is defined
as
he diagonal terms of the ma'trix
zyxwvu
(k)
when calculations are performed
as
in
a
systolic arcliitecture
[l].
This value is calculated by By using dehitions
13)
and
(10)
and neglecting second higher order terms, it follows that and
zyxwvut
14)
By using the following approximation
i
1
E{[Aaij(k)]'}
U:
zyxwvutsrqpon
~E{[ACOSB,(~)]~}
0
i
1
Aim
{E{
uLj(k)}E{[Asin
Bm(k)]'}
+
m=O
E{[Aumj(k)l2}(1
A)}
+
i
1
it
follows that
{E{uLj(k)}(l
A)
D
ean Square Value
of
Asin
O, k)
For a division operation, the following approximation is valimd for small
Aa
B.
Mean Squared Value
of
Abi(lC)
1
1
a+Aa
a
The quantities
bi(k)
are the values of the first element of the intermediate vectors that result from the application of Givens rotations to the vector
d2(lc).
The general expression for
bi(k)
is
given by Using the above relation on the definition of sin
B,(k)
a good approximation results. Using definitions
1) 4)
on this approximation and considering only the Gstorder terms, it fcdlows Substituting definitions
13)
and
10)
in the above equation and using only the first order terms
as
an approximation, it follows that
E{
[Asin
(k)]'}
=
E{[Aa,,,(k)lZ){(l
)'
E{[Ac,(k)] }A (l
A)'
;
U;

1
E{[AcosB,(k)]'}
t
nt=O
E
I
j O
E{[A~inBj(k)]~}
E{[A j(k)J'}(l
A)}
+
99
E.
zyxwvutsrqp
ean Squared Value
of
4
cos
Oi(k)
zyxwvut
of
COS
zyxwvutsrqponml
(k)
it follows Using the approximation
19)
in the following definition
H. Mean Squared Value
of
Ae(k)
The error signal in the infmite precision implementation is given by Substituting in the resulting equation the dehitions
(1)
zyxwv
(4)
and
considering only the order terms, it follows Using definitions
1) 4)
in the above equation, and consid ering only the resulting first order terms, it follows
X(1
X)3E{
Aci(k)] }
zyxwvutsrq
X(lX)a'f,
l+X
zyxwvutsrqpon
X
F.
Mean Squared Value
of
414
zyxw
k)
For the infinite precision implementation
of
the
QRRLS
algorithm, the elements of the triangularized matrix
U k)
axe calculated by
7rtJ(k)
=
X'/ trw(k
)cmB, k)
+
av k)sinB, k)
zyxwvutsr
24)
Substituting definitions (1)(4) in the above equation, and considering only the resulting first order terms, it follows that
I.
Mean Squared Value
of
Aw, k)
The taps vector
w(k)
is calculated through the
buck
sub stitution algorithm
[l],
nd the mean squared value of its deviation was obtained on
[6]
as
; E{
[Asin
Oi(k)] }
[
1
1+X
G.
Mean Squared Value
of
Adz,i(k)
The elements
of
the vector
d2(k)
re resultant
of
the
ap
plication of
N
+
1
Givens rotations to
X1I2d2 k
l ,
that is The expeted value
of
norm of the deviation vector
Aw(lc)
can be obtained by using
,,(k)
=
X1r2%,,(k
1)cm
(k)
+
b,(k)
sin
O,(k)
26)
Using the same procedure for calculating
nu,
(k),
t follows
111.
SIMULATION
ESULTS
ND
CONCLUSION
A
program in
C
was written in order to simulate the
QR
RLS
algorithm implemented with fixedpoint arithmetic to
27)
zyxwvut
99
verify the accuracy of the formulas presented.
In
all exam ples, the experimental set up consisted of the system identi fication application where both the input signal and mea, surement noise are pseudorandom sequences with normal distribution and zeromean.
zyxwvutsrq
movingaveraging processes was utilized
zyxwvutsr
s
the unknown system in the simulations. The order
of
the unknown system was 4 in all experiments. In the examples presented, the QRRLS algorithm ran for 1500 iterations and the simulation results are obtained by aver aging the results of
zyxwvutsrqp
00
independent runs. InGnite precision simulations are actually executed with 64 bits floatingpoint arithmetic. In the kedpoint implementation, the quantities are represented by numbers with magnitude less than uni ty. Frequent overflow is avoided by choosing the input signal variance appropriately.
In
the simulations to be presented, we measure the mean squared value of the deviation in the prediction error (excess
of
MSE) and the mean squared norm
of
Aw(k),
and compare with the results obtained by using Table
I.
The validity of the proposed formulas was tested by changing some of the adaptive filtering environment param eters. Except for where it is explicitly indicated, the simula, tions use the same values for input signal variance
(30
dB), precision (15 bits), forgetting factor
zyxwvutsr
A
=
0.95) and reference signals (an MA process derived from the input signal, added with
a
measurement noise of variance equal to
zyxwvutsrq
70dB).
Analysing tables 11IV we can verify that the procedure shown on table
I
and the developed formulas are a powerful tool for predicting the numerical errors due to the recursive nature
of
the QRDRLS algorithm.
30 32 35
eferences
zyxwvu
l]
S.
Haykin,
Adaptive FiEter Theory,
PrenticeHall, EnglewoodCliffs, New Jersey, 1991.
[2]
S.
Leung and
S.
Haykin, “Stability of recursive QRD LS algorithms using finiteprecision systolic array im plementation”,
IEEE
Trans. on Acoust., Speech, Signal Processing,
vol. 37, pp. 760763, May 1989. [3] M.
G.
Siqueira and
P.
S.
R. Diniz, “Infinite precision analysis
of
the QRDRLS algorithm”,
IEEE Interna tional Symposium on Circuit and Systems,
Chicago

USA,1993. [4]
P.
S.
R. Diniz and M.
G.
Siqueira, “Fixedpoint error analysis of the QR recursive least squares algorithm”, submitted to
IEEE
Transactions on Circuits and Systems.
[5]
C. Samson and
V.
Reddy, “Fixed point error andy sis
of
the normalized ladder algorithm”,
IEEE Trans. on Acoust., Speech, Signal Processing,
vol. 31, p. 11771191, October 1983. [6] M.
G.
Siqueira and
P.
S.
R. Diniz, “Stabilty Analysis of the QRRecursive Least Squares Algorithm”,
IEEE Midwest Conference on Circuit and Systems,
De troit

USA ,1993.
I
64.7 64.3 92.6 92.2
I
62.6 62.3 92.5 92.2 59.7 89.4 92.5 92.2
TABLE
C
PSEUDOCODE
OR
COMPUTATION
F
zyxw
{
[AW~(~)]~) for
i=O;
i
5
N;
++)
I
Calculate
E{[Au~~(~)]’}
hrough
14)
Calculate
E{[AC~(~)]~}
hrough
(18)
Calculate
E([A
in
zyxw
, k)]’}
and
E{
[ACOS
Z k)lz}
hrough (21)
and
(23) for
+i+l;
j
_
N;
++
i
Calculate
E{
[Aa,
k)]’}
through
14)
Calculate
E{[AU,~(~)]~}
hrough
25)
X
0.90
Calculate
E([Abi(k)]*}
hrough
15)
Calculate
E{
[A ,i(k)]’}
hrough
27)
Calculate
E{[Awi(k)Iz}
hrough (30)
E{llak)ll;}
dB)
EiIWk)l2}
dB) Simulated
I[
Calculated Simulated Calculated 68.2
11
67.7 93.7
II
93.2 TABLE
I1
SIMULATIONS
OR
VARIOUS
zy
:
Et
IP k)ll3
dB)
i ”
dB)
’
Simulated Calculated Simulated Calculated 2.5 69.7 69.S 92.5 92.2 27 67.7 67.3 92.6 92.2
I
\
,,,L,
\
I
LL
\
I‘
I
\
I
i ”
dB)
’
Simulated Calculated Simulated Calculated 2.5 69.7 69.S
I
92.5 92.2 27 67.7 67.3 92.6 92.2
TABLE
11
SIMULATIONS FOR VARIOUS TABLE
IV
SIMULATIONS
OR
DIFFERENT
RECISIONS
68.2 64.3
Ij
91.7 92.2 124.9 124.6
/I
152.8 152.4
50

154.0
1.54.7
I[
182.8 182.5 94.8 94.4
[
122.3 122.:3
993