Elektronika 2010-11.pdf - Instytut Systemów Elektronicznych ...

Nevertheless the product of algorithm in such case is numerical
value of
⎛ y
z = arctg 0
, estimated by cumulated sum
⎠ ⎞
n
⎝ x 0
of angles (+/- for left/right) applied for consecutive rotations.
Singular Value Decomposition of a matrix consists in finding
a series of singular values σ 1
, σ 2
..., σ l
which simplify inversion
of matrix. For each matrix M ∈ R m, n there exist orthogonal
matrices U ∈ R m, m and V ∈ R n, n , for which
T
m
,
n
U
MV
= Σ = diag( σ
,σ
K
,σ
)
∈
(5)
1 2
l
R
where l = min(m,n), and for r = rank(A) the diagonal values
fulfill conditions
σ
≥
σ
≥
K
≥
σ
0
(6)
1 2
r
>
σ
σ
=
K
=
σ
0
(7)
r
+ 1 = r
+
2
l
=
CORDIC architecture
Two variants of CORDIC architectures are presented in Fig. 1
and 2. Both solutions are full-synchronous with single clock. In
the first – sequential approach, same arithmetic modules are
reused in consecutive iterations. Intermediate results are fed
back via the registers and the appropriate angles are delivered
to arithmetic units by the muxes. Control is provided by iteration
counter. Another concept is pipelined architecture presented in
Fig. 2. Schematic shows a hardware providing 3 consecutive iterations.
Arithmetic blocks are replicated for each iteration, thus
the data flow may form a pipeline. This solution provides much
faster throughput, lower flexibility and needs more hardware resources.
On the other hand the control circuitry is more simple
for this solution, leading to some savings and much higher clo-
nreset
A pseudo-inverse matrix M + may be determined by
M
+
+
=
V
Σ
U
where Σ + is a pseudo-inverse of diagonal matrix, i.e. it is diagonal
matrix formed by inverted (when non-zero) values of.
SVD is currently classified among the most efficient numerical
methods leading to matrices inversion σ 1
, σ 2
..., σ l
. SVD may be
performed by the appropriate rotation of a matrix. For a basic
2 × 2 matrix M
= ⎡
a
b
⎤ the rotation angle is .
⎣
c
d
⎛ c + b
arctg
⎞
⎦
⎝
d
−
a
⎠
This operation may be done by double use of CORDIC
in two modes. First the appropriate angles are determined
and then the rotations are performed. Due to the properties of
CORDIC the iterations may be described by combinations of
adding/subtracting and shifts of bits:
x
i
+ 1 =
x
i
+
δ
i
⋅
SHIFT i
(
y
i
)
(8)
y
i+ 1 =
y
i
− δ i
⋅
SHIFT i
(
x
i
)
(9)
+
δ
where δ i
= +/-1 denotes left or right shift. Eventually hardware
implementation of CORDIC consists of adders, subtractors
and muxes
T
xin
yin
zin
i
i
i
rotation
angle
clk
nreset
clk
nreset
clk
i
shift
i
shift
i
+1
"0000 "
±
di
±
di
±
di
start, i
Fig. 1. CORDIC – sequential architecture
Rys. 1. Architektura CORDIC w wersji sekwencyjnej
i
i
i
nreset
enable
clk
nreset
enable
clk
nreset
enable
clk
nreset
clk
xout
yout
zout
i
nreset
nreset
nreset
xin
±
±
±
xout
clk
clk
shift >> 1
clk
shift >> 2
d
d
d
nreset
nreset
shift >> 1
nreset
shift >> 2
yin
±
±
±
yout
clk
nreset
d
clk
d
clk
d
zin
nreset
nreset
clk
rotation
angle 1
±
clk
rotation
angle 2
d
d
d
Fig. 2. CORDIC – pipelined architecture Rys. 2. Architektura CORDIC w wersji potokowej
±
clk
rotation
angle 3
±
zout
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