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6.4. PROJECTION TO STETTER-STRUCTURE 93
domain):
⎧
⎪⎨
⎪⎩
log(ˆυ 1 )= 0
log(ˆυ 2 ) = log(ˆυ 2 )
log(ˆυ 3 ) = 2 log(ˆυ 2 )
log(ˆυ 4 ) = log(ˆυ 4 )
log(ˆυ 5 ) = log(ˆυ 2 ) + log(ˆυ 4 )
log(ˆυ 6 ) = 2 log(ˆυ 2 ) + log(ˆυ 4 )
log(ˆυ 7 ) = 2 log(ˆυ 4 )
log(ˆυ 8 ) = log(ˆυ 2 ) + 2 log(ˆυ 4 )
log(ˆυ 9 ) = 2 log(ˆυ 2 ) + 2 log(ˆυ 4 )
(6.35)
The entries in the vector ˆυ satisfy the following relations in order to exhibit Stetter
structure with respect to multiplication:
⎧
⎪⎨
⎪⎩
ˆυ 1 = 1
ˆυ 2 = ˆυ 2
ˆυ 3 = ˆυ 2
2
ˆυ 4 = ˆυ 4
ˆυ 5 = ˆυ 2 · ˆυ 4
ˆυ 6 = ˆυ 2 2 · ˆυ 4
ˆυ 7 = ˆυ 4
2
ˆυ 8 = ˆυ 2 · ˆυ 4
2
ˆυ 9 = ˆυ 2 2 · ˆυ 4
2
(6.36)
Also the projected vector ˆψ exhibits Stetter structure with respect to addition (because
this vector is also in the logarithmic domain). To show this, we use the leastsquares
solution for φ in Equation (6.24) denoted by φ LS . We know from (6.25) that
the following holds:
ˆψ = Wφ LS − 2π k. (6.37)
The vector φ LS contains the two elements φ LS
1 and φ LS
2 which yields the following
for ˆψ:
⎛
ˆψ =
⎜
⎝
0 0
1 0
2 0
0 1
1 1
2 1
0 2
1 2
2 2
⎞
⎛
⎝
⎟
⎠
φ LS
1
φ LS
2
⎞
⎠ − 2π
⎛
⎜
⎝
k 1
k 2
k 3
k 4
k 5
k 6
k 7
k 8
k 9
⎞
. (6.38)
⎟
⎠
From the equation above we can derive the relations φ LS
1 = ˆψ 2 +2πk 2 and φ LS
2 =
ˆψ 4 +2πk 4 . Substituting this back into the latter equation yields the structure in the