This document presents two methods for blind identification of finite impulse response (FIR) systems using a greatest common divisor (GCD) approach. It begins by introducing the blind system identification problem and GCD methods. It then outlines the main result that the impulse response h can be obtained from the GCD of the z-transforms of the output signals from two experiments with different input signals. The document proceeds to describe Method 1 and Method 2 for computing the GCD and recovering h. Both methods have a computational complexity of O(T3y) where Ty is the length of the signals. Future work is proposed to extend the GCD approach to noisy data and multivariate systems.

1. Blind FIR system identification
using a greatest common divisor
approach
Mayank and Ivan Markovsky
Dept. ELEC, Vrije Universiteit Brussel
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2. Outline
• The blind system identification problem
• Greatest common divisor methods
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3. Blind system identification
• Finite impulse response (FIR) system
y = u ∗ h
y(t) = (u ∗ h)(t) =
n
τ=0
h(τ) u(t − τ), t = 0, . . . , Ty
• Blind system identification: given y, find u and h
• Ill-posed problem: there are infinitely many solutions
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4. Two experiments
20 40 60 80
−1
0
1
10 20 30 40
−1
0
1
20 40 60 80
−0.2
0
0.2
10 20 30 40
−1
0
1
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5. Question: (y1
, y2
) → h
20 40 60 80
−1
0
1
20 40 60 80
−0.2
0
0.2
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6. Answer: GCD(Y1
, Y2
) ↔ h
20 40 60 80
−1
0
1
20 40 60 80
−0.2
0
0.2
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7. Z-transform
U(z) = Z(u) =
Tu
t=0
u(t)z−t
Convolution property. Let U, H and Y be the Z-tranform
of signals u, h and y, respectively
y = u ∗ h ⇐⇒ Y = U H
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8. Greatest common divisor (GCD)
Let
Y1
=
T1
i=0
y1
i zi
and Y2
=
T2
i=0
y2
i zi
The GCD of Y1
and Y2
is a highest possible degree
polynomial, that is a factor of both Y1
and Y2
Example 1. Let
Y1
= z3
− 1 = (z − 1)(z2
+ z + 1),
Y2
= z3
+ 2z2
+ 2z + 1 = (z + 1)(z2
+ z + 1)
GCD(Y1
, Y2
) = z2
+ z + 1
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9. Main result
Theorem. Let
yi
= ui
∗ h, for i = 1, 2,
where inputs u1
and u2
have finite support. Let U1
, U2
and Y1
, Y2
are the Z-transform of the input signals u1
, u2
and output signal y1
, y2
respectively. If input polynomials
U1
and U2
have no common roots, then
h = α Z−1
GCD(Y1
, Y2
) , α ∈ R
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10. Link to linear algebra: Multiplication matrix
U H = Y















u0
u1 u0
... u1
...
un−1
...
... u0
un un−1
... u1
un
...
...
... un−1
un















Mn+1(U)





h0
h1
...
hn





=





y0
y1
...
yTy





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11. Vandermonde matrix
• For the vector z = (z1, . . . , zn)
VT (z) =





(z1)0
(z2)0
· · · (zn)0
(z1)1
(z2)1
· · · (zn)1
...
...
...
...
(z1)T
(z2)T
· · · (zn)T





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12. Shifting property of Vandermonde matrix





(z1)0
· · · (zn)0
(z1)1
· · · (zn)1
...
...
...
(z1)T−1
· · · (zn)T−1





VT (z)



z1
...
zn



diag(z)
=





(z1)1
· · · (zn)1
(z1)2
· · · (zn)2
...
...
...
(z1)T
· · · (zn)T





VT (z)
VT (z) diag(z) = VT (z)
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13. Sylvester matrix
S (Y1
, Y2
) = MT2
(Y1
) MT1
(Y2
)
=















y1
0 y2
0
y1
1 y1
0 y2
1 y2
0
... y1
1
...
... y2
1
...
y1
n−1
...
... y1
0 y2
n−1
...
... y2
0
y1
n y1
n−1
... y1
1 y2
n y2
n−1
... y2
1
y1
n
...
... y2
n
...
...
... y1
n−1
... y2
n−1
y1
n y2
n















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14. Lemma 1.
• deg GCD(Y1
, Y2
) = T1 + T2 − rank S(Y1
, Y2
)
dim ker(S(Y1,Y2))
Lemma 2. If ker S(Y1
, Y2
) = span(w1, . . . , wn),
there exist z = (z1, . . . , zn) such that,
• w1 · · · wd
W
G = VT (z), for some nonsingular G
• GCD(Y1
, Y2
) =
n
i=1
(z − zi)
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15. Method 1: [w1 · · · wn] → (z1, . . . , zn)
• Let ker S(Y1)
, Y2
) = span(w1, . . . , wn)
• Define W = [w1 · · · wn], WG = VT (z)
• WG diag(z) = WG ← Shift-property
• WG diag(z) = WG
• G diag(z) G−1
= W†
W
• eig(W†
W) = (z1, . . . , zn) ← roots of the GCD
• H =
n
i=1
(z − zi)
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16. Source code: Method 1
function H = fir_deconv1(Y1, Y2)
W = null (Sylvester_matrix(Y1, Y2));
W1 = W(1:end − 1, :);
W2 = W(2: end, :);
roots = eig(W1 W2);
H = poly(roots );
end
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17. Method 2
Let H = GCD(Y1
, Y2
) of degree n and the degree of Y1
and Y2
are T1 and T2 such that
• Y1
= U1
H and Y2
= U2
H
•
Y1
U1
=
Y2
U2
= H
• Y1
U2
− Y2
U1
= 0
• MT2−n+1(Y1
) MT1−n+1(Y2
)
U2
−U1 =
0
0
• Y2
= U2
H ⇐⇒ Y2
= Mn+1(U2
)H
• H = Mn+1(U2
)
†
Y2
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18. Source code: Method 2
function H = fir_deconv2(Y1, Y2)
M_Y1 = Multiplication_matrix (Y1, T2 − d + 1);
M_Y2 = Multiplication_matrix (Y2, T1 − d + 1);
M = [M_Y1 M_Y2];
z = null (M);
n = T2 + T1 − rank(Sylvester_matrix(Y1, Y2));
U2 = z(1:T2 − n + 1);
M_U2 = Multiplication_matrix (U2, n + 1);
H = M_U2 Y2;
end
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19. Elapsed time: O(T3
y )
0 200 400 600 800 1000
−1
0
1
2
3
4
5
length of the signals Ty
Time(sec)
Method 1
Theoretical time
0 200 400 600 800 1000
−1
0
1
2
3
4
length of the signals Ty
Time(sec)
Method 2
Theoretical time
5
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20. Future work
• Extend the GCD approach for the Blind FIR system
Identification to
• Noisy data
• Multivariate systems
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