In this presentation, we focus on the problem of blind joint calibration of multiband transceivers and time-delay (TD) estimation of multipath channels. We show that this problem can be formulated as a particular case of covariance matching. Although this problem is severely ill-posed, prior information about radio-frequency chain distortions and multipath channel sparsity is used for regularization. This approach leads to a biconvex optimization problem, which is formulated as a rank-constrained linear system and solved by a simple group Lasso algorithm. % This method is general and can be also applied for calibration of sensors arrays and in direction of arrival estimation. Numerical experiments show that the proposed algorithm provides better calibration and higher resolution for TD estimation than current state-of-the-art methods.

1. Joint Blind Calibration and Time-Delay
Estimation for Multiband Ranging
Tarik Kazaz
Collaborators: Mario Coutino, Gerard J.M. Janssen and Alle-Jan van der Veen
Email: t.kazaz@tudelft.nl
Circuits and Systems group, Faculty of EEMCS
Delft University of Technology
ICASSP 2020, Barcelona, Spain

2. How often do you ask yourself questions like:
Where is my bike, wallet or keys?
Where is my child or pet?
Where are my parents or grandparents, and are they well?
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3. Imagine that using wireless signals:
We are able to detect fall of a person.
Vehicles are able to position themselves within lane-level accuracy.
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4. For many interesting applications, current positioning approaches
are not accurate enough!
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5. Wireless positioning typically requires range estimation
Ranging: exchange probing signals ⇒ channel estimation
Channel = RF chain effects + multipath propagation
h(t)
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6. Parametric Channel Model
Tx
Rx
h(t)
t
𝛼1
𝜏1 𝜏2 𝜏3 𝜏4
𝛼2
𝛼3
𝛼4
h(t)
The multipath channel with K propagation paths is defined by
h(t) =
K
k=1
αkδ(t − τk) and H(ω) =
K
k=1
αke−jωτk
, (1)
αk ∈ R and τk ∈ R+ - the gain and time-delay of the kth resolvable path.
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7. The Channel Estimation Problems
Channel estimation for communications vs localization
𝒉𝒊(𝒕)
τ1, α1
τK, αK
τ2, α2s
s(t) 𝑦𝑖(𝑡)
gRx,i(t)
𝑒−𝑗𝜔 𝑖 𝑡
𝑒−𝑗𝜔 𝑖 𝑡
𝒄𝒊 𝒕 = 𝒉𝒊(𝒕) ∗ gTx,i(t) ∗ gRx,i(t)
gTx,i(t)
𝒚𝒊
ci(t)
hi(t)
gi(t) = gTx,i(t) ∗ gRx,i(t)
- the compound channel impluse response,
- the multipath channel impluse response,
- the compound response of RF chains.
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8. Problem Formulation
Our objective is to jointly:
estimate parameters α = [α1, . . . , αK]T ∈ CK and
τ = [τ1, . . . , τK]T ∈ RK
+ , and
calibrate RF chains, i.e., estimate gi(t),
by probing the channel with known wideband OFDM signal s(t).
𝒉𝒊(𝒕)
τ1, α1
τK, αK
τ2, α2s
s(t) 𝑦𝑖(𝑡)
gRx,i(t)
𝑒−𝑗𝜔 𝑖 𝑡
𝑒−𝑗𝜔 𝑖 𝑡
gTx,i(t)
𝒚𝒊
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9. Signal Model
The OFDM probing signal is given by
s(t) =
N−1
n=0 snejωscnt, t ∈ [−TCP , Tsym]
0, otherwise ,
and it covers L separate frequency bands with bandwidth B and central
frequencies ωi, i = 0, · · · , L − 1.
s = [s0, . . . , sN−1]T ∈ CN
TCP and Tsym = 2π/ωsc
ωsc
- the known pilot symbols,
- the cyclic prefix and OFDM symbol duration,
- the subcarrier spacing.
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10. Signal Model
𝒉𝒊(𝒕)
τ1, α1
τK, αK
τ2, α2s
s(t) 𝑦𝑖(𝑡)
gRx,i(t)
𝑒−𝑗𝜔 𝑖 𝑡
𝑒−𝑗𝜔 𝑖 𝑡
gTx,i(t)
𝒚𝒊
The signal received at the ith band after conversion to the baseband and
low-pass filtering is
yi(t) = s(t) ∗ ci(t) + wi(t) , (2)
where ci(t) = gi(t) ∗ hi(t), wi(t) is low-pass filtered Gaussian white noise.
The corresponding CTFT of the signal yi(t) is
Yi(ω) =
S(ω)Ci(ω) + Wi(ω), ω ∈ [−B
2 , B
2 ]
0, otherwise ,
(3)
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11. Data Model
𝒉𝒊(𝒕)
τ1, α1
τK, αK
τ2, α2s
s(t) 𝑦𝑖(𝑡)
gRx,i(t)
𝑒−𝑗𝜔 𝑖 𝑡
𝑒−𝑗𝜔 𝑖 𝑡
gTx,i(t)
𝒚𝒊
The discrete frequency domain data model of the signals received during P
probing intervals can be written as
yi(p) = diag(s)ci(p) + wi(p) , p = 1, . . . , P , (4)
where ci(p) = diag(gi)hi(p).
s
ci(p), gi, hi(p)
- are the known pilot symbols,
- are the samples of Ci(ω), Gi(ω) and Hi(ω) at the
subcarrier frequencies, respectively,
wi(p) - is zero-mean white Gaussian distributed noise.
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12. Data Model
Consider that bands are laying on the discrete grid ωi = ω0 +niωsc, ni ∈ N,
and ω0 is the lowest frequency considered during probing.
Then, the samples in hi, can be written as
Hi[n] =
K
k=1
αke−jniωscτk
e−jnωscτk
, n = −
N
2
, . . . ,
N
2
. (5)
The multipath components introduce band-dependent and
subcarrier-dependent phase shifts in the frequency response.
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13. Data Model
The channel vector hi(p) satisfies the model
hi = Mdiag(θi)α(p) , (6)
where M ∈ CN×K is a Vandermonde matrix
M =





1 1 · · · 1
Φ1 Φ2 · · · ΦK
...
...
...
...
ΦN−1
1 ΦN−1
2 · · · ΦN−1
K





. (7)
Φk = e−jφk , φk = ωscτk are subcarrier-dependent phase shifts, and
θi = [θi,1, . . . , θi,K]T ∈ CK, θi,k = Φni
k are band-dependent phase shifts.
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14. Data Model
Each ci is estimated from yi by deconvolution as
ci(p) = diag−1
(s)yi(p) .
The deconvolved measurements satisfy the model
ci(p) = diag(gi)Mdiag(θi)α(p) + wi(p) . (8)
Let, c(p) = [cT
1 (p), . . . , cT
L(p)]T ∈ CNL collect multiband measurements,
then it satisfies the model
c(p) = diag(g)A(τ)α(p) + w(p) , (9)
where A(τ) = [a(τ1), . . . , a(τK)] ∈ CNL×K,
A(τ) =





M
Mdiag(θ1)
...
Mdiag(θL−1)





, g =





g1
g2
...
gL





, w =





w1
w2
...
wL





.
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15. Joint Blind Calibration and Time-Delay Estimation
Our objective is to estimate g and τ from the measurement matrix
C = [c(1), . . . , c(P)] ∈ CNL×P , which satisfies
C = diag(g)A(τ)X + W , (10)
where X = [α(1), . . . , α(P)] ∈ CK×P , and W collects w(p) ∀ p.
g and τ can be estimated by solving the following optimization problem
ˆg, ˆτ, ˆX = min
g,τ,X
C − diag(g)A(τ)X 2
F , (11)
where · F is the Frobenius norm of a matrix.
This problem is ill-posed and non-linear, and therefore impossible to solve
without further assumptions.
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16. Joint Blind Calibration and Time-Delay Estimation
Assumptions:
The entries of g are slowly changing, and it can be approximated as:
g = Bp ,
where B ∈ CNL×R as columns has R known basis functions and p
are unknown calibrating parameters
The multipath channel is sparse and the time-delays lay on a uniform
grid of M NL delays, i.e., τk ∈ T = {0, τmax
M , . . . , τmax(M−1)
M }
Then (11) can be reformulated to the biconvex optimization problem
ˆp, ˆXs = min
p,Xs
C − diag(Bp)ADXs
2
F + λ XT
s 2,1 , (12)
where AD = [a(t0), . . . , a(tM−1)] ∈ CNL×M is a dictionary, tm = m
M τmax
and Xs ∈ CM×P is a row sparse matrix.
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17. Joint Blind Calibration and Time-Delay Estimation
The problem in (12) can be solved using alternating minimization, but there
is convergence guarantees and it is computationally complex, especially in
multiple measurement scenarios, i.e. P > 1.
Key idea:
Formulate joint blind calibration and time-delay estimation as a sparse
covariance matching optimization problem.
The covariance matrix of c(p) is given by
Rc := E{c(p)cH
(p)} ∈ CNL×NL
,
= diag(g)A(τ)ΣαAH
(τ)diag(¯g) + σ2
wINL ,
(13)
where Σα = diag(σα), σα = [σ2
α,1, . . . , σ2
α,K]T , and Σw = σ2
wINL.
Vectorized form is
rc = diag(g ⊗ g)K(τ)σα + rw , (14)
where K(τ) = A(τ) ◦ A(τ) ∈ C(NL)2×K
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18. Joint Blind Calibration and Time-Delay Estimation
We can write
rc = diag(Dz)KDrα + rw , (15)
where D = B ⊗ B, z = p ⊗ p, KD = AD ◦ AD is a dictionary matrix and
rα ∈ RM is a K sparse vector that collects the powers of the MPCs.
Then, z and rα can be estimated by solving biconvex optimization problem
ˆz, ˆrα = min
z,rα
˜rc − diag(Dz)KDrα
2
2 + λ rα 1 , (16)
where ˜rc = ˆrc −rw, · 2 denotes the 2-norm of the vector, λ > 0 controls
the level of sparsity of rα, and · 1 is the 1-norm of a vector.
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19. Joint Blind Calibration and Time-Delay Estimation
To alleviate difficulties arising from the biconvexity of the objective function,
we reformulate (16) using lifting techniques on the unknown variables.
The elements of rc can be written as
[rc]n = [Dz]nkT
n rα + [rw]n = dT
n zrT
αkn + [rw]n , ∀ n, (17)
where dT
n and kT
n denote the nth row of D and KD, respectively.
Therefore, rc can be written as
rc = A(Q) + rw := vec({dT
n Qkn}NL
n=1) + rw , (18)
where Q := zrT
α and A : CR2×M → CNL is the linear operator.
Given that dT
n Qkn = (kn ⊗ dn)T vec(Q) ∀ n, (18) becomes
rc = Γq + rw , (19)
where the nth row of Γ ∈ CNL×R2M is γT
n = (kn ⊗dn)T , and q = vec(Q).
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20. Joint Blind Calibration and Time-Delay Estimation
The problem of estimating z and rα then reduces to finding a rank-1
matrix Q satisfying the set of linear constrains (19), and solution can
be found by solving
ˆQ = min
Q
˜rc − Γvec(Q) 2
2 + λ Q ∗ , (20)
where ˜rc = ˆrc − rw, · ∗ is the nuclear norm of a matrix.
The matrix Q is not only rank-1 but also column sparse. Since for any
matrix L, L 2,1 > L ∗ holds, we can reformulate (20) to simpler form
ˆq = arg min
q
˜rc − Γvec(Q) 2
2 + λ Q 2,1 , (21)
where the regularization parameter λ > 0 is set to be proportional to the
noise power σ2
w. This problem is convex and can be identified as a
group Lasso problem, which can be solved efficiently.
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21. Joint Blind Calibration and Time-Delay Estimation
Input: {T , B, Γ, ˜rc, λ}
ˆq ← arg minq ˜rc − Γvec(Q) 2
2 + λ Q 2,1;
ˆQ ← unvec(ˆq);
{ˆz, ∼, ˆrs} ← svds(Q, 1);
Z ← unvec(ˆz);
{∼, ∼, ˆp} ← svds(Z, 1);
ˆg ← Bˆp;
{ˆσα, indxSet} ← find(ˆrs ∼= 0);
ˆτ ← T (indxSet);
Output: {ˆp, ˆg, ˆσα, ˆτ}
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22. Numerical Simulations
Scenario:
The multipath channel has eight dominant MPCs, i.e., K = 8,with
gains distributed according to a Rician distribution,
The continuous-time channel is modeled using a 2 GHz grid, with
channel tap delays spaced at 500 ps,
The receiver estimates the channel frequency response in four
frequency bands, i.e., L = 4, using a probing signals with N = 64
subcarriers and a bandwidth of B = 20 MHz,
The central frequencies of the bands are {10, 70, 130, 280} MHz,
The gain errors, i.e., elements of g, are drawn uniformly from the
interval of [−3, 3]dB, considering that gain variations are smooth over
subcarriers.
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23. Performance comparison
100 200 300 400 500 600 700 800 900 1000
10
-1
10
0
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24. Performance comparison
-10 -5 0 5 10 15 20 25 30
10
-1
10
0
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25. Performance comparison
100 200 300 400 500 600 700 800 900 1000
0
1
2
3
4
5
6
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26. Performance comparison
-10 -5 0 5 10 15 20 25 30
0
1
2
3
4
5
6
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27. Thank You!
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