Constrained Pole-Zero Linear Prediction:

Constrained Pole-Zero Linear Prediction:

Optimization of Cascaded Biquadratic Notch Filters for Multi-Tone Frequency Estimation

Toon van Waterschoot and Marc Moonen

ESAT/SCD

Abstract

Constrained pole-zero linear prediction (CPZLP) is a new method for parametric frequency estimation of multiple real sinusoids in noise. The method is based on a signal model that consists of a cascade of constrained biquadratic models, thereby exploiting the linear prediction property of sinusoidal signals. The signal model is parametrized directly with the unknown frequencies, which are then estimated using a numerical optimization approach. By independently optimizing each biquadratic stage in the cascade model, a computationally efficient algorithm is obtained which has linear complexity. The linear complexity allows for using relatively long data records, leading to high accuracy even at low signal-to-noise ratios (SNR).

Multi-Tone Frequency Estimation

Problem definition: Given a length-L data sequence y (t) =

N X n=1

α _n cos(ω _n t + φ _n ) + r(t), t = 1, . . . , L

estimate the unknown frequencies ω _n , n = 1, . . . , N .

State-of-the-art:

Non-parametric methods:

• typically FFT-based

• limited accuracy: resolution ∼ L

• reasonable complexity: O(L log L) Parametric methods:

• ML, nonlinear LS, total LS, subspace (MUSIC, ESPRIT), ...

• high accuracy: CRLB is achieved ∀ L > L _min (SNR)

• high complexity: O (L ² ) or higher

Proposed method:

• parametric method based on constrained pole-zero model

• reasonable accuracy: CRLB is approached ∀ L > L _min (SNR)

• low complexity: O (L)

Constrained Pole-Zero Linear Prediction

Signal model: cascade of N constrained biquadratic models y (t) =





 N Y n=1

1 − 2ρ cos θ _n z ^{− 1} + ρ ² z ^{− 2} 1 − 2 cos θ _n z ^{− 1} + z ^{− 2}





 e(t)

Goal:

θ _n → ω _n , n = 1, . . . , N Global objective:

min θ

1 L

L X t=1

e ² (t, θ)

Decoupled objectives:

min θ _n

1 L

L X t=1

e ² _n (t, θ _n ), n = 1, . . . , N

Im j

− 1

− j

e ^jθ

ⁿ

1 Re θ _n

ρe ^jθ

ⁿ

e ^{− jθ}

ⁿ

ρe ^{− jθ}

ⁿ

with θ _n = ^h θ ₁ . . . θ _n ^{i T} and e _n (t, θ _n ) =





 n Y l=1

1 − 2 cos θ _l z ^{− 1} + z ^{− 2} 1 − 2ρ cos θ _l z ^{− 1} + ρ ² z ^{− 2}





 y(t)

Decoupled optimization: If N subproblems are solved consec- utively, then estimates of θ ₁ , . . . , θ _n−1 are available when solving subproblem n ⇒ each subproblem reduces to a scalar problem!

Line search algorithm for nth subproblem:

θ _n ^(k+1) = θ _n ^(k) + µ _k p ^(k)

• step length µ _k ⇒ backtracking with Armijo’s sufficient de- crease condition

• search direction p ^(k) ⇒ quasi-Newton with damped BFGS up- dating:

p ^(k) = −B _k ^{− 1} ∂

∂θ _n V _n ^ θ ˆ ^(k)

n



B _{k +1} = max

 v _k

s _k , γB _k



= scalar BFGS update

with s _k = ˆ θ _n ^(k+1) − ˆ θ _n ^(k) and v _k = _∂θ ^∂

n V _n ^ θ ˆ ^(k+1)

n

 − ^∂

∂θ _n V _n ^ θ ˆ ^(k)

n



Performance Evaluation

Computational complexity: number of multiplications M _BFGS = ¯ κN ^h (13 + 3 ¯ β )L + (17 + 5 ¯ β ) ⁱ

¯

κ: average no. iterations per subproblem

β ¯ : average no. backtracking steps per iteration per subproblem

Frequency variance:

averaged over 100 Monte Carlo runs

10

²

10

³

−120

−100

−80

−60

−40

−20

Frame length N (samples)

F re q u en cy va ri an ce (d B )

θ

1

θ

2

θ

3

CRLB( ω

₁

) CRLB(ω

₂

) CRLB( ω

₃

)

−10 −5 0 5 10 15 20 25 30 35 40

−140

−120

−100

−80

−60

−40

−20 0

SNR (dB)

F re q u en cy va ri an ce (d B )

θ

₁

θ

₂

θ

₃

CRLB(ω

₁

) CRLB(ω

2

)

CRLB(ω

₃

)