1. Signal processing and applications
University of Donja Gorica
Marko Simeunović, PhD

2. Outline Problems in the PPS estimation
Improved PD-based PPS estimators
2D PPS estimators
Non-uniform signal sampling and impulsive noise
Refining PPS parameter estimates
The QML approach
Applications

3. Polynomial phase signals - Applications

4. Signal model The PPS model can be described as: s(n) – Pth order PPS
ai – phase parameters
A – amplitude
v(n) – zero-mean white Gaussian noise
The goal is to estimate parameters of s(n) from noisy observations x(n).

5. Maximum-likelihood estimation
The maximum-likelihood estimation is done by performing P- dimensional search:
The amplitude and initial phase are estimated after dechirping of higher-order phase parameters.
Pros:
high accuracy close to the CRLB;
low SNR threshold.
Cons:
P-dimensional search;
limitation to lower order PPSs.

6. Phase differentiation
The phase differentiation (PD) based approach is proposed for reducing estimation complexity:
where and are lag parameters.
High-order ambiguity function (HAF) is defined as:
HAF peaks at:

7. HAF and PHAF The HAF has following issues:
number of noisy and interference terms increases as P does;
high SNR threshold for relatively low order PPSs;
error propagation;
problem with cross-terms for multi-component signals.
Product HAF (PHAF) is solution for multi-component PPS:

8. CPF
The cubic phase function (CPF) is proposed for the estimation of cubic phase signals (P=3):
The CPF reaches maximum at the second order phase derivative.
CPF pros with respect to the HAF:
6dB lower SNR threshold;
reduced effect of the error propagation.
Cons:
it can only estimate cubic phase signals;
fast realization.

9. Our research directions
The general conclusion is:
A compromise between calculation complexity
and accuracy has to be made
Our research directions:
reducing number of auto-correlations, i.e. the order of the PD operator;
search optimization in procedure for locating peak of estimator function;
parameter refinement procedures;
parameter estimation in impulsive noise and missing sample case;
estimation of multi-dimensional signals;
applications of PPS estimators.

10. Hybrid CPF-HAF approach
The HAF/PHAF performance can be improved by reducing number of PDs and modification of the auto-correlation function.
The hybrid CPF-HAF (HCPF-HAF) approach can be described as:
HCPF-HAF vs HAF:
lower number of noise and interference terms;
for 9dB lower SNR threshold;
reduced error propagation effect.

11. HCPF-HAF vs HAF

12. Product HCPF-HAF approach
The product HCPF-HAF is introduced for the estimation of multi- component PPSs:
performance of mono-component PPS

13. Two component PPSs

14. HO-CPF-WD approach
The HO-CPF-WD approach is proposed for the estimation of higher- order PPSs.
It consists of three steps procedure:
Step 1: estimation of odd order phase parameters:
Step 2: estimation of even order phase parameters:
Step 3: estimation of initial phase and amplitude.

15. Genetic algorithm
Efficient maximization of the HO-CPF-WD and ML approach is challenged task.
Problem: Large numbers of local optima that increases as P does.
Solution: Search optimization using genetic algorithm.

16. GA setup Problem: in 1% of cases divergence occurs.
Max generation
300
Stall generation 20
Cost function
ML/HO-WD/HO-CPF
Population
size
5x30
coding
real numbers
Init. population
distribution
uniform
range
[0,50]
Selection
roulette wheel
Elit count 2
Crossover
type
heuristic
ratio
1.2
Mutation
Gaussian
init. variance
0.8
Migration
Direction
both
interval
number of population
20%
Problem: in 1% of cases divergence occurs.
Solution: divergence detection
if Of≥εME GA convergence,
if Of<εME GA divergence.
Divergence occurs – rerun the algorithm.
Pros:
reduced complexity with respect to the direct search;
real-time estimation of PPSs up to 10th order.

17. GA validation

18. Non-uniform sampled CPF
Two reasons for realization of the CPF using the FT:
reducing the estimation complexity;
applying advanced techniques for search optimization.
The non-uniform sampled CPF (NU-CPF) is defined as:
Peak position: second-order phase derivative.
Complexity: same as for the HAF.
Interpolation is used for the determination of signal samples at non-integer time instants.

19. NU-CPF vs CPF

20. 2D PPS 2D PPS can be described as
The PD operator defined for 2D PPS is of the following form:
It consists of phase differentiations along n and m coordinates.

21. 2D PPS estimators
The 2D versions of PD-based estimators are given in the following:
2D HAF (FFM):
2D PHAF:
2D CPF:

22. 2D HAF vs 2D HCPF-HAF

23. HOCPF-WD with nonuniform sampled PPS
In our research, we considered scenario where signal is nonuniformly sampled at time instants tk, where total number of samples K is below that required by the sampling theorem, i.e. K≤N.
Two cases are considered:
Case 1: signal is symmetrically sampled with respect to t=0, i.e. tk= -tK-k-1;
Case 2: signal sampling is not symmetric with respect to t=0.
Solution for Case 1: since pairs x(tk) and x(tK-k-1) are known, the HOCPF- WD is calculated for available samples:

24. HOCPF-WD with nonuniform sampled PPS
Solution for Case 2: interpolation of missing samples from the pairs x(tk) and x(tK-k-1).
Interpolation steps:
Step 1: Interpolation by the DFT with factor F=4 or F=8 to obtain signal
Step 2: Calculation of missing sample by linear interpolation using two closest samples from the refined grid, and

25. Results for Case 1

26. Results for Case 2

27. Missing samples case
Assume that sampling recordings can be described as (NM – position of missing samples):
Interpolation in auto-correlation domain is more convenient that interpolation in FT domain since higher-order PPS is more concentrated.
Interpolated auto-correlation function:
is evaluated from using interpolation in time-lag domain – Matlab interp1 function.
is now used in the PD operator as 1st order PD.

28. Missing samples case - evaluation

29. Mixed Gaussian and impulsive noise
The impulsive noise should be recognized and samples corrupted by impulses should be eliminated from the signal and treated as missing ones.
Recognition of samples corrupted by impulsive noise can be performed using hard thresholding as
Threshold is chosen based on assumed percentage of sampled corrupted by impulsive nose:
Optimal threshold can be selected performing PPS estimation for various percentages q∈Q and choosing optimal one by maximization of the cost function (ML).

30. Simulations

31. Parameter refinemet
Due to errors in the estimation process signal after dechir-ping is still Pth order PPS
The O’Shea based refinement estimates errors made in coarse estimation pushing estimation MSE to the CRLB.
The O’Shea approach consists of the following steps:
Step 1: Signal filtering by moving average filter
Step 2: Phase unwrapping

32. Parameter refinemet
Step 3: Polynomial regression
The O’Shea refinement is generalized to both multi-component and multi-dimensional PPSs.

33. QML algorithm PPS estimators are generally unbiased and nonlinear.
Biased estimators coupled by O’Shea refinement is able to handle high-order PPS.
The QML algorithm can be described using following steps:
Step 1: Calculate the STFT for various window widths h
Step 2: Estimate IFs from the STFT
Step 3: Coarse parameter estimation using polynomial interpolation
Step 4: Parameter refinement using O’Shea approach to obtain

34. QML algorithm The QML algorithm can be used also for:
Step 5: Select optimal estimates by maximizing the criteria function
The QML algorithm can be used also for:
PPS when P is not known in advance;
for signals with non-polynomial modulation.
The QML algorithm is extended to:
2D PPSs;
multi-component PPS;
compound/combined models;
aliased signals.

35. QML in action
Hybrid sinusoidal FM + PPS
7th order PPS

36. QML in action
6th order PPS – aliased signal

37. Applications - sonars
In sonars, signal received by the mth sensor can be modeled as
where d distance between sensors, c speed of light i θ direction of arrival.
Parameters of this signal can be estimated from the PD
Estimation of aP i ψ: from FT-a with respect to n and m, respectively.
sinusoid with respect to n
sinusoid with respect to m

38. Applications - sonars

39. Applications - radars
Radars return can be modeled as
2D DFT is standard radar image.
Ideal radar image – FT of the first-order 2D PPS signal.
Focusing: estimation and dechirping of higher order PPS parameters.
movement
stationary target
moving
nonuniform movement
complicated movement
rotation and vibration

40. Applications - radars

41. Conclusions

42. Thanks!