Parameter estimation of chirp signal, such as instantaneous frequency (IF), instantaneous frequency rate (IFR), and initial phase (IP), arises in many applications of signal processing. During the phasebased parameter estimation, a phase unwrapping process is needed to recover the phase information correctly and impact the estimation performance remarkably. Therefore, we introduce support vector regression (SVR) to predict the variation trend of instantaneous phase and unwrap phases efficiently. Even though with that being the case, errors still exist in phase unwrapping process because of its ambiguous phase characteristic. Furthermore, we propose an SVRbased joint estimation algorithm and make it immune to these error phases by means of setting the SVR's parameters properly. Our results show that, compared with the other three algorithms of chirp signal, not only does the proposed one maintain quality capabilities at low frequencies, but also improves accuracy at high frequencies and decreases the impact with the initial phase.
Chirp signals, that is, secondorder polynomial phase signals, are common in various areas of science and engineering. For example, in a synthetic aperture radar (SAR) system, when the target is moving, the regulated signals will change into chirp ones after being reflected [
By introducing structural risk minimization (SRM) principle, support vector regression (SVR) exhibits excellent capabilities for generalizing and learning. From the viewpoint of the quadratic relationship between absolute signal phase and time series, therefore, this paper employs SVR to unwrap phases and estimate IF and IFR recursively. We avoid taking the rationality of phase noise model into account by not making any approximations in it. At one time, we reduce the estimation performance’s dependence on phase unwrapping process with a proper choice of SVR’s parameters. It has been shown that, except for the property of low sensitivity to initial phase and closely approaching the CramerRao lower bound (CRLB) at low frequencies, the proposed algorithm improves its estimation performance at high frequencies.
The signal model used here is similar to that in [
Here,
When
A more reasonable model proposed in [
In (
At first, we present a line
Next, we assume
According to (
In fact, fitting errors larger than
Equation (
Substituting (
We obtain
Not only does (
Recursive implementation of SVRbased algorithm.
Setting SVR’s parameters is a difficult problem, but has a pronounced impact on SVR’s performance, for example, insensitive loss coefficient
When
Intuitively, insensitive loss coefficient
Penalty factor
We have compared the proposed algorithm entitled as SVR estimator with the other three: the HPF estimator proposed in [
Because HPF estimator does not need the phase unwrapping process, Figure
(a) Arbitrary phase unwrapping processes with
Figures
(a) MSE of IF with
(a) MSE of IFR with
It is shown that whether
Everything is as in Figure
MSE of IF of SVR estimator with
MSE of IFR of SVR estimator with
Everything is as in Figure
(a) MSE of IF with
(a) MSE of IFR with
Because we translate SVR into QP problem and need to search the minimums during the process, we can not derive the explicit form of the computational complexity of SVR estimator. So everything is as in Figure
Consuming time with different
Algorithm 





HPF  380  942  3721  10654 
DK  203  476  1098  3711 
LFK  299  784  2362  8643 
SVR  437  1107  4805  13074 
We can see that SVR estimator’s consuming times are more than the others’, especially when
Phase unwrapping process is a key point in phasebased frequency estimation of chirp signal. Firstly, we adopt SVR to learn the unwrapped phases at previous time points, predict the variation trend of phase efficiently, and derive the estimation value for the next time point. Once acquired, in terms of relationship between absolute signal phase and time series, we address a simple and effective frequency estimation algorithm of chirp signal. The proposed algorithm completely exhibits its advantages of higher estimation accuracy, lower sensitivity of frequency, and initial phase, by sacrificing more consuming times.
Because SVR predicts the curve’s variation trend merely in terms of training set which consists of previous points’ values, we even can estimate the frequency of chirp signal under the nonGaussian condition by the same way.
Stressing that the proposed algorithm learns training set and gets the approximate values of SVR’s insensitive loss coefficient
This research is supported by the National Natural Science Foundation of China (Grant no. 61201380).