SDPLL-Based Frequency Estimation of a Sinusoid in Colored Noise

The problem of frequency estimation of a single sinusoid observed in colored noise is addressed. Our estimator is based on the operation of the sinusoidal digital phase-locked loop (SDPLL) which carries the frequency information in its phase error after the noisy sinusoid has been acquired by the SDPLL. We show by computer simulations that this frequency estimator beats the Cramer-Rao bound (CRB) on the frequency error variance for moderate and high SNRs when the colored noise has a general low-pass filtered (LPF) characteristic, thereby outperforming, in terms of frequency error variance, several existing techniques some of which are, in addition, computationally demanding. Moreover, the present approach generalizes on existing work that addresses different methods of sinusoid frequency estimation involving specific colored noise models such as the moving average (MA) noise model. An insightful theoretical analysis is presented to support the practical findings.


INTRODUCTION
Rapid frequency estimation of a sinusoid in the presence of noise is a problem that is frequently encountered in signal processing and communications with applications varying from radar, sonar, signal interception and detection, carrier synchronization and many others.Many methods of sinusoidal frequency estimation have been developed especially for the white noise case.The maximum likelihood (ML) estimator involving the location of the peak of the periodogram is well-known.The performance of the ML estimator in terms of estimator error variance achieves the CRB at high signal-to-noise ratios (SNR's), but at the expense of a large computational complexity, even when the fast Fourier transform (FFT) is used (Fu H. et al, 2007).The ML estimator is also a batch processing technique starting the processing only after all samples have been received (Richard Brown III D. et al, 2010).Subsequent to the introduction of the ML method, several fast and accurate sinusoidal frequency estimators in the presence of white noise have been reported (Kay S.,1989) attaining the CRB on variance for high enough SNR.
Frequency estimation techniques driven by a frequency tracking loop such as the PLL have also been reported in the literature in the context of operation in a white noise environment (Sithamparanathan K., 2008).The advantage of such an approach is that the PLL acts as a dynamic band-pass filter (BPF) to track the frequency, which improves the SNR due to reduced bandwidth (BW) once the signal is acquired.This improvement in SNR certainly leads to an improvement in the frequency error variance for a given system.Indeed, it has been shown that such a frequency estimator using an arc-tan DPLL (Sithamparanathan K., 2008) beats the CRB for low numbers of estimation samples.The appeal of DPLLs is also evident in that they offer low-complexity sample-bysample operation suitable for real-time applications (Richard Brown III D. et al, 2010).
In this paper, we improve on the work in (Sithamparanathan K., 2008) to include the case where the accompanying noise is colored.This case occurs in many applications and is definitely more practically relevant (Stoica P. et al, 1997) In our work, however, we choose to investigate the SDPLL as a frequency estimator in colored noise.Among DPLL's, the uniform sampling SDPLL's are particularly popular, as they are simple to implement and suitable for relatively wide locking ranges (Hussain Z. M. et al, 2011).(Elasmi-Ksibi R. et al, 2010) have recently treated the case of frequency estimation of a sinusoid in colored noise exploiting highorder lags of the autocorrelation function (ACF) of the noisy sinusoid, and consequently restricting the investigation to MA noise models but achieving low computational complexity.The present SDPLL-based work compares favorably with (Elasmi-Ksibi R. et al, 2010) as regards computational complexity, the extension of the MA case to generally any LPF characteristic, in addition to attaining and even surpassing the CRB associated with the white noise case for a wide range of SNR values extending from moderate to high values.A highpass filtered (HPF) characteristic of noise, however, is shown to render the system incapable of attaining the CRB.Obviously, however, long settling time and the presence of overshoot are drawbacks of PLL frequency estimation (Saber M. et al, 2011).
The rest of the paper is organized as follows: Section 2 summarizes the SDPLL analysis and explains the design methodology.Sections 3 and 4 focus on the SDPLL as a frequency estimator in white and colored noise respectively, analytically highlighting the improvement on the error variance that is achieved with a lowpass filtered noise characteristic.Section 5 presents simulation results of the proposed system.Finally, Section 6 concludes the paper.

ANALYSIS AND DESIGN OF THE NOISE-FREE SDPLL
SDPLLs are non-uniform sampling sinusoidal PLLs that have the advantage of being simple to implement and having relatively wide locking ranges.The block diagram of this system is shown in Fig. 1.The SDPLL consists of a sampler-ADC unit which serves as a phase detector, a digital LPF (DF), and a digital voltage-controlled oscillator (DCO).The latter provides constant-amplitude but variablefrequency output pulses that control the sampling instants of the sampler-ADC unit (Hussain Z. M. et al, 2011).
The DF output is input to the DCO to vary its phase and hence sampling instant of the input analog signal x(t).When the signal is absent, the DCO runs at its free-running frequency which we call o f .When the sampler takes a sample of x(t) at the kth sampling instant, the ADC converts the analog value of the input into a digital value that the DF uses at its input to yield y(k) at its output.This digital y(k) is then input to the DCO.As the DCO input changes, so does the sampling period T(k) of the ADC.The sampling process is non-uniform, i.e. the sampling frequency is not constant.Under certain conditions, namely, the locking conditions, the convergence or locking occurs such that T(k) approaches the inverse of i f , the input sinusoid frequency.Fig. 2 shows the waveforms associated with the SDPLL.
For the dual purpose of clarity and symbols unification, we find it convenient to summarize the elementary analysis of the SDPLL already present in the literature such as in (Hussain Z. M. et al, 2011).
Assume the sinusoidal input is given by: where A is the signal amplitude, ) θ is the information-bearing phase given by: After sampling, the input signal at the kth sampling instant t(k) will take on the following form: Now we can define the input phase at the kth sampling instant as: The sampling interval of the DCO at the kth sampling instant is given by: ) To is the DCO free-running period.This shows that the kth output sample of the DF, y(k), effectively determines the sampling period T(k+1).Eq. ( 4) reveals how the DCO operates; it decreases its period (increases its frequency) as the DF output increases.
From the above findings, the phase difference equation of the system is given by (Hussain Z. M. et al, 2011): Let us assume that the digital filter transfer function is H(z)=G1 where G1 is a constant.This yields a first-order SDPLL.Then, the above equation is used to modify the difference equation to: Defining K1 to be The parameters K1 and W control the locking range.The locking conditions have been derived in (Hussain Z. M. et al, 2011).We state here the results of the derivation in (Hussain Z. M. et al, 2011) The above two equations specify a range of K1 that ensures that the input frequency is within the frequency locking range of the first-order SDPLL.
According to the above inequalities, we may design our first-order SDPLL.For example, if we choose a DCO with , then clearly W=1.2.And from the locking conditions of eq. ( 5), K1 can be safely chosen as K1=1.7.

THE SDPLL AS A FREQUENCY ESTIMATOR OF A NOISY SINUSOID
The SDPLL can detect the input frequency in the presence of noise, even for low SNRs.
Having designed the first-order SDPLL in Section II, we now explain the frequency estimator driven by the SDPLL when the input sinusoid is contaminated by additive white Gaussian noise (AWGN).The noisy input sinusoid may be written as: where n(k) is additive noise.We are interested in estimating the frequency of the noise-free input sinusoid which we call i f .The SDPLL is a computationally efficient sample-by-sample method to extract i f from noisy observations.The PLL acts as a dynamic BPF that tracks the input frequency improving the SNR by bandwidth reduction.The SNR improvement leads to an improvement in frequency error variance.Using the symbols in Section 2, the sequence t(k) can be used to generate a frequency sequence

THE SDPLL FREQUENCY ESTIMATOR OF A SINUSOID IN COLORED NOISE
In this paper, we wish to investigate the colored noise case.In (Elasmi-Ksibi R. et al, 2010), a novel method of frequency estimation of a sinusoid in colored noise using multiple autocorrelation lags is presented.However, the method is applicable only to environments where the colored noise is of finite memory, i.e. moving average (MA) noise models.The method features low computational complexity and compares favorably with its forerunners.We show that frequency estimation in colored noise is possible with different types of lowpass-filtered noise, including the MA noise model, whereas the method in (Elasmi-Ksibi R. et al, 2010) is applicable only to the MA model.
In particular, for the SDPLL, we find that lowpass-filtered colored noise, added to the input sinusoid whose frequency is to be estimated, significantly enhance the SNR when compared to the white noise case, whereas a high-passfiltered noise has the adverse effect.We have also found that this is true for any kind of lowpass-filtered noise.We simulate the colored noise by passing AWGN through a first-order IIR LPF first, then a FIR LPF of a MA type.We can explain this favorable behavior of the SDPLL as follows: Since the output pulses from the DCO control the sampling of the sine-pluscolored-noise signal, the digital LPF sampling frequency has, on average, a value of i f =0.83 Hz according to the design of Section 2. Therefore, the folding frequency (corresponding to the digital frequency of π radians) is The CRB is given by: The probability density function The CRB in our work is the lower bound of the frequency error variance as in eq. ( 7).Analytical evaluation of the CRB (Vaseghi S.V., 2006) is usually extremely difficult, if not impossible; therefore we resort to computer simulation to find it (Noels N. et al, 2002).

SIMULATION RESULTS
The first-order SDPLL is simulated in MATLAB 7. The program for the white noise case is listed below in pseudo-code; the symbols are as presented in Section 2. We assume that the input noise power (variance) is equal to unity.Therefore, to compare between white and colored noise effects, we have to equate the powers of the two.Thus, we multiply the colored noise sequence by 2 1 a − before adding it to the sine wave.If we take a=0.9, then we have to multiply by 19 .0 .After making the necessary modification to the above pseudocode and the corresponding MATLAB code, we plot the variance versus SNR on the same graph as that for the white noise case as in Fig. 6.The variance versus SNR for the corresponding HPF filtered colored noise case is also shown in the figure .Case II: We also test low-pass filtered noise using a MA process.We choose the FIR MA filter transfer function as: Also assuming unity input noise power, we have to multiply the filter output by 5 for a meaningful comparison.The resulting curve is also included in Fig. 6, and is shown to coincide nearly with IIR filter case.The conclusion is that for LPFs, in general, the frequency error variance is improved (decreased).The Cramer Rao bound (CRB) is a lower bound on the error variance of any unbiased estimator and serves as an important benchmark to compare practical estimators (Noels N. et al , 2002), (Vaseghi S.V., 2006).Fig. 6 is extended to include high SNRs.The CRB is the asymptote to the curves as SNR increases.Therefore, we can find it practically as the figure shows.Fig. 7 is a zoom-in to the moderate-to-high SNR region of Fig. 6.It can be seen from Fig. 7 that when the additive noise is a low-pass-filtered one, we gain an improvement over the white noise case as regards the variance of the frequency estimate for moderate and high SNRs.

CONCLUSIONS
An SDPLL-based single-sinusoid frequency estimator in white and colored noise has been simulated and tested.It has been shown that when the additive noise to the sinusoid has a LPF characteristic we gain an improvement of the estimator performance in terms of reduced frequency error variance regardless of the type of LPF whether IIR or FIR.This work is an extension of related PLL-based methods reported in the literature for the white noise case, and of specifically MA-noise-model non-PLL-based frequency estimators.We have shown through simulation that our estimator beats the CRB of the white noise case for moderate and high SNRs.Error variance (dB) SNR (dB)

SYMBOLS AND ACRONYMS
frequency, and o θ is a constant phase.The locking range of the loop is the range of instantaneous frequencies that the loop can track.Since the locking range is dependent on the deviation of ω from the loop center frequency o ω , we can then write eq.(1) as: at approximately f= i f .The variance of this frequency estimator is a function of the SNR; it decreases as the SNR increases.Fig.
.415 Hz.The periodic frequency transfer function of this digital LPF has a typical form shown in Fig. 4. It is clear that this filter has high gain around 0.83 Hz.This means that LPF colored noise boosts the SNR thereby improving the frequency error variance for the SDPLL acting as a frequency estimator.On the other hand, a HPF for producing colored noise has the typical transfer function shown in Fig 5.The presence of a peak at half the input frequency, i.e. at 0.415 Hz, results in a degradation of the performance of the SDPLL as a frequency estimator and the frequency error variance increases.As in any estimation problem, the idea is to produce from a noisy observation vector n

. Prof. Nuha Abdul Sahib Alwan SDPLL-Based Frequency Estimation Asst. Lect. Zainab Hassan Fakhri of a Sinusoid in Colored Noise
as these results are key attributes in SDPLL design.