频率估计常用方法.pdf

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A Review of the Frequency Estimation and Tracking Problems P.J. Kootsookos CRC for Robust and Adaptive Systems DSTO, Salisbury Site Frequency Estimation and Tracking Project February 21, 1999 Abstract This report presents a concise review of some frequency estimation and frequency tracking problems. In particular, the report focusses on aspects of these problems which have been addressed by members of the Frequency Tracking and Estimation project of the Centre for Robust and Adaptive Systems. The report is divided into four parts: problem speciﬁcation and discussion, associ- ated problems, frequency estimation algorithms and frequency tracking algorithms. Part I begins with a deﬁnition of the various frequency estimation and tracking problems. Practical examples of where each problem may arise are given. A com- parison is made between the frequency estimation and tracking problems. In Part II, block frequency estimation algorithms, fast block frequency estimation algorithms and notch ﬁltering techniques for frequency estimation are dealt with. Frequency tracking algorithms are examined in Part III. Part IV of this report examines various problems associated with frequency esti- mation. Associated problems include Cram´er-Rao lower bounds, theoretical algo- rithm performance, frequency resolution, use of the analytic signal and model order selection.

Contents I Frequency Estimation and Tracking Problems 1 The Frequency Estimation Problem 1.1 Single Tone Frequency Estimation . . . . . . . . . . . . . . . . . . . . . . . 1.2 Multi-harmonic Frequency Estimation . . . . . . . . . . . . . . . . . . . . 1.3 Multi-tone Frequency Estimation . . . . . . . . . . . . . . . . . . . . . . . 1.4 Frequency Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Frequency Tracking Problem 2.1 Demodulation of Digital Signals . . . . . . . . . . . . . . . . . . . . . . . . 2.2 FM Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Tracking in High Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Frequency Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II Frequency Estimation Algorithms 4 4 5 5 6 6 6 7 7 8 8 9 1 Block Frequency Estimators 9 1.1 The Maximum Likelihood Estimator of Frequency . . . . . . . . . . . . . . 9 1.2 Approximate Maximum Likelihood Techniques . . . . . . . . . . . . . . . . 10 1.2.1 The Maximiser of the Periodogram . . . . . . . . . . . . . . . . . . 10 1.3 Fourier Coeﬃcient Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.1 Fourier Coeﬃcient Interpolation Methods . . . . . . . . . . . . . . . 11 . . . . . . . . . . . 11 1.3.2 The Generalised Phase Interpolation Estimator 1.4 Sample Covariance Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Signal Subspace Methods . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4.1 1.4.2 Noise Subspace Methods . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Fast Block Frequency Estimators 14 3 Notch Filtering Techniques 15 3.1 Oﬀ-line Filtering Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1.1 The Technique of Fernandes, Goodwin and de Souza . . . . . . . . 15 . . . . . . . . . . . . . . . 16 3.1.2 The Technique of Quinn and Fernandes 3.2 On-line Filtering Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2.1 The Hannan-Huang Estimator . . . . . . . . . . . . . . . . . . . . . 18 . . . . . . . . . . . . . . . . . 18 3.2.2 The Technique of Nehorai and Porat 4 Summary of Frequency Estimation Algorithms 20 4.1 Block Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Fast Block Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3 On-Line Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 III Frequency Tracking Algorithms 22 1 On-line Frequency Trackers 22 1.1 A Simple Filtering Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.2 The Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3 The Gaussian ‘Sum’ Approach . . . . . . . . . . . . . . . . . . . . . . . . . 24 2 Hidden Markov Model Approaches 3 Summary of Frequency Tracking Algorithms IV Associated Problems 1 Cram´er-Rao Lower Bounds 25 25 26 26 2 Performance Indicators 27 2.1 Performance of Frequency Estimation Algorithms . . . . . . . . . . . . . . 27 2.1.1 Approximate Maximum Likelihood Frequency Estimator Performance 28 . . . . . . . . . 31 2.1.2 Weighted-Phase Frequency Estimator Performance 3 The Analytic Signal 4 Frequency Resolution 5 Model Order Selection 6 Summary of the Associated Problems 31 32 33 33

Part I Frequency Estimation and Tracking Problems Frequency estimation and tracking problems, algorithms and related topics are discussed in this report. The aim is to present a concise sketch of these problems, describe current techniques and indicate loose ends. In this Part, several estimation and tracking problems are speciﬁed and examined. The two problems to be given most scrutiny in the remainder of the report are single-tone frequency estimation and single-tone frequency tracking. We discuss the reasons for this emphasis. Part II of this report describes several block-processing frequency estimation algo- rithms, starting with the standard Gaussian, white noise maximum likelihood approach. Other approaches discussed include the periodogram maximiser, Fourier coeﬃcient in- terpolation and and sample covariance techniques. Details are then given for some fast block-processing algorithms which are of interest because of their computational simplic- ity. The Part concludes with the examination of several notch ﬁltering techniques for frequency estimation. The last of these techniques, that of Nehorai and Porat [1986] may be used as a frequency tracking algorithm, and so this naturally leads on to Part III where several frequency tracking algorithms are examined. Further issues such as use of the analytic signal, frequency resolution and model order reduction are canvassed in Part IV. Rather than list the possible future directions for research on these topics out of context, areas where further work is required are indicated throughout the report. 1 The Frequency Estimation Problem There are three main parameter estimation problems which involve frequency estimation: • Single tone frequency estimation: where the signal is a single, constant-frequency sinusoid. This is the simplest frequency estimation problem. • Multi-harmonic frequency estimation: where the signal is composed of the sum of harmonically related sinusoids. This case is important because rotational or periodic phenomena rarely generate sinusoidal waveforms. • Multi-tone frequency estimation: where there are several tones of unrelated fre- quency present. This problem occurs in the analysis of signals containing emissions from more than one target. Each problem assumes a diﬀerent signal model is to be ﬁtted to measured data. Due to the large amount of literature available, this discussion will be conﬁned to the ﬁrst problem, that of estimating the parameters of a single tone in noise. 4

1.1 Single Tone Frequency Estimation The single tone in noise parameter estimation problem is deﬁned as follows. Problem 1: Single Tone in Noise Let {yt} be generated by the model yt = µ + ρ cos (ω0(t − ν) + φ) + εt (1) where µ is the mean value, ρ is the signal amplitude, ω0 is the frequency of the signal, ν = T−1 2 , φ is the initial phase and εt is some zero-mean random noise sequence with variance σ2. ω0, φ and σ2 given only the measurements {yt : t = 0 . . . T − 1}. The single tone in noise problem is then to estimate the real-valued parameters µ, ρ, 2 A related problem is estimation of the parameters in the following signal model: t zt = µ + ρ exp i (ω0(t − ν) + φ) + ε where the parameters are as before, except that both µ and ε t are complex-valued. Rife and Boorstyn [1974] examined this case. For the majority of this report, however, we shall conﬁne our discussion to the real signal model yt of (1). Remark 1: The signal model (2) is sometimes referred to as the analytic signal (Ville [1948]) model. 2 In all of these problems, algorithm selection depends on whether the sample size T is ﬁxed or increasing. If T is ﬁxed, block-processing algorithms are considered. For T increasing, on-line algorithms are of interest. (2) 1.2 Multi-harmonic Frequency Estimation Where frequency information is to be gleaned from acoustic sources such as rotating machinery, non-linear eﬀects within the generating system often give rise to harmonics and sub-harmonics of the fundamental mode of interest. In these situations, (1) does not model the physical situation well, so a signal model which accounts for the added harmonics should be used. Such a signal model is given below. Problem 2: Harmonically-related Tones in Noise Let {yt} be a multi-harmonic signal modelled by p yt = µ + ρj cos (jω0(t − ν) + φj) + εt (3) j=1 where µ, ν, εt and σ2 are deﬁned as in Problem 1 and ω0 is now the fundamental frequency of the signal, ρj is the amplitude of the jth harmonic and φj is the initial phase of the jth harmonic. µ, ρj, ω0, φj, σ2 and p given only the measurements {yt : t = 0 . . . T − 1}. The multiharmonic-related tones in noise problem is then to estimate the parameters 2 The papers by Barrett and McMahon [1987], James, Anderson and Williamson [1991a] and James, Anderson and Williamson [1991b] examine the multiharmonic frequency es- timation problem. 5

1.3 Multi-tone Frequency Estimation In certain environments, several tonal sources of diﬀering frequencies may be present in the one signal. While, in some cases, it may be possible to apply single-tone techniques in this situation, it is more desirable to account for the extra problem complexity by altering the signal model. The multiple tones in noise problem is deﬁned as follows. Problem 3: Multiple Tones in Noise Let {yt} be a multi-tonal signal modelled by p yt = µ + ρj cos (ωj(t − ν) + φj) + εt (4) j=1 where µ, ν, εt and σ2 are deﬁned as in Problem 1 and ωj is now the frequency of the jth signal component, ρj is the amplitude of the jth tone and φj is the initial phase of the jth tone. σ2 and p given only the measurements {yt : t = 0 . . . T − 1}. The multiple tones in noise problem is then to estimate the parameters µ, ρj, ωj, φj, 2 Hannan [1973] has examined this case and Rife and Boorstyn [1976] have examined the equivalent complex signal model problem. 1.4 Frequency Estimation In this report, we shall only examine in detail algorithms for the frequency estimation problem. The main reason for this is that the added complexity introduced by signal models (3) and (4) obfuscates some of the key issues by either introducing new problems or by increasing the dimensionality of the problem. For instance, use of model (4) instead of (1) means that model order selection now becomes an issue. We shall discuss this and other issues further in Part IV. 2 The Frequency Tracking Problem The frequency tracking problem is somewhat more complicated than the estimation prob- lem. Three physical situations where frequency tracking is of interest are 1. decoding digital information from a frequency-shift keyed bit stream, 2. in demodulation of an FM radio signal and 3. tracking the revolutions per minute of the engine of a manœuvring vessel via acoustic data. Each of these three problems may be described by the following general problem state- ment. 6

Problem 4: Frequency Tracking Let {yt} be modelled by yt = µ + ρ cos t k=0 ωk + φ + εt (5) where ωk is called the instantaneous frequency of the signal and µ, ρ, ν, φ, εt and σ2 are as deﬁned in Problem 1. σ2 and the sequence {ωk} given only the measurements {yt : t = 0 . . . T − 1}. The frequency tracking problem is then to estimate the signal parameters µ, ρ, φ and 2 Boashash [1992a] gives a general discussion of the problem of instantaneous frequency estimation. Remark 2: The most notable diﬀerence between this problem and Problems 1 to 3 is that the estimation of µ, ρ, φ and σ2 is a parameter estimation problem, whereas estimation of the sequence {ωk : k = 0 . . . T − 1} is a state estimation problem. 2 Remark 3: The deﬁnition of this tracking problem does not immediately suggest an ap- propriate error measure. In parameter estimation problems it is common to use the least square error criterion. However, in tracking problems, concepts such as ‘loss of track’ or ‘escape time’ may be of more importance. 2 The escape time for the phase locked loop frequency tracker has been examined by Dupuis and Kushner [1987]. The three physical situations above may be examined by suitable selection of various parameters in Problem 4. Because of this diﬀerent parameter selection, some algorithms may be more appropriate to apply to certain problems than others. 2.1 Demodulation of Digital Signals For example, when a frequency-shift keyed bit stream is to be decoded, ωk will be at known constant values for some duration, which is also perhaps known. While the constant frequencies are known, the change from frequency to frequency is generally stochastic. Remark 4: In this situation, hidden Markov models are probably the most appropriate approach to take due to the discrete nature of process of interest. 2 2.2 FM Demodulation For the FM demodulation problem, set ωk = ωc + λk with ωc λk where ωc is the carrier frequency and λk is the speech signal to be demodulated. The fre- quency variation λk may possibly be modelled as an auto-regressive (or linear predictive) process. In these cases, the noise is generally fairly small. For instance, because of the eﬀect of threshold which is characterised by a severe degradation in reception below some noise level, the signal (or carrier) to noise ratio (SNR) for a commercial (stereo) FM station is usually required to be 10 log10 ρ 2σ2 > 20dB. 7

Remark 5: A major diﬀerence between this problem and that of digital signal demodulation is that here the frequency variation itself is modelled as being stochastic. 2 2.3 Tracking in High Noise For the manœuvring vessel problem, ωk will be slowly time-varying while, in general, the noise is large. For the underwater case, the operating signal to noise ratio is generally 10 log10 ρ 2σ2 < −20dB. Fortunately, because of a priori knowledge of the working environment, only a small band of known frequencies is generally of interest. Remark 6: The frequency tracking problem in this situation is thus further constrained to take place within a known ‘gate’ of possible frequencies. 2 Remark 7: A problem that also needs attention is that of deciding whether or not an interesting signal is present given knowledge of the background noise. The issues are called track initiation or detection and track termination and they are of major concern. 2 2.4 Frequency Tracking Part III of this report examines some particular frequency tracking algorithms, and some other aspects of the problem are examined in Part IV. Further Work 1: Methods for allowing a priori information about the instantaneous frequency law ωk to be used should be examined. Further Work 2: Block estimators of frequency may be used as frequency trackers, provided the frequency variation is slow enough over the length of the data. Bounds on the performance of block and on-line techniques, for a given amount of frequency variation (speciﬁed either stochastically or deterministically), would allow direct comparisons between such techniques to be made. Multi-harmonic and multi-tone extensions of Problem 4 may also be deﬁned, however they are not discussed here as this will overly complicate the problem. 8

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