Tracking Performance of Costas Loops with Hard-Limited In-Phase Channel

It is becoming increasingly popular in the design of suppressed carrier receivers, which employ Costas loops for earrier reconstruction, to hard-limit the output of the in-phase channel. Doing so allows replacement of the analog multiplier, which forms the loop error signal, with a chopper-type device which typically exhibits much less dc offset. The false lock behavior of such a hard-limited loop was recently investigated and shown to be quite different from that of the conventional Costas loop without the hard limiter. This paper presents the companion, analysis of the tracking performance of the hard-limited loop and assesses the penalty, if indeed it is a penalty rather than an improvement, in this performance relative to the conventional Costas loop with an analog third multiplier. In particular, for the case ofRCarm filters and NRZ data, the squaring loss (or equivalently the linear loop tracking jitter) is evaluated and illustrated as a function of the ratio of arm filter bandwidth to data rate and data signal-to-noise ratio. Superimposed on these numerical results will be the corresponding ones for the conventional Costas loop. As a finale, the equivalence in operation of the Costas loop with hard-limited in-phase channel and a baseband modulation carrier reconstruction loop referred to as a demod/ remod loop is discussed.     

Theory of False Lock in Costas Loops

This paper is concerned with new results on data related false lock of Costas loops caused by arm filter distortion of the data. The false lock phenomena is demonstrated to occur for random as well as periodic data and the relative levels of lock are predicted. It is shown that, for the case of random data with Manchester symbols, strong lock points exist when the incoming carrier frequency and the loop VCO frequency differ bynR_{s}/2 (n = 1, 2, 3, ...)Hz where Rsis the data symbol rate. In addition to the loop error signal the lock detection signal(I^{2}Q^{2})is also analyzed for false lock effects. The theory also applies to the case of NRZ data symbols with the main result being strong false lock points also occur at VCO frequency offsets ofnR_{s}/2 (n = l, 2, 3, ldot)but at different levels than the Manchester data. The results of these analyses lead to schemes capable of detecting false lock.     

False lock in quadriphase receivers with unequal I- andQ-channel data rates

Results are presented concerning the noise-free false lock problem in quadriphase receivers with unequal I-channel and Q-channel data rates. It is shown that false lock may occur when the incoming carrier frequency and the loop VCO (voltage-controlled oscillator) frequency differ by an amount that is a linear combination of a quarter of I-channel and Q-channel data rates. General expressions for the false lock performance of quadriphase receivers are given for random as well as periodic data patterns. Specific closed-form expressions for the false lock margin are derived when the arm filters of the quadriphase receivers are single-pole Butterworth filters, and the format of the random data is NRZ (nonreturn-to-zero). Numerical evaluations of false lock margin as a function of the product of 3 dB arm-filter bandwidth and symbol duration are carried out for random data using various arm filters and for different periodic I-channel and Q-channel data patterns     

Decision-directed coherent delay-lock tracking loop forDS-spread-spectrum signals

The authors present a nonconventional joint data demodulation-pseudo-noise (PN) code tracking scheme for direct sequence (DS) spread-spectrum (SS) signals which solves problems of component imbalance and sensitivity with hardware simplicity and no performance degradation. An integrate-and-dump Costas loop is used for carrier recovery and data demodulation of the SS signal. Both data and carrier are then used to derive the baseband error signal of the code tracking loop. Moreover, a single passband correlator is used to perform the early-late correlation, leading to a hardware complexity equivalent to that of the tau-dither scheme, but without its loss in performance. Results of a thorough theoretical analysis of the system in an additive Gaussian noise (AWGN) environment are reported. They provide performance curves in terms of steady-state jitter and mean time to first lock loss. A superior jitter performance for low values of E_b/ N₀ with respect to a traditional noncoherent delay lock loop (DLL) is shown, along with the potential gain of Manchester coding upon the more usual NRZ format     

A new pattern jitter free frequency error detector

One method to enable fast acquisition of a phase-locked loop (PLL) in spite of large frequency offsets and small PLL bandwidths is to use an additional AFC (automatic frequency control) loop. A suitable frequency error detector (FED) for large frequency offsets is the well-known balanced quadricorrelator. This FED is shown to produce great pattern jitter if it is to work with digital modulated MQAM and MPSK signals (M>2) and random data. The authors describe how this pattern jitter for MQAM and MPSK signals can be overcome completely. Another understanding of the balance quadricorrelator is also given     

QPSK扩频调制信号载波跟踪环路设计

扩频接收基带通常需要载波跟踪环来完成本地载波与接收信号载波的同步,科斯塔斯环是常用的非相干载波相位跟踪环,具有较好的鉴相特性。本文基于常规的单路单载波解调的科斯塔斯环原理,对科斯塔斯环做了改进,提出了对双通道平衡QPSK扩频基带所用载波跟踪环科斯塔斯环的改进方法,并推导出环路误差鉴相信号,用Matlab进行了仿真实验,结果证明利用本环路可实现载波稳定跟踪。     

Asymptotic performance of M-ary orthogonal modulation ingeneralized fading channels

The asymptotic (M→∞) probability of symbol error P_e,_m for M-ary orthogonal modulation in a Nakagami-m fading channel is given by the incomplete gamma function P(m, mx) where x=In 2/(E_b/N₀) and E_b is the average energy per bit. For large signal-to-noise ratio this leads to a channel where the probability of symbol error varies as the inverse mth power of E_b/N₀. These channels exist for all m⩾1/2. The special case of m=1 corresponds to Rayleigh fading, an inverse linear channel     

A Generalized "Polarity-Type" Costas Loop for Tracking MPSK Signals

A generalized tracking loop is proposed for tracking MPSK signals at high SNR. It is similar in form to the polarity-type Costas loop used for tracking QPSK signals, in which limiters in the arms provide the data estimates needed for data wipeoff in forming the loop error signal. For the MPSK signal, the hard limiters are replaced by "multivalued limiters" withM/4 (M geq 4)positive and negative values. The outputs of the multivalued limiters in each arm are theIandQprojections of the data. TheScurve is computed for this MPSK loop, and plotted for different SNR's. A comparison is made to theScurve for theMth power loop (orM-phase Costas loop), which is optimum for low SNR. From this comparison, a rough value of the "boundary" SNR can be obtained; this value may be used by the designer to determine which loop structure should be used. The advantage of the high SNR structure proposed here is that it is much simpler to implement than the corresponding low SNR structures.     

Novel p.s.k. tanlock loop

A new loop for coherent demodulation of suppressed carrier phase-shift-keyed (p.s.k.) signals is presented. The p.s.k. tanlock loop (p.s.k.-t.l.l.) has a wider tracking range and faster phase acquisition than the Costas or squaring loops usually used for p.s.k. suppressed carrier tracking, but it has a greater tendency to false lock.     

A new carrier recovery circuit for land mobile satellitecommunication

A DCF (dual carrier filter) reverse-modulation-type carrier recovery circuit is proposed to achieve a low carrier skipping rate and satisfactory phase tracking performance for coherent detection of PSK (phase shift keying) signals in fast Rician fading channels. The proposed scheme employs both narrow and wide bandwidth carrier filters simultaneously for the reverse-modulation-type carrier recovery circuit. It is clarified by computer simulation that the Pe performance of a QPSK (quadriphase shift keying) modem employing the proposed scheme shows an improvement of 1.5 dB in required E_s/N_O at Pe=10⁴ (after Viterbi decoding (R=7/8, K=7), C/M (direct-to-multipath signal power ratio)=10 dB, interleaving size=64×64), compared with conventional coherent detection employing the reverse modulation tank-limiter scheme or the Costas loop scheme     

Optimum Design and Performance of Costas Recievers Containing Soft Bandpass Limiters

This paper investigates several key questions concerning the mechanization and design of a Costas receiver for reconstruction of a carrier from a suppressed carrier signal. For baseband NRZ encoded data symbols and a soft bandpass limiter preceding the loop, several design issues which are considered herein and which affect acquisition and tracking performance are: (1) The choice of an IF bandwidth. (2) The optimum choice of the Costas arm filter bandwidths as well as the spectral roll-off characteristics. (3) The optimum choice of loop bandwidth to data rate ratio for a given signal-to-noise ratio. (4) The signal suppression factor and the combined limiter-squaring loss. (5) The variations in loop bandwidth and damping with signal level. (6) The choice of the limiter transfer characteristic. (7) Performance degradation due to the presence of a limiter. Various new results in system design are presented and typical numerical results are given and graphically demonstrated in SNR regions of practical interest. The theory is applicable to the design of carrier reconstruction loops required in the implementation of spread spectrum communication receivers.     

基于PLL环路的卫星QPSK载波调制信号相位跟踪算法

卫星信号接收端天线接收的卫星信号经过射频前端处理后会变成数字中频信号,而相位跟踪是中频信号处理的重要环节,准确的相位跟踪可以为卫星信号解码、测距和定位等功能的准确性提供保障.基于此,该文提出了基于锁相环（Phase Locked Loop,PLL）的卫星QPSK载波调制信号相位跟踪算法.该算法利用多路正交载波和二象反正切鉴相器获取精确的相位差信息,形成闭合回路锁定载波相位以实现跟踪.实验结果表明,多路异相载波的方法可实现更快速的相位锁定,与传统Costas环路和松尾环相比,进入稳态时间分别减少了20%和24%以上,稳态方差分别减小21%和32%.     

I/Q channel reversal correcting properties of anSQPSK Costas loop with integrated symbol synchronization

An integrated-carrier loop/symbol synchronizer, using a digital Costas loop with matched arm filters to demodulate staggered quaternary phase-shift keyed (QPSK) signals, is analyzed. An expression is derived for the S curve, parameterized by bit synchronization error. This result suggests that the demodulator structure offers an inherent I/Q channel reversal correcting capability. Computer simulation results are presented that support this conclusion, and suggest that ambiguity resolution performance depends on the ratio of carrier and synchronization loop bandwidths     

Acquisition performance of various QPSK carrier tracking loops

The frequency and phase acquisition performance of three quadrature phase shift keying (QPSK) carrier tracking loops, the MAP estimation loop, the Costas crossover loop, and the generalized Costas loop, is described. Acquisition time and probability of acquisition as a function of both loop signal-to-noise ratio and frequency offset to loop bandwidth ratio are obtained via computer simulations for type II and III loops. It is shown that the MAP loop, which results in the smallest squaring loss for all signal-to-noise ratios, is sometimes outperformed by the other two loops in terms of acquisition time and acquisition probability     

On the Calculation of Squaring Loss in Costas Loops with Arbitrary Arm Filters

The calculation of the optimum performance of suppressed carrier receivers with Costas loop tracking is directly related to evaluating the loop's so-called squaring loss. Recent work by the author and others presented specific numerical results for this loss when the input data were biphase-L(Manchester coded) and the Costas loop arm filters were of then-pole Butterworth type. These results were largely obtained by numerical integration on a digital computer. This paper presents a partial fraction expansion technique for arriving at closed form expressions for squaring loss for Costas loops with arbitary arm filters and NRZ as well as Manchester coded data. Specific closed form results are given for one and two pole Butterworth filters as examples.     

The channel capacity of discrete time phase modulation in AWGN

The capacity of a bandlimited phase-only modulated system in additive white Gaussian noise is calculated, where the phase is not restricted to a finite alphabet and the demodulator supplies only phase measurements. The results match a known asymptote at high signal-to-noise ratios and form an upper limit on the capacities of MPSK for finite M     

一种扩展载波环频率捕获范围的方法

在MPSK数字载波恢复中,可以通过锁频锁相环捕获大范围频率偏移。低信噪比下,锁相环的频率捕获范围较小,而工程中锁频环剩余频偏通常较大,因而需要扩展锁相环的频率捕获范围。描述了频差估计以及COSTAS环中相位检测技术的实现方法,分析了COSTAS环的相位检测特性,并根据这种算法的鉴相特性提出了一种扩展频率捕获范围的载波恢复方法。以QPSK为例,通过MATLAB仿真了这种技术在低信噪比情况下对频率偏移的检测与跟踪性能,仿真结果表明,提出的方法适合低信噪比下大范围的频率捕获。     

Lock detection in Costas loops

Previous analyses of lock detector algorithms for Costas loops have ignored the effects of the inherent correlation between samples of the phase error process. In this work, analysis and simulations are used to quantify the effects of phase correlation on lock detection for the `square law' and `absolute value' type detectors. Results which depict the lock detection probability as a function of loop signal-to-noise ratio for a given false alarm rate are obtained. It is shown that the square law detector experiences less degradation due to phase jitter than the absolute value detector and that the degradation in detector signal-to-noise ratio is more pronounced for squarewave than for sinewave signals     

Hybrid Carrier and Modulation Tracking Loops

The tracking loops presented herein exploit the coherency of the observed data sidebands at the carrier frequency for improvements in phase-coherent tracking, telemetry, and command system performance. The loop is motivated by one aspect of generalized harmonic analysis, viz., the so-called cross spectrum. The basic idea of the loop centers around using the power in the composite received signal (sidebands including cross-modulation components) to enhance the signal-to-noise ratio in the carrier tracking loop, thereby reducing the noisy reference loss and ultimately the subcarrier reference jitter, the symbol sync jitter, and the error probability of the data detector. The modulation from the extracted sidebands is removed in such a way that the coherence power in the data sidebands can be used for tracking at the carrier frequency.     

Eigenvalues and eigenvectors of covariance matrices for signalsclosely spaced in frequency

The eigenstructures of common covariance matrices are identified for the general case of M closely spaced signals. It is shown that the largest signal-space eigenvalue is relatively insensitive to signal separation. By contrast, the ith largest eigenvalue is proportional to δω^2(i-1) or δω^4(i-1), where δω is a measure of signal separation. Therefore, matrix conditioning degrades rapidly as signal separation is reduced. It is also shown that the limiting eigenvectors have remarkably simple structures. The results are very general, and apply to planar far-field direction-finding problems involving almost arbitrary scenarios, and also to time-series analysis of sinusoids, exponentials, and other signals