Symbol timing recovery in digital subscribers loops in the presenceof residual echo and crosstalk

A general analysis is presented for the jitter performance of a common symbol timing recovery (STR) system used in a digital subscriber loop (DSL) transceiver that uses adaptive echo cancellation for high-speed digital communications. Exact analytical expressions for the mean and variance of the timing wave and for the RMS phase jitter are derived as a function of the bandwidth of the postfilter for a given set of input parameters representing a particular digital subscriber loop and its noise environment, including such effects as residual echo and crosstalk. Numerical results show that the presence of these disturbing signals can substantially degrade the STR performance. The effect of the excess bandwidth factor of the prefilter on this degradation is investigated     

Statistical Properties of Timing Jitter in a PAM Timing Recovery Scheme

A new technique is presented for evaluating the performance of a popular type of timing recovery circuit for baseband synchronous pulse amplitude modulation (PAM) data signals. The timing circuit consists of a square-law device followed by a narrowband filter tuned to the pulse repetition frequency along with provision for reshaping the pulses entering the timing path (prefiltering). The output of the timing circuit is a nearly sinusoidal timing wave whose zero crossings indicate the appropriate sampling instants for demodulation of the PAM signal. For a random data sequence, the timing wave exhibits phase fluctuations which are strongly dependent on the pulse shapes entering the timing path and the passband shape of the narrow-band filter. Expressions for rms phase fluctuation in the timing wave as a function of the prefiltering and postfiltering characteristics of the filters preceding and following the square-law device are presented. These expressions have a form which is especially suitable for studying the case where the baseband PAM signal is band-limited to frequencies less than the pulse repetition frequency. A condition on prefiltering and postfiltering which gives error-free timing recovery is presented. Results obtained from some specific examples serve to illustrate several aspects of the timing recovery problem.     

Presence of residual echo and impulsive noise

A general analysis is presented for the jitter performance of a common type of symbol timing recovery (STR) system employed in a digital subscriber loop (DSL) transceiver for high-speed digital data transmission over twisted copper pair cables in the presence of residual echo and impulsive noise (IN). Numerical results, obtained for an experimental study of a timing recovery system, show that the presence of these disturbing signals can substantially degrade the STR performance     

Nonlinear transformations of random processes

This paper provides a general method of calculating the mean-square bandwidth (and other spectral moments) of an arbitrary zero-memory nonlinear transformation of a stationary random process. The method is valid when the original process is an arbitrary combination of other random processes. It can be used to determine the mean-square bandwidth (or the spectral moments) of the transformed process either before or after that process is passed through a bandpass filter. Five examples of the application of this method are provided simplifying and generalizing known results, as well as providing new results.     

Analysis of a serial symbol timing recovery technique employingExclusive-OR circuit

An analysis is presented of the performance of a serial symbol timing recovery (STR) circuit which employs an Exclusive-OR circuit for conventional coherent digital modulated communication systems. The output of the timing circuit is a nearly sinusoidal wave whose zero crossings indicate the appropriate sampling instants for extraction of the data. Assuming that the data pulses entering the timing path are even symmetric, exact analytical expressions for the mean and mean-squared values of the timing wave and for the RMS phase jitter are derived as a function of various system parameters such as channel band limiting, postfiltering, delay element, and power spectral density of noise. Numerical results, also checked by computer simulations, show that considerable improvement can be obtained in jitter performance, in addition to the advantages over other STR techniques of lower cost and simpler hardware implementation     

A General Spectral Analysis of Time Jitter Produced in a Regenerative Repeater

A general theory is presented for evaluating the spectral density of time jitter produced in a self-timed regenerative repeater. The timing circuit consists of a general memoryless nonlinear device followed by a narrow-band tank circuit tuned as close as possible to the symbol rate; the signal (timing wave) at the output of the circuit has both amplitude and phase modulations that cause the timing error (jitter). The theory includes the cases of arbitrary pattern statistics, pulse waveform, and nonlinear processing, in contrast to previous work which is strongly limited to the assumptions of pattern symbol independence and particular nonlinearities. The time jitter is treated as a discrete-parameter random process and its spectral density is finally related to the input message statistics, pulse waveform, and timing circuit parameters. In most cases the spectral density turns out to be a rational function of exp (j2pifT),fbeing the frequency andTthe symbol period. Hence, a spectral factorization can be obtained that leads to a useful linear equivalent circuit of the timing (nonlinear) circuit (NLC). This last feature is illustrated by three examples.     

Stochastic analysis of the LMS algorithm with a saturationnonlinearity following the adaptive filter output

This paper presents a statistical analysis of the least mean square (LMS) algorithm with a zero-memory scaled error function nonlinearity following the adaptive filter output. This structure models saturation effects in active noise and active vibration control systems when the acoustic transducers are driven by large amplitude signals. The problem is first defined as a nonlinear signal estimation problem and the mean-square error (MSE) performance surface is studied. Analytical expressions are obtained for the optimum weight vector and the minimum achievable MSE as functions of the saturation. These results are useful for adaptive algorithm design and evaluation. The LMS algorithm behavior with saturation is analyzed for Gaussian inputs and slow adaptation. Deterministic nonlinear recursions are obtained for the time-varying mean weight and MSE behavior. Simplified results are derived for white inputs and small step sizes. Monte Carlo simulations display excellent agreement with the theoretical predictions, even for relatively large step sizes. The new analytical results accurately predict the effect of saturation on the LMS adaptive filter behavior     

Design of optimal interpolation filter for symbol timing recovery

We propose an optimal interpolation filter for symbol timing recovery in a digital receiver where the input analog-to-digital (A-D) conversion sampling clock is not synchronized to the transmitter symbol clock. The optimal filter is designed by minimizing the mean-square error (MSE) at the output of the receiver, assuming that the correct timing difference between the nonsynchronized input samples and the correct strobe synchronized to the symbol is given. The MSE minimization procedure results in a system of linear equations which can easily be solved to yield the optimal filter coefficients. We also analyze the symbol tracking performance of a symbol synchronizer that employs the proposed optimal interpolation filter. Although the proposed optimal filter is designed assuming the availability of the correct timing difference, we show that it minimizes the timing estimation error variance even when the computed timing difference values deviate from the ideal, we also show that the timing estimation is virtually unbiased if the length of the interpolation filter is greater than or equal to 4. Simulation results are included to show the performance improvement realizable by employing the optimal interpolation filter     

Quantization

Quantization is the process of replacing analog samples with approximate values taken from a finite set of allowed values. The approximate values corresponding to a sequence of analog samples can then be specified by a digital signal for transmission, storage, or other digital processing. In this expository paper, the basic ideas of uniform quantization, companding, robustness to input power level, and optimal quantization are reviewed and explained. The performance of various schemes is compared using the ratio of signal power to mean-square quantizing noise as a criterion. Entropy coding and the ultimate theoretical bound on block quantizer performance are also compared with the simpler zero-memory quantizer.     

Performance analysis of an optical serial-to-parallel converterwith erbium-doped fiber amplifier

In this paper, we develop a performance analysis of an surface-emitting second harmonic generation (SESHG) optical serial-to-parallel converter using an erbium-doped fiber amplifier (EDFA) as a preamplifier. The analysis is complicated by the fourth-order nonlinearity that acts on the signal plus amplified spontaneous emission (ASE) noise to create many beat noises at the binary decision device. However, we demonstrate that Gaussian approximation for the beat noise statistics is reasonable. We calculate the BER of the system as a function of the SHG nonlinear cross section (A^NL), EDFA gain, the bandwidth of the optical filter that band-limits the ASE noise, and the timing pulse-to-data pulse power ratio. We find that for reasonable values of these and other parameters, the EDFA/SESHG serial-to-parallel converter combination should be able to operate at or below a BER of 10^-12. We find that small increments (0-2 dB) in the signal power that is input to the EDFA are enough to compensate the effects of ASE noise for most of the parameter variations we consider. From this point of view, the ASE noise has little effect on system performance. However, when the input power is fixed, we show evidence in terms of BER that the ASE noise plays a significant role, particularly in the high A^NL, high gain case. Also in this case, we show that the optimal timing pulse-to-data pulse power ratio is somewhat different from the value that is optimal for the system without an EDFA     

Minimum Mean-Square Error Timing Recovery Schemes for Digital Equalizers

Two algorithms are presented for optimum timing recovery in digitally implemented equalizers. The first one is a polarity-type algorithm based on the conventional minimum mean-square error criterion. A theoretical analysis is made to characterize the algorithm phase detector and evaluate its steady-state phase jitter variance. Influence of various channel and system design parameters on the algorithm performance is illustrated using phase jitter probability densities obtained by means of computer simulations. Interaction of the algorithm with decision-directed carrier recovery is also examined. It is shown that interaction with carrier recovery may considerably degrade the timing acquisition performance, and a second algorithm is then presented which eliminates this interaction. The second algorithm is based on the minimization of a modified mean-square error criterion which provides a measure of the intersymbol interference, independently of the carrier phase. Decision-directed timing and carrier recoveries are thus decoupled and the system startup period is considerably reduced. Phase detector characteristic and steady-state jitter performance of the second algorithm are evaluated by analytical means and computer simulations, as in the first algorithm.     

All-optical TDM data demultiplexing at 80 Gb/s with significant timing jitter tolerance using a fiber Bragg grating based rectangular pulse switching technology

We demonstrate the use of fiber Bragg grating based pulse-shaping technology to provide timing jitter tolerant data demultiplexing in an 80 Gb/s all-optical time division multiplexing (OTDM) system. Error-free demultiplexing operation is achieved with /spl sim/6 ps timing jitter tolerance using superstructured fiber Bragg grating based 1.7 ps soliton to 10 ps rectangular pulse conversion at the switching pulse input to a nonlinear optical loop mirror (NOLM) demultiplexer comprising highly nonlinear dispersion shifted fiber (HNLF). A 2-dB power-penalty improvement is obtained compared to demultiplexing without the pulse-shaping grating.     

Characterization of All-Optical Regeneration Potentials of a Bistable Semiconductor Ring Laser

All-optical pulse reamplification, reshaping (2R), and retiming (3R) using a monolithic bistable semiconductor ring laser (SRL) is demonstrated for the first time. The regeneration performance of the SRL is characterized with an all-optical settable switching threshold, achieving significant increases in the extinction ratio (ER) of the output pulse for input ER as low as 1 dB. For retiming, a rectangular retiming window generated by a clean clock signal is used to eliminate the timing jitter in the input pulse. For input pulse with peak-to-peak timing jitter as high as $sim {hbox {12}}%$ of the bit period, the timing jitter in the retimed output pulse is reduced to $ ≪ 2%$ of the bit period. The pulsewidth of the final regenerated data can be controlled by changing the width of the retiming window. The SRL is, therefore, shown to have a “hard” digital performance in both amplitude and time domain suitable for all-optical 3R.      

A Simulation Study on Clock Recovery of a Minimum Bandwidth Signal

Jitter and BER performance of non-linear clock recovery circuits are evaluated for a minimum bandwidth signal. To investigate the effect of bandwidth in optical transmission, BER performance of MB810 and NRZ signals with various types of timing recovery circuits are compared for a 40 Gbit/s optical link. Among the systems adopting non-linear timing recovery circuits, the MB810 signal employing an absolute value rectifier shows superior performance. We show that timing recovery without a non-linear circuit is also possible, and compare its BER performance with that of the others. Jitter performance of the fourth-law rectifier and that of the absolute-value rectifier is made for a minimum bandwidth signal. The mathematical derivation of the timing wave is also carried out for the clock recovery of the fourth-law rectifier. The result shows that the derived timing wave is expressed as a function of a pulse shape entering the timing path and the band-pass filter tuned to the pulse repetition rate.     

Slot clock recovery in optical PPM communication systems withavalanche photodiode photodetectors

Slot timing recovery in a direct-detection optical PPM communication system can be achieved by processing the photodetector output waveform with a nonlinear device whose output forms the input to a phase-locked loop. The choice of a simple transition detector as the nonlinearity is shown to give satisfactory synchronization performance. The RMS phase error of the recovered slot clock and the effect of slot timing jitter on the bit error probability were directly measured. The experimental system consisted of an AlGaAs laser diode (λ=834 nm) and a silicon avalanche photodiode photodetector. The system used Q =4 PPM signaling and operated at a source data rate of 25 Mb/s. The mathematical model developed to compute the RMS phase error of the recovered clock is shown to be in good agreement with results of actual measurements of phase errors. The use of the recovered slot clock in the receiver resulted in no significant degradation in receiver sensitivity compared to a system with perfect slot timing. The system achieved a bit error probability of 10^-6 at a received optical signal energy of 55 detected photons per information bit     

Maximum Likelihood (ML) Symbol Timing Recovery (STR) Techniques for Reconfigurable PAM and QAM Modems

A fast symbol timing recovery (STR) scheme suitable for flexible-symbol-rate, burst-mode M-ary QAM (M-QAM) and PAM (M-PAM) demodulators used in broadband communications is proposed. The maximum- likelihood (ML) estimation is used to derive a feed-forward STR strategy to achieve fast and accurate derivation of the optimum timing instant with a very short training preamble for applications requiring short packet transmission. In addition, the signal sample at the optimum timing instant is reproduced by interpolation. Analysis to derive the ML-based timing estimation algorithm and STR structure is presented. Analytical models are developed to evaluate the effects of imperfect interpolation and channel additive white Gaussian noise (AWGN) on the timing estimation performance. The derived expressions can be used for optimum interpolation filter design and selection of key parameters to achieve the required mean squared timing error. Simulation results are in good agreement with the analytical prediction.     

Ultrafast soliton-trapping AND gate

An ultrafast, all-optical, soliton-trapping AND gate that consists of a birefringent optical fiber followed by a frequency filter is demonstrated. The gate is sensitive to the timing of the input pulses and provides an output with a large energy contrast. The performance of the gate is characterized by varying the total input energy, the ratio between the energies of the two input pulses, and the arrival-time difference between the input pulses. It is shown that the gate efficiency (characterized by its ON-OFF contrast ratio) increases with increasing pulse energy up to the limit where Raman effects become dominant in the fiber, and the optimal performance of the gate is obtained with two input solitons having equal energies. The gate efficiency degrades with increasing difference of arrival time of the two input pulses, but a contrast ratio of 5:1 can still be obtained for a full pulse width of timing mismatch. The experimental results are in agreement with numerical simulations using the coupled nonlinear Schrodinger equations     

Word timing recovery in direct detection optical PPM communicationsystems with avalanche photodiodes using a phase lock loop

A technique for word timing recovery in a direct detection optical pulse position modulation (PPM) communication system is described. It tracks on back-to-back pulse pairs in the received random PPM data sequences with the use of a phase locked loop. The experimental system consisted of an AlGaAs laser diode transmitter (λ=833 nm) and a silicon avalanche photodiode photodetector, and its used Q=4 PPM signaling at a source data rate of 25 Mb/s. The mathematical model developed to characterize system performance is shown to be in good agreement with the experimental measurements. Use of this recovered PPM word clock, along with a slot clock recovery system described previously, caused no measurable penalty in receiver sensitivity when compared to a receiver which used common transmitter/receiver clocks. The completely self-synchronized receiver was capable of acquiring and maintaining both slot and word synchronizations for input optical signal levels as low as 20 average detected photons per information bit. The receiver achieved a bit error probability of 10^-6 at less than 60 average detected photons per information bit     

A Digital Approach to Symbol Timing Recovery Systems

An analysis of a symbol timing recovery (STR) technique using digital processing is presented. The ratio of the discrete spectral power at the symbol rate frequency to the nearby continuous spectral power is used as a criterion of STR performance. It is shown that this power ratio equals the quality factor of the narrow bandpass filter and that it does not depend on the value of the delay element. The performance of the STR subsystem is consequently determined by the quality factor of the bandpass filter rather than by the specific delay. In addition to this, some experimental evidence that the additive channel noise has little effect on the power ratio is given. A modified phase-locked loop with anLCprefiiter is proposed to extract the symbol timing clock. This prefilter improves the acquisition and synchronization performance of the PLL. The STR technique presented here has the advantages of lower cost and simpler hardware implementation over other serial STR techniques.     

Simultaneous amplification and compression of ultrashort solitons in an erbium-doped nonlinear amplifying fiber loop mirror

A simple technique for simultaneous amplification and compression of ultrashort fundamental solitons is proposed. It is based on an erbium-doped nonlinear amplifying fiber loop mirror. Numerical simulations show that, unlike conventional erbium-doped fiber amplifiers in which nonlinear effects lead to serious degradation of pulse quality, the proposed device performs efficient high-quality amplification and compression of ultrashort solitons while nearly preserving the soliton nature of the input pulses. We have also studied the effects of loop characteristics, nonsoliton input pulses, and higher order fiber effects on the device performance and show that the proposed scheme is fairly insensitive to small variations in both the loop and input pulse parameters.