A Characterization of "Cycle" Slipping in a Digital PLL Based MPSK Demodulator

Michael P. Fitz School of Electrical Engineering

Purdue University West Lafayette, IN 47907-1285

Abstract Digital communication demodulators generate a phase

reference ambiguity. For architectures that use unique word decoding, a slip of the phase reference from one lock point or attractor to an adjacent attractor can cause a burst of decoding errors. This paper presents a characterization of the first-order. decisiondirected digital phase-locked loop as ,a discrete time Markov chain. Traditional Markov analysis techniques utilizing absorbing states are used to analyze the mean time to slip between attractor points. The mean time is numerically evaluated for the first-order loop with unmodulated, BPSK, QPSK, and 8PSK modulated signals.

1. lntroductbn Cycle slipping is a characteristic of the phase-locked

loop (PLL) [4, 61. The PLL is a recursive estimator of the phase of a signal (usually a sinusoidal signal). The cyclo- stationary nature of a sinusoid and the nonlinear form of the PLL produce multiple lock or attractor points. These attractor points for an unmodulated sinusoid are separated by 27c radians. For an unmodulated sinusoid, cycle slipping results when noise causes the recursive estimate to leave the vicinity of one attractor point and settle near another attractor point. When a cycle slip occurs the reference either adds a cycle or deletes a cycle compared to the signal being tracked. Cycle slipping in a PLL for an unmodulated signal degrades the frequency estimation capabilities of a PLL and produces clicks in FM demodulation.

A similar slipping phenomenon is apparent in PLL structures used for phase estimation of digitally modulated sinusoids (e.g.. a Costas loop). The attractor or lock points are closer together than 2rc radians. The closer spacing of the attractor points is due to the nonlinear processing in loops for modulated signals and the rotational symmetry of most practical digital constellations. This paper considers MPSK modulation which has lock or attractor points separated by 2xFI radians. Note that for modulated signals adjacent attractors are not separated by a full cycle but the notation "cycle" slipping is still used because of the similarity to the cycle slipping in loops with unmodulated sinusoids. Since the phase shift of the received signal induced by the channel is normally unknown a priori, the multiple lock points produce a phase ambiguity during demodulation. This ambiguity is resolved by either the use of differential encoding or unique word decoding. While differential encodingldecoding is not extremely sensitive to the slipping between attractors, the bit error probability (BEP) performance is degraded by 1-3dB compared to ideal coherent demodulation. Communication systems that attempt to operate

R. Jean-Marc Cramer* TRWInc.

Redondo Beach, CA 90278 * Work supported by a TRW Fellowship

with near ideal coherent performance (e.g., satellite communications) usually use unique word decoding.

Slipping in a PLL-based demodulator with unique word decoding can be a significant source of errors in demodulation. The unique word decoding resolves the demodulator phase ambiguity so that near coherent performance is possible. If after the ambiguity is resolved, a slip of the phase reference to an adjacent attractor point produces a burst of errors. This burst is caused by the slip producing a rotation in the demodulated constellation in comparison to the decision rules, which produces a large error rate. The assessment of the probability of slipping between occurrences of the unique word is important to the system architect.

Cycle slipping is well characterized for an analog PLL with an unmodulated input [4,6]. but no analytical results have yet appeared for slipping in loops for modulated signals. This problem remains open for the analog loops with modulated signals, but digital PLL (DPLL) are characterized using the theory of Markov chains [3, 5, 81. This paper analyzes slipping in a decision-directed (DD) DPLL, but the extension to other loop configurations (e.g., generalized Costas loops) is straightforward. Section I1 reviews the chain characterizing the DPLL and presents a theorem for assessing the mean time to slip, Section 111 presents the detailed analytical models for the DD DPLL and the numerical results, and Section IV concludes.

II. Cycle Slipping Markov Model This paper analyzes "cycle" slipping in the ideal pulse

shape matched filter, decision-directed DPLL-based demodulator for MPSK modulation in additive white Gaussian noise. Figure 1 is the block diagram of a decisiondirected DPLL (first- order) based MPSK demodulator. Assuming that the pulse shape satisfies Nyquist's criterion for zero ISI, the matched filter output has the form

where vn=8T(n)+80. BT(n) is the transmitted symbol phase, 80 is the unknown carrier phase, Es is the energy per symbol, and /n is a zero-mean, delta correlated, discrete time complex Gaussian random process with a variance of No. The output of the matched filter and the current phase estimate are input into a phase detector (PD) which generates both a signal proportional to the phase error after decision-directed processing and the symbol decisions. The measured phase error, $n, output from the PD, is the innovation for the loop. Figure 2 is the noiseless S-curve for the decision-directed PD. This sawtooth PD produces an ideal measure of the phase error and provides better acquisition performance than more conventional phase

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92CH3132-810000-0380 $3.00 0 1992 IEEE ICC '92

71

Input - Matchec- - Filter

X, Detector A/D= and -

1 2 - 1 1 Figure 1. The demodulator block diagram

Figure 2. Noiseless phase detector characteristic (S-curve)

detectors (e.g., a Costas loop). An MPSK phase estimation architecture has M stable lock points or attractors in [-n,x] due to the transmitted constellation symmetry. A slip is defined as a phase najectory which travels from one atrractor to an adjacent attractor. By appropriate modification of the Markov chain presented in [3], "cycle" slipping can be numerically analyzed.

The phase process in the DPLL is a first-order Markov chain. The Chapman-Kolmogorov (CK) equation governing the operation of the DPLL is

Pn+1 ( i n c l ) = I p a [ i n + 1 J i n ] ~ n ( i n ) din e. (2)

where:

Pn('n) = probability density function (pdf) of the phase estimate at time n

PI(+) = state transition pdf of the DD loop Bn = The probability space of the phase estimate

In this model the noise corrupting the demodulator is stationary so that the chain is time-homogeneous. If the phase process is limited to a finite number of states (necessary for a digital implementation) the discrete time Markov process becomes a chain and the CK integral equation becomes the matrix equation

at time n.

where S is the state transition matrix and the column vector p(Bn) is the distribution of the chain at time n. For the S-curve given in Figure 2 the difference equation governing the DPLL is

A

(4) 6(n + 1) = K&n) + (1 - K)i(n).

is the phase after decision-directed processing and K is the loop gain. The elements of the state transition matrix are functions of the matched filter output phase and the phase reference at time n. If the phase quantization step size is small, the elements of the state transition matrix are accurately approximated as [3]

(5 )

where A is a constant proportional to the phase step size in the chain that guarantees that the columns of S sum to unity. The

pdf of 6 is discussed in Section I11 for BPSK, QPSK, and 8PSK. This Markov chain permits a complete characterization of the demodulator's performance.

The use of absorbing states in the Markov chain permits an accurate characterization of the slipping phenomena. In the numerical work, without any loss in generality, 80 is set to 0 radians. Consequently Pn is equivalent to the phase error, (Pn, defined modulo-2n. The phase error process, due to the assumed lack of ISI, is independent of the transmitted symbols, so an all ones transmitted sequence is assumed for the derivation in this paper. The DPLL model in [3] has M attractors in this 2n interval; the attractors of the chain are located at q=O, +n/M, f2x/M, --. b. Figure 3 shows a resulting steady-state density function for BPSK modulation. Cycle slipping is characterized by keeping the same loop model near one of the attractors (cp=O) and placing absorbing states at the chain boundaries in the vicinity of the adjacent attractors (cp=b/M). The state transition matrix with absorbing boundaries, Sa, is then given as

= O elsewhere

s, = Pa, ST P., [ a 0, :I (6)

where ST is the reduced state transition matrix for the transient states and Pa; is the absorption probability vector for the absorbing state i. ST is just a subpartition of the state transition matrix generated in [3], while the components of Pai are

- 1

pi, = ~(i,,, = ailen = e:).

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2 . I

1.5

h

, 4 1 a

0.5

0

Phase Error cp (Degrees) Figure 3 BPSK modulation for Et,/Ng=6.OdB.

Steady-state phase error density function of

With this modified Markov chain, the slipping in a DPLL is analyzed with classical Markov techniques [2,7]. The chain has two recurrent classes (the absorbing states) and one transient class. The mean time to a slip is now equivalent to the mean time to absorption in a recurrent class. TBaeoneW For a finite state Markov chain with a state transition matrix S containing a set of recurrent classes R and a transient class T, the mean time to absorption in one of the recurrent classes is given as

E[T,] = lTII -S,]-'p,

where ST is the reduced state transition matrix that corresponds to only transient states, I is the identity matrix, l=[l,l,....,l]T. and po is the initial distribution of the chain. Proof? A proof is found in [7].

Consequently, the slipping performance of a DPLL demodulator for MPSK is characterized if the state transition matrix (5 ) is derived. The abovementioned theorem provides the vehicle to analyze the mean time to slip.

111. Loop Models In this section, four specific signal cases are

considered: an unmodulated sinusoid, BPSK, QPSK, and 8PSK modulation. The unmodulated cycle slipping results are compared to the theory derived for the analog PLL [4,6]. The modulated signal results are new. The state transition pdf or matrix, (5 ) is a nonlinear function of both On and 8n. This pdf is derived by a transformations of random variables on On, so the pdf of 8, is of fundamental importance and is given as

where On is the phase of the matched filter output and Wn is the signal (mean) phase of the matched filter output defined in (1).

Unmodulated Simal, A DPLL for an unmodulated signal has been a subject

of frequent study [5]. Again without loss of generality, assume \Yn=Oo. The uniformly sampled DPLL has a state transition pdf, (3, characterized with

Figure 4 shows plots of the normalizedl mean time to cycle slip obtained by evaluating (8). For comparison, the analog PLL mean time to cycle slip, given as [4,61

where a is the loop SNR and BL is the noise equivalent bandwidth of the loop, is also plotted in Figure 4.

Two points are of interest. First, the performance of the DPLL is not independent of the loop bandwidth as with the analog loop. This characteristic is due to the step size restrictions imposed by reducing the loop bandwidth (i.e., smaller K). The second characteristic is that the mean time to slip for the DPLL reaches an asymptotic lower bound as the S N R becomes small. This again is in contrast to the analog loop and is due to the DPLL making only one update per symbol. This discrete innovation time and the restricted step size limit how fast the loop can slip even with a very noisy input. It should also be pointed out that the smaller the value of K, the closer the high S N R performance of the DPLL is to that of the analog PLL. This is because the difference equations governing the DPLL converge to the differential equations characterizing the analog PLL as K becomes smalI [l].

BPSK Modulated Signals, Numerical results for slipping with a BPSK modulated

signal input to the DD loop are generated in an analogous fashion as for the loop with an unmodulated input. In general

&n) is characterized as

with O(n) characterized by (9) and the decoded phase being a

function of O(n) and 6(n). A density function of 6(n) conditioned on &n) and y(n) is fundamental to the Markov analysis developed in this paper. This conditional density

1 Normalized by the loop bandwidth, i.e., E(T,BN).

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function is now derived for BPSK modulation only, but the generalization to QPSK and 8PSK is straightforward. For BPSK decision rules

- n - x if e(n)- - 5 e(n) I e(n) + - 2 2

O,(n)=O

(13) 6,(n)= x otherwise

which, used in (12), produces

and a Conditional pdf given as [3]

-n x pr(a) = p,(aly, = O)+p,(a+ x]yn = 0) - s a 5 -

= O elsewhere. (15) 2 2

The two terms in (15) correspond to the two decisions possible for BPSK. Note that (15) is equivalent to

-x R pr(a) = pe(alyrn = 0) + pe(alyn = n)

= O elsewhere. (16)

Using (15) or (16) and (9) in (5) generates the state transition matrix. Figure 5 shows plots of the normalized mean time to cycle slip obtained by evaluating (8) for BPSK. The DPLL loop for BPSK is much more likely to slip than the loop for unmodulated signals. This characteristic is caused by the attractor points being closer together. Since the attractor points are closer, a lower "potential" barrier exists between the attractor points, making it easier for noise to cause phase error trajectories to exceed these barriers.

OPSK Modulated Signals, Numerical results for slipping with a QPSK modulated

signal input to the DD loop are generated in the same manner as for BPSK modulation. The pdf of the phase of the matched filter output after DD processing with QPSK modulation is (using a form like (16)) given as [3]

- I a I - 2 2

= O elsewhere. (17)

The four terms in (17) correspond to the four decisions possible for QPSK. Using (17) and (9) in (5) generates the state transition matrix. Figure 6 shows plots of the normalized mean time to cycle slip obtained by evaluating (8) for QPSK. Again QPSK is more likely to slip than BPSK due to the lowering of the "potential" barrier between the attractor points.

BPSK Modulated SimalL Numerical results for slipping with an 8PSK

modulated signal input to the DD loop are again generated in the same manner. The pdf of the phase of the matched filter output after DD processing is given as

-x n -Sa<- 8 8

= O elsewhere. (18)

The eight terms in (18) correspond to the eight decisions possible for 8PSK. Using (18) and (9) in (5) generates the state transition matrix. Figure 7 shows plots of the normalized mean time to cycle slip obtained by evaluating (8) for 8PSK.

IV. Conclusions Quantifying cycle slipping is important for

demodulators that use unique word decoding. This paper demonstrates that the mean time to slip is generated with Markov analysis techniques. The higher the order of the modulation, the greater the chance of a slip. This characteristic is produced by moving the attractor points closer together which reduces the "potential" barrier between the attractor points. Noise is then more likely to cause a phase error trajectory to exceed this barrier, producing a slip.

Loop SNR,dB

Figure 4. The normalized mean time to cycle slip for the first-order DPLL and first-order analog PLL versus loop SNR. Sawtooth PD, absorbing boundary=+345".

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_ . . . . . . . . - , . . . t . . . _ . . , . . . . . - , . . . . . . . . . . . . . . . , . , . . . . . _ . . . . . . . . - , . . . . . . . , . . . . . . . . . . . . . . . - . , . . . . . , - . . . . . . . . . . . . . . . .

_ . . . . . . . . _ - , , , . . . . . -

1 ' 1 ' 1 ' I 1 1 '

- + K=0.125

. . . _ . . . . . . . . . . . . _ . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . .

l o , ' ' , ! , , , ! ' , , ! ' ' , ! , , , ! , ' ' ! , , , ! I I I ! " ' '

-6 -2 2 6 10 Eb/No, dB

Figure 7. The normalized mean time to cycle slip for the first-order DPLL for 8PSK modulation versus loop SNR. Sawtooth PD, absorbing bo~ndary&35~.

References

A. Benveniste, M. Metivier and P. Priouret. Adaptive Algorithms and Stochastic Approximations, Springer- Verlag, New York, 1990. E. Cinlar, Introduction to Stochastic Processes, Prentice- Hall. Englewood Cliffs, NJ, 1975. R. J. Cramer and M. P. Fitz, "The Bit Error Probability Performance for a Digital PLL Based MPSK Demodulator," 1991 IEEE Military Communication Conference, Washington, D.C., pp. 5.1.1-6. W. C. Lindsey, Synchronizat ion Systems in Communication and Control, Prentice-Hall, Englewood Cliffs, NJ, 1972. W. C. Lindsey and C. M. Chie. "A Survey of Digital Phase-Locked Loops,'' Proc. IEEE, Vol. 69, April 1981, pp. 410-431. H. Meyr and G. Ascheid, Synchronization in Digital Communications, John Wiley 8c Sons, New York, 1990. H. M. Taylor and S. Karlin, An Introduction to Stochastic Modeling, Academic Press, Inc., Orlando, FL, 1984.

[8] A. Weinberg and B. Liu, "Discrete Time Analyses of Nonuniform Sampling First- and Second-Order Digital Phase Locked Loops," IEEE Trans. Commun., Vol. COM- 22, February 1974, pp. 123-137.

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