SYMBOL TIMING RECOVERY FO A CLASS OF CONSTANT ENVELOPE SIGNALS

Edgar H. Satorius' and James J. Mulligan

US Army CECQM Center for Signals Warfare

Vint Hill Farms Station Warrenton. VA 221 86-51 00

ATTN : AM S E L- R D-S W-TRS (M ulligan)

Abst rac t

A critical component in any communication system is timing recovery at the receiver. In this paper, we examine different techniques for recovering symbol timing with special emphasis on the class of constant envelope, continuous phase FSK (CPFSK) signals. These signals are of interest because they are bandwidth efficient and do not impose high peak power transmitter requirements. Although techniques for equalizing these types of signals have been extensively investigated (e.g., constant modulus blind demodulation), symbol timing recovery has not been as thoroughly analyzed. In this paper, we present a performance analysis of envelope-derived timing recovery as applied to CPFSK modulation. The results of this analysis will provide useful guidelines for the development of constant envelope, timing recovery systems.

1. Introduction

Rhodes (Ref. 1) was one of the first to clearly show that the reduction of phase discontinuities in phase-modulated signals results in desirable spectral characteristics (lower spectral sidelobes) when the signals are passed through a bandlimiting non-linearity. Based on these results, one might be led to conclude that continuous phase modulated signals passed through nonlinearities would be practically devoid of high frequency spectral content. This, in turn, would imply that simple, envelope-derived timing recovery techniques are not effective for recovering timing information from continuous phase modulated signals thereby forcing the application of more complex techniques.

To the contrary, we have found that envelope-derived timing recovery (EDTR) can be effectively applied to continuous phase modulated signals. This is significant since EDTR techniques are simple and are effective for a wide variety of modulation formats (Ref. 2-7). Furthermore,

' Currently on leave from the California Institute of Technology, Jet Propulsion Laboratory, Pasadena, Calif.

EDTR techniques are highly desirable especially since they can be implemented as a stand-alone system independent of carrier recovery.

In this paper, we focus on a special class of continuous phase modulation: CPFSK. Aside from being bandwidth efficient, this modulation actually encompasses a wide class of continuous phase modulation formats including MSK. The general model for our analysis is depicted in Figure 1 . We assume that the complex basebanded input, z(t), is a noise- free, unfiltered CPFSK waveform:

where: t

~ ( t ) = 5 d(z) dz , T = symbol interval,

h = modulation index ,

T O

cc

and: d(z) = In g(z - nT) , In = n-th data symbol. n = - m

Data Demodulator

Symbol EDTR subsystem Timing

Figure 1. System model.

For CPFSK modulation, g(t) is a rectangular pulse

Acknowledgment: The research described in this paper was carried out by the Jet Propulsion Laboratory, California institute of Technology, under a contract with the National Aeronautics and Space Administration.

423 24ACSSC-1219010423 $1.00 (c, 1990 MAPLE PRESS

which is unity for 0 < t 5 T and zero otherwise. More general types of continuous phase modulation signals can be easily derived from CPFSK simply by controlling g(t) (Ref. 8). The index, h , conveniently parametrizes the structure (bandwidth) of the CPFSK modulation. A small modulation index, 0 < h < 0.5, corresponds to a lowpass spectral shape whereas larger values of h gives rise to spectral peaks at the keying frequencies (some examples are given in the next section, see also Ref. 9, chp. 3). A value of h = 0.5 corresponds to MSK modulation which is the most popular type of CPFSK.

The EDTR subsystem depicted in Figure 1 is a simplified version of an actual EDTR circuit. Such a circuit would typically incorporate a phase locked loop, in addition to the narrow bandpass post-filter, for further reducing timing jitter in the envelope-squared output, e(t). In this paper, we ignore timing jitter in our assessment of system performance and instead focus on the magnitude of the baud rate line component in e(t). As noted in Ref. 4 , this is the primary indicator of the effectiveness of EDTR. (The effects of timing jitter on EDTR performance for a wide class of modulation formats including CPFSK are being considered in Ref. IO.)

ysfem Performance Evaluation

In our analysis, we have focused on the role of the lowpass pre-filter in maximizing the amplitude of the baud rate line component in the output, e(t), of the envelope- squaring detector. Note that the signal excess bandwidth, a , available io the EDTR subsystem is determined by the lowpass pre-filter bandwidth, B:

a = 2BT - 1 .

As 01 decreases to zero, the baud rate line component vanishes (higher order nonlinearities are required for timing recovery in this case as discussed, e.g., in Ref. 6-7). The baud rate line component also vanishes as a increases due to the constant envelope property of z(t). Thus, tFere is generally an optimal pre-filter bandwidth, B , tha! maximizes the baud rate line magnitude. As will be shown, B is remarkably insensitive to the modulation index, h.

To help interpret our performance analysis results, we first present in Figure 2 CPFSK spectra corresponding to h = 0.25, 0.5 and 0.75. Also indicated in Figure 2 are the pre- filter cutoff frequencies corresponding to different values of the signal excess bandwidth: a = 0.1 (10% excess bandwidth), 0.5 (50%) and 1.5 (150%). As is seen, 50% excess bandwidth precisely captures the mainlobe energy for MSK (h = 0.5) but does not completely encompass the h = 0.25 mainlobe and encompasses a little more than the mainlobe for h = 0.75. Similarly, a = 10% excess bandwidth only encompasses part of the mainlohe for all three values of

h whereas CI = 150% encompasses all of the mainlobe and approximately all of the first sidelobe.

The significance of this becomes clearer when we examine the magnitude of the baud rate line component in e(t). Specifically. i t is shown in Ref. 1 0 that the mean envelope function can be expressed as a harmonic series:

n = -- where E{.} denotes expectation over the random data symbols and the amplitude of the baud rate line, C,, can be expressed in the form:

cc a . -

- m i 2 -20 I"-

- 5 0 1

-601

Figure 2. CPFSK spectra.

In this integral expression for C1, HpRE( f ) denotes the frequency response of the pre-filter and S(f) is a quadratic function of the spectrum of the data pulse, g(t). Assuming H p R E is an ideal, lowpass filter (1-sided bandwidth = B), then the expression for C, becomes:

(1 + a ) / 2

c l = q T2 ( 1 - a ) / 2 dx S(x/T)

From this expression it becomes clear that as the excess bandwidth, a , vanishes, the baud rate line component also vanishes. This occurs when BT = 0.5. With reference to Figure 2, it is seen that this condition (BT = 0.5) corresponds to filtering out all hut a fraction of t h e mainlobe energy in the CPFSK signal spectrum. As ct increases, more of the signal's mainlobe is passed through the pre-filter and the magnitude of the baud line component, ICl/, increases. This increase in IC, I continues until the signal sidelobes are passed through the pre-filter (see Figure 2) in which case positive-negative cancellations occur in the integral (defining C1) which forces IC1/ back to zero.

We have simulated the envelope output corresponding to a 2-level (In = & I ) , CPFSK input signal. Typical envelope

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spectra are presented in Figures 3-5 corresponding to h = 0.25, 0.5 and 0.75. These data were generated using 64 samples per baud interval, T, and a 6-th order, Butterworth lowpass pre-filter. A 1024 point FFT was used to generate the envelope spectra.

To reduce timing jitter generated by the random data patterns, 20 successive FFTs were coherently averaged. In spite of this averaging, jitter noise is clearly present -- dominating the lower frequency portion of the envelope spectra. The baud rate line is also clearly visible for all values of the modulation index and signal excess bandwidth. Note that it increases monotonically with h which, in turn, corresponds to a more frequency-compressed mainlobe in the signal spectrum (see Fig. 2). This increasing concentration of signal energy with increasing h results in a larger baud rate line component.

The spectral level of the baud rate line also varies with excess bandwidth. As seen in Figs. 3-5, the largest baud rate line is achieved at 50% excess bandwidth, for all values of the modulation index. This is more clearly illustrated in Figure 6 where the magnitude of the baud rate line component is plotted versus excess bandwidth.

-f- alpha = 10%

---e- alpha = 50% -t- alpha = 150%

Baud rate line 2o t

-- 5 1 5 25

Freq cell

Figure 3. Envelope spectra for h = 0.25.

3. Conclusions

The results of this analysis clearly show the importance of the pre-filter bandwidth in optimizing EDTR system performance for GPFSK (the same conclusion was made in Ref. 3 with regards to QAM). The important point, however, is that the optimal pre-filter bandwidth is remarkably insensitive to the modulation index, at least over the range considered here (0.25 6 h 5 0.75). Thus the implementation of an EDTR system for CPFSK can be carried out without detailed knowledge of the modulation index.

The logical proaression of this work is towards the analysis of a complete demodulation system incorporating timing recovery as well as channel equalization. The primary performance metric for this type of analysis is, of course, bit

error rate. In various system the subject of

this way, the impact of tradeoffs between parameters can be readily addressed. This is

a report currently in preparation (Ref. IO).

-f- alpha = 10%

alpha = 50%

I alpha = 150% f

I -15 5 1 5 25

Freq ce l l

27.5

a U

a, 5 , a, J

-

- 2 ij 7.5 a, Q fn

-12.5 5

Figure 4. Envelope spectra for h = 0.5.

---f- alpha = 10% - alpha = 50% - alpha = 150%

\ I

1 5 Freq ce l l

25

Figure 5. Envelope spectra for h = 0.75.

References

1. Rhodes, S., "Effects of Hardlimitino on Bandlimited Transmissions with Conventional and Offset QPSK Modulation," Proc. Nat. Telecommun. Conf., Houston, TX, 1972.

425

5 0 100 150 200

a lpha (%)

Figure 6. Magnitude of the baud rate line.

2. Ready, J., Gooch, R., "Comparison of Nonlinearities for Symbol Timing Recovery," T w e n t y - f i r s t A s i l o m a r Conference o n Signals, Systems & Computers, Pacific Grove, CA, 2-4 November 1987.

3. Gitlin, R., Hayes, J., "Timing Recovery and Scramblers in Data Transmission," Bel l Syst. Tech. J., vol. 54, pp. 569-593, March 1975.

4. Franks, L., "Synchronization Subsystems: Analysis and Design," chp 7 in: D ig i ta l Commun ica t i ons , ed. by K. Feher, Prentice-Hall, Inc. 1983.

5. Jablon, J., Farrow, C., Chou, S-N., "Timing Recovery for BI i n d Eq u a I iz a t i o n , I ' A s i I o m a r Conference o n Signals, Systems & Computers, Pacific Grove, CA, 31 October-2 November 1988.

6 . Mazo, J., "Jitter Comparison of Tones Generated by Squaring and by Fourth-Power Circuits," Bell Syst. Tech. J., vol. 57, pp. 1489-1498, May-June 1978.

7. Reed, D., Wickert, M., "Spread Spectrum Signals with Low Probability of Chip Rate Detection," IEEE J. on Selected Areas in Comm., v. 7, pp. 595-601, May 1989.

8. Anderson, J., Aulin, T., Sundberg, C-E., Digital Phase Modulat ion, Plenum Press, 1986.

9. Proakis, J., Digital Commun ica t i ons , McGraw-Hill, 1983.

T w e n t y - s e c o n d

10. Satorius, E., Mulligan, J., "Analysis of Timing Recovery and Blind Demodulation Systems," report in preparation, 1990 .

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