1302 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-28, NO. 8, AUGUST 1980

Synchronization for QPSK Transmission via Communications Satellites

Absfrucf-Carrier phase and symbol timing synchronization per- formances are determined for QPSK transmission over the nonlinear communications satellite channel with several typical modem filtering configurations. A computer simulation is used that models con- ventional or offset QPSK transmission, the modem filters, the nonlinear satellite TWTA, and the receive modem synchronizers. These simulations are used to obtain errors in carrier phase and symbol timing estimates caused by both thermal noise and the modulation-pattern-induced jitter. The statistics of these errors are obtained for several channel filtering combinations and TWTA backoff levels. These experiments are extended to examine the overall effect of synchronization errors on bit-error probability for one particular implementation of the synchronizers.

I. INTRODUCTION

C URRENTLY, digital data transmission via communications satellite channels uses quaternary phase-shift keying

(QPSK) almost exclusively. QPSK offers the advantage of rela- tively high bandwidth efficiency (approximately 1.5 bits/s/Hz) when operating with large earth stations. It is easily combined with various forms of error-control coding to provide band- width and power efficiency when small earth stations partici- pate in the satellite network.

In the design of QPSK transmission links, several factors can affect transmission perqormance. These include the choice of modem filters and the effects on the transmitted signal-of the earth station high-power amplifier (HPA) nonlinearity, the satellite input and output filters, and the satellite final power amplifier nonlinearity. In addition to these “noise-free’’ characteristics, there are the effects of thermal noise experi- enced on both the up- and downlink and potential interference due’ to other signals that share the channel or occupy adjacent channels. A final effect depends on the implementation of the receiving demodulator including the means of synchronization to achieve coherent detection.

When modem filtering is selected to achieve the sometimes conflicting goals of minimizing intersymbol interference and maintaining acceptable adjacent channel (or adjacent trans- ponder) interference levels, a remaining problem is the impact of filtering and the channel nonlinearity on synchronization errors. This paper primarily focuses on the latter area where the AM/AM and AM/PM characteristics of the helix-type traveling-wave tube amplifier (TWTA) in the satellite are the dominant channel nonlinearity. A secondary goal is to deter- mine the effect of these synchronization errors on detection performance.

Stated qualitatively, variations in both the estimates of car-

Manuscript received July 25 , 1979; revised April 17, 1980. This paper is based upon work performed at COMSAT Laboratories under the sponsorship of the Communications Satellite Corporation.

The authors are with COMSAT Laboratories, Clarksburg, MD 20734.

rier phase and symbol timing can be largesources of degradation for QPSK signaling over the nonlinear satellite channel. These fluctuations, or synchronizer jitter, come from two sources: thermal noise, which ultimately limits channel performance, and data-dependent self-interference caused by the fact that the channel is bandwidth limited and nonlinear. The latter would be present even for operation at very large Eb/No values.

11. THE CHANNEL MODEL

Fig. 1 shows the overall channel model used to study syn- chronizer errors. This model includes the type of modula- tion used (i.e., conventional or offset QPSK); the modem transmit filter, including modulating pulse shaping; channel nonlinearity; thermal noise experienced on the downlink; the receive modem filter; and synchronizers which extract carrier phase and symbol timing estimates from the received signals. .

Both conventional and offset forms of QPSK modulation were simulated. If the channel bandwidth is not too narrow, offset QPSK modulation experiences less spectral regrowth than conventional QPSK after passage through a channel non- linearity [ l ] . This feature may be advantageous in certain applications.

Two types of channel filtering were used in the model. In the first, cosine rolloff Nyquist filtering [2] was used with binary modulation applied as plus o r minus impulses to the ideal square-root Nyquist filter shape at the transmitter. The same filter function was employed at the receiver, giving an overall (linear) channel filter response equal to the Nyquist response. For the second filter combination, delay-equalized Butterworth filters were used at both the transmitter and re- ceiver, and the B3Ts product (product of 3-dB bandwidth and symbol duration) of the transmitter filter was varied from 1.25 to 1.5. (The receive B3Ts product was maintained at 1.1 .) With Butterworth filters, only rectangular NRZ input wave- form shapes were used.

Transmission over the communications satellite channel can encounter nonlinearities in both the earth station HPA and in the satellite TWTA. Generally, both elements are memoryless peak-power devices which can be characterized by curves of output envelope and output phase shift versus input envelope (AM/AM and AM/PM characteristics, respectively). For these investigations, only one of these nonlinearities, the satellite TWTA, has been included in the simulation model, with AM/ AM and AM/PM characteristics as shown in Fig. 2. The opera- ting point of the nonlinearity, as measured by the output backoff relative to saturation, was used as a variable in the sim- ulations. The nonlinear characteristics are typical of those en- countered in the INTELSAT IV-A transponders. The final

0090-6778/80/0800-1302 $00.75 0 1980 IEEE

PALMER et al. : QPSK TRANSMISSION 1303

A - SHAPE 4 THERMAL

NOISE - DATA

TWTA RECEIVE

FILTER (NONLINEAR) MODEM EZ

SHAPE B CHAN - (DELAY

----)

DATA FOR ---L

OFFSET)

SYMBOL TIMING RECOVERY

COHERENT DEMODULATION

CARRIER . PHASE

ESTIMATION

+ I IMPULSE OR SIN ( 1

RECTANGULAR

Fig. 1. Channel model.

0 NONLINEARITY CURVES

r BO

- 60 * ---

c 6 . . w I , D - 5 -20

I

I I'

I ,

I / -40 B

I 0

. - I

(L z 0 . -

- 20

- 4 0 L " " " ' ' " ' ' ' ~ " " " " " ' ' " ' 0 -20 -15 -10 - 5 0 5 IO

P IN (dB)

Fig. 2. TWTA nonlinear characteristics. (Solid line for AM/AM; dashed line for AM/PM.)

narrow-band filtering. For QPSK, modulation removal may be affected by frequency quadrupling, which rotates all four phase angles until they are coincident. Fig. 3 illustrates this form of carrier synchronizer. A tuned filter centered at the fourth harmonic of the carrier frequency f, is used to average the noise so that a high signal-to-noise ratio (SNR) is achieved. Frequency division by a factor of four is then used to obtain a coherent reference from the filter output.

The fourth-power nonlinearity (frequency quadrupling) creates intermodulation products between the signal and noise terms. For rectangular waveforms and additive white Gaussian noise, the degradation in SNR associated with the extra cross products can be shown to be [3]

elements in the model of Fig. 1 are the synchronizers, which are described in the next section.

111. SYNCHRONIZER MODELS AND PREDICTED PERFORMANCE

The synchronizer models that have been used in the simula- tins are described in this section. Theoretical performances of carrier phase and symbol timing synchronizers are also reviewed for a channel that is ideal except for additive white Gaussian noise. These performance estimates serve as useful bounds for comparing the simulation results. In the actual channel, band- width limitations and nonlinearities create self-interference that degrades synchronization relative to the ideal case. The computer simulation results given in Section IV show the ef- fect of the channel on synchronizer performance.

These results are restricted to synchronizer implementations that are commonly used for conventional QPSK which re- quires rapid acquisition. However, since this acquisition was not examined, the results are applicable only to continuous systems. Also, although synchronizers for offset QPSK modu- lation could (ideally) be modified to optimize performance, the same models were used for both modulation formats.

A . Carrier Synchronization A carrier reference for coherent demodulation of QPSK sig-

nals can be obtained from a fully modulated or suppressed- carrier transmission by modulation removal and subsequent

where p denotes the SNR at the input to the frequency quad- rupler as given by

The loss y is irrecoverable and can only be minimized by re- ceiver filtering that is matched to the transmitted pulse shape, in which case the noise bandwidth Bin of the receiver filter equals the quaternary symbol rate R,. In (2) , C represents un- modulated carrier power, No is the single-sided noise density, and Eb is the signal energy per bit.

The SNR per symbol interval 2yEb/No is enhanced by a factor of R,/BN with additional filtering that has a noise band- width of BN. Therefore, the SNR of the carrier reference is given for an ideal AWGN channel by

s 2Eb R, C _ - -Y- - = y . NO BN NOBN

Only the quadrature noise contributes significantly to tracking errors. Hence, the steady-state tracking variance for carrier synchronization may be approximated as the inverse of 2S/N:

1304 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-28, NO. 8, AUGUST 1980

FILTER TO SIGNAL BANDWIDTH TO IMPROVE SNR NARROWBAND PRIOR TO FILTERING TO FREQUENCY OBTAIN HIGH QUADRUPLING

RECEIVED SNR AT 4f,

COHERENT REFERENCE

TRANSMISSION AT CARRIER FREQUENCY

22 x4 D fc

BANDWIDTH Bin fc

ADJUST PHASE FOR COHERENCE

Fig. 3. Carrier synchronizer with frequency quadrupling for modula- tion removal.

In the actual band-limited channel with nonlinearity, the phase angle of the transmission is disturbed by varying amounts from its steady-state value by the random modulation. There- fore, frequency quadrupling does not convert all of the signal energy into an unmodulated replica at 4fc. There are two effects of the failure to completely remove the modulation: a reduc- tion in the average value of the in-phase component at 4fc, and generation of a quadrature component at 4fc with a magnitude that is modulation-sequence-dependent. The quadrature self- noise at 4fc results in a random phase jitter at the input to the carrier phase estimator.

Table I gives the theoretical tracking performances for a linear AWGN channel with E,/Nd values of 6, 9, and 12 dB, and Bin = R, for Nyquist filtering and 1.1 R, for Butterworth filtering. The synchronizer bandwidth BN is constant at R,/60. In terms of 3-dB bandwidth B3 and time constant 7, the noise bandwidth of the tuned single-pole filter is BN = (r/2)B3 =

Although the phase jitter caused by white noise is seen to be proportional to noise bandwidth, this is not the case for the quadrature component at 4fc that results from intersymbol in- terference. This self-noise has a spectral density that may vanish at the center frequency. Gardner has investigated the quadra- ture spectrum for one specific pulse shape when carrier syn- chronization for QPSK was recovered by frequency quadru- pling [4] . It was shown that the spectrum for the quadrature self-noise vanished at 4fc and peaked at approximately R, Hz away from the center frequency for the pulse shape con- sidered. The output phase jitter caused by self-noise for this case would be proportional to BN2 for a simple tuned filter and to B N 3 for a filter with a sharp cutoff response.

B. Symbol Synchronization

1 /(2+

Symbol timing is required at the receiver to determine the appropriate sampling times for bit detection. The symbol synchronizer obtains the required timing reference from transi- tions in the symbol sequence. Because of self-interference re- sulting from the bandwidth restriction and nonlinearities of the channel, the zero crossings associated with these bit transi- tions are displaced from their mean locations as a function of the modulation. Additive noise in the channel also perturbs the transition points. The symbol synchronizer utilizes narrow- band filtering to average the locations of zero crossings over a rather large number of transitions. As a result, the synchroni- zer provides a timing reference with only a small phase variance.

TABLE I

SYNCHRONIZATION FOR IDEAL CASE OF LINEAR AWGN CHANNEL WITHOUT SIGNAL DISTORTION FROM BAND

LIMITING

THEORETICAL PERFORMANCE OF CARRIER

2 . 4 0 2 4 . 4 2 I 1 I 2 . 2 0 I 2 4 . 6 1

2 8 . 5 5

2 8 . 6 6

3 2 . 1 8

N 0 . 5 9 3 2 . 2 4

'B for Butterworth and N for Nyquist.

(deg )

0.00181

0.00173

0.000697 1 . 5 1

0 . 0 0 0 6 8 1 1 . 5 0

0 . 0 0 0 3 0 3 0 . 9 9 7

0 . 0 0 0 2 9 9 0 . 9 9 0

A second-order nonlinearity suffices to obtain a timing sig- nal from the modulated transmission. The QPSK transmission may be multiplied by a delayed version of itself, which is a method known as delay and multiply (D&M). Such a form would be appropriate for QPSK transmissions with rectangular pulses (i.e., no band limiting). A special D&M technique refer- red to as squaring can ais0 be employed with zero delay. Squaring is effective whenever a QPSK transmission has enve- lope variations, which is true for any band-limited signal. Gardner [SI has shown that the clock component of the mul- tiplier is maximized when zero delay is used with Nyquist pulses. For the simulation results in the following section, performance was insensitive to the delay value used.

In the computer simulations for the D&M symbol syn- chronizer, the delay chosen was TJ2 , which maiimizes the SNR for rectangular pulses. Usually, this synchronizer is implemented at IF (see Fig. 4) with a delay of approximately T,/2 that is finely adjusted to yield a phase shift 'of ?n; rad.

Equivalent baseband circuits may be employed to represent the IF operations of D&M and squaring. For multiplier inputs with subscripts 1 and 2 denoting undelayed and aelayed-com- ponents, respectively, the D&M operation yields a baseband voltage of w(t) = X 1 X 2 + Y , Y , where X and Y are the two quadrature components of the QPSK transmission.

This baseband equivalent model for D&M and squaring (see Fig. 5) was used in the computer simulations. For offset QPSK, the staggering of binary components requires the use of the difference X , X 2 - Y , Y2 rather than the sum of these terms. Since there is no convenient method at- IF to obtain this

PALMER et al. : QPSK TRANSMISSION 1305

FILTER TO SIGNAL BANDWIDTH PRIOR TO MULTIPLICATION

RECEIVED

TRANSMISSION

(AT IF) f,=fc

NARROWBAND TUNED FILTER CENTEREDAT SYMBOL RATE

SYMBOL TIMING

f,=R, REFERENCE

T,/2 WITH FINE ADJUSTMENT DELAY OF APPROXIMATELY

FOR PHASE COHERENCE

Fig. 4. Symbol synchronizer for QPSK that employs a delay-and- multiply at IF.

PLUS SIGN FOR

RECEIVED CONVENTIONAL

OR 0-QPSK TRANSMISSION

(AT IF1 i) I+ COHERENT

CARRIER REFERENCE

E%$%F:R r q x 2 ‘1 D=T,/Z E++ TIMING

WAVEFORM

L&+JLb& X 1

DEMODULATORS

Fig. 5. Symbol synchronizer for QPSK with low-pass implementation of delay-and-multiply.

difference, the low-pass implementation is a necessity for off- set QPSK.

For Butterworth filtering, timing performance is estimated under idealized conditions based upon the D&M synchronizer. Rectangular pulses (no band limiting) and additive white Gaus- sian noise are assumed.’ For Nyquist filtering, it is possible to improve the theoretical prediction by including the bandwidth restriction in terms of the “excess bandwidth” factor r. In this Nyquist case, the synchronizer model employs squaring. Ran- dom bit patterns with transition probabilities of p t = 0.5 are assumed. for both cases.

For a symbol synchronizer with either D&M or squaring, the timing component at a frequency of R, is obtained from the product of the signal with itself. The major noise term out of the multiplier results from the cross products of signal and input noise. A small additional noise in the output results from the product of input noise with itself. This small noise output is responsible for a “squaring loss” in the expression for synchronizer SNR.

The output of either the D&M or the squarer i s input to a narrow-band tuned filter with a center frequency of R,. For this filter, the noise bandwidth BN is set at a value much smaller than R, to average the timing waveforms over many symbol intervals and thereby achieve a high SNR. Theoretical SNR expressions for the filter outputs have been derived for both the cases of D&M with rectangular pulses and squaring with Nyquist pulses. For brevity, only the results are included here and the lengthy derivations are omitted.

For rectangular pulses and no filtering. the D&M model for

random bit patterns yields the following output SNR from the tuned filter:

where p1 is the squaringloss factor given by

1 I ’ P1 = 1 + (4Eb/No)-’ ’

In terms of the rolloff factor or excess bandwidth r, the output SNR for Nyquist filtering is

S N - = P 2 ( ; ) ( $ ) ( 3

where

1

1 + (2Eb/N0)- ’ P 2 =-

Timing error variance in radians squared is obtained from the inverse of 2S/N. The factor of r /4 in the timing SNR when Nyquist filtering is employed corresponds to a factor of 4/n2 for rectangular pulses. Hence, the bandwidth constraint of Ny- quist filtering lowers the available SNR by a factor of 16/(n2r). The synchronizer SNR is lowered similarly for the bandwidth constraint imposed by Butterworth filtering; however, this loss is not included in ( 5 ) .

Theoretical predictions of symbol synchronization errors due to white noise were obtained using (5) and (6), and the results are given in Table 11. These predictions do not include the jitter caused by the self-noise from intersymbol interfer- ence. Franks [ 6 ] and Bubrouski ‘[7] have shown that the ef- fect of this seif-noise can be reduced by using a compensating filter prior to the squarer that modifies the transfer function to yield even symmetry about R,/2. Gardner [8] has verified by computer simulations that such a prefilter will greatly re- duce timing jitter from self-noise when the channel is linear. It was also shown that this compensation was not nearly as ef- fective in reducing timing jitter when the channel contained a nonlinearity. This prefiltering was not employed in the compu- ter simulations reported in Section V.

IV. SIMULATION RESULTS

A . Method of Simulation

The investigations of the effect of synchronization errors on channel performance have had two objectives. The first was to simulate specific ‘synchronizer implementations and obtain experimental statistics on phase and symbol timing errors. A second, more difficult, objective was to determine the effect of these synchronizatian errors on overall channel perform- ance as indicated by bit error probability. To achieve both of these goals, the approach has been to augment an existing time-domain channel simulation mogram r91 that was used to

1306 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-28, NO. 8, AUGUST 1980

TABLE I1 THEORETICAL TIMING PERFORMANCES FOR LINEAR AWGN

CHANNEL WITH B N = R,/60

TABLE 111

SYNCHRONIZERS (DEG) PEAK-TO-PEAK PHASE AND TIMING JITTER INTO

Pulse Shape (r)

a Eb/No Bandwidth (deg)

S/N (dB) (dB)

Excess

- 6 4.24 19.61

Rectangular 2.96 22.74 9 - - 12 2.08 25.80

0.4

4.58 18.95 0.7 Roll-Off

6.06 16.52 0.4 Nyquist With Cosine

9

6.66 15.70 6 0.7

8.80 13.27 6

0.4 I 1: 4.22 19.65

1 0.7 I 12 I 22.08 I 3.19 I I I I I I

produce noise-free records of randomly modulated signals at the input to the receive-modem filter. This record was then used repeatedly, after noise was added in the form of Gaussian ran- dom numbers, in a separate simulation of the synchronizers themselves. Finally, the resultant statistics of phase and timing errors were used, along with an empirically derived function for the sensitivity of bit error rate to static synchronization er- rors, to obtain an overall average error probability. This tech- nique has given accurate results when compared to hardware simulation results [ 101 .

B. General Conditions for the Experiments The variables for the computer simulation experiments

were the choice of modem filters, Eb/No, the operating point (i.e., input backoff) of the channel nonlinearity, and the type of modulation (conventional or offset QPSK). Other param- eters in the model were kept fixed for all of the experiments; for example, the synchronizer bandwidths were kept constant at R,/60. For most runs, the symbol synchronizer utilized a delay of TJ2; however, additonal simulation runs were made with other delay values, including zero delay, to check the sensitivity to this value.

Four different transmit/receive modem filter combinations were used in the simulation model shown in Fig. 1 :

1) square-root, cosine rolloff Nyquist filters with 70-percent rolloff; impulse data

2) square-root, cosine rolloff Nyquist filters with 40-percent rolloff; impulse data

3) transmit filter six-pole Butterworth, B3Ts = 1.25; re- ceive filter four-pole Butterworth, B3 Ts = 1 .I ; NRZ data

4) transmit filter six-pole Butterworth, B3Ts = 1.5; re- ceive filter four-pole Butterworth, B,Ts = 1.1 ; NRZ data.

Simulations were made for Eb/N0 values of 6, 9, and 12 dB, and TWTA input backoff levels were varied from 0 (satu- rated), 4, 7, and 10 dB. Even at the latter value, the trans- ponder still cannot be considered as a linear device because of significant AM/PM at this operating point.

C. Qualitative Indicators o f Synchronization Errors A general indicator of the magnitude of the pattern-induced

phase and timing spread that appears at the input to the syn- chronization circuits can be obtained from the simulated eye

Nyquist Filters

70% Roll-off Saturated

Input Backoff 4 dB

10 dB 7 dB

40% Roll-off Saturated

InDut Backoff dB

7 dB 10 dB

Butterworth Filters BTS = 1.25

Saturated

InDut Backoff 4 dB

10 dB 7 dB

ET, = 1.5 Saturated

Input Backoff 4 dB

10 dB 7 dB

Conventional OPSK

Timing

47

47 49 45

124

112 108 101

54

59 59 60

43

48 48 44

-r Phase

9

8 7 7

11

10 10 9

14

12 10 10

10

8 7 8

Offset

Timing

58

49 54

43

146

128 112 106

86

80 79 61

59

54

47 56

IPSK

'hase

15

11 10 8

22

19 15 12

19

16 13 12

15

10 10

9

1

and scatter diagrams in Fig. 5 . The waveforms and the aggregate pairs of sample points indicate approximately the raw jitter. These symbol-to-symbol variations are smoothed or averaged by the synchronizer filters.

Table I11 summarizes the peak-to-peak magnitude of this input jitter. The timing errors are measured directly from the eye diagrams and represent the peak-to-peak spread in the zero-crossing jitter that results from self-interference caused by channel bandwidth restrictions and the TWTA nonlinearity. Note, for example, that this spread is quite small for the wide- bandwidth Butterworth transmit filter and is unaffected by the channel nonlinearity. In contrast, the peak-to-peak timing jitter is much larger for the narrower bandwidth, 40-percent rolloff Nyquist filters, and the extent of these variations tends to increase as the channel becomes more nonlinear.

The peak-to-peak input phase jitter 6, is read directly as the angular spread of points in the scatter diagrams shown at the right of Fig. 6. It should be noted that the phase estimator processes the continuous waveforms. Nevertheless, the scatter in the pairs of samples on each symbol approximately indicates the raw phase fluctuations that must be smoothed by the phase synchronizer.

For offset QPSK, a sample point for either of the quadra- ture bit streams correspond to a point midway between sam- ples for the other bit stream. Thus, zero crossings at bit transi- tions of one bit stream occur at the sample points for the other and cause a *7r/4 shift in phase. Hence, offset QPSK tends to have an eight-phase nature for scatter diagrams based upon the values of the staggered bit steams at the same time instants. The staggered bit streams have therefore been realigned in plotting the scatter diagrams for offset QPSK. Consequently, these diagrams evidence a four-phase nature similar to that for conventional QPSK. It should be noted that the synchronizer

. PALMER et al. : QPSK TRANSMISSION 1307

I: I ~

t

1 I

1

1 i I

i 1

I i

1308 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-28, NO. 8, AUGUST 1980

i

----I-- h j P --"+ a

I ---"+- a

--$+ a

I 1

---"+ a

' t c

PALMER ef al. : QPSK TRANSMISSION 1309

r Offset QPSK Conventional OPSK . .. ,

f Phase Timing Phase Timing Nyquist Filters ~

Input Backoff (dB) - 7 4 0

5.5

2.5 2.6 2.6 3.6(4.7)‘ 3.7 3.7 5.4 5.5

9.5

4.5 4.7 4.8 6.0 6.5 6.7 8.8 9.3

10.6

3.5 4.2 4.5 ! 5.2 6.2 6.8 1

8.3 9.6

i

2.9 3.2 3.4 ! 4.5 4.9 5.2 ! 7.4 8.0 8.4

- Inr 10

)ut Backoff (dB) )ut Backoff (dB) I Inuut Backoff (dB) I - - 10

2.7 1.6 1.2

3.1 1.9 1.5

3.4 1.8 0.9:

3.0 1.7 0.8t -

- 7

- 7 -

2.9 1.8 1.3

3.2 2.0 1.7

4.7 2.6 1.1

4.2

0.91 2.2

-

- 4 -

3.1 1.9 1.4

3.5 2.2 1.8

5.0 2.8 1.1

4.5

0.9 2.3

-

- 4 -

7.0 5.4 4.1

10.0

6.6 7.8

8.5 6.4 4.3

7.3 5.6 3.9: -

- 7 -

2 . 7 , 1.6; 1.2

3.1 1.9 1.5

3.5 1.8 3.9d

3.0

3.8’ 1.7

-

- 4 -

2.7 1.. 6 1.3

3.2 1.9 1.5

3.6

0.9’ 1.9

3.1 1.7 0.8’ -

0 0 0 10

0% Roll-Off I 6.7 I 2; 2.8

L.8 1.3

3.1 2.0 1.6

4.5 2.4 1.1

4.0 2.1 0.95

3.3 2.1(2.5)* 1.5

4.0 2.6 2.2

6.0 3.3 1.3

2.5 5.1

1.1

6.8

4.0 5.2

10.0 7.7 6.5

8.7 6.6 4.6

7.2 5.t 4.1 -

2.9

1.4 1.8(2.1)*

3.2 2.0 1.7

3.9

1.0 2.1

3.4

0.92 1.8

5.6 3.8 2.6

9.6 6.8 4.9

11.2 6.9 4.2

8.t 5.2 3.2 -

7.4

4.4 5.7(5.3)‘

10.2 7.9 6.8

7.9 5.9 3.8

7.2 5.4 3.5

10% Roll-Off I 10.0 7.7 6.4

3utterworth Filters

BT, = 1.25

9 12

BTQ = 1.5

I 7.2 9

12 5.5 3.9

*Numbers in parenthesis represent a repetition of the simulation run for 10 times the original length.

(7) would be independent of thermal noise (Eb/No), but would show some dependence on the synchronizer bandwidths. Be- cause of the highly correlated nature of the raw input syn- chronization errors, u2 would not vary directly as bandwidth, but rather as a power of this bandwidth [5], [ 6 ] . Also, these variances would be expected to show some proportionality to the raw input errors in Table 111.

For the simulation experiments, the synchronizer band- widths were kept constant at R,/60. Thus, the functional dependence on bandwidth cannot be observed in the results. However, performance variations for the different filtering combinations and for the different backoff conditions should be observable.

Table IV summarizes the simulation results and gives rms phase and timing errors in degrees for the various cases. These results were obtained for different input backoffs from TWTA saturation. Also, the simulations were performed for .!?,/No values of 6 , 9, and 12 dB where E, is defined in terms of un- modulated carrier power C and modulation bit rate R b as

Although each run consisted of many repetitions of the original data record with different random numbers added for each repetition, the resulting number of independent samples of the synchronizer outputs was not large enough to precisely estimate synchronizer errors. Confidence intervals in the esti- mated errors were perhaps 20-40 percent. However, the corre- sponding simulations for different backoffs and modulation types were made with identical bit and noise sequences so that relative results are meaningful, despite large uncertainties in the absolute results. Note, for example, the relatively small

Eb/No = C/NoR b .

inputs did not have this realignment because these devices must operate on the offset QPSK transmission with its staggered bit alignments.

Both the raw zero-crossing range from the eye diagrams and the symbol-by-symbol phase jitter from the scatter diagrams are highly constrained processes with a strong dependence upon the modulation patterns. Consequently, the raw jitter may be highly correlated and cannot generally be treated as an equiva- lent “white noise” to be smoothed by the synchronizer filters.

D. Synchronizer Output Errors Although the simulation results reveal qualitative evidence

of varying degrees of raw input synchronization errors with different channel conditions, it is the output synchronization errors that actually degrade detection performance. These out- put errors in both phase and symbol timing would be expected to include two components: one resulting from thermal noise at the input to the receive modem, and another caused by the input pattern jitter. Each error component has been reduced in magnitude by averaging over many symbol intervals. For both the phase and timing estimates, the rms output error u would be governed by functional relationships of the general form

u2 = [ { term due to } + { term due to]] (7) thermal noise pattern jitter

where u2 represents the variances in either timing or carrier phase. The first component would vary inversely as Eb/No and directly as the synchronizer bandwidth. The second term in

1310 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-28, NO. 8, AUGUST 1980

16

IO 15 8

g 5 0 4

2 E 5 2

E

E 25

1 .o 35

0.5 6 9 12 6 9 12 6 9 12

Eb /No (dB) Eb/No (dB) Eb/No (dB) 70% ROLLOFF

NYQUIST 40% ROLLOFF

NYQUIST BUTTERWORTH

TRANSMITTER BT, = 1.25 AT

(a)

n - 4 K

0.5

16

16

10 15 8 d

m 0

- m 9

W

4 z 25 5

2 Q Z

Ln 12

1 35 B ! Z

2 6 9 1 2 v

I Z

Eb/N, (dB) t, BUTTERWORTH

BT,= 1.5 AT TRANSMITTER

6 9 12 6 9 12 6 9 12 6 9 12

Eb/No (dB) & / N o (dB) Eb/No (dB)

NYQUIST NYQUIST 70% ROLLOFF 40% ROLL OFF

Eb/No (dB)

TRANSMITTER BTs= 1 5 AT

BUTTERWORTH

TRANSMITTER BTs = 1.25 AT

BUTTERWORTH

(b) Fig. 7. Synchronizer errors versus Eb/No. (a) Input backoff 10 dB.

(b) TWTA saturated (theoretical predictions for thermal noise alone shown as dotted lines).

variations in synchronizer errors that accompany variations of the TW.TA input backoff from power saturation. However, there is a clear correspondence between increasing Eb/No and a reduction in tracking errors for both carrier phase and sym- bol timing as shown in Fig. 7(a) and (b). The rms errors for the case of 10-dB TWTA input backoff are plotted in Fig. 7(a), and the rms errors for the case of TWTA power saturation or zero backoff is shown in Fig. 7(b). Pattern-induced jitter from self-noise is proportional to a power of the noise band- width B N . Thus, the use of a narrow bandwidth of B N = RJ60 for the synchronizer filters caused the effect of self-noise to be relatively small compared to that of additive Gaussian noise.

The theoretical values obtained in Secti,on 111 for syn- chronization errors by thermal noise alone are also plotted in Fig. 7(a) and (b) as dotted lines. These predictions show rough quantitative agreement with the simulation results, although the pattern jitter appears to increase the synchronization errors. This increase is most evident for carrier phase errors at the higher Eb/No values with Nyquist filtering and for timing er- rors with 40-percent rolloff Nyquist filtering.

For the cases examined by simulation, differences in rms

synchronization errors for the two modulation types were quite small under the same filtering conditions. Actually, the timing performances should have been identical for offset QPSK and QPSK if the channels were linear. Because of the AM/PM conversion characteristic (see Fig. 2), this linear condition was not obtained at the highest input backoff (10 dB) used in the simulations. For all of the synchronizer performance comparisons, the differences in performance are relatively small and are subject to statistical variations. In Table IV, certain simulation runs were repeated for a much longer duration to spotcheck problems with statistical con- vergence. No large discrepancies were noted.

An additional set of simulation runs was made for con- ventional QPSK at Eb/No = 6 dB where the delay element in the model of the symbol synchronizer was reduced to 3/16 T, and then to 0. The rms timing errors achieved for both de- lay values were extremely close to those obtained for a delay of TJ2; the largest difference was less than five percent.

Carrier synchronizer performance was not greatly affected by the channel bandwidth restriction. Thus, the rms synchro- nizer errors are only slightly worse for the narrower filtering

PALMER et al .: QPSK TRANSMISSION

cases. However, bandwidth restriction has two significant ef- fects upon symbol synchronization. First, the clock compo- nent obtained from a second-order nonlinearity is related in magnitude to the spectral content of the filtered signal in the near region of ItrR,/2 away from the carrier frequency. Narrow filtering thus reduces the available clock component and.in- creases the self-noise from intersymbol interference. Because of these effects of bandwidth restriction, the rms jitter for symbol synchronization shown in Fig. 7(a) and (b) is much larger for Nyquist filtering with 40-percent rolloff than for 70- percent rolloff.

For certain pulse shapes, staggered timing results in a con- centration of spectral energy at f, f R,/2 when bit transitions occur if the envelope is held constant. In fact, offset QPSK with envelope limiting will approximate three-tone digital FM, with tones at f, - R J 2 , f,, and f, + R,/2. The concentration of energy at the tones f, - R,/2 and f, + R,/2 results in a greater clock component at R,. Therefore, timing performance for offset QPSK improves slightly as the TWTA is operated closer to power saturation, which corresponds roughly to hard limiting of the envelope.

In a comparison of the phase estimation errors in Table IV, it is noted that for either QPSK format, a bit transition in only one of the quadrature bit streams results in a phase shift of n/2 rad. If the channel is tightly band-limited and has a hard- limiting nonlinearity, then this phase shift will occur at a fairly uniform rate, thus approximating a frequency shift. The fourth-order nonlinearity used to extract a carrier component at 4f, will multiply the phase and frequency shifts by a factor of 4. Therefore, the phase out of the fourth-order nonlinearity will be almost uniformly distributed over 27r rad in the region of a single bit transition under these conditions. It follows that the recovered carrier power is reduced for cases of tight filtering and operation of the TWTA near power saturation. These phenomena are evidenced in Table IV by an increase in rms carrier phase errors as the TWTA backoff is reduced.

Because of its staggered timing for the quadrature bit streams, offset QPSK has more n/2 phase transitions than QPSK. For many forms of filtering, the greater number of 7r/2 transitions results in larger synchronization errors in carrier phase, especially for operation of the TWTA near saturation. This is evidenced in Table IV for the case of Butterworth filtering. However, the carrier synchronizer per- formance for Nyquist filtering are approximately the same for offset QPSK and QPSK. The reason for this equal performance for Nyquist pulse shapes is not well understood by the authors at this time.

E. Impact of Synchronization Errors on Pb Detection results were obtained for all of the simulations to

determine the effect of synchronization errors on bit error probability Pb. Results were obtained for coherent detection of the data bits for both perfect carrier and symbol timing references (i.e., with no jitter) and for references perturbed by the experimentally observed jitter. Table V lists the proba- bility of bit error at the three values of Eb/No for all simula- tions. (These results include the effect of synchronizer jitter.)

Detection performance is perhaps best explained by com- paring the results with ideal coherent QPSK detection. In this

1311

comparison, performance loss is defined as the necessary in- crease in Eb/No for detection over a nonideal channel to yield the same probability of bit error as ideal coherent detection. Table VI lists the losses for the simulated cases evaluated at three different bit error probabilities, and

According to the table, detection with noisy references for carrier phase and symbol timing had a reasonably insignificant effect upon performance for the cases investigated. The ad- ditional detection loss caused by noisy, rather than perfect, references was at most only about 0.1 dB. Such a small degra- dation in performance caused by synchronizer jitter requires an exglanation. First, the phase synchronizer bandwidth used in the simulations was adjusted to yield an SNR in excess of 13 dB at the fourth harmonic of the carrier. This allows carrier synchronization with filtering centered at 4fc to have a long expected time between catastrophic failures such as cycle slips. Consequently, the SNR at the fundamental carrier fre- quency f, is very high (approximately 25 dB) for the carrier synchronizer. This same bandwidth was also used for the sym- bol synchronizer. Thus, both synchronizer references had very small rms errors, even at the lowest Eb/NO value used in the simulations.

Although the rms jitter was small for both carrier and sym- bol synchronization errors, it is possible for either synchroni- zer to experience significant biasing error. If such a bias exists, the synchronizers would ordinarily be adjusted to minimize the bit error rate. This manual adjustment compensates for the fact that the synchronizers do not necessarily select the reference phase or timing epoch that minimizes error rate. In the simulation results, any bias was removed by searching for the minimum error rate condition. Without such bias correc- tion, however, significant degradation in detection perform- ances might occur. For example, Fig. 8 depicts the increase in bit error probability for conventional QPSK detection for either a timing or carrier phase bias. Curves are given for all four filter configurations. The effect of a biasing error is much more pronounced at the lower bit error probabilities.

The detection results of Table VI are not intended to de- termine the best type of modulation or filtering. However, they do show detection trends for two types of modulation for different filter configurations. The detection results are re- stricted to single transmissions so that there is no adjacent channel interference. Consequently, results are uniformly bet- ter for the wider transmit filter bandwidths, Nyquist with 70- percent rolloff, and Butterworth with BT, = 1.5. With adjacent channel signals, the detection results for these wide bandwidth cases could be worse than those for the narrower bandwidths. The adjacent channel interferences would increase detection losses for all cases, but most significantly for those with wide transmit filter bandwidths.

These are some interesting trends that can be observed from the Pb results of Table VI. With Butterworth filtering, detec- tion performance was improved by operating the satellite TWTA at power saturation. Apparently, this improvement was caused by an increase in downlink power with the pulse shapes being restored to almost rectangular by the envelope limiting at saturation. However, the same phenomenon is destructive with respect to the desired Nyquist shaping and, therefore, the detection performance was worse for the Nyquist cases when

1312 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-28, NO. 8, AUGUST 1980

I ,

0 1 1 0 o m m e

I 1 d m

I 1 4 0

I 1 0 0 0 0

I 1 0 0

I 1

rl r l r l r l d r l r l 0 0

d r l 0 0

X X x x x x x x x x

r l r l

x x 0 . ID N r l o n r - m N N ID*

m a r - m

. . . . . . o m e n I D N d m 1 0 3 o m

. . . .

m n I 1

n o I 1

m e 0 0

I 1 0 0 0 0

d m I 1 I 1 I 1

r l d 0 0

d r l 0 0

d r l d d 0 0

x x x x x x x x r l d d d

x x x x I D m I D N

N d

O N m 3

m r - r - m

. . d n . . d o

n n r l m N O d m

. . . . . . d r l

. .

o n m n I 1

0 0

r - m 0 0

I 1 0 0

d d r l d d d 0 0 0 0 0 0 d d

x x d r l

x x d d

x x x x x x x x m m m I D 3 Q m N o m d 3

m m 1 1 I 1 1 1 “p d e

. . . . . . . . ~n m n d m N ” 4 3 m m

. . . . m n n n

1 1 I 1 m m I 1

d d

0 0 I 1

0 0

m r - I 1

0 0

r - r - I 1

0 3 r l d r l d r l d r l d

0 0 r l r l

0 0

x x d r l

x x x x x x x x x x r - 0 3 N N r - O d r - d m m N d Q d d m n d id r - 3 . . . . . . . . . . . .

m m I 1

m n 1 1

m m d d m r - I 1

r - P I 1

0 0 I 1

0 0 I 1

4 7 . d d 0 0 0 0 d d

0 0 r l r l

0 0 d d d r l

x x x x x x x x x x x x N O . . o r - . . ~m m m m m m o n d d m a m 3 “ m r l d r l

. . . . . . . .

o n o n m m I 1 I 1 I 1 7 7 c o r - r - P

0 0 I 1

0 0 I 1

0 0 r l r l r i d

0 0 0 0 d d

0 0 d r l d d d r l

x x x x x x x x x x x x P O 0 0 m e n o m e

N O d e m m 3 3 d “ N 3

m a . . . . . . . . . . . .

m n n o m m I 1

d d I 1 1 1

r - r - I 1

0 0 I 1

0 0 I 1

r l r l id 0 0 0 0 r l r l r l r l

0 0 d r l

0 0 r l 3

x x x x x x x x x x x x

“ N 3 N N d a m N I D

~n m d *ID N 3 N I D d - i

m m

. . N d . . . . . . * . . .

m m n o l n m d d m m I 1 I 1 I 1 I 1 0 0

I 1 0 0 0 0

r l r l d r l 3 4 4 3 0 0 0 0 0 0

d r l r l d

x x x x x x x x x x x x

r - r - I 1

a n -co m m m o d d

N O l n d m m N ” d m I D D . . Q d . . . . . . . . . .

PALMER e t al. : QPSK TRANSMISSION 1313

lodulation

QPSK

OQPSK

TABLE VI SUMMARY OF DETECTION LOSSES (dB)

Filter Type

NY-40

NY-40

NY-70

NY-70

BU-1.25

BU-1.25

BU-1.50

BU-1 .50

NY-40

NY-40

NY-70

NY-70

BU-1.25

BU-1 .25

BU-1.50

BU-1.50

lackof! Input

~

l o 0

10

0

10

0

10

0

10

0

10

0

I. 0

0

10

0 -

s t a t i c Loss (dB)

- 10-3 - 0.24

0.43

0.12

0.20

0.98

0.63

0.85

0.35

0.42

1.14

0.03

0.20

1.0

0.60

0.70

0.35 ~

at Ph - 10-4 - 0.25

0.50

0.14

0.27

1.05

0.68

0.85

0.57

0.52

1.12

0.04

0.25

1.05

0.70

0.70

0.42 -

__ 10-5 - 0.33

0.57

0.22

0.32

1.16

0.77

0.95

0.66

0.62

1.20

0.05

0.30

1.15

0 . 8 0

0.76

0.58 -

r; ittered Loss ( d B )

- 10-3 __ 0.26

0.53

0.16

0.24

1.0

0.65

0.90

0.60

0.45

1.2

0.05

0.25

1.10

0.75

0.72

0.41 -

- t pb

- 10-4

0.27

0.60

0.18

0.31

1.03

0.70

0.88

0.60

0.55

1.2

0.06

0.20

1.15

0 . 8 0

0.72

0.45 -

1.36

3.67

3.26

3.36

1.2

0.80

1.0

0.68

0.65

1.25

0.07

0.35

1.22

0.93

0.78

0.60 -

1

the TWTA was operated at power saturation. For most cases, there were no significant differences in detection performances for conventional QPSK and offset QPSK. However, the wider bandwidth requirement of offset QPSK under nonlinear con- ditions was evident from the greater detection losses for off- set QPSK with Nyquist filtering that has only a 40-percent excess bandwidth. This increased loss of offset QPSK was most pronounced when the TWTA was operated at saturation.

V. CONCLUSIONS

This paper has presented computer simulation results of the synchronization errors experienced by QPSK and OQPSK signals transmitted over the nonlinear communications circuit. One set of synchronizer implementations was modeled along with several modem filtering configurations, and synchroniza- tion results were obtained for various backoff levels of the nonlinearity and Eb/No values.

In -addition to obtaining the statistics of synchronization errors, the simulation process has been extended further to determine the impact of synchronization errors on overall link performance as measured by bit error probability. This last step was the original motivation for this work; it was necessary to include synchronization errors with the other sources of overall communications link degradation to obtain good agree- ment between software and laboratory (hardware) simulations. Although this additional source of degradation can be large in certain situations, for the cases examined in this paper, the ad- ditional losses in detection performance are quite small.

A more sensitive indicator of synchronizer performances is provided by the rms errors in the carrier phase and symbol timing estimates. It was postulated that these errors would consist of two parts, one component due t o thermal noise, and a second due to modulation-pattern-induced waveform distortion. The latter would be expected to be dependent upon modulation type, modem filtering, the operating point (backoff) of the channel nonlinearity, and the exact syn-

:016128 4 ’0 ’612 24 18 30

48

PHASE BIAS TIME BIAS B T - 1 2 5

(DEG) BUTTERWORTH( B T s = 1.m PEG) ..?..,, ........

l - 2 - - - - -

Fig. 8. Sensitivity of QPSK detection to synchronization biasing errors when TWTA is operated at saturation (parametric in three values of EbfNo).

chronizer implementation. Some of these dependencies were observed in the simulation results, but in some cases, a relative insensitivity was noted.

REFERENCES [ I ] S . A . Rhodes, “Effects of hardlimiting on bandlimited trans-

missions with conventional and offset QPSK modulation,” in 1972 Nat. Telecommun. Conf. Rec. , Houston, TX, Dec. 4-6, 1972, pp. 20F- I -?OF-7.

[2 ] R . W. Lucky, J . Salz. and E. J . Weldon, J r . , Principles of Data Communications. New York: McGraw-Hill, 1968, pp. 45-54.

[3] J . J . Stiffler. Theory of Synchronous Communications. Englewood Cliffs, NJ: Prentice-Hall. 1971, pp. 243-247.

141 F. M. Gardner, “Self-noise in synchronizers,” this issue, pp. 1159-1163..

[SI --, “Carrier and clock synchronization for TDMA digital

TM-169 (ESTEC), pp. 232-233, Dec. 1976. communications.” European Space Agency Tech. Memo. ESA

[61 L. E. Franks, “Carrier and bit synchronization in data com- munications-A tutorial review,” this issue, pp. 1107-1 121.

1314 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-28, NO. 8, AUGUST 1980

[7] L. E. Franks and J . P. Bubrouski, “Statistical properties of timing jitter in a PAM timing recovery schemc,” IEEE Trans. Commun.,

[ 8 ] F. M. Gardner. “Clock and carrier synchronization: Prefilters and anti-hangup investigations,” European Space Agency Tech. Memo.

[9] W. L. Cook, “Interactive computer simulation of satellite transmission systems,” in Proc. 5th Annu. Pittsburgh Conf. Modeling and Simulation, Apr. 1974.

[ I O ] L. C. Palmer and S. Lebowitz, “Including synchronization in time domain simulations.” COMSAT Tech. Rev., vol. 7 , pp. 475-526, Fall 1977.

VOI. COM-22, pp. 913-920. July 1974.

ESA CR-984, p. 5.42, NOV. 1977.

* Larry C. Palmer (“55) was born in Washington. DC. on December 2 5 . 1932. He received the B.S. degree from Washington and Lee University, Lexington, VA, the B.E.E. degree from Rensselaer Polytechnic Institute, Troy, NY, both in 1955. and the M.S. and Ph.D. degrees in electrical engineering from the Uni- versity of Maryland, College Park, in 1963 and 1970, respectively.

From 1955 to 1957 he served in the U.S. Army, Signal Corps, as an Electronics Officer.

Following this. he spent six years with the Radcom-Emertron Division of Litton Systems, Silver Spring, MD, where he was involved in the design and development of airborne electronic systems including radar altimeters and ECM equipment. I n 1963 he joined ITT INTELCOM, Falls Church, VA. and worked on the military communications satellite system. After the acquisition of INTELCOM by the Computer Sciences Corporation in 1965. he was involved in several projects concerning systems analysis, modeling. and simulation related to communications and navigation

COMSAT Laboratories, Clarksburg, MD. in 1974 as a Senior Staff satellite systems, and digital signal processing systems. He joined

Scientist in the Transmission System Laboratory. He is responsible for systems analysis related to future communications satellite systems with emphasis on modulation/multiple access techniques and computer simulation modeling to predict transmission impairments. Since 1979 he has served as Program Manager for COMSAT’s participation i n the ARPA Atlantic Packet Satellite Network (SATNET).

Dr. Palmer is a member of Tau Beta Pi and Eta Kappa N u . He is Past Chairman of the IEEE Information Theory Group in the Washington. DC area.

Smith A. Rhodes (”65) was born in Richmond, VA, on October 2, 1929. He received the B.S. degree in electrical engineering from Virginia Polytechnic State University, Blacksburg. in 1959 and the M.S. degree in electrical engi- neering from North Carolina State University, Raleigh, in 1962.

From 1959 to 1965 he was engaged in various communications system studies at Bell Labor- atories. During 1965 to 1968 at Page Com- munications his primary task was the analysis of

digital communications. From 1968 to 1974 he was employed by the Computer Science Corporation. Falls Church, VA. where he performed analyses and studies of satellite communications. Since 1974 he has been engaged in R&D studies of digital satellite communications at COMSAT Laboratories, Clarksburg, MD. His major interests lie in the application of statistical communications theory to the performance evaluations of digital communications systems by either an analytical approach or by computer simulation. His topics of special interest include modulation, synchronization. detection, error-correction coding. and modeling of communications for computer simulations.

Mr. Rhodes is a member of Tau Beta Pi, Eta Kappa Nu, and Phi Kappa Phi. He is also a member of the IEEE Communications Society and the IEEE Information Theory Group.

* Sheldon H. Lebowitz received the B.E.E. degree from the City College of New York. New York. NY. in 1964.

He has worked at the U.S. Navy Marine Engineering Laboratory, the Electromagnetic Compatibility Analysis Center, Norden Division of United Aircraft, and Computer Sciences Corporation. I n 1974 he joined COMSAT Laboratories, Clarksburg, MD. where he is now a Staff Scientist under the Director of the Transmission Systems Laboratory. He is re-

sponsible for the design, development, and implementation of various digital computer simulation programs. He has been responsible for the development and upgrading of COMSAT’s Channel Modeling Program (CHAMP 2) . He is also responsible for the design and development of the FEC Coding Simulation Program (CODSIM). which simulates the performance of Viterbi and threshold decoders, and the Detailed Receiver Simulation Program (RECEIVER), which simulates the carrier and symbol timing recovery circuit acquisition performance in both the steady- state and burst modes and also evaluates the bit-error rate of the modem.