Carrier-phase synchronisation in the demodulation of UQPSK

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Carrier-phase synchronisation in the demodulation of UQPSK signals

Prof. A.P. Clark, MA, PhD A. Aftelak, BSc

Indexing terms: Signal processing, Demodulation

Abstract: The paper describes a novel carrier- phase synchronisation process that has been developed for use in a digital communication system operating over a satellite link and employing unbalanced quaternary phase shift keyed (UQPSK) signals. The satellite is not geostationary and may travel close to the surface of the earth over at least a part of its orbit. Correct carrier phase synchronisation needs to be achieved here as soon as the satellite rises above the horizon, during the time it is transmitting data signals, and therefore without the use of a known training signal. Under these conditions the digital signal is received at a low signal to noise ratio and in the presence of a severe Doppler shift. Correct carrier phase synchronisation must then be maintained as the satellite passes overhead and until it approaches the horizon again. Thus the synchronisation circuits must tolerate also a high rate of change of Doppler shift, albeit at a higher signal to noise ratio.

The synchronisation algorithm is in two separate parts that operate sequentially and are both non-data aided. The first part removes the Doppler shift from the received signal and the second part then achieves carrier phase synchronisation, which is maintained over the rest of the transmission.

The paper describes the new algorithms and presents the results of a series of computer simulation tests to assess their performances.

1 Introduction

In digital communication systems operating over satellite links, increasing use is now being made of unbalanced quaternary phase shift keyed (UQPSK) signals [l-71. A UQPSK signal normally has a constant or near constant envelope, and is formed by the sum of two constant or near constant envelope binary phase shift keyed (BPSK) signals, with the same carrier frequency and in phase quadrature (at 90" or n/2 radians). The two BPSK signals carry independent messages, originating from different sources. They may have similar or widely differing bit

Paper 68791 (E8), first received 14th October 1988 and in revised form 13th July 1989

The authors are with the Department of Electronic and Electrical Engineering, Longborough University of Technology, Longborough, Leicestershire LE1 1 3TU, United Kingdom

I E E PROCEEDINGS, Vol. 136, Pt. I , No. 5 , O C T O B E R 1989

(element) rates, which are not in any way related, and their relative amplitudes are typically in the range 1 : 1 to 1 : 3. The bit rate in either channel may lie in the range 40 kbit/s to 4 Mbit/s. Finally, the receiver must be flex- ible and capable of handling a wide range of different UQPSK signals.

Any movement of the satellite relative to the earth sta- tions introduces a Doppler shift into the received signal.

This is a time modulation of the received signal, which introduces a frequency offset (change in frequency) into the signal carrier and also slightly changes the signal element rate

[SI.

Whereas a geostationary satellite does not usually introduce serious Doppler shifts, satellites that travel close to the surface of the earth over at least a part of their orbit, can introduce very considerable Doppler shifts. The frequency shift introduced into the received signal can now exceed 800 kHz, with a maximum rate of change around 11 kHz per second.

Because the change in signal element rate, normally introduced by a Doppler shift, is small enough not to significantly influence the systems studied here, it can be ignored. The term 'Doppler shift' is therefore used here to refer exclusively to the frequency offset that results from the time modulation.

In order to achieve the best available tolerance to noise, coherent demodulation is used at the receiver [8].

The coherent demodulator can now itself be used both to remove the Doppler shift and to correct the residual phase error in the complex values demodulated baseband signal. Once correct carrier-phase synchronisation has been obtained, it is relatively straightforward to achieve element time-synchronisation and hence the detection of the received data symbols.

The initial synchronisation of the receiver on to the received signal must be achieved as the satellite rises above the horizon and is transmitting data. Under these conditions there is a high (but effectively constant) Doppler shift and a low signal to noise ratio. Further- more, the receiver has no prior knowledge of the received data symbols.

The paper is an initial feasibility study of some novel techniques of Doppler shift and carrier phase correction, that are non-data aided and have been developed for use under the conditions just described. The techniques differ from those conventionally used and are potentially capable of achieving a much better performance with UQPSK signals. In order to restrict the investigation to within reasonable limits, various idealised assumptions have been made and only certain particular signals have been tested. The latter have, however, been selected to be as representative as possible of the more typical signals likely to be received.

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2 Modulator, satellite link and demodulator A UQPSK signal can, in principle, be generated by the linear modulator shown in Fig. 1. The data symbols so,

I lowpass

f i l t e r A

bandpass f i I t e r

f i c o s Znf, t

lowpass f i l t e r B

=--P

^-fisin ^Zfif,t

Fig. 1

Lowpass filter A ^~ 1 so. Mt - hT,) Lowpass filter B -

1

s , , , & t - kT,)

Linear modulator for U Q P S K signal

h

t

c. UQPSK s i g n a l

and sl, k have the possible values f 1, and E h so, h d ( t - hTo) and c k sl, k d ( t ^-kTl) are streams of binary polar signal elements, regularly spaced at time intervals of To and T, seconds, respectively. To and Tl may have any two values in the range typically from 0.25 to 25 pms. d(t) is a unit impulse at time t = 0, and

f,

Hz is the signal carrier frequency. The lowpass filter A has an impulse response

do O < t < T o

and the lowpass filter B has an impulse response d , O < t < T l

The in-phase and quadrature baseband signals in Fig. 1 are

h g O ( t ^-hTO) and sl, k g l ( t - kTl)

respectively, and could be typically as in Fig. 2.

The sum of the two BPSK signals in Fig. 1 forms a UQPSK signal with a constant envelope, and this signal is fed through a bandpass filter, which does not distort, attenuate or delay the signal. The filter is needed in practice to remove spurious frequency components generated in the modulator. It is assumed throughout thatf, 9 _l/To andf, 5> l/Tl. The modulator does not change the signal energy.

The output UQPSK signal from the modulator is fed to an up-converter (frequency translation equipment) needed to change the carrier frequency to that required

for transmission to the satellite. With the constant envelope signal used here, the satellite transponder does not distort the signal [SI. The satellite retransmits the signal to the receiving earth station, where it is down- converted to a carrier frequency that is

f,

Hz, in the absence of any Doppler shift. An approximate prior knowledge of the Doppler shift is employed in the down- converter to correct the majority of the Doppler shift, leaving a residual frequency offset of up to f 5 kHz in the UQPSK signal at the input to the demodulator in Fig. 3.

r . _ f L \

signal

I

Fig. 3 Demodulator and sampler

t :IT

It is assumed that, in transmission over the satellite link, the signal carrier phase is advanced by p ( t ) radians, and the signal is multiplied by a positive real-valued constant c. The latter represents the attenuation in transmission which is effectively held constant by the automatic gain control circuits that precede the demodulator. The Doppler shift in the received modulated-carrier signal is given by the rate of change with time of p(t). The wide- band noise that is in practice added to the received signal [9] is taken to be stationary white Gaussian noise with zero mean and a two-sided power spectral density of + N o . It is assumed here that there is no adjacent channel or cochannel interference in the received signal [9].

Associated with the coherent demodulator in Fig. 3 is a voltage controlled oscillator (VCO) and phase shifter, whose output signals are the two reference carriers 4 2 cos [27tf, t

+

~ ( t ) ] and - J2 sin [27& t

+

~ ( t ) ] in Fig.

3. The residual Doppler shift (frequency offset) at the output of the demodulator is the rate of change with time of p ( t ) - q ( t ) . To reduce the latter to zero the receiver must feed a control signal to the VCO, such that

At)

- V ( t ) = 9 ( 3 )

where 9 is a constant phase angle. When there is no Doppler shift in the received UQPSK signal, p(t) is constant, and when a zero control voltage is fed to the VCO,

~ ( t ) is constant, so that there is now no residual Doppler shift in the demodulated signal.

The bandpass filter in Fig. 3 does not distort, attenu- ate or delay the received UQPSK signal, but is needed in practice to prevent (or at least reduce) overloading by noise from the following multipliers. The two lowpass filters here each have a transfer function

1 -1/2T < f < 1/2T

J(f) = {O elsewhere ⁽⁴⁾

where T seconds is the sampling interval of the following samplers. The filters are not physically realisable, but can be made so for practical purposes, if they introduce a sufficient delay. The cut-off frequency, 1/2T Hz, of each lowpass filter is greater than the highest frequency component in the in-phase and quadrature demodulated baseband data signal waveforms (the wanted signals) at the inputs of the two lowpass filters, such that the latter

IEE PROCEEDINGS, Vol. 136, Pt. I , No. 5 , O C T O B E R 1989

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Table 1 : Parameter values of signals A and B

Signal a b To T, Bit rate Bit rate E, (/I

of in-phase of quadrature (dB) channel channel

(kbit/s) (kbit/s)

A 2 1 51 T 71

r

^313.7 ^225.4 ⁷¹T 10 log,,(35.5/u2) B 1 1 51 T 71 T 313.7 225.4 51 T 10 log,,(25.5/u2)

two waveforms are passed through the respective lowpass filters with no distortion, attenuation or delay.

The in-phase and quadrature demodulated waveforms ro(t) and r , ( t ) are sampled simultaneously (in synchrony) at the time instants {iT), where T = 0.0625 x giving 16 x 10" samples per second.

This is the Nyquist rate for each waveform. The resulting samples {rei} and { r l , i ) , where = ro(iT) and r l , i =

r,(iT), contain all the useful information in the received bandpass signal [lo, 111. The samples are fed to the appropriate matched filters and detectors (not shown in Fig. 3). Since we are not here concerned with the detec- tion process, but rather with the use of the { r o , i } and

{ I , , i } to achieve correct carrier-phase synchronisation of the receiver, we shall not consider further the matched filters and detectors.

Strictly speaking, the lowpass filters in the demodulator produce a rounding of the rectangular baseband signals (Fig. 2). However, for the values of To, T, and T considered here, where T To and T

<

T,, the rounding is not significant and can be ignored.

Let

ri = ro,

+

j r l . ( 5 )

where j =

J

^-1, then it may be shown [7, 121 that, for practical purposes,

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ri = (aso, h

+

j b s , , ^{k )}exp ( j 4 i )

+

^wi

where a = cd,, b = cd,,

4i

= p(iT) - q(iT), and h and k are appropriate integers. Eqn. 6 holds quite accurately, provided that the Doppler shift in the received signal (Fig. 3 ) is much smaller than the bandwidth of the data signal. The quantity w i is a complex valued Gaussian noise component originating from the additive white Gaussian noise. The real and imaginary parts of the { w i } are statistically independent Gaussian random variables with zero mean and variance, which can be shown [8,

1&12] to be

c2 = No/2T (7)

With ideal carrier-phase synchronisation, q(iT) = p(iT), so that exp ( j 4 J = 1, and now, in the absence of noise,

(8) ri = + a f j b

In the presence of a constant residual Doppler shift (frequency offset) in the demodulated waveform r(t),

4i

increases or decreases at a constant rate with i (when not expressed modulo-2.n), the rate being proportional to the Doppler shift. In the absence of noise and Doppler shifts over the duration of successive signal elements with the same values of so, h and the same values of sl, k , the corre- sponding { r i } are all the same. For the satisfactory oper- ation of the receiver, it is sufficient that

4;

is held constant at one of the four values 0, & n/2 or .n.

The signal to noise ratio in the received signal is taken to be I) dB, where

*

⁼10 log10(EdNo) (9)

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and E , is the average received signal energy per bit, for the one of the two BPSK signals for which this is the smallest. From eqn. 7, No = 2a2T.

Two particular UQPSK signals are studied in the computer simulation tests and these are known here as signals A and B. The important properties of the two signals are given in Table 1. For signal A , the quadrature BPSK signal has the smaller value of E , , which is b2Tl.

For signal B, the in-phase BPSK signal has the smaller value of E , , which is a2To. The system is required to operate with values of I) down to -1.5, which corre- sponds to values of a2 up to 50.15 for signal A and up to 36.02 for signal B.

3 Measurement of Doppler shift

When operating at low signal to noise ratios, to achieve either Doppler-shift correction or carrier-phase synchronisation, as much linear filtering as possible must be carried out before any nonlinear processing, in order to maximise the signal to noise ratio at the input to the nonlinear processor. Thus, rather than operate on the individual signals { r o , i } and { r l , i } in Fig. 3, these are combined or averaged in groups of n samples, where n is an integral power of 2, with a value in the range 1 to 2048. The resulting signals are

1 " - 1

1 -

f O . i = i x r O . n i - h h = O

and

1 n - 1

fl,

ⁱ⁼ I l , n i - h (1 1)

h = O

In the absence of noise or of any significant variation of

f$,,-h with h (see eqn. 6), and if no transitions in the signal waveforms of Fig. 2 occur during the determination (generation) off,,; andf,. i , the n signals ( r o , n i - h } in eqn. 10 are all the same, and the n signals { T , , ~ ~ - , , ) in eqn. 11 are all the same. The complex-valued signal

now takes on one of the four possible values of rni - h , as given by eqn. 6 with i replaced by ni - h. Clearly, one restriction on the maximum value of n is that 4 , , - h

should not vary significantly as h varies over the range 0 to n - 1, which, for a given n, places an upper limit on the initial Doppler shift that can be corrected in the demodulated signal. Another most important restriction on n is that nT, the effective period involved in the deter- mination off,, should be less than both To and TI, so that there cannot be more than one transition of either of the two modulating waveforms (Fig. 2) during the determination of f, . This limits the average error in

fi

caused by transitions.

By operating on the {f,} in place of the {ri}, the effective sampling rate of the demodulated signal waveform has been reduced by n, which has the advantage not only of increasing the signal to noise ratio, but also of

353

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reducing the number of operations required in the sub- sequent signal processing.

Before using any

f,

in a synchronisation process,

f,

is checked to ensure that

If,l

⁽¹³⁾

where K is an appropriate positive real valued constant. If eqn. 13 is not satisfied, the correspondingf;, is rejected as unreliable, since there may be a serious error in its phase angle.

Assume first that the signalf, is accepted as valid for further processing. The basic technique adopted here is to map the signalf, into an appropriate segment of the line containing all real numbers, in such a way that the result- ant real valued signal does not vary too much when I , , ~ - ~

moves from one of its four possible values (determined from eqn. 6) to any other possible value. The signalf;, is first expressed in polar form as

(14) where

1 f, I

is the absolute value (modulus) off,, and yi is the phase angle off, . Assume the absence of noise and no transitions of the modulating waveforms during the determination off,. The four possible values off, are then the same as the four possible values of I , , - , , , for the given phase rotation

4,,i

^-^h, provided only that

4ni ,,

^{does not}

vary significantly as h varies over the range 0 to n - 1.

The phase angle y i , measured in radians, is now used in place off, and has a value in the range 0 to 27c. It is then expressed modulo-p, where p = 7c or 4 2 . Let

(15) f, =

I

f;,

I

~ X P (jyi)

Ai = yi modulo - p

Ai = yi - k p (16)

O < A i < p (17)

so that

where k is the appropriate integer such that

The value of p in eqns. 15-17 is determined by aJb, as follows, where a and b are the corresponding constant parameters in eqns. 6 and 8, and a 2 b.

When 1.3 2 aJb 2 1.0, p = 4 2

Otherwise p = 7c.

In the case of a QPSK signal, p = 4 2 , and when aJ b > 1.3, the signal is treated as a BPSK signal. Since the receiver has prior knowledge of a and b, p can be chosen before the start of operation, and the in-phase BPSK signal is taken as that with the larger magnitude, thus ensuring that a 2 b. At high signal to noise ratios, a greatly improved performance can be achieved by setting p to nJ4, when 3.0 2 aJb > 2.0, and to nJ3, when 2.0 2 aJ b > 1.3. However, at the low signal to noise ratios studied here, serious noise enhancement is introduced by reducing p to nJ4 or nJ3, so that these arrangements are not considered.

In the absence of noise, Ai is not seriously affected by any one of the four possible values taken on byf, , so that in the presence or absence of noise, Ai can be used as a measure of the phase rotation $ , , - h (eqn. 6), without requiring any knowledge of the data symbols { s , , ~ } and {sl, k } . Thus non-data-aided operation can be achieved by operating on Ai. Now,

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&. =

A.

-

A.

I I I - m

where ti is a measure of the rate of change of phase

354

(frequency offset) and therefore of the Doppler shift in the demodulated signal. m is a suitable positive integer that gives a convenient phase change over the interval of mnT seconds, and is measured in arbitrary units of frequency (not in Hz), which, for convenience, are taken to be radians per mnT seconds. It is, of course, assumed here that the signalf,-, was accepted as valid.

4 Doppler shift correction

The Doppler shift in the received modulated carrier signal is corrected by appropriately adjusting the instan- taneous frequency of the VCO that generates the reference carriers

J2 cos C2.L t

+ v(OI

- J2 sin [27$ t

+

~ ( t ) ] and

in the coherent demodulator of Fig. 3, such that eqn. 3 is satisfied. For convenience, the sensitivity of the VCO to its control signal is assumed to be such that for a frequency change in its output signal of ui, i - l units of frequency, a control signal of ui, i p must be fed to the VCO.

Suppose now that the Doppler shift in the received modulated-carrier signal is yi units of frequency, where yi is the rate of change with time of p(t) (see Section 2) in radians per mnT seconds. Then the residual Doppler shift in the demodulated baseband signal is

zi = yi - vi, i - 1

radians per mnT seconds. With perfect tracking of the Doppler shift, vi, ^- = yi , so that zi = 0. Clearly zi is the error in ui, i - 1, and ci in eqn. 18 is a noisy measurement of zi . Under ideal conditions, ei = zi .

The quantity u i , i - l is the one-step prediction of y i , determined after the receipt off,- by means of a degree- 0 polynomial filter. The degree-0 least-squares fading memory prediction of y i + 1, determined by the polynomial filter after the evaluation off,, is now given by

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U i + l , i = U i . j - 1

+

⁽¹^-^U)Ei ⁽²⁰⁾

where ^Uis a positive real-valued constant in the range 0 to 1, normally close to 1. At the start of operation, ui, i -

is set to zero.

The arrangement so far described for determining ci in eqn. 18 has a serious weakness, which has, for convenience, been temporarily ignored. Because Ai is determined modulo-p (eqns. 15-17), it is quite possible that different values of the integer k (eqn. 16) are used for Ai and Ai-,,, , thus introducing the corresponding error into ei (eqn. 18). Provided that the Doppler shift is not too large for the given time interval mnT between Ai and Ai ^-,,,, the measured Doppler shift can be taken as

ci = Ai ^-Ai-,,, or Ai ^-Ai-,,, ^-^por Ai - Ai-,,,

+

^p(21) and then selecting the one of the three possible values that has the smallest magnitude. At high signal to noise ratios, this technique operates well. However, at the low signal to noise ratios of interest here, it has been found that the best performance is achieved by employing eqn.

18 and then limiting the magnitude of E ~ , as follows.

When ci >

5,

ei is set to 5, and when .si < - 5 , .si is set to - 5 , where

5

is a constant in the range 0 to pJ2. The limiting of .si, just described, is employed here in all computer simulation tests.

IEE PROCEEDINGS, Vol. 136, Pt. I , N o . 5, O C T O B E R I989

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When k successive { & i + h } , for h = 1, 2 , ..., k, cannot be evaluated because of the rejection of some of the corresponding { f ; + h } and ( f i - , , , + h } , eqn. 18 is not employed during this period but is replaced by

z

L

⁰^15-

* i + h , i + h - 1 = 'i, i- 1 ( 2 2 )

for h = 1, 2, . . . , k. Then, on the receipt of A i + k + l from which E ~ + ~ + is evaluated, the algorithm given by eqn. 18 is resumed, as before.

It is evident that when ci is determined according to eqn. 18, and the magnitude of ci is such that > p / 2 when measured in the absence of noise or errors caused by the modulo-p boundaries (different values of k in eqn.

16 for Ai and Ai_,,,), then a false lock is likely to be obtained in the Doppler shift correction algorithm just described. Since

I

ci

I

increases both with the magnitude of the Doppler shift and with the value of m, the greater the initial value of the Doppler shift to be corrected, the smaller the value of m that can be used. Unfortunately, the smaller the value of m the smaller also is the accuracy to which the Doppler shift is corrected. Thus, as soon as the algorithm has reduced the residual Doppler shift to a sufficiently small fraction of its original value, the integer m is increased (typically quadrupled or doubled) and the other adjustable parameters of the algorithm are adjusted where necessary. The process just described is then repeated several times to give the corresponding sequential synchronisation processes, each with a different set of parameter values, at the end of which the Doppler shift has effectively been reduced to zero. The number of sequential processes required is dependent on the UQPSK signal and the signal to noise ratio. In practice, of course, the receiver does not know the value of the residual Doppler shift, so that each of the separate stages of the process has a given duration that is selected appropriately.

To assist in the design of the multi-stage synchro- nisation process just described, Table 2 shows the relationship between m and the maximum Doppler shift that can be tolerated without false locking of the algorithm. The Doppler shifts are here measured in Hz, and the signals A and B are as given in Table 1.

Many different Doppler shift correction algorithms have been tested, in addition to that just described, several of these being phase-locked loops. The latter are generally superior to the technique described, once they

Table 2: Relationship between m and the maximum Doppler shift tolerated by t h e system, for signals A and B, w i t h n = 32

m Signal A with p = n Signal B with p = 4 2

4 8 16 32 64 128 256 51 2 1024 2048

Maximum Doppler shift

(Hz) 31,250 15,625 781 2 3906 1953 976 488 244 122 61

Maximum Doppler shift with safety margin (Hz) 20,000 10,000 5000 2500 1200 600 300 150 75 40

Maximum Doppler shift

(Hz) 15,625 781 2.5 3906.25 1953 976 488 244 122 61 30

Maximum Doppler shift with safety margin (Hz) 10,000 5000 2500 1200 600 300 150 75 40 20

~~ ~

IEE PROCEEDINGS, Vol. 136, Pt. I , No. 5 , O C T O B E R 1989

have achieved correct phase lock. The disadvantage of the systems is that they require an excessive time to acquire phase lock, for which reason they have been rejected.

5

The nonlinearities involved in generating Ai (eqn. 15) and ci (eqn. 18) enhance the noise level in

fi

(eqn. 14), and some idea of the magnitudes of the noise enhancement is given by Figs. 4-6, which show the probability density functions of Ai and c i . Figs. 4 and 5 show the probability

Probability density functions of I , and ^E,

I L

^,v=ll ⁵²

I 1

,5 50

0

0 0 5 1 0 1 5 2 0

A , ^Irad

Fig. 4 Probability density function of Ai when expressed modulo-n/2

0 30- al U

2 5 0 25-

U U

0

,$ 0 20-

U

-

0

s!

0 15-

0 IO-

0 05-

o & / , 1

0 0 5 1 0 1 5 2 0 2 5 3 0 3 5 A , , r a d

Fig. 5 Probability density function of Ai when expressed modulo-n 355

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density functions for

Ai,

with p = n/2 and n, respectively, and with signal A, no transitions, no Doppler shift and at various signal to noise ratios. Clearly, no data are transmitted here. In each case the correct phase angle is p/2. It is evident from Fig. 4 that, when the signal to noise ratio (1,9 dB) drops to -6.5 dB, all information on the phase angle has effectively been lost, so that the correct phase angle cannot now be determined. However, when p is doubled to n, as in Fig. 5, adequate information is available to determine the required phase angle, at least approximately, at the given signal to noise ratio.

Figure 6 shows the probability density functions of ci (eqn. 18), for various signal/noise ratios with signal A,

0 18-

0 I€-

0 1L-

g

⁰^12-

;

⁰^10-

E!

0

z

aJ ⁰^08-

-

E!

0 06-

0 04-

0 02-

0.

- 1 5 -1 0 -0.5

'\

0 5 1 0 1 5 E , , rad

Fig. 6 Doppler sh$t

Probability density function of ci, with /A = x and a 5 k H z

when there is a Doppler shift of 5 kHz, p = n, m = 16, n = 16, ci is limited with

5

= 1.5, and there are no tran- sitions (no data are transmitted). The correct value of ci is 0.5 radians. As the signal to noise ratio reduces, so the probability density function becomes flattened and its peak moves towards zero. At the lowest signal to noise ratio, much of the required information has been lost although there is still enough for correct operation to be achieved.

6 Carrier phase correction

Once the Doppler shift has been effectively eliminated from the demodulated signal, by appropriately adjusting the frequency of the VCO, the Doppler-shift correction algorithm is replaced by a phase-locked loop. In principle, the latter could operate on the {Ai> (eqn. 15). Now, with a BPSK signal, p = n and the average value of

Ai

is set to n/2, whereas, with a QPSK signal, p = 742 and the average value of

Ai

is set to x/4. Such systems can be made to operate well [13-181. As before, when 1.3 2 a/

b

>

1.0, the UQPSK signal is best treated as a QPSK signal, and when a/b > 1.3, the UQPSK signal is best treated as a BPSK signal. Unfortunately, when the data symbols ( S 0 . h ) and { S i & } in a UQPSK signal are not 356

well scrambled but appear as repetitive sequences, a significant jitter may now be introduced into the

{Ai},

even in the absence of noise, leading to an appreciable degra- dation in tolerance to any additive noise that may be present. Thus it is highly desirable that the operation of the carrier phase correction algorithm be made independent of the {s0,h} and ( s l , k } . The method of measuring the phase error in a received signal f,, that has been adopted for the carrier phase correction algorithm, achieves this result.

The signalf, is now mapped into the upper half of the complex-number plane, to give the signalf: , and it is also mapped into the right half of the complex number plane, to give the signal

f y

. The mapping off, is carried out as follows. When the imaginary part off, is positive,f! =f,, and when it is negative,f; = -f,. When the real part off, is positive,fy

=fi,

and when it is negative,f; =

-A.

The receiver then forms the signal

mi = b Re (f:) - a Im (f:) (23)

and uses m i in place of Ai as a measure of the phase angle o f f ; . In eqn. 23, Re ( f : ) is the real part o f f and Im (f;) is the imaginary part of f;, where a and b are the constant parameters in eqns. 6 and 8, as before. The phase angle of f ; lies in the range 0 to ^71,whereas the phase angle o f f lies in the range - n/2 to 4 2 . When the phase angle of f : lies in the range 0 to 4 2 , f:

=fy,

otherwise

f:

=

-f;.

It can be seen that, in the absence of noise, as the phase angle o f f : increases from 0 to n/2, so mi decreases from b(a2

+

b2)'12 to -a(a2

+

b2)'12, and as the phase angle off: increases from 4 2 to n, so mi decreases from a(a2

+

b2)"2 to -b(a2

+

b2)'l2. There is a discon- tinuity in wi whenever the phase angle yi off, has one of the values 0, 4 2 , n or 3x12. Whenf, has any of the four possible values of ri in eqn. 8, that is, f a f jb, then m i = 0, and whenf;, is close to any of these points, there is only a small change in mi asf, moves from one of its four possible values to any other. The absence of both noise and transitions in the modulating waveforms is assumed here, and the four possible values off, correspond to the four possible combinations of values of the associated data symbols ^{s 0 , h}and sl,k (eqn. 6 with

bi

⁼0 and wi =

0).

Suppose now that

f, = (as,, h

+

jbs,, k ) ~ X P (jbi)

where there is no noise and

bi

is the phase error in the demodulated baseband signal (as in eqn. 6, but with the subscript of

bi

here related to that off, instead of ri). If so, = s ' , ~ = 1 and

bi

⁼0, then f , = a

+

jb, wi = 0, and both b Re (f:) and a Im (f;) are positive quantities. An increase (positive movement) in

bi

rotates bothf: and

f;

anticlockwise. This reduces the magnitude of b Re ( f : ) and increases the magnitude of a Im

(fy),

in eqn. 23, which causes wi to become negative. Similarly, a decrease (negative movement) in

bi

rotates bothf: andfy clock- wise. This increases the magnitude of b Re (f:) and reduces the magnitude of a !m

(fy),

which causes m i to become positive. It can readily be shown that a similar behaviour is obtained for each of the other three possible values of as,, h

+

^jbs,,^{k .}

Thus mi can be used as a control signal to hold

bi

at its desired value of 0 or n. With a random sequence of equally likely data symbol values, only the two given values of

bi

are stable points, except for the particular case where a = b, when there are four stable points such that

bi

⁼0, f n / 2 or n. Where the data symbol values are

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not equally likely, it is possible for a temporary lock to occur with the wrong phase angle, but this is unlikely to persist for long.

The preferred arrangement of carrier-phase correction uses mi and operates with the following algorithm.

(24) (25)

(26)

(27) where u i + ^1,is the control signal fed to the VCO, and

fl

is a real valued constant in the range 0 to 1, usually close to 1. The parameters U : + and U;+ 1, are measures of the rate of change and acceleration, respectively, of the phase error that is being controlled by ui+ i . Further details of these are given elsewhere [19]. The particular virtue of this algorithm is that, contrary to the case with the previous phase measuring algorithm (eqn. 15), it introduces no phase jitter, regardless of the value of a/b, and the noise enhancement due to the nonlinearities in the algorithm is no greater than that in the previous algorithm with p = 71/2. When a/b > 3, the UQPSK signal is best treated as a BPSK signal. Now eqn. 23 is replaced by

e . _I= 0. _I- 0.

1 - 1

~ : l + ~ , ~

⁼^u:i-l

+

^OS(1^-^/3)’ei

u : + 1 , i = u ; , i - 1

+

2u;+1,i

+

^lS(1

+

^BX1^-^B)2ei

u i + l , i = U i . i - 1

+

u : + l , i

- U;+ 1,

+

(1 - fi3)ei

mi = Re (f:) (28)

and mi is used in eqns. 24-27, as before.

The carrier phase control algorithm can track accurately a constant rate of change of Doppler shift, so that it can handle both the case where the satellite is just over the horizon, with a large but slowly varying Doppler shift, and also the case where the satellite is approximately overhead, with a small but rapidly varying Doppler shift. The acceleration in the Doppler shift does not seem to be sufficient to degrade the performance of the algorithm. When the satellite is nearly overhead, the signal to noise ratio is close to its maximum value for the given transmission, thus the parameter values of the algorithm could well be adjusted (for this part of the transmission) by reducing the value of

/3.

This increases the bandwidth of the loop, so that it responds more rapidly to changes but is, of course, more seriously affected by noise.

7 Assessment of systems

The purpose of the Doppler shift corrector is to reduce the Doppler shift to a level at which it is within the lock range of the carrier phase corrector, such that the latter will synchronise within a reasonably short period of time.

The lock range of the carrier phase corrector is in theory infinite, provided that no limit is placed on the time required to achieve lock. Thus the lock range has been redefined as the maximum Doppler shift for which carrier-phase synchronisation is achieved, within a given period of time and for a given residual phase jitter. The time allowed is 3.84 seconds and the level of the residual phase jitter is taken as the peak jitter, in radians, as measured over a period of 0.32 seconds. Five different values of the peak residual phase jitter, given by 0.265, 0.18, 0.145,O.l and 0.05 radians, have been studied in the tests, but only the results for 0.265, 0.145 and 0.05 radians are given here. The parameter values employed in the

IEE PROCEEDINGS, Vol. 136, Pt. I , No. 5 , O C T O B E R 1989

Doppler-shift corrector are changed at appropriate time intervals, without however interrupting the processing of the signal, but only one set of parameter values are employed in the carrier phase corrector.

Figs. 7-10 show the performance of the Doppler shift corrector with signal A, for different values of the initial Doppler shift. Throughout these tests, $ = -3.5, p = ^71, n = 32, ( = 1.0 and ^IC= 0.4. In each figure the Doppler shift is reduced to a value below the starting value of the

- 1 0 0 0 1 I I 1 f

0 0 1 0 2 0 3 0 4 0 5 0 6 0 7

Doppler shift correction for signal A , with residual Doppler shift time in seconds f r o m s t a r t u p

Fig. 7 initially at 5 k H z . JI = - 3 . 5 , ~ = H andm = 16

1200

1000

800

N I c -

+ _- 600

L m a,

-

L

g 400 n ⁰

- 0

;

²⁰⁰

m P

0

- 200

0 0 1 02 0 3 0 4 0 5 06 0 7 time i n seconds f r o m s t a r t u p Fig. 8

shqt initially at 1200 H z .

Doppler-shift correction for signal A , with residual Doppler

= - 3 . 5 , ~ = nandrn = 64

351

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Doppler shift in the next figure, so that the separate operations can be carried out in sequence to give the required low Doppler shift at the end of the complete operation.

Figs. 11-14 show the corresponding performance of the Doppler shift corrector with signal B. Throughout these tests, 1/1 = -0.94, p = 7~12, n = 32 and K = 0.4. The different curves in each of Figs. 11-14 correspond to different random sequences for both the data-symbol values and the noise components.

0: I 0 9999995

-1 00-1

Y

-1 50

I

0 0 1 0 2 0 3 0 4 0 5 0 6 07 time in seconds f r o m s t a r t u p Fig. 9

initially at 300 Hz.

$ = - 3 S , p = n a n d r n = 2 5 6

Doppler shift correction for signal A , with residual Doppler shift

80

,&=O 9999999

lo

99999995 I

g 60 0 99999997

- 1 0 I Î Î I Î ¹ Î

0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 time i n seconds from s t a r t u p Fig. 10

shift initially at 75 Hz.

$ = -3.5, p = n and rn = 1024 358

Doppler shift correction for signal A , with residual Doppler

Figs. 15 and 16 show the performance of the Doppler shift corrector, for signals A and B, respectively, where the parameters of the Doppler shift corrector are adjusted after 0.32 and 0.96 seconds, in the case of signal A, and after 6.4, 12.8, 19.2 and 25.6 seconds, in the case of signal B. Thus, with signal A, the Doppler shift corrector oper- ates in three successive stages (different modes), whose durations are 0.32, 0.64 and 0.96 seconds, respectively, and, with signal B, it operates in five successive stages,

6000-

5 0 0 0 .

N

I

5 4000

c L - L m

:

L 3 0 0 0 a n

Cl 0

-

;

2000

? f ^m

1 0 0 0

0

- 1 o o c

0 1 2 3 4

t i m e i n seconds from s t a r t u p Fig. 11

shift initially at 5 kHz.

$ = -0.94, p = 4 2 , a = 0.999997, 5 = 0.4 and rn = 8

Doppler shift correction for signal B, with residual Doppler

1 6 0 0 1 4 0 0

1 N

I C

- 1

L

zooi

_{O !} ^h

- 2 00

i

- 4 00

0 1 2 3 4

time in seconds from startup Fig. 12

shift initially at 1200 Hz.

JI = -0.94, p = nl2. ^U= 0.999999,

Doppler shift correction for signal B, with residual Doppler

= 0.45 and rn = 32

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each having a duration of 6.4 seconds. With signal A , the Doppler shift corrector is replaced by the carrier-phase corrector after 1.92 seconds, and the carrier phase corrector then reduces the peak residual phase jitter to below 0.145 radians after another 1.28 seconds. With signal B, the Doppler shift corrector is replaced by the carrier phase corrector after 32 seconds, and the carrier phase corrector then reduces the peak residual phase jitter to below 0.265 radians after another 6.4 seconds. In Fig. 15,

800

700

600

N

I G Z - 500

L VI L

- a,

400 n 0

5

³⁰⁰

L -

0

200

1

oc

^~~

I 1 2 3 4

t i m e in seconds from startup Fig. 13

shft initially at 600 Hz.

I/, = -0.94, p = n/2, a = 0.9999995, 5 = 0.45 and m = 64

Doppler shift correction for signal B, with residual Doppler

3501

’”1

⁰¹₀ ₁^I

^{\ d} ^V

^’ ¹

2 3 4

time in seconds f r o m s t a r t u p Fig. 14

shgt initially at 300 H z .

JI = -0.94, p = 4 2 , a = 0.9995’991,l = 0.45 and m = 128

IEE PROCEEDINGS, Vol. 136, Pt. I , No. 5, OCTOBER 1989 Doppler shift correction for signal B, with residual Doppler

I+9

= -3.5, p = n, n = 32,

5

= 1.0 and K = 0.4, whereas, in Fig. 16,

I+9

= -0.94, p = n/2, n = 32, = 0.4 and IC = 0.4.

For stages 1, 2 and 3 in Fig. 15, rn takes on the values 16, 64 and 256, respectively, whereas a takes on the values 0.999997,0.9999995 and 0.9999999. For stages 1-5 in Fig.

16, m takes on the values 8, 32, 64, 128 and 256, respectively, whereas a takes on the values 0.999997, 0.999999, 0.9999995,0.9999997, and 0.99999995.

Further tests on signal A , have shown that, with an

50001~

r,

3 0 0 0 2500 2000 1500

I

c c

Q

500-

- 5 0 0

I

Î Î Î

0 0 5 1 0 1 5 2 0

time in seconds f r o m s t a r t u p

Fig. 15 Multi-stage Doppler shift correction for signal A , with residual Doppler sh$t initially at 5 k H z .

I/, = -3.5 and p = IT.

“ O 0 1

3000-

\

-

Q Q 0

<

^2000-

0

2

VI

1000-

0 ⁿ_\ _,J-- ^-

-1000 I Î Î , Î

0 5 10 15 20 25 30 35 time i n seconds f r o m startup

Multistage Doppler shift correction for signal B, with residual Fig. 16

Doppler shft initially at 5 k H z . I/, = -0.94 and p = 4 2

359

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initial Doppler shift of 5 kHz and $ = -3.5, carrier phase synchronisation, for a peak residual phase jitter of only 0.05 radians, can be achieved in under 5 seconds.

Again, with signal A and the carrier phase corrector, the time required to reduce an initial Doppler shift of 100 Hz to a peak residual phase jitter of 0.145 radians, is 3.3 seconds, and the time required to reduce an initial Doppler shift of 15 Hz to a peak residual phase jitter of 0.05 radians is 3.7 seconds. Thus the carrier phase corrector can achieve lock in the presence of appreciable levels of Doppler shift. Once lock has been achieved, the system can tolerate considerable variations in Doppler shift of the received signal.

8 Conclusions

Both the Doppler shift and carrier phase correction algo- rithm have been shown to operate well with the signals A and B, when the signal is received in the presence of a large Doppler shift and high level additive white Gauss- ian noise. The techniques described here therefore appear to be well worth further study, particularly under more practical conditions where there is adjacent or cochannel interference and where limited arithmetic accuracy is involved in processing the signal at the receiver.

9 Acknowledgment

The investigation has been partly sponsored by Satellites International Ltd., Hambridge Road, Newbury, Berk- shire, as part of a contract for the development and pro- duction of a variable rate demodulator, for the European Space Agency.

10 References

1 WEBER, C.L.: ‘Candidate receivers for unbalanced QPSK, ITC Proc., XII, 1976, pp. 455464

2 SIMON, M.K., and ALLEM, W.K.: ‘Tracking performance of unbalanced QPSK demodulators. Part I - Biphase Costas loop with passive arm filters’, I E E E Trans., COM-26, 1978, pp. 1147- 1156

3 SIMON, M.K., and ALLEM, W.K.: ‘Tracking performance of unbalanced QPSK demodulators. Part I1 - Biphase Costas loop with active arm filters’, IEEE Trans., COM-26, 1978, pp. 1157-1166 4 BRAUN, W.R., and LINDSEY, W.C.: ‘Carrier synchronisation

techniques for unbalanced QPSK signals - Part 1’, IEEE Trans., 5 BRAUN, W.R., and LINDSEY, W.C. : ‘Carrier synchronization

techniques for unbalanced QPSK signals - Part II’, I E E E Trans., COM-26, 1978, pp. 1325-1333

COM-26. 1978. DD. 1334-1341 6

7

8

9 10 1 1

12 13 14 15 16

17 18

SIMON,’ M.K. :“Error probability performance of unbalanced QPSK receivers’, I E E E Trans., COM-26, 1978, pp. 139G1397 CLARK, A.P.: ‘Carrier phase synchronization in the demodulation of UQPSK signals’, Private Communication, Loughborough Uni- versity, July 1986, pp. 1-80

CLARK, A.P. : ‘Principles of digital data transmission’, 2nd ed., Pentech Press, 1983

EVANS, B.G. (ed.): ‘Satellite communication systems’, Peter Per- egrinus, 1987

WOZENCRAFT, J.M., and JACOBS, I.M.: ‘Principles of communi- cation engineering’, Wiley, 1965

LATHI, B.P.: ‘Random signals and communication theory’, Inter- text Books, 1968

CLARK, A.P.: ‘Adaptive detectors for digital modems’, Pentech Press, 1989

GARDNER, F.M.: ‘Phaselock techniques’, Wiley, 1966

LINDSEY, W.C.: ‘Synchronization systems in communications’, Prentice Hall, 1972

GUPTA, S.C.: ‘Phase-locked loops’, Proc. IEEE, 63, 1975, pp. 291- 306

FRANKS, L.E.: ‘Carrier and bit synchronisation in data communication - a tutorial review’, I E E E Trans., COM-28, 1980, pp. 1107- 1120

POKLEMBA, J.J.: ‘A joint estimator detector for QPSK data transmission’, C O M S A T Tech. Rev., 14, 1984, pp. 211-259 KAM, P.Y.: ‘Maximum likelihood carrier phase recovery for linear suppressed-carrier digital data modulations’, I E E E Trans., COM-34.1986. DD. 522-527

19 MORRISON, N:: ‘Introduction to sequential smoothing and pre- diction’, McGraw-Hill, t969

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