DOQPSK - Differential Demodulation of Filtered Offset QPSK

Christoph G. Gkther* and Joachim Habermam** * ASCOM Tech Ltd., Gewerbepark, 5506 Migenwil, Switzerland ** Facbhochschule Giessen-Friedberg, 61 169 Friedberg, Germany

Ahtmcf: Offset QPSK (OQPSK) and xl4-QPSK have been studied intensively over the last years for applica- tion in digital mobile radio systems. Though OQPSK shows a better performance over nonlinear channels than x/QQPSK, the latter is preferred, since it allows for differential demodulation, which is an attractive combined channel estimation and demodulation techni- que on fast fading channels. In the present paper, we describe a receiver for the differential demodulation of differentially modulated OQPSK signals. The bit error rate of the new receiver is compared with x/QDQPSK on AWGN and Rayleigh fading linear ckurnclr, with transmitter and receiver raised m i n e pulse shaping being assumed in all cases. On these linear channels and a roll of factor r = 0.5, a loss of less than 0.5 dB of DoQPsK with respect to n/QDQPSK is found. Therefore, the advantage of OQPSK over x/QQPSK on non-linear channels will essentially be maintained with differential demodulation.

I. INTRODUCTION

For several years, an ongoing discussion about the modulation format for emerging digital cellular mobile radio systems has taken place (see for instance 111 and [2]). The secpnd generation American Digital Cellular

same modulation format is also used by the Japanese Digital Cellular (JDC) System and for the Trans European Trunked Radio (TETRA). The choice of x/4-QPSK is motivated by two observations: Firstly, the envelope of nI4-QPSK has no zeros. This reduces the generation of harmonics and the distortion of the signal by amplifier non-linearities. The former property is important for the overall system performance and the latter one for the link performance. Secondly, a differential precoding of nI4-QPSK signals allows for simple differential demodula- tors, which is a highly desirable property when the carrier phase of the received signal changes rapidly, as it is the case on fading mobile radio channels. Since OQPSK shows further r e d u d envelope fluctuations (see Fig.1, from [3]), it outperfoms x/4-QPSK on non-linear chan- nels. The non-availability of a differential demodulator for OQPSK, however, led to a clear preference for nl4-QPSK.

Standard (ADC-U.S. TIA 45.3) spe~ifies n/4-QPSK. The

The difficulty of differential demodulation is due to the time overlap of the I- and Qsigoals. Our objective is to show a differential demodulator which can cope with this.

Fig. 1: Complex phase plots of x/4QPSK and OQPSK for r-0.4

The paper is organized as follows: Section II describes the system model. Section Iu gives a derivation of the differential demodulator for raised cosine filtered DOQPSK. Section IV, finally, describes simulation r d t s which show d losses of DOQPSK on liuear channels.

11. MODEM COIWIGURATION

A. Transmitter

The model of the DOQPSK tranSmiaer is shown in Fig.2. It diffem from standard OQPsK by including a differential encoder.

Differen- xlk tial ,Pulse Shaping Encoder LPF

serial data - input

Fig. 2 DoQPSK Traasmitter Model

1542 0-7803-1927-3/94/$4.00 @ 1994 IEEE

_ c ~

The detailed description is as follows: the serial stream of bits (ao, Po, alr pl, ...) of rate l/Tb is split into an I- and Q-stream txk and p k , respectively. The bits are enco- ded differentially and the Q-stre!am is shifted by Td2. We thus have:

These baseband signals are filtered by square root raised cosine filters (including x/sin(x) aperture equalizers) and are input to quadrature a modulator.

B. Receiver

Fig. 3 shows the block diagram of the receiver.

cos(wt)

-sin( 0 t) Fig. 3: D o Q P S K Receiver Model

The received signal r(t) is down-converted and low pars fdtered with square root raised w i n e filters. The I- and Q-components of the resulting complex baseband signal are sampled at the rate 2(Ts. In the case of a roll off factor 1, the received signal becomes

where zl, and nl denote the transmitted complex base- band signal, the slowly varying phase, and the noise, respectively. The latter is typically assumed to be white Gaussian. In the case of interference, square root raised Cosine filtered white Gaussian noise has also to be consi- dered.

TIE differential phase preprocessing rlq2* then elimina- tes the unknown phase and provides the input for the metric computation of the trellis de- algorithm. This part is discussed in the following section.

111. DOQPSK DEMODULATOR

A . General Remarks

Let us shortly reconsider the difficulties with differenti- al demodulation of OQPSK. In OQPSK, the I- and Q- channels are staggered, i.e., time shifted by half a symbol period. With coherent demodulation, the I- and Q-compo- nents are recovered and the time shift is of no relevance. In a differential scheme, however, this time shift leads to a mutual interference of the I- and Q-components. Conven- tional differential demodulators, like those described in 141, can no more be applied. The interference needs to be resolved. The situation is similar to the differential demo- dulation of MSK, which can be seen as a modified form of OQPSK (see 151). Essentially, with MSK, sinusoidal pulses are being used instead of rectangular ones with OQPSK. For MSK it is known that the error performance with differential demedulation and symbol by symbol decision is poor (see [6]).

B. Demodulator Structure for Nyquist Pulse Shaping

The structure of the demodulator is the same for any type of Nyquist filtering. Let us consider the I-channel, for example. Due to the Nyquist criterim, there is no interfe- rence from other Ichannel symbols. However, there may be an infinite number of Q-channel symbols which interfe- re with the I-channel symbol considered. Amongst the possible Nyquist pulse shapes, we shall restrict ourselves to raised cosine filtered signals with a roll off r=l. In this case, only the previous and next Q-channel symbol interfe- re on a given I-channel symbol (see (2)). This is shown in Fig. 4.

-1

Fig. 4: I- and Qchannel signals which satisfy Nyquist 1 and 2

The interference from Q- to I-channel needs to be

Let us first consider the output of the preprocessing at resolved using a trellis.

the sampling values 1=2k, when no noise is considered:

1543

and then at the sampling values 1=2k+I:

1 1 1 1 1 1 1 1 r2 -3 - 1 1 1 1 1 1 1-1 -rrrT 1 1 1-1 1 1-1 1 prp- 1 1 1-1 1 1-1-1 T p - 1 1-1 1 1-1 1 1 r 1 - T 1 1-1 1 1-1 1-1 p-p- 1 1-1-1 1-1-1-1 (-1p- 1 1-1-1 1-1-1 1

If we could somehow obtain estimates of x and y, differential decoding also provides estimates of a and 9:

11 1-1 1 1

TABLE 1

PULSE SHAPING TRELLIS FOR DoQPSK WITH NYQUIST 1 AND 2

- 1 1 1 1 r p -

-1-1-1 1 Fir -1-1-1-1 T T -

Similarly, we need to consider transitions from the lto the final state y2k- initial state

1X2ky2k+lX2k+2 at tbe odd Samphg pOhb. NOW, it is htereshg to Ime that & + l = q k and Y=+i"Yzk When the initial and final states are coosidered at odd and even sampling points. This means that the resulting 16 state trellis with two transitions leaving and entering each node is essentially time invariant. In Table 1, we show the transitions and their associated values of X and Y. Since both X and Y are invariant when we complement the initial and f m l state, only the fmt 8 transitions are in- cluded.

Note that the output of the differential preprocessing unit X and Y can take 4 and 5 values, respectively, instead of the familiar 3 values of DQPSK. 'Ihese values can be observed at the sampling points of the eye diagram shown in Fig. 5. (In Fig. 5 R(t) is identical with X(t).)

2.

4 R(t)

1.

0.

-1 .

Fig. 5: Eye diagrams of the signals R(t)-X(t) and Y(t) for 1-1

h order to complete the description of the demodulator, we need to indicate the metric that shall be used. Under the usual assumption of additive white Gaussian noise (AWGN), this metric is the squared Euclidean distance, i.e., the metric increment is the squared Euclidean distance between the samples (X,Y) and the values from Table 1. Obviously, this is an approximation even in the case of AWGN channel noise, since the square root raised cosme filtering at the receiver destroys the whiteness and since

1544

the non-linear preprocessing also destroys the property of being Gaussian. The departure from Gaussian noise can be neglected at high signal to noise ratios. The detailed im- pact of the approximation is assessed through the simula- tions in the next session.

IV. SIMULATION RESULTS

The computer simulation was based on a discrete time complex baseband signal representation, except for the filter sections where frequency domain representations were used. The block diagrams of Fig. 2 and Fig. 3 show the high level description of the simulation Root raised cosine film with a roll off factor r are used for pulse shaping. The trellis decoder is as specified by the state diagram of Table 1. The metric is the squared Euclidean dismce.

Remember that this setting has been derived to be used with a roll off factor r= 1. Another value of r would have led to a different trellis. The interference of an infinite number of Q-channel samples on a given I-channel would have required a reduced state decoding sheme. The eye diagrams for X(t)==R(t) and Y(t) in the case of r=0.5, shown in Fig. 6, suggest that the r=l trellis may lead to a r e d d state decoding with an acceptable performance. This is what we have investigated in more detail.

i . 0.00 'Yk . 0.125

Fig. 6 Eye diagrams of the signals R(t)-X(t) and Y(t) for r0.5

Fig. 7 shows results for DOQPSK on an AWGN channel for different roll off factors. In the same figure, results for unfiltered DQPSK (plaine line) are also inclu- ded. They can be considered as representative for Unfilte- red n/CDQPSK, if the differential encoding is over two symbols, i.e., if the I-component xI satisfies the relation X2kxZk-2=ak, where ak is the k-th I-channel bit and if the Q-component satisfies a similar relation. The unfiitered n/CDQPSK results are also representative for the filtered case as long as the the filtering satisfies the Nyquist criterion.

The curve for DOQPSK and r=l, plotted as a dotted line, shows a certain degradation at small values of This is due to the use of an approximation for the metric in the detection algorithm (see above). However, for higher E p o , the curve for DOQPSK converges to that of DQPSK (n/CDQPSK). Furthermore, the degradation of the performance in the case of r-0.5 (dashed line) remains surprisingly low, namely around 0.5dB in the range of interest.

0 2 4 6 8 1 0 1 2 1 Fig. 7: BER versus E& on an AWGN channel

At r=0.3 (dash-dotted line), the increased intersymbol interference leads to an irreducible error rate that starts to become perceptible. The actual irreducible error rate could not be determined due to the excessive duration of the simulation At r=0.3, the degradation is still acceptable for the "mis s ion of digitized speech. In the case of compu- ter data, however, thinking about a more adapted reduced state trellis starts to become worthwile.

In a narrowband mobile radio communications system, transmission channels subject to Rayleigh fading with a Jakes Doppler spectrum are commonly considered. We, cmespondingly, ran simulations for such channels. The product of Doppler shift fd and symbol duration T, was chosen to be 0.007.

At 900 MHz and a data rate of 36 kbivs, this corre- sponds to a vehicle speed of 1501rm/h. Fig. 8 shows the

1545

results for fdtered DOQPSK with r=0.5, and r-0.3 as well Systems," IEEE Trans. Veh Technd., vol. 40, pp. 355-365, May 1991.

[3] European Te1eco"unicatioas St.ndards IastitutiOa ETSI, "Bit Ermr Performance of n/4-QPSK and OQPSK," ElS&TC-RES

as the results for n/4-DQPSK described in [7]. As expec- ted, DOQPSK again shows small losses when compared to nl4-DQPSK. 6.2(9Z-Z45), 1991.

lo-1,

1 o-2

-3 10-

BER

[4] J.G. Proakis, "Digital Cinnmunicatians," McGmw- Hill, 1989. [S] S. Gmemeyer, A. McBride, ' 'MSK cud offsa QPSK Modulgtion,"

[6] J. Andetson, T. Aulm, C.-E. Snndberg, ' 'Digital phae Modulrtion," IEEE Tram. on Commun, pjk 809-819, August 1976.

Plenum Press. 1986. --. 171 European Telecommunications Stpndprds Institatioa ETSI, ' *Sitnula-

t i ~ n Results fOr x14-DQPSK and OQPSK," nS@TC-RES 62 (91-141), 1991. r 0 . 3

- 4 4 - DQPSK E& /No [dBl

I I

10 12 14 16 18 20 22 2 Fig. 8: BER vetsus E&,, Rayleigb fading channel with

f, . T, - 0.007 (f,: maximum Doppler frequency) V. SUMMARY AND CONCLUSIONS

A receiver s t ~ a m for the differential &modulation of filtered DOQPSK signals was derived. Its essential ele- ments are a differential pr- and a trellis decoder. In the specific case of raised cosine pulse shaping with a roll off factor t -1 , the trellis has 16 states. On the linear channels considered, the performane degradation with respect to DQPSK was found to be itsignificant. When the Same simple t-l-demodulator was used for signals filtered with a roll off factor d . 5 ( J X system for instawe), a degradation of around 0.5 di3 in the range of interest was found. This holds for both AWGN and Rayleigh fading channels.

The new receiver structure disproves the frequent belief that OQPSK and different demodulation cannot be combi- ned. The trade off between the reduced complexity of the power amplifier and the maeased complexity of the baseband processing must clearly be made on a case by case basis. It is, however, certainly worthwile to use this additional degree of freedom in the design of future mobile radio ~ u n i c a t i o n system.

VI. REFERENCES

[l] H. Fuukawa, K. Matsuyama, T. Sato, T. Takaaka. and Y. Takeda, ' 'A xi4 Shifted DQPSK Demodulator for Pemd Mobile Commu- nications System." PIMRC'92. pp. 618622. Bostoa, MA, Oct. 1992.

[Z] K. Feher, . 'MODEMS for Emerging Digital Cellular-Mobile Radio

1546