Diversity for Star 16QAM on Fading Channels Arne Svensson Chalmers University of Technology Department of Information Theory S-412 96 Gothenburg, Sweden phone +46 31 772 17 51 or +46 70 565 48 20 fax +46 31 772 17 48 e-mail [email protected] ABSTRACT Star 16QAM is a modulation method that transmits 4 bits per symbol and has the advantage that it may be differentially encoded and detected. It is very robust to fast multiplicative Rayleigh fading and is suitable for mobile telephone systems and personal communication networks. In this paper we study maximum likelihood differential detection for bit decisions using L diversity branches as well as much simpler post detection combining techniques. The bit error probability is simulated for both diversity detectors on a multiplicative Rayleigh fading channel with additive white Gaussian noise. Both detectors are found to perform almost identically. Large performance gains are obtained with diversity.

1. Introduction Future personal communication systems require bandwidth efficient digital modulation schemes due to the limited radio spectrum available. At the same time the bit error probability of the modulation method must be small also in fading. Differential encoding and detection is well-known to be very robust to fast fading and has the advantage that no carrier phase tracking is needed. In order to improve the bandwidth efficiency as compared to BPSK and QPSK, a 16 point constellation that may be differentially encoded and detected, has recently gained much attention [1]-[7]. This scheme is both referred to as star 16QAM (quadrature amplitude modulation), 16DAPSK (differential amplitude phase shift keying) and 2DASK/8DPSK (differential amplitude shift keying, differential phase shift keying). Star 16QAM use a combination of independent 2DASK and 8DPSK. Star 16QAM is proposed as a modulation method on a Rayleigh fading channel for future mobile radio communications and PCNs (personal communications networks) in [1] (and its references). The bit error probability of star 16QAM is analysed on AWGN and Rayleigh fading channels in [2] and [3]. A simple diversity scheme is proposed and theoretically analysed in [4]. An improved differential detector that employs decision feedback in order to improve the reference signal is proposed and analysed in [5] and [6]. Finally a multiple-symbol differential detection scheme is proposed in [7]. In [5]-[7] fading is not considered but only AWGN channels. In this paper we study two different diversity detectors, that

both are based on the decision variables used by the single branch detector [2], [3], i.e. essentially phase difference and amplitude ratio between two consecutive received symbols. The main contribution is the derivation of the likelihood functions for independent 8DPSK and 2DASK bit decisions based on these decision variables and simulation of the bit error probability of this ML (maximum likelihood) detector. For the decisions on 8DPSK we consider both using amplitude and phase information as well as using only phase information for the decision. The ML detector is quite complex to implement, so we also consider an easily implementable diversity detector that combines the 8DPSK and 2DASK single branch detector decision variables from the branches, respectively, using weights that are easily derived from the received signal strength (RSS). Again, for 8DPSK we consider two alternative ways of making decisions. One of them was used already in [8] for DPSK, while the other is a simple generalization of a combing scheme used in [4]. The diversity combing rule for amplitude decisions is novel. We refer to this detector as RSS diversity combining. By using simulated bit error probability we show that the bit error probability of the RSS detector is almost equal to the bit error probability of the ML detector and we conclude that the performance of ML detection may be obtained with simple implementations. The detector proposed in [4] uses a slightly different technique, but yet simple, for obtaining a decision variable for the 2DASK decision.

2. Transmission model and background In this paper we assume perfect sample timing synchronization and that the overall filter response is an ISI-free Nyquist filter response that is shared equally by the transmit and receive filters. The channel is assumed to be a multiplicative Rayleigh fading channel with additive white Gaussian noise [9]. 3 bits of the 4 bit star 16QAM symbol are mapped to the differential phase ∆φ k = φ k – φ k – 1 according to the Gray encoding rule given in Table 1. Here φ k ∈ { mπ ⁄ 4 ; m = 0, …7 } is the transmitted phase at time t = kT where T is the symbol time. In this paper we refer to the 8DPSK individual bits as the most significant bit (msb), the middle bit and the least significant bit (lsb) in the order as they appear in Table 1. The fourth bit of the star 16QAM symbol is mapped onto an amplitude level change

To be presented at The Sixth International Symposium on Personal, Indoor and Mobile Radio Communications, PIMRC´95, Toronto, Canada, September 1995.

mum bit decisions. Due to page limitation we need to be brief here, see [10] for more details.

Table 1 Gray encoding rule between bits and phase increments codeword

∆φ k

codeword

∆φ k

000

0

110

π

A. 2DASK decision

001

π/4

111

5π/4

011

π/2

101

3π/2

010

3π/4

100

7π/4

First we consider decisions on the bit carried in the amplitude of the star 16QAM symbol. We now derive the ML detector that uses the ratios z l, k λ l, k = -------------(2) l = 1 , …, L z l, k – 1

such that a transmitted “1” changes the amplitude from a L to a H or vice versa, while a transmitted “0” does not change the amplitude. Here a k ∈ { a L, a H } denotes the amplitude of the transmitted signal. Following e.g. [9], the received sample in receiver branch l at time t = kT is given by z l, k = a k e

jφ k

g l, k + γ l, k .

(1)

The fading is here represented by g l, k which is a zero-mean complex Gaussian random variable with 〈 g l, k 2〉 = 1 , where we use the notation 〈 x〉 to denote expected value. The fading correlation between consecutive fading samples is assumed to be η = 〈 g l, k g l, k – 1∗〉 = J 0 ( 2πf d T ) , where J 0 ( x ) is the zeroth order Bessel functions of the first kind and f d is the so called maximum Doppler frequency [9]. The fading is assumed to be independent between receiver branches, i.e. 〈 g l, k g m, n∗〉 = 0 for all l ≠ m and all k and n , where g m, n∗ denotes the complex conjugate of g m, n . The star 16QAM jφ symbols a k e k are assumed equally likely. The noise is represented by γ l, k which is a zero-mean complex Gaussian random variable with variance σ γ2 = N 0 ⁄ T . The noise sequence is assumed to be white. The mean energy per symbol per receiver branch is given by 2 ) ⁄ 4T . We use E ⁄ N as a measure of signal E s = ( a L2 + a H b 0 to noise ratio, where E b = E s ⁄ 4 is mean received energy per bit per receiver branch and N 0 is the single-sided power spectral density of the Gaussian noise. A single branch differential detector ([1]-[4]) use (here we drop the diversity branch index for simplicity) arg ( z k z k – 1∗ ) as a decision variable for the 3 bit 8DPSK decision. Here arg x denotes the phase of the complex argument with phase defined in ( – π, π ] . The detector is a traditional 8PSK detector. The decision variable for the 2DASK bit is given by z k ⁄ z k – 1 , where x denotes the absolute value of the argument. The decision rule is: decode “0” when β L < z k ⁄ z k – 1 < β H and “1” otherwise. Here β L and β H are properly chosen thresholds.

at time t = kT in order to make an optimum decision on the 2DASK bit. Following the approach by Adachi [9], we find that the conditional probability density function (pdf) is Ricean [11] and is given by p 2DASK ( λ l, k z l, k – 1, a k, a k – 1 ) =

,

(3)

λ l, k λ l, k M λ + --------- ⋅ exp – ------------------------ I ----------------- Vλ 2V λ 0 V λ λ l2, k

M λ2

where (arguments of M λ and V λ omitted)

σ ( ak) M λ = ρ ( a k, a k – 1, 0 ) ----------------------- , σ ( ak – 1)

(4)

σ ( ak ) 2 2 -------------------- (1 – ρ (a , a Vλ = k k – 1, 0 ) ) , z l, k – 1 2

(5)

σ ( ak) =

a k2 N ----- + -----0- , 2 T

(6)

η a k a k – 1 exp ( j∆φ k ) ρ ( a k, a k – 1, ∆φ k ) = --2- ---------------------------------------------- , ) σ (a ) σ (a k

(7)

k–1

and I 0 ( x ) is the zeroth order modified Bessel functions of the first kind. From eq. (4) we can easily derive the likelihood functions for transmitted “0” and “1” respectively, when the fading in the two branches are assumed uncorrelated. The likelihood function for decision “0” becomes Γ 2DASK ( 0 ) =

L

∏ l=1

∑

x ∈ { a L, a H }

p 2DASK ( λl, k z l, k – 1, x, x ) ,(8)

where we have omitted a constant factor which appears in both likelihood functions. In the likelihood function for decision “1”, the sum runs over the 2 terms with different amplitude levels.

3. Maximum likelihood detectors In this paragraph we derive the likelihood functions for making independent 8DPSK and 2DASK decisions, respectively, by using a differential detector with L branch diversity. In [9] Adachi derives the likelihood function for making optimum post detection symbol decisions on DPSK. We apply his technique to star 16QAM in this paper but we concentrate on opti-

B. 8DPSK bit decision using amplitude and phase We now turn to the decision of the 3 bits transmitted by using 8DPSK. First we study a detector based on the variable ∗ (9) = z z l = 1 , …, L , v l, k

l, k l, k – 1

i.e. we use all information available in the received samples.

4. Simple diversity combining schemes

The pdf needed for this case is given in [9] and is given by p 1 ( v l, k z l, k – 1, a k, a k – 1, ∆φ k ) =

,

(10)

v l, k – M v 2 1 ------------- ⋅ exp – -------------------------- 2πV v 2V v where

A. Diversity combining for 2DASK decisions M v = z l, k – 1

σ ( ak) 2ρ ( a , a ----------------------, ∆φ ) k k–1 k σ (a ) k–1

(11)

and V v = σ ( a k ) 2 z l, k – 1 2 [ 1 – ρ ( a k, a k – 1, ∆φ k ) 2 ] .

(12)

In order to obtain the likelihood function for a symbol decision on each 8DPSK symbol we average the pdf in eq. (11) over the amplitude levels A = { a L, a H } , leading to p 8DPSK ( v l, k z l, k – 1, ∆φ k ) = 1 --4

In this section we consider a much simpler diversity combining technique, which will be referred to as RSS combining detectors. These detectors are partly quite simple generalizations of detectors and ideas proposed in [4] and [8].

∑ ∑

ak ∈ A ak – 1 ∈ A

.

(13)

p 1 ( v l, k z l, k – 1, a k, a k – 1, ∆φ k )

where λ l, k is given in eq. (3). The decision rule is now given by 0 if β L < Λ k < β H . (17) 1 elsewhere In eq. (17), r is a parameter which is chosen to optimize the performance. A similar, but different, decision rule is used in [4] and we will use this as a comparison.

B. Vectorial diversity combining for 8DPSK decisions

This pdf may be used as a likelihood function for making optimum symbol decisions of the 8DPSK symbols. The likelihood function for making optimum bit decisions on each bit in the 3 bit 8DPSK symbol is obtained by averaging over the phase increments corresponding to bit “0” and “1”, respectively. For the most significant bit (msb) we obtain from Table 1, (14)

Γ msb,8DPSK ( 0 ) =

∑

The decision variable we propose for 2DASK is given as 1 - ( z rλ + z rλ ) , Λ k = ---------------------------------(16) r 2, k 2, k z 1, k + z 2, k r 1, k 1, k

∆φ k ∈ { 0, π ⁄ 4, π ⁄ 2, 3π ⁄ 4 }

p 8DPSK ( v l, k z l, k – 1, ∆φ k )

V k = arg ( z 1, k q v 1, k + z 2, k q v 2, k ) ,

(18)

where v l, k is defined in eq. (10) and q is a parameter that should be selected such that the performance is optimized. The symbol decision rule is now a traditional 8PSK detector for Gray encoded symbols. This decision variable is a simple generalization of the rule proposed in [4]; we have included the RSS weights.

C. Phase diversity combining for 8DPSK decisions

for decision “0” and a similar expression with a sum over the other four phases for decision “1”. With the help of Table 1 it is now straight forward to derive the likelihood functions for the other bit decisions.

C. 8DPSK bit decisions using phase only It is possible to decode the 8DPSK bits by using phase information only. This could e.g. be used if a limited receiver is applied for the 8DPSK decision. Still, however, a non-limited receiver must be used for the 2DASK decisions. A receiver equiped with a logarithmic amplifier is such an example. An alternativ decision variable in this case for the 2DASK decision is to directly use the amplitude in logarithmic scale, which is outside the scope of this paper. The decision variable is now v˜ l, k = arg v l, k ,

For 8DPSK we propose a vectorial addition of the decision variables from the branches given by

(15)

with v l, k given as in eq. (10). The derivation of the likelihood functions for this case follows the previous section except for the pdf in eqs. (11)-(13). Instead we need the pdf of the phase of a complex Gaussian variable and it can be found in e.g. [12]. The interested reader is referred to [10] for the details.

The order of addition and evaluating phase can be exchanged in eq. (19) above. This decision variable is used in [8] for DPSK and is given by 1 o v˜ - z + , (19) W k = ----------------------------------o z 1, k + z 2, k o 1, k 1 , k v˜ 1, k – v˜ 2, k + π z 2, k o v˜ 2, k + 2π ---------------------------------2π where v˜ l, k is defined in eq. (16), x is the largest integer less than or equal to x and the last term is needed due to the modulo 2π definition of phase [10]. The exponent o is a parameter to be optimized.

5. Simulated bit error probability A. Prerequisites In this section we present some simulated bit error probabilities. We will compare the performance of the proposed detectors with the detector proposed in [4], which is as simple as the RSS detectors considered in this paper.

100

100 ML, no diversity ML, 8DPSK phase only ML, 8DPSK amplitude and phase

10-1

Chow, Nix, McGeehan detector

10-2

bit error probability

bit error probability

10-1

10-3

10-4 RSS, no diversity RSS, 8DPSK vectorial combining RSS, 8DPSK phase combining RSS, selection combining

10-5

10

15

20

25

10-3

10-4

10-5

10-6 5

10-2

ML, no diversity ML, 8DPSK phase only ML, 8DPSK amplitude and phase Chow, Nix, McGeehan detector RSS, no diversity RSS, 8DPSK vectorial combining RSS, 8DPSK phase combining RSS, selection combining

30

35

40

10-6 10-4

mean received Eb/N0 per branch Figure 1 Bit error probability versus mean received signal to noise ratio per receiver branch when the maximum Doppler frequency is zero.

All simulations for the RSS combining detectors and the detector proposed in [4] are performed with 1 million independent trials. The simulations for the ML detectors are performed with fewer trials, but the confidence is still good at the bit error probabilities considered.

B. Coefficients used in the simulations Numerous simulations have been performed for the RSS combining detectors in order to find the best choice of parameters. The chosen parameters are ring ratio aH/aL=2.0, upper threshold βH=1.5, lower threshold βL=0.7, r=2.7, q=0 and o=2.4. It should also be mentioned that small changes in the parameters do not change the performance significantly.

C. Simulated average bit error probability In Figure 1 we show simulated average bit error probability versus mean received E b ⁄ N 0 per receiver branch, for the ML detector, the RSS combining detector and the detector proposed in [4]. The maximum Doppler frequency is here 0, i.e. the results are for very slow fading. We also show the performance for selection combining. The conclusion is that from a performance point of view the RSS combining detectors use the decision variables in an almost optimum way. The performance degradation due to a detector that use only phase information for its decisions on the 8DPSK bits is less than 0.1 dB. Selection combining leads to a performance penalty of around 1.5 dB. The performance gain due to diversity is almost 8 dB at a bit error probability of 1

10-3

10-2

10-1

normalized maximum Doppler frequency fdT Figure 2 Bit error probability versus normalized maximum Doppler frequency when the mean received signal to noise ratio approaches infinity.

percent. In Figure 2 we show the bit error probability versus normalized maximum Doppler frequency when E b ⁄ N 0 → ∞ . The conclusions are here the same as above. We can note that the diversity detector can sustain almost 3 times as large Doppler frequency as the single branch detector at a bit error probability of 1 percent.

D. Subchannels with different bit error probabilities Star 16QAM offers subchannels with different bit error probabilities, which may prove useful when transmitting bits with different sensitivity to errors. In Figure 3 we show the bit error probability for each individual bit in the 16QAM symbol and the average bit error probability of the bits mapped on 8DPSK. From this figure it is clear that the bit error probability of the least significant bit in the 8DPSK symbol is about a factor of two larger than the bit error probability of the other two bits in the 8DPSK symbol. The reason for this is that the least significant bit is always different for one of the neighbouring 8DPSK symbols while the probability for this to happen for the other two bits is 1 ⁄ 2 and these are the most likely errors which occur. It is also very clearly seen that the bit error probability for the 2DASK bit is another factor of two larger than the bit error probability of the least significant bit in the 8DPSK symbol. The difference between the smallest and largest bit error probability is therefore about a factor of 4. By changing parameters such as ring ratio, thresholds and exponents, these factors

with about 4 dB lower E b ⁄ N 0 both with and without diversity. With a real channel estimation algorithm we expect a few decibel degradation in slow fading and more in fast fading, which makes star 16QAM an attractive modulation method.

100 msb, 8DPSK middle bit, 8DPSK lsb, 8DPSK 2DASK average, 8DPSK

bit error probability

10-1

10-2

The maximum likelihood detector studied in this paper makes independent decisions on the bits mapped to phase and amplitude, respectively, in the star 16QAM symbol. Furthermore it is restricted to use the decision variables at the output of a differential detector in each diversity branch, which are used for the decisions by a single branch detector. There is no guarantee that this is the optimum diversity detector for star 16QAM with differential detection, and in fact the detector proposed in [4] has a slightly smaller bit error probability. It is likely that further performance improvements may be obtained by other diversity detectors for star 16QAM.

no diversity

10-3 diversity

10-4

10-5

7. References -6

10

5

10

15

20

25

30

35

40

mean received Eb/N0 per branch Figure 3 Bit error probability of the different star 16QAM subchannels versus mean received signal to noise ratio per receiver branch when the maximum Doppler frequency is zero.

could be changed to a certain extent [10]. In this way, star 16QAM offers at most 3 subchannels with different bit error probabilities, which may be used for unequal error protection.

6. Discussion and conclusions In this paper we study two methods for diversity reception of differentially encoded and detected star 16QAM. The channel is a multiplicative Rayleigh fading channel with additive white Gaussian noise. By using computer simulation, we show that the bit error probability of the maximum likelihood differential detector may be obtained in practice by a simple post diversity combining detector. The diversity gain is around 8 dB at a bit error probability of 1 percent where the signal to noise ratio is defined as mean received energy per bit per noise spectral density per diversity branch. The diversity detector is much less susceptible to maximum Doppler frequency. The error floor is almost a factor of 3 lower for the diversity detector when the bit error probability is in the order of 1 percent. Star 16QAM is also shown to offer at most 3 subchannels with different bit error probabilities, which may prove useful in communication systems which transmits bit of different importance. Some speech encoding methods have this property. Star 16QAM with differential encoding and detection offers a bit error probability of 1 percent at an E b ⁄ N 0 of around 21 dB without diversity and around 13 dB per receiver branch with 2 branch diversity. Assuming perfect channel estimation, including both phase and amplitude, with traditionally rectangular 16QAM, we find that the same bit error probability is obtained

[1] W.T. Webb, and L. Hanzo, Modern Quadrature Amplitude Modulation: Principles and Applications for Fixed and Wireless Channels, Pentech Press, 1994. [2] F. Adachi and M. Sawahashi, “Performance analysis of various 16 level modulation schemes under Rayleigh fading”, Electronics Letters, Vol. 28, No. 17, 13th August 1992, pp. 1579-1581. [3] Y.C. Chow, A.R. Nix and J.P. McGeehan, “Analysis of 16-APSK modulation in AWGN and Rayleigh fading channel”, Electronics Letters, Vol. 28, No. 17, 13th August 1992, pp. 1608-1610. [4] Y.C. Chow, A.R. Nix and J.P. McGeehan, “Diversity improvement for 16-DAPSK in Rayleigh fading channel”, Electronics Letters, Vol. 29, No. 4, 18th February 1993, pp. 387-389. [5] M. Sawahashi and F. Adachi, “Decision feedback differential detection of multilevel amplitude/phase shift keying”, Proceedings JTC-CSCC’93, July 26-28, 1993, Nara, Japan, pp. 217-222. [6] F. Adachi and M. Sawahashi, “Decision feedback differential detection of 16-DAPSK signals”, Electronics Letters, Vol. 29, No. 16, 5th August 1993, pp. 1455-1457. [7] T. Suzuki and T. Mizuno, “Multiple-symbol differential detection scheme for differential amplitude modulation”, Proceedings IZS’94, Zurich, Switzerland, March 1994, pp. 196-207. [8] M. Ikura and F. Adachi, “Postdetection phase combining diversity”, IEEE Transactions on Vehicular Technology, Vol. 43, No. 2, May 1994, pp. 298-303. [9] F. Adachi, “Postdetection optimal diversity combiner for DPSK differential detection”, IEEE Transactions on Vehicular Technology, Vol. 42, No. 3, August 1993, pp. 326-337. [10] A. Svensson, “On differentially encoded star 16QAM with differential detection and diversity”, To appear in IEEE Transactions on Vehicular Technology. [11] J.G. Proakis, Digital communications, 3rd edition, McGraw-Hill 1995. [12] R.F. Pawula, S.O. Rice and J.H. Roberts, “Distribution of the phase angle between two vectors perturbed by Gaussian noise”, IEEE Transactions on Communications, Vol. COM-30, No. 8, August 1982, pp. 1828-1841.

1. Introduction Future personal communication systems require bandwidth efficient digital modulation schemes due to the limited radio spectrum available. At the same time the bit error probability of the modulation method must be small also in fading. Differential encoding and detection is well-known to be very robust to fast fading and has the advantage that no carrier phase tracking is needed. In order to improve the bandwidth efficiency as compared to BPSK and QPSK, a 16 point constellation that may be differentially encoded and detected, has recently gained much attention [1]-[7]. This scheme is both referred to as star 16QAM (quadrature amplitude modulation), 16DAPSK (differential amplitude phase shift keying) and 2DASK/8DPSK (differential amplitude shift keying, differential phase shift keying). Star 16QAM use a combination of independent 2DASK and 8DPSK. Star 16QAM is proposed as a modulation method on a Rayleigh fading channel for future mobile radio communications and PCNs (personal communications networks) in [1] (and its references). The bit error probability of star 16QAM is analysed on AWGN and Rayleigh fading channels in [2] and [3]. A simple diversity scheme is proposed and theoretically analysed in [4]. An improved differential detector that employs decision feedback in order to improve the reference signal is proposed and analysed in [5] and [6]. Finally a multiple-symbol differential detection scheme is proposed in [7]. In [5]-[7] fading is not considered but only AWGN channels. In this paper we study two different diversity detectors, that

both are based on the decision variables used by the single branch detector [2], [3], i.e. essentially phase difference and amplitude ratio between two consecutive received symbols. The main contribution is the derivation of the likelihood functions for independent 8DPSK and 2DASK bit decisions based on these decision variables and simulation of the bit error probability of this ML (maximum likelihood) detector. For the decisions on 8DPSK we consider both using amplitude and phase information as well as using only phase information for the decision. The ML detector is quite complex to implement, so we also consider an easily implementable diversity detector that combines the 8DPSK and 2DASK single branch detector decision variables from the branches, respectively, using weights that are easily derived from the received signal strength (RSS). Again, for 8DPSK we consider two alternative ways of making decisions. One of them was used already in [8] for DPSK, while the other is a simple generalization of a combing scheme used in [4]. The diversity combing rule for amplitude decisions is novel. We refer to this detector as RSS diversity combining. By using simulated bit error probability we show that the bit error probability of the RSS detector is almost equal to the bit error probability of the ML detector and we conclude that the performance of ML detection may be obtained with simple implementations. The detector proposed in [4] uses a slightly different technique, but yet simple, for obtaining a decision variable for the 2DASK decision.

2. Transmission model and background In this paper we assume perfect sample timing synchronization and that the overall filter response is an ISI-free Nyquist filter response that is shared equally by the transmit and receive filters. The channel is assumed to be a multiplicative Rayleigh fading channel with additive white Gaussian noise [9]. 3 bits of the 4 bit star 16QAM symbol are mapped to the differential phase ∆φ k = φ k – φ k – 1 according to the Gray encoding rule given in Table 1. Here φ k ∈ { mπ ⁄ 4 ; m = 0, …7 } is the transmitted phase at time t = kT where T is the symbol time. In this paper we refer to the 8DPSK individual bits as the most significant bit (msb), the middle bit and the least significant bit (lsb) in the order as they appear in Table 1. The fourth bit of the star 16QAM symbol is mapped onto an amplitude level change

To be presented at The Sixth International Symposium on Personal, Indoor and Mobile Radio Communications, PIMRC´95, Toronto, Canada, September 1995.

mum bit decisions. Due to page limitation we need to be brief here, see [10] for more details.

Table 1 Gray encoding rule between bits and phase increments codeword

∆φ k

codeword

∆φ k

000

0

110

π

A. 2DASK decision

001

π/4

111

5π/4

011

π/2

101

3π/2

010

3π/4

100

7π/4

First we consider decisions on the bit carried in the amplitude of the star 16QAM symbol. We now derive the ML detector that uses the ratios z l, k λ l, k = -------------(2) l = 1 , …, L z l, k – 1

such that a transmitted “1” changes the amplitude from a L to a H or vice versa, while a transmitted “0” does not change the amplitude. Here a k ∈ { a L, a H } denotes the amplitude of the transmitted signal. Following e.g. [9], the received sample in receiver branch l at time t = kT is given by z l, k = a k e

jφ k

g l, k + γ l, k .

(1)

The fading is here represented by g l, k which is a zero-mean complex Gaussian random variable with 〈 g l, k 2〉 = 1 , where we use the notation 〈 x〉 to denote expected value. The fading correlation between consecutive fading samples is assumed to be η = 〈 g l, k g l, k – 1∗〉 = J 0 ( 2πf d T ) , where J 0 ( x ) is the zeroth order Bessel functions of the first kind and f d is the so called maximum Doppler frequency [9]. The fading is assumed to be independent between receiver branches, i.e. 〈 g l, k g m, n∗〉 = 0 for all l ≠ m and all k and n , where g m, n∗ denotes the complex conjugate of g m, n . The star 16QAM jφ symbols a k e k are assumed equally likely. The noise is represented by γ l, k which is a zero-mean complex Gaussian random variable with variance σ γ2 = N 0 ⁄ T . The noise sequence is assumed to be white. The mean energy per symbol per receiver branch is given by 2 ) ⁄ 4T . We use E ⁄ N as a measure of signal E s = ( a L2 + a H b 0 to noise ratio, where E b = E s ⁄ 4 is mean received energy per bit per receiver branch and N 0 is the single-sided power spectral density of the Gaussian noise. A single branch differential detector ([1]-[4]) use (here we drop the diversity branch index for simplicity) arg ( z k z k – 1∗ ) as a decision variable for the 3 bit 8DPSK decision. Here arg x denotes the phase of the complex argument with phase defined in ( – π, π ] . The detector is a traditional 8PSK detector. The decision variable for the 2DASK bit is given by z k ⁄ z k – 1 , where x denotes the absolute value of the argument. The decision rule is: decode “0” when β L < z k ⁄ z k – 1 < β H and “1” otherwise. Here β L and β H are properly chosen thresholds.

at time t = kT in order to make an optimum decision on the 2DASK bit. Following the approach by Adachi [9], we find that the conditional probability density function (pdf) is Ricean [11] and is given by p 2DASK ( λ l, k z l, k – 1, a k, a k – 1 ) =

,

(3)

λ l, k λ l, k M λ + --------- ⋅ exp – ------------------------ I ----------------- Vλ 2V λ 0 V λ λ l2, k

M λ2

where (arguments of M λ and V λ omitted)

σ ( ak) M λ = ρ ( a k, a k – 1, 0 ) ----------------------- , σ ( ak – 1)

(4)

σ ( ak ) 2 2 -------------------- (1 – ρ (a , a Vλ = k k – 1, 0 ) ) , z l, k – 1 2

(5)

σ ( ak) =

a k2 N ----- + -----0- , 2 T

(6)

η a k a k – 1 exp ( j∆φ k ) ρ ( a k, a k – 1, ∆φ k ) = --2- ---------------------------------------------- , ) σ (a ) σ (a k

(7)

k–1

and I 0 ( x ) is the zeroth order modified Bessel functions of the first kind. From eq. (4) we can easily derive the likelihood functions for transmitted “0” and “1” respectively, when the fading in the two branches are assumed uncorrelated. The likelihood function for decision “0” becomes Γ 2DASK ( 0 ) =

L

∏ l=1

∑

x ∈ { a L, a H }

p 2DASK ( λl, k z l, k – 1, x, x ) ,(8)

where we have omitted a constant factor which appears in both likelihood functions. In the likelihood function for decision “1”, the sum runs over the 2 terms with different amplitude levels.

3. Maximum likelihood detectors In this paragraph we derive the likelihood functions for making independent 8DPSK and 2DASK decisions, respectively, by using a differential detector with L branch diversity. In [9] Adachi derives the likelihood function for making optimum post detection symbol decisions on DPSK. We apply his technique to star 16QAM in this paper but we concentrate on opti-

B. 8DPSK bit decision using amplitude and phase We now turn to the decision of the 3 bits transmitted by using 8DPSK. First we study a detector based on the variable ∗ (9) = z z l = 1 , …, L , v l, k

l, k l, k – 1

i.e. we use all information available in the received samples.

4. Simple diversity combining schemes

The pdf needed for this case is given in [9] and is given by p 1 ( v l, k z l, k – 1, a k, a k – 1, ∆φ k ) =

,

(10)

v l, k – M v 2 1 ------------- ⋅ exp – -------------------------- 2πV v 2V v where

A. Diversity combining for 2DASK decisions M v = z l, k – 1

σ ( ak) 2ρ ( a , a ----------------------, ∆φ ) k k–1 k σ (a ) k–1

(11)

and V v = σ ( a k ) 2 z l, k – 1 2 [ 1 – ρ ( a k, a k – 1, ∆φ k ) 2 ] .

(12)

In order to obtain the likelihood function for a symbol decision on each 8DPSK symbol we average the pdf in eq. (11) over the amplitude levels A = { a L, a H } , leading to p 8DPSK ( v l, k z l, k – 1, ∆φ k ) = 1 --4

In this section we consider a much simpler diversity combining technique, which will be referred to as RSS combining detectors. These detectors are partly quite simple generalizations of detectors and ideas proposed in [4] and [8].

∑ ∑

ak ∈ A ak – 1 ∈ A

.

(13)

p 1 ( v l, k z l, k – 1, a k, a k – 1, ∆φ k )

where λ l, k is given in eq. (3). The decision rule is now given by 0 if β L < Λ k < β H . (17) 1 elsewhere In eq. (17), r is a parameter which is chosen to optimize the performance. A similar, but different, decision rule is used in [4] and we will use this as a comparison.

B. Vectorial diversity combining for 8DPSK decisions

This pdf may be used as a likelihood function for making optimum symbol decisions of the 8DPSK symbols. The likelihood function for making optimum bit decisions on each bit in the 3 bit 8DPSK symbol is obtained by averaging over the phase increments corresponding to bit “0” and “1”, respectively. For the most significant bit (msb) we obtain from Table 1, (14)

Γ msb,8DPSK ( 0 ) =

∑

The decision variable we propose for 2DASK is given as 1 - ( z rλ + z rλ ) , Λ k = ---------------------------------(16) r 2, k 2, k z 1, k + z 2, k r 1, k 1, k

∆φ k ∈ { 0, π ⁄ 4, π ⁄ 2, 3π ⁄ 4 }

p 8DPSK ( v l, k z l, k – 1, ∆φ k )

V k = arg ( z 1, k q v 1, k + z 2, k q v 2, k ) ,

(18)

where v l, k is defined in eq. (10) and q is a parameter that should be selected such that the performance is optimized. The symbol decision rule is now a traditional 8PSK detector for Gray encoded symbols. This decision variable is a simple generalization of the rule proposed in [4]; we have included the RSS weights.

C. Phase diversity combining for 8DPSK decisions

for decision “0” and a similar expression with a sum over the other four phases for decision “1”. With the help of Table 1 it is now straight forward to derive the likelihood functions for the other bit decisions.

C. 8DPSK bit decisions using phase only It is possible to decode the 8DPSK bits by using phase information only. This could e.g. be used if a limited receiver is applied for the 8DPSK decision. Still, however, a non-limited receiver must be used for the 2DASK decisions. A receiver equiped with a logarithmic amplifier is such an example. An alternativ decision variable in this case for the 2DASK decision is to directly use the amplitude in logarithmic scale, which is outside the scope of this paper. The decision variable is now v˜ l, k = arg v l, k ,

For 8DPSK we propose a vectorial addition of the decision variables from the branches given by

(15)

with v l, k given as in eq. (10). The derivation of the likelihood functions for this case follows the previous section except for the pdf in eqs. (11)-(13). Instead we need the pdf of the phase of a complex Gaussian variable and it can be found in e.g. [12]. The interested reader is referred to [10] for the details.

The order of addition and evaluating phase can be exchanged in eq. (19) above. This decision variable is used in [8] for DPSK and is given by 1 o v˜ - z + , (19) W k = ----------------------------------o z 1, k + z 2, k o 1, k 1 , k v˜ 1, k – v˜ 2, k + π z 2, k o v˜ 2, k + 2π ---------------------------------2π where v˜ l, k is defined in eq. (16), x is the largest integer less than or equal to x and the last term is needed due to the modulo 2π definition of phase [10]. The exponent o is a parameter to be optimized.

5. Simulated bit error probability A. Prerequisites In this section we present some simulated bit error probabilities. We will compare the performance of the proposed detectors with the detector proposed in [4], which is as simple as the RSS detectors considered in this paper.

100

100 ML, no diversity ML, 8DPSK phase only ML, 8DPSK amplitude and phase

10-1

Chow, Nix, McGeehan detector

10-2

bit error probability

bit error probability

10-1

10-3

10-4 RSS, no diversity RSS, 8DPSK vectorial combining RSS, 8DPSK phase combining RSS, selection combining

10-5

10

15

20

25

10-3

10-4

10-5

10-6 5

10-2

ML, no diversity ML, 8DPSK phase only ML, 8DPSK amplitude and phase Chow, Nix, McGeehan detector RSS, no diversity RSS, 8DPSK vectorial combining RSS, 8DPSK phase combining RSS, selection combining

30

35

40

10-6 10-4

mean received Eb/N0 per branch Figure 1 Bit error probability versus mean received signal to noise ratio per receiver branch when the maximum Doppler frequency is zero.

All simulations for the RSS combining detectors and the detector proposed in [4] are performed with 1 million independent trials. The simulations for the ML detectors are performed with fewer trials, but the confidence is still good at the bit error probabilities considered.

B. Coefficients used in the simulations Numerous simulations have been performed for the RSS combining detectors in order to find the best choice of parameters. The chosen parameters are ring ratio aH/aL=2.0, upper threshold βH=1.5, lower threshold βL=0.7, r=2.7, q=0 and o=2.4. It should also be mentioned that small changes in the parameters do not change the performance significantly.

C. Simulated average bit error probability In Figure 1 we show simulated average bit error probability versus mean received E b ⁄ N 0 per receiver branch, for the ML detector, the RSS combining detector and the detector proposed in [4]. The maximum Doppler frequency is here 0, i.e. the results are for very slow fading. We also show the performance for selection combining. The conclusion is that from a performance point of view the RSS combining detectors use the decision variables in an almost optimum way. The performance degradation due to a detector that use only phase information for its decisions on the 8DPSK bits is less than 0.1 dB. Selection combining leads to a performance penalty of around 1.5 dB. The performance gain due to diversity is almost 8 dB at a bit error probability of 1

10-3

10-2

10-1

normalized maximum Doppler frequency fdT Figure 2 Bit error probability versus normalized maximum Doppler frequency when the mean received signal to noise ratio approaches infinity.

percent. In Figure 2 we show the bit error probability versus normalized maximum Doppler frequency when E b ⁄ N 0 → ∞ . The conclusions are here the same as above. We can note that the diversity detector can sustain almost 3 times as large Doppler frequency as the single branch detector at a bit error probability of 1 percent.

D. Subchannels with different bit error probabilities Star 16QAM offers subchannels with different bit error probabilities, which may prove useful when transmitting bits with different sensitivity to errors. In Figure 3 we show the bit error probability for each individual bit in the 16QAM symbol and the average bit error probability of the bits mapped on 8DPSK. From this figure it is clear that the bit error probability of the least significant bit in the 8DPSK symbol is about a factor of two larger than the bit error probability of the other two bits in the 8DPSK symbol. The reason for this is that the least significant bit is always different for one of the neighbouring 8DPSK symbols while the probability for this to happen for the other two bits is 1 ⁄ 2 and these are the most likely errors which occur. It is also very clearly seen that the bit error probability for the 2DASK bit is another factor of two larger than the bit error probability of the least significant bit in the 8DPSK symbol. The difference between the smallest and largest bit error probability is therefore about a factor of 4. By changing parameters such as ring ratio, thresholds and exponents, these factors

with about 4 dB lower E b ⁄ N 0 both with and without diversity. With a real channel estimation algorithm we expect a few decibel degradation in slow fading and more in fast fading, which makes star 16QAM an attractive modulation method.

100 msb, 8DPSK middle bit, 8DPSK lsb, 8DPSK 2DASK average, 8DPSK

bit error probability

10-1

10-2

The maximum likelihood detector studied in this paper makes independent decisions on the bits mapped to phase and amplitude, respectively, in the star 16QAM symbol. Furthermore it is restricted to use the decision variables at the output of a differential detector in each diversity branch, which are used for the decisions by a single branch detector. There is no guarantee that this is the optimum diversity detector for star 16QAM with differential detection, and in fact the detector proposed in [4] has a slightly smaller bit error probability. It is likely that further performance improvements may be obtained by other diversity detectors for star 16QAM.

no diversity

10-3 diversity

10-4

10-5

7. References -6

10

5

10

15

20

25

30

35

40

mean received Eb/N0 per branch Figure 3 Bit error probability of the different star 16QAM subchannels versus mean received signal to noise ratio per receiver branch when the maximum Doppler frequency is zero.

could be changed to a certain extent [10]. In this way, star 16QAM offers at most 3 subchannels with different bit error probabilities, which may be used for unequal error protection.

6. Discussion and conclusions In this paper we study two methods for diversity reception of differentially encoded and detected star 16QAM. The channel is a multiplicative Rayleigh fading channel with additive white Gaussian noise. By using computer simulation, we show that the bit error probability of the maximum likelihood differential detector may be obtained in practice by a simple post diversity combining detector. The diversity gain is around 8 dB at a bit error probability of 1 percent where the signal to noise ratio is defined as mean received energy per bit per noise spectral density per diversity branch. The diversity detector is much less susceptible to maximum Doppler frequency. The error floor is almost a factor of 3 lower for the diversity detector when the bit error probability is in the order of 1 percent. Star 16QAM is also shown to offer at most 3 subchannels with different bit error probabilities, which may prove useful in communication systems which transmits bit of different importance. Some speech encoding methods have this property. Star 16QAM with differential encoding and detection offers a bit error probability of 1 percent at an E b ⁄ N 0 of around 21 dB without diversity and around 13 dB per receiver branch with 2 branch diversity. Assuming perfect channel estimation, including both phase and amplitude, with traditionally rectangular 16QAM, we find that the same bit error probability is obtained

[1] W.T. Webb, and L. Hanzo, Modern Quadrature Amplitude Modulation: Principles and Applications for Fixed and Wireless Channels, Pentech Press, 1994. [2] F. Adachi and M. Sawahashi, “Performance analysis of various 16 level modulation schemes under Rayleigh fading”, Electronics Letters, Vol. 28, No. 17, 13th August 1992, pp. 1579-1581. [3] Y.C. Chow, A.R. Nix and J.P. McGeehan, “Analysis of 16-APSK modulation in AWGN and Rayleigh fading channel”, Electronics Letters, Vol. 28, No. 17, 13th August 1992, pp. 1608-1610. [4] Y.C. Chow, A.R. Nix and J.P. McGeehan, “Diversity improvement for 16-DAPSK in Rayleigh fading channel”, Electronics Letters, Vol. 29, No. 4, 18th February 1993, pp. 387-389. [5] M. Sawahashi and F. Adachi, “Decision feedback differential detection of multilevel amplitude/phase shift keying”, Proceedings JTC-CSCC’93, July 26-28, 1993, Nara, Japan, pp. 217-222. [6] F. Adachi and M. Sawahashi, “Decision feedback differential detection of 16-DAPSK signals”, Electronics Letters, Vol. 29, No. 16, 5th August 1993, pp. 1455-1457. [7] T. Suzuki and T. Mizuno, “Multiple-symbol differential detection scheme for differential amplitude modulation”, Proceedings IZS’94, Zurich, Switzerland, March 1994, pp. 196-207. [8] M. Ikura and F. Adachi, “Postdetection phase combining diversity”, IEEE Transactions on Vehicular Technology, Vol. 43, No. 2, May 1994, pp. 298-303. [9] F. Adachi, “Postdetection optimal diversity combiner for DPSK differential detection”, IEEE Transactions on Vehicular Technology, Vol. 42, No. 3, August 1993, pp. 326-337. [10] A. Svensson, “On differentially encoded star 16QAM with differential detection and diversity”, To appear in IEEE Transactions on Vehicular Technology. [11] J.G. Proakis, Digital communications, 3rd edition, McGraw-Hill 1995. [12] R.F. Pawula, S.O. Rice and J.H. Roberts, “Distribution of the phase angle between two vectors perturbed by Gaussian noise”, IEEE Transactions on Communications, Vol. COM-30, No. 8, August 1982, pp. 1828-1841.