Successive Interference Cancellation Schemes in Multi-Rate DS/CDMA Systems Ann-Louise Johansson and Arne Svensson Chalmers University of Technology Department of Information Theory S-412 96 Gothenburg Sweden e-mail
[email protected]/
[email protected] Abstract In this paper we propose successive interference cancellers which will work in a variable data rate environment. Firstly, a successive interference cancellation (IC) scheme for M-level rectangular QAM in DS/CDMA systems is analysed. Secondly, the performance under Rayleigh fading is analysed for single modulation systems and systems employing different methods for handling multiple data rates. One suggestion for handling multi-rate systems is to use mixed modulation, which imply that each user chooses modulation format according to need. Another suggestion is to use parallel channels, where each user transmits over one or several synchronous channels. We show that the successive interference canceller, for mixed modulation or parallel channel systems, gives a considerable increase in performance and flexibility compared to both single modulation systems and mixed modulation or parallel channel systems employing a conventional detector.Introduction
1 Introduction In the future users will demand mobile telephone systems to be able to handle many more services than speech, like e.g. facsimile, Hi-Fi audio and computer data, which is not possible today. To achieve this we need to use a multiple-access method which is flexible and has the prospect of capacity increases and being able to handle variable data rates. Recently Code Division Multiple Access (CDMA) has been suggested to be a multiple-access method able to stand these requirements. The first company to propose this technique for a practical system is Qualcomm Inc in the USA, who are working on a CDMA digital cellular radio communication system. A Direct-Sequence CDMA (DS/CDMA) system has several unique features. Some of them are spectrum sharing, rejection of multipath signal components or possibility to utilize them for recombining [1] and frequency reuse factor of one in a cellular scenario [2]. These features are highly desirable, though a CDMA system employing a conventional, matched filter, detector is interference limited, which directly determines the system capacity. The conventional detector is optimal in a single-user channel corrupted only by additive white Gaussian noise (AWGN). The presence of a number of users in the system introduces multiple-access interference (MAI), since the signature sequences used are not perfectly orthogonal, which leads to an irreducible error probability. In a mobile radio scenario the transmitters move in relation to the
receiver and the energies of the received signals are to be neither equal nor constant. In this situation the conventional detector fails to demodulate weak users, even when the cross-correlation between the signals is relatively low. This is known as the near/far problem. One way to combat this problem is to use stringent power control [2]. Another approach is to use more sophisticated receivers which are near/far resistant. Recently a lot of attention has been directed to the area of multi-user detectors, which has the prospect of mitigating the near/far problem and by cancelling the MAI also increasing the total system capacity. The research in this area was trigged by Verdu [3], who worked on the optimal multi-user detector. This detector is very complex and that has initiated further research in the area of suboptimal, lower complexity, detectors both using parallel detection [4]-[8] and serial or successive interference cancellation (IC) [9]-[13]. The motivation for our work is to evaluate an efficient detector for a multi-user and multi-rate DS/CDMA system. A method to handle multiple data rates is to let the users use different forms of modulation [14]. A user could, depending on the need, choose between using e.g. BPSK, QPSK or 16-QAM modulation. Therefore, an IC scheme for M-ary QAM is analysed in this paper, based on the IC scheme for coherent BPSK modulation derived by Patel and Holtzman [9]-[11]. The analysis is extended to cover systems where the users employ different QAM formats. Another approach to handle multiple data rates is to let each user transmit over one or several parallel channels according to requirements [14]. This can of course also be used in combination with different modulation formats. The operation of the IC scheme is made in the following manner: The signal amplitude of each user is estimated from the output of a linear correlator, where we correlate the received signal with each user’s signature sequence. It should be noted that this will not give a perfect estimate of the amplitude though the scheme will be simple. We then rank the users in decreasing order of their received powers, which are obtained using these correlator outputs. The user’s signals are decoded and cancelled from the received signal successively starting with the strongest user. If we consider an uplink, corresponding to a situation when we are interested in all received signals, then we will use a scheme where all the users are decoded and cancelled. On the other hand, if we consider a downlink we would perform the successive IC scheme until the desired signal had been decoded. Due to limited space we will only give the main performance results in this paper. All derivations may be found in [15] for the interested reader. We will consider the coherent case of demodulation and frequency-nonselective fading. The performance measure used is average bit error probability. We assume perfect knowledge of the phase, the time delay and the channel gain for each signal. Accordingly, we also assume perfect power ranking in the performance analysis.
2 System Model and Decoder Structure We consider a system model for a system utilizing square lattice QAM, where the received signal, for a K user system, is
K
∑ αk
r ( t) =
k=1
2E 0 I --------- d kI ( t – τ k ) c k ( t – τ k ) cos ( ω c t + φ k ) + T (1)
2E Q α k --------0- d kQ ( t – τ k ) c k ( t – τ k ) sin ( ω c t + φ k ) + n ( t ) T which is the sum of all the transmitted signals embedded in AWGN. d kI / Q ( t ) is a sequence of A kI ,/ Q l amplitude, rectangular pulses of duration T. T is the inverse of the symbol rate, assumed to be equal for all users. A kI ,/ Q l is the information-bearing signal amplitude of the quadrature carriers for the k th user’s l th symbol element, which together generate M equiprobable and independent symbols. They take the discrete values (2)
A kI ,/ Q l ∈ { – M + 1, – M + 3, …, M – 1 }
since it requires M signal amplitude levels for the In-phase (I) and Quadrature (Q) components, respectively, to form a signal constellation for M-ary QAM. 2E 0 is the energy of the signal with lowest amplitude. c kI / Q ( t ) is the k th user’s signature sequence, which is used for spreading the signal in the in-phase or the quadrature branch. It consists of a sequence of, antipodal, unit amplitude, rectangular pulses of duration T c . The period of all the users’ signature sequences are N = T ⁄ Tc , so there is one period per data symbol. We assume without loss of generality that the signature sequences have power equal to one. τ k is the time delay and φ k is the phase of the k th user. In the asynchronous, though symbol-synchronous, case they are i.i.d. uniform random variables over [ 0, T ) and [ 0, 2π ), respectively. Both parameters are assumed to be known in the analysis. ω c represents the common centre frequency, α k represents the channel gain and n ( t ) is the AWGN with two-sided power spectral density of N 0 ⁄ 2 . c iI ( t – τ i ) cos φ i cos ω c t
LPF
δI ( t)
Z iII, l + 1 τi + T --- ∫τ dt T i – c iQ ( t – τ i ) sin φ i
+
S iI, l
IQ 1 τ +T --- ∫τi dt Z i, l T i I c i ( t – τ i ) sin φ i
r ( t)
LPF
δQ ( t)
1 τ +T --- ∫τi dt Z QI T i i, l Q c i ( t – τ i ) cos φ i
sin ω c t
τi + T dt τi
1 --- ∫ T
Z iQQ ,l
Decision Device
( )2
( S i, l ) 2
+ ( )2 + +
+
S iQ, l
Decision Device
Figure 1. M-ary QAM DS/CDMA receiver
Figure 1 shows the structure of the i th user’s receiver, when detecting the l th symbol. Let us consider symbol zero of the first user. Combining the correlator outputs we obtain the decision
variables S 1I , 0 and S 1Q, 0 for the I and Q components as [15] S 1I , 0 = ( Z 1II, 0 – Z 1QI , 0) =
E -----0-α 1 A 1I , 0 + N 1I 2T
IQ S 1Q, 0 = ( Z 1QQ , 0 + Z 1, 0 ) =
E -----0-α 1 A 1Q, 0 + N 1Q 2T
(3)
where N 1I / Q is a noise term of the decision variable including both Gaussian noise and noise caused by interference. For convenience and reasons explained later we will write N kI / Q instead of N kI ,/ Q 0 for the noise term of the decision variable of the zeroth symbol. It can be shown [15], [16] with the help of trigonometric functions that N 1I / Q is given by I N1 = NQ = 1
E -----02T
K
1
∑ IkII, 1 ( AkI , AkQ, τk, 1, φk, 1 ) + --2- [ n1II – n1QI ]
k=2
E -----02T
(4)
K
1
I Q - [ n QQ – n 1IQ ] ∑ IkQQ , 1 ( A k , A k , τ k, 1, φ k, 1 ) + -2 1
k=2
where the n 1 components are uncorrelated Gaussian random variables and the sums of the I kII, 1 and I kQQ , 1 components are interference due to the resulting K-1 users. The components caused by the k th user in the expression above are defined by Q I kII, 1 ( A kI , A kQ, τ k, 1, φ k, 1 ) = Γ kII, 1 ( A kI , τ k, 1 ) cos φ k, 1 + Γ kQI , 1 ( A k , τ k, 1 ) sin φ k, 1
(5)
I Q IQ I QQ Q I kQQ , 1 ( A k , A k , τ k, 1, φ k, 1 ) = Γ k, 1 ( A k , τ k, 1 ) ( – sin φ k, 1 ) + Γ k, 1 ( A k , τ k, 1 ) cos φ k, 1
(6)
and
where φ k, i = φ k – φ i , τ k, i = τ k – τ i , A kI = [ A kI , 0 , A kI , 1 ] and A kQ = [ A kQ, 0 , A kQ, 1 ] . The two functions Γ kII, 1 and Γ kQI , 1 are easily shown to be [15], [16] T
αk 1 II I - ∫ ∑ A kI , l p T ( t – τ k, 1 – lT ) c kI ( t – τ k, 1 ) c 1I ( t ) dt Γ k, 1 ( A k , τ k, 1 ) = ----T 0l = 0 T 1 α Γ QI ( A Q, τ ) = -----kA Q p ( t – τ k, 1 – lT ) c kQ ( t – τ k, 1 ) c 1I ( t ) dt k, 1 k k, 1 T ∫ ∑ k, l T
(7)
0l = 0
where p T ( t ) is a unit amplitude, rectangular pulse of length T and the delay τ k is assumed to QQ be shorter than τ 1 . The reason for this is explained later. Γ kIQ , 1 and Γ k, 1 are obtained from similar expressions with small changes in indices [15], [16].
3 Successive Interference Cancellation Scheme with M-ary QAM A receiver with IC is shown in Figure 2. The received signal is correlated with each signature
sequence and the correlator outputs are used both for deciding which user is the strongest and in the cancellation of that users signal. The strongest user, the one with the largest α k , is decoded first and cancelled at baseband from the received signal. The detector is a coherent detector and it is assumed to perform ideally. Subsequently all the users are decoded and cancelled in decreasing order of their powers. Let us assume we have the means of deciding which user is the strongest and therefore also the one most likely to be decoded correctly. A suggestion of such a scheme is given in [11], [15].
c iQ ( t – τ i ) sin φ i
+
cos ω c t
LPF
δI ( t)
c iI ( t – τ i ) cos φ i
–
+ +
S 1I , 0
Detector 1 S 1Q, 0
r ( t)
Detector 2
LPF sin ω c t
δQ
S KI , 0
( t)
+ +
–
Detector K S KQ, 0
+
S iI, 0 S iQ, 0
Select Max & Decode S iI, 0
S iQ, 0 c iI ( t – τ i ) ( – sin φ i ) c iQ ( t – τ i ) cos φ i
Figure 2. M-ary QAM receiver with Interference Cancellation
Without loss of generality we consider the decision of A kI ,/ Q 0 at time τ k + T corresponding to symbol element 0. All the symbol elements prior to the zeroth have already been decoded. We assume also without loss of generality that user 1 has the strongest signal and, accordingly, it is decoded first and cancelled from the composite signal. Assume that we have a situation where the strongest user, in this case user 1, has a time delay τ 1 which is shorter than any other user’s time delay as e.g. user 2’s in Figure 3. Then it would not be enough to cancel only symbol element 0 of user 1 from the composite signal to reduce the noise caused by interference since the decision variable of user 2’s zeroth symbol will also include interference from user 1’s first symbol. To overcome this we decode and cancel user 1’s first symbol element too before we continue the IC scheme for the second strongest user. Therefore, without loss of generality, we assume in the analysis τ 1 > τ 2 > … > τ K . In this symbol-synchronous case we are decoding symbol element 0 and since we assume τ k < τ 1 , the interference from the k th user in [ τ 1 , τ 1 + T ) originate from symbol element 0 in [ τ 1 , τ k + T ) and symbol element 1 in [ τ k + T, τ 1 + T ) . We use the decision variables S 1I , 0 and S 1Q, 0 defined in Eq. (3) to estimate user 1’s baseband signal for symbol element 0 and cancel it from the composite signal. This cancellation is not
0
0
1
T
τ1 + T
τ1
User 1
τ2
2
2T
3T
τ 1 + 2T
τ2 + T
τ 2 + 2T
User 2 τK
τK + T
τ K + 2T
User K Figure 3. Asynchronous system
perfect since the signature sequences are not orthogonal and the correlator output also contains Gaussian noise. Considering the general case, the resulting baseband signal for the I channel after the h th cancellation will be as follows [15], [16] δ hI , 0 ( t ) = δ hI – 1, 0 ( t ) – S hI , 0 p T ( t – τ h ) c hI ( t – τ h ) cos φ h – S hQ, 0 p T
(t –
τ h ) c hQ
(8)
( t – τ h ) sin φ h
When h is 1 the term δ 0I , 0 ( t ) corresponds to the remaining baseband signal after cancellation of all symbol elements prior to the zeroth, and consequently we will get δ 1I , 0 ( t ) after cancelling user 1’s zeroth symbol. Let user 2 be the second strongest user and also the user with the second longest time delay. Now there are only K-2 interfering users left since the strongest user has been cancelled from the composite signal. We proceed in the same manner for the second strongest user as well as for the subsequent users. The decision variable before the h th cancellation is then given by [15], [16] S hI ,/ Q0 =
E -----0-α h A hI ,/ Q0 + N hI/ Q 2T
(9)
Hence, there are now h-1 cancelled and K-h+1 remaining symbol elements 0. Therefore, the total noise component for the h th user in the I channel is N hI
=
E -----02T
K
∑ k = h+1 h–1
–
∑ NjI JjII, h ( τj, h, φj, h ) + NjQ JjQI, h ( τj, h, φj, h )
j=1
and
1 I kII, h ( A kI , A kQ, τ k, h, φ k, h ) + --- [ n hII – n hQI ] – 2 (10)
N hQ
E -----02T
=
K
∑ k = h+1 h–1
–
1 QQ I Q - [ n – n hIQ ] – I kQQ , h ( A k , A k , τ k, h, φ k, h ) + -2 h (11)
∑ NjI JjIQ, h ( τj, h, φj, h ) + NjQ JjQQ , h ( τ j, h, φ j, h )
j=1
where the first sum consists of noise caused by the remaining interfering users, the second term is white Gaussian noise and the last sum is the resulting noise caused by imperfect cancellations. The correlation terms J 1II, 2 and J 1QI , 2 , in Eq. (10) above, are defined as T
J 1II, 2
1 ( τ 1, 2, φ 1, 2 ) = --- ∫ c 1I ( t – τ 1, 2 ) c 2I ( t ) cos ( φ 1, 2 ) dt T 0 T
(12)
1 Q - c ( t – τ 1, 2 ) c 2I ( t ) sin ( φ 1, 2 ) dt J 1QI , 2 ( τ 1, 2, φ 1, 2 ) = -T∫ 1 0
In these expressions the correlation of c 2I ( t ) is with noise caused by imperfect cancellation of symbol element -1 and 0 of the first user, since τ 2 < τ 1 . See Figure 4, where shaded lines indicate cancelled symbol elements.
-T
-1
0
0
τ1 – T
User 1
τ1 τ2
τ2 – T
1
T
2T
τ1 + T τ2 + T
User 2 Figure 4. Cross-correlation between users
We are also considering a frequency-nonselective, slowly fading channel which implicate that the interference power can be regarded as equal for two consecutive symbol elements. Therefore we do not distinguish between N 1I , –1 and N 1I , 0 in Eq. (10) and (11). QQ Similarly the correlation terms, J 1IQ , 2 and J 1, 2 , in Eq. (11) above are defined as T
J 1IQ ,2
1 ( τ 1, 2, φ 1, 2 ) = --- ∫ c 1I ( t – τ 1, 2 ) c 2Q ( t ) ( – sin ( φ 1, 2 ) ) dt T 0 T
1 Q - c ( t – τ 1, 2 ) c 2Q ( t ) cos ( φ 1, 2 ) dt J 1QQ , 2 ( τ 1, 2, φ 1, 2 ) = -T∫ 1 0
(13)
3.1 Ranking of the users In this paper we do not consider algorithms for ranking the power of users. Instead throughout the analysis we assume perfect ranking of the users. A suggestion of how to do the ranking in the detector is found in [11] and [15].
4 Performance Analysis on a Stationary Channel Evaluating the performance of the IC scheme will be done by applying Gaussian approximation [9]. The noise components, caused by interference from the other users in the system, in Eq. (10) are therefore modelled as independent Gaussian noise. We have chosen to use the Gaussian approximation partly since it is commonly used and partly because we did not know any other practical way to evaluate the performance. By using a Gaussian approximation an increase in noise and interference variance immediately leads to an increase in error probability. It is likely that this will occur also for the true distribution and we are therefore convinced that relative performance will not change significantly. Absolute performance is likely to be too optimistic though [17]. Additionally, since we do not consider any fading in this section, the channel gain, α k , will be constant. We also assume that power control is used for distance and shadow fading and consequently, in this case, all the α k ‘s will be equal.
4.1 Performance Analysis of QAM IC Scheme We start with calculating the variance of the I and Q channel decision variables conditioned on α k , i.e. η hI/ Q = Var N hI/ Q α h
(14)
It is easy to show that all the random variables in N hI/ Q are independent and with zero mean. Consequently we can model N hI/ Q as independent Gaussian random variables with zero mean and variance η hI/ Q . When using the Gaussian approximation it is easy to obtain the probability of error from the theory of single transmission of QAM signals over a AWGN channel [1]. The symbol error rate with ideal coherent detection can then be expressed as Peh = 1 – ( 1 – PehI ) ( 1 – PehQ )
(15)
where PehI/ Q is the probability of error for the transmission over the I channel, respectively the Q channel, given by [1] M–1 PehI/ Q = 2 ------------------ Q ( ρ hI/ Q ) M
(16)
where the Q-function is the complementary Gaussian error function. ρ hI/ Q in Eq. (16) is a signal-to-noise ratio for the I and Q channel defined as
ρ hI/ Q
E -----0-α h 2T = ----------------η hI/ Q
(17)
for the h th user in the ideal coherent case. The expression for the variance in Eq. (17) depends on which kind of sequences we use. For deterministic sequences the variance in Eq. (14) yields η hI
E0 N M–1 = -----0- + -------------- ⋅ --------------3 4T 12N 3 T
K
∑
α k2 ( r kII, h + r kQI , h) +
k = h+1
(18)
h–1
1 ---------3 ∑ η jI r jII, h + η jQ r jQI ,h 6N j=1
where η hQ will have a similar expression with some small changes in indices. The different r k, h terms in Eq. (18) are average interference defined in [19]. For random sequences [18] we get η hI
N M – 1 E0 = -----0- + -------------- ⋅ ----------- 3NT 3 4T
h–1
K
∑ k = h+1
α k2
2 + ------- ∑ η jI 3N
(19)
j=1
where we have used the equality η jI = η jQ . The ranking of the users is completely random since all the α k ‘s are equal. We then obtain the average probability of symbol error for each user if we take the average of all symbol error rates (Eq. (15)) calculated for each stage of cancellation, i.e. 1 Pe = ---K
K
∑ Pek
(20)
k=1
To be able to compare the results obtained for QAM users with BPSK users we need to calculate the bit error rate (BER). We assume a Gray-encoded version of M-ary QAM. When the probability of symbol error is sufficiently small, the probability of mistaking a symbol for the adjacent one vertically or horizontally is much greater than any other possible symbol error [20]. Under this assumption the BER is easily derived from the symbol error rate, stated in Eq. (20), through 1 Pb QAM ≈ ------------- Pe log M
(21)
where log is logarithm base two. This approximation of the BER is very good for E b ⁄ N 0 ≥ 10dB , where E b is energy per bit, which is the region of interest.
4.2 Numerical Examples The performance of systems over a Gaussian noise channel is of minor interest, since the basis of the scheme is that the users are to be ranked in order of their powers. Assuming a stationary channel results in a system with equally strong users. The order of cancellation is therefore performed randomly. Although, consider a group of pure BPSK, QPSK and 16-QAM systems where the performance is measured in average BER, as a function of E b ⁄ N 0 . If we compare groups of systems with the same total transmitted bit rate, for example 20 BPSK, 10 QPSK and 5 16-QAM users, we would find that the performance of 16-QAM systems on a stationary
channel is considerably poorer than for BPSK and QPSK systems, especially for a low number of users [15], [16]. Though, if we compare the results of three single user systems with the same throughput, both in the case of IC and when employing a conventional matched filter detector, all systems employing IC show a reduction of BER by a factor of 5-10 times.
5 Performance Analysis of QAM IC Scheme under fading In the case of fading, we consider a system where the K users are received through independent, frequency-nonselective slowly fading channels. This is a suitable model for systems where the information bandwidth is smaller than the coherence bandwidth, which specifies the bandwidth over which all the frequency components behave similarly. For a wideband signal the model apply in sub-urban areas where the local mean of the mobile radio signal can be said to be constant over a small area, with slow variations when the receiver moves [21]. When the symbol period, T, is less than the coherence time, the multiplicative process, caused by fading, may be regarded as constant during at least a few signal intervals. This can be considered the condition for the term slowly fading [1]. The channel has to fade sufficiently slow for making it possible to estimate the phase shift, φ k , from the received signal of the k th user adequately. This is a requirement for coherent detection [1]. Operating an IC scheme for M-ary QAM also demands that the fading is slow, since the channel gain, α k , has to be estimated for correct decoding of the symbols and the ranking of the users. No instantaneous power control is applied, only average power control which takes care of shadowing and distance attenuation.
5.1 IC Scheme for QAM under Single-Path Rayleigh Fading In this paper we will use the same method, for analysing the single-path Rayleigh fading channel, as was used in [11] for BPSK modulation. The equations for the noise variance and error probabilities obtained in the previous section were all conditioned on α k , which was assumed to be perfectly estimated. The amplitudes decide the order of cancellation, thus the order will change continuously with fading. The unconditioned error probability of the I channel for each stage of cancellation is easily obtained from the conditioned probability of error in Eq. (16) as follows ∞
Pˆ e hI =
∫ PehI fα
h
( x ) dx
(22)
0
where f α ( x ) is the distributions of the ordered amplitudes. The disordered amplitudes are h assumed to be Rayleigh distributed with unit mean square value. That is, the average received power from all the users is equal. This is the consequence of assuming perfect power control for shadowing and long term fading. From this assumption f α ( x ) is obtained by using order h statistics [22]. In the same manner as in Eq. (17) we here define the signal-to noise ratio, ρ hI , for the h th user in the I channel as E -----0-α h 2T ρ hI = -------------------------E α [ η hI ] k
(23)
where E α [ η hI ] is the expected value with respect to α k of the conditioned variance, stated in k Eq. (19) for random sequences. The variance is approximated as a sum of independent Gaussian random variables with zero mean and the expected variance is given by N M – 1 E0 = -----0- + -------------- ⋅ ----------- 3NT 3 4T
E α [ η hI ] k
h–1
K
∑
E α [ α k2 ] k
k = h+1
2 + ------- ∑ E α [ η jI ] k 3N
(24)
j=1
The amplitudes are ordered in non-decreasing order and K is the total number of users. For random sequences Pˆ e hI = Pˆ e hQ and the unconditioned symbol error rate can be written as in Eq. (15). Taking the average of all symbol error rates at all stages of cancellation yields the average probability of symbol error in the same way as given in Eq. (20). This is a proper performance measure of each user, since, as stated earlier, the order of cancellation will change with fading.
5.2 Numerical Examples The average BER of different single user systems, 40 BPSK, 20 QPSK and 10 16-QAM users, with the same throughput under Rayleigh fading is shown in Figure 5. The length of the ran0
10
Processing Gain = 127 Random Sequences −1
10
Conv
Pb −2
10
BPSK (40) QPSK (20) 16−QAM (10) Single BPSK Single 16−QAM
IC
−3
10
0
5
10 15 Eb /N0 (dB)
20
25
Figure 5. Performance of single modulation systems with and without IC under Rayleigh fading
dom sequences are 127. The result is compared to both a single user BPSK system and a single user 16-QAM system under fading and corresponding systems employing a conventional detector. The performance of the BPSK and QPSK systems are almost identical and it is difficult to distinguish between the different curves. The performance of the 16-QAM system is inferior compared to the BPSK and QPSK systems, but the performance should be compared with the single 16-QAM user bound, which the 16-QAM IC system is close to. Overall, the performance for all the systems using IC is considerably better compared to the systems employing the conventional detector. The results also show that up to about 18 dB of E b ⁄ N 0
the QPSK and BPSK systems perform almost equally well as in the case of transmission of a single BPSK user, but thereafter the curves level out.
6 Performance Analysis of Mixed Modulation Systems One way to handle multi-rate systems is to let each user choose a modulation format in correspondence with required transmitted data rate [14]. In the sequel we will evaluate the performance of a system where the users employ different forms of modulation, e.g., a combination of BPSK, QPSK and 16-QAM users.
6.1 DS/CDMA Mixed System Model Description We consider a system where we have K 1 BPSK, K 2 QPSK and K 3 16-QAM users. To make a comparison between different forms of modulation we let the transmitted bit energy, E b , be equal for all users independent of the modulation format used. We then rewrite the energy E 0 as a function of E b , given as 3 log M E 0 = E b -----------------------2 ( M – 1)
(25)
which is valid for M-ary QAM. For BPSK E 0 = E b . If we define M 1 = 4 (QPSK) and M 2 = 16 (16-QAM) we can express the signal-to-noise ratio, ρ hI , conditioned on α k , for the h th QPSK user as follows E b α h2 3 log M 1 ------------- ⋅ -------------------------2T 2 ( M 1 – 1 ) I ρ h = -----------------------------------------------η i + η hI + η mI
(26)
where i, h-1 and m specifies the number of cancelled BPSK, QPSK and 16-QAM users, respectively. η i is the expression for the variance of the BPSK users employing random sequences [9] and η hI and η mI is the variance for QPSK, respectively 16-QAM, users defined in Eq. (19). Rewriting Eq. (26) we get
ρ hI
= αh
M1 – 1 N0 2 ---------------- ----------------------- + ------2 3 E b log M 1 3N 2 ( M1 – 1) 1 ------------------------- ⋅ ------3 log2 M 1 3N
∑
α k2
j = i+1
2 ( M 1 – 1 ) log2 M 2 1 -------------------------------------------- ⋅ ------3N 3 log2 M 1
h–1
2 + ------- ∑ ( ρ jI ) –2 + 3N
α k2
k = h+1
K1
∑
K2
j=1
i 1 2 ( M1 – 1) + ------- ⋅ ------------------------ρ –2 + 3N 3 log2 M 1 ∑ j
(27)
j=1
K3
∑ k = m+1
( M 1 – 1 ) log2 M 2 2 m 2 - ⋅ ------( ρ jI ) –2 α k + ---------------------------------------( M 2 – 1 ) log2 M 1 3N ∑
–1 / 2
j=1
where we find the noise caused by interference from QPSK users on the first line, from BPSK users on the second line and from 16-QAM users on the last line of the equation. The expression for BPSK users is easily obtained in the same manner [15] and for 16-QAM users the
expression will be similar to the one in Eq. (27) with some changes in indices.
6.2 Performance Analysis on a Stationary Channel To evaluate the performance of a mixed modulation system we use the Gaussian approximation. We assume that the noise N hI/ Q of the QPSK user is Gaussian with zero mean and variance η hI/ Q and analogous for the BPSK and 16-QAM users. To calculate the total BER for the system we first calculate the BER for each user. To get the total BER for the whole system we weight together each users’ BER in the following way
∑ PbiBPSK R1 + ∑ PbhQPSK R2 + ∑ PbmQAM R3
i h m Pbtot = -----------------------------------------------------------------------------------------------------------K1 R1 + K2 R2 + K3 R3
(28)
where R 1 , R 2 and R 3 are the data rates for the BPSK, QPSK and 16-QAM users respectively.
6.3 Mixed Modulation Systems under Rayleigh Fading In the case of Rayleigh fading we follow the same procedure using order statistics as in “IC Scheme for QAM under Single-Path Rayleigh Fading” in section 5.1. The difference is that now we have a mixed system and the amplitudes are ordered independent of modulation format. Using Eq. (16), (22) and (23) together with Eq. (27) we can calculate the performance for each QAM user of the system. Calculating the performance for BPSK users is done in a similar manner [15].
6.4 Numerical Examples The average performance of a mixed system with 20 BPSK, 10 QPSK and 5 16-QAM users with and without IC is shown in Figure 6. The length of the random sequences are 127. The amplitudes in each mixture have been distributed among the users independently of modulation format and the result is an average over 100 mixtures. The average BER for each kind of users is also shown in the graph together with the single user BPSK bound. Figure 7 depicts the same mixed system as in Figure 6, but in this graph we compare the result with three single modulation systems with the same throughput. Inspecting the two graphs we can see that the average performance for the QPSK and BPSK users is slightly better in the case of 35 mixed users compared to their corresponding single modulation systems. On the other hand, the average performance of the 16-QAM users is better for the single modulation system than for mixed system. This can be explained by the fact that the corresponding number of users in the mixed system is lower compared to the BPSK and QPSK single modulation systems but higher for the 16-QAM systems. (This is if we count QAM users as two because of the I- and Q-branch.) Furthermore we have to contemplate the fact that the average is made over a finite number of mixtures, because of the very time consuming calculations, which may slightly affect the result.
0
10
Processing Gain = 127 Random Sequences −1
Mix Conv
10 Pb
−2
10
IC
Av. BPSK Av. QPSK Av. 16−QAM Mix (20/10/5) Single BPSK
−3
10
0
5
10 15 Eb /N0 (dB)
20
25
Figure 6. Average BER of a mixed modulation system with and without IC under Rayleigh fading 0
10
Processing Gain = 127 Random Sequences −1
10
Mix Conv
Pb −2
10
BPSK (60) QPSK (30) 16−QAM (15) Mix (20/10/5) Single BPSK
IC
−3
10
0
5
10 15 Eb /N0 (dB)
20
25
Figure 7. Average BER of a mixed system with and without IC and single modulation systems with IC under Rayleigh fading
7 Performance Analysis of Parallel Channel Systems We have previously discussed the possibility to handle multi-rate systems by the means of employing higher modulation formats. Another alternative is to use parallel channels for transmission of information [14]. We simply let a user send simultaneously over as many channels
as required for a specific data rate. In this kind of system there are going to be a larger number of interfering signals, though the synchronous signals corresponding to each user will have considerably lower cross-correlation than asynchronous signals.
7.1 Parallel Channels We consider first a system with purely synchronous transmission. We have ∆ parallel channels with exactly the same channel parameters since they belong to the same user. In other words, the relative time delay and phase between the channels are equal to zero. For a detailed analysis of the following steps see [15]. We use the expression for the total noise component given in Eq. (10) for M-ary QAM, in the case when τ k, h and φ k, h both are zero. Taking the variance we get η hI
N M – 1 E0 = -----0- + -------------- ⋅ ------------2- 3 4T 2TN
∆
h–1
∑
α i2
( θ iII, h
i = h+1
( 0) ) 2
1 + -----2- ∑ η jI ( θ jII, h ( 0 ) ) 2 N
(29)
j=1
where θ iII, h is the periodic cross-correlation function [19]. In the case of BPSK modulation the variance of the total noise component will be as follows Eb N η h = -----0- + ------------24T 2TN
∆
∑
h–1
α i2 θ i2, h
1 ( 0 ) + -----2- ∑ η j θ j2, h ( 0 ) N
(30)
j=1
i = h+1
The variance stated in Eq. (29) and Eq. (30) are for deterministic sequences. We obtain the equations for random sequences using the fact [17] that E [ θ i2, h ( 0 ) ] = N . These can be found in [15] and [9].
7.2 Combination of Synchronous and Asynchronous Transmission We consider now a system where we have K users and where each user, k, transmits over ∆ k channels. Thus, we will have a system with a total number of information-bearing channels equal to K
κ =
∑ ∆k
(31)
k=1
Tests have shown that successive cancellation among synchronous signals, when employing deterministic sequences, does not improve the performance very much. Hence the term that specifies the noise due to cancellation of parallel signals may be excluded and instead we consider a receiver where each user’s parallel signals are decoded and cancelled simultaneously. The complexity of the receiver will not increase considerably since the decision variables for all signals have to be generated anyway to determine which signal is the strongest. Futhermore the channel parameters, τ h and φ h , are equal for parallel channels. Combining Eq. (18) and Eq. (29) we can write the noise variance for the g th signal of the h th user in a M-ary QAM system with both asynchronous and synchronous transmission as
η gI , h
E0 M–1 = -------------- ⋅ --------------3 12N 3 T M – 1 E0 - ⋅ ------------2- ------------3 2TN
κh – 1
κ
∑
k = κh + 1 ∆h
∑ i = 1 i≠g
1 η I r II + η jQ r jQI α k2 ( r kII, h + r kQI ,h + , h ) + --------6N 3 ∑ j j, h j=1
N α i2 ( θ iII, h ( 0 ) ) 2 + -----04T
(32)
where κ h – 1 is the total number of interfering asynchronous signals, relative to the signals transmitted by user h , which have been cancelled, i.e. h–1
κh – 1 =
∑ ∆k
(33)
k=1
κ – κ h are the remaining interfering asynchronous signals.
7.3 Synchronous and Asynchronous Transmission under Rayleigh Fading When we transmit information on parallel channels under Rayleigh fading the parallel channels will be affected by the same channel parameters. That is the delay, the phase shift and the amplitude will be the same. Since synchronous channels interfere less with each other we have not considered successive interference cancellation of parallel channels; instead they are all detected and cancelled simultaneously. The performance of a system with parallel channels is analysed using the same method with order statistics as described earlier. The difference in this case is that we order the K user, each one with ∆ k parallel channels, and then assign the same pdf f α ( x ) and mean square value E α [ α k2 ] to all the ∆ k channels of user k. k
k
7.4 Numerical results The performance of systems employing parallel channels under Rayleigh fading are shown in Figure 8. Systems with K users and 2, 3 or 4 parallel channels per user are compared with asynchronous systems with 2 ⋅ K , 3 ⋅ K and 4 ⋅ K users. K is in this case equal to 10. One user’s all parallel signals are detected and cancelled simultaneously and the results show that the performance of the parallel channel systems employing Gold sequences is very close to the performance of asynchronous systems (the curves are almost not distinguishable). This is caused by the fact that the interference from the strongest interferer will be larger in the case of parallel channels since that user will send over ∆ k channels instead of one. This will counterbalance the improvement caused by the good cross-correlation properties of Gold sequences in the synchronous case. In Figure 9 we compare the average performance of 15 QPSK users with two parallel channels each, with that of 15 16-QAM users’. The result show that QPSK users employing parallel channels perform better than 16-QAM users.
8 Discussion and Conclusions We have demonstrated the use of M-level rectangular QAM with successive IC in single mod-
0
10
Processing Gain = 127 Gold Sequences −1
10
−2
Pb 10
Single BPSK QPSK (20,30,40) QPSK (10), P=2,3,4
−3
10
−4
10
0
5
10 15 Eb /N0 (dB)
20
25
Figure 8. Average BER of parallel channel systems and asynchronous QPSK systems (employing Gold sequences) with IC under Rayleigh fading 0
10
Processing Gain = 127 Gold Sequences Conv
−1
10
−2
IC
Pb 10
−3
10
QPSK (15), P=2 QPSK (30) 16−QAM (15) Single BPSK
−4
10
0
5
10 15 Eb /N0 (dB)
20
25
Figure 9. Average BER of a parallel channel QPSK system and an asynchronous 16-QAM system (employing Gold sequences) with IC under Rayleigh fading.
ulation systems and compared the average BER between systems with the same throughput. Two different methods, mixed modulation and parallel channels, for handling multiple data rates have been analysed. These systems have been studied in the case of Rayleigh fading and they have been compared with single modulation systems. The conclusion is that the successive IC scheme has relatively low complexity even in the case
of higher modulation and yields considerable increase in performance compared to conventional matched filter detection. Mixed modulation systems offer more flexibility at the cost of a slight decrease in average performance compared to pure asynchronous QPSK systems. The 16-QAM users in the mixed system, who are most sensitive to noise, will have the highest average BER and the QPSK and BPSK users will have about the same average BER as for each respective single modulation system. IC, however, significantly improves the performance of mixed modulation systems as compared to using single user detectors [14]. On the other hand, if we instead have a QPSK system where the users employ parallel channels the average performance is almost equal for all the users. (It would be equal if all the users in the system use the same number of parallel channels.) Though, to achieve good performance in a parallel channel system it is important to use signature sequences with good cross-correlation properties to make use of the advantages with synchronous channels. The drawback with parallel channels is that sooner or later we run out of signature sequences and in some cases it would be better to add a 16-QAM user, if the user can accept a small decrease in performance, instead of a QPSK user with two parallel channels. The largest flexibility is obtained by a combination of parallel channels and mixed modulation. Future work within this project will be to analyse mixed systems where the users have different powers. We have seen from the results that the BER for 16-QAM users is higher than for BPSK and QPSK users. So by increasing the E b for the 16-QAM users we could decrease the BER to meet the performance of the BPSK and QPSK users. Additionally we will study multistage IC schemes, where we decrease the interference further and increase the capacity by employing a modified version of the IC scheme in several stages. Multipath fading channels with RAKE combiner will also be included in the analysis, as will channel coding. Two different IC schemes, resulted from work independent from ours, have been proposed recently in [23] and [24]. Though their work have only minor similarities to ours. In both papers they consider solely BPSK systems and Gaussian channels. Hence, no fading is taken into consideration. In [23] they evaluate the average BER for a multistage successive IC scheme under the assumption of unequal power control. Also in [24] they consider a multistage IC scheme, but in this case parallel detection is employed.
9 Acknowledgement This work was supported by the Swedish National Board for Industrial and Technical Development (NUTEK), project number 9303363-2.
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