A Novel Ternary CDMA Code for TPSK Modulation Scheme ...

Wireless Pers Commun (2011) 58:315–335 DOI 10.1007/s11277-009-9893-y

A Novel Ternary CDMA Code for TPSK Modulation Scheme Chee Kyun Ng · Nor Kamariah Noordin · Borhanuddin Mohd Ali · Sudhanshu Shekhar Jamuar

Published online: 12 December 2009 © Springer Science+Business Media, LLC. 2009

Abstract In code division multiple access (CDMA), two or more chips are grouped together to form symbols and each symbol is transmitted during the symbol period. The phase shift keying (PSK) modulation techniques map the digital baseband data into two or more possible signals by varying the phase of a radio frequency (RF) carrier. The recently proposed PSK scheme called ternary PSK (TPSK) scheme can convey three possible symbols. In this paper, a novel ternary based CDMA sequence so-called large area synchronous even ternary (LAS-ET) sequence is introduced to increase spectrum efficiency in TPSK scheme. Its sequence duty ratio and cross-correlation are analyzed. The performance analysis of this sequence is compared with the large area synchronous (LAS) sequence in term of symbol error rate and chip error rate (CER) over various channel models. It is shown that TPSK scheme in LAS-ET sequence outperforms LAS sequence in terms of CER evaluation. At the same time, the spectrum efficiency is doubled when a pair of chips in LAS-ET sequence is mapped into one symbol. Keywords

CDMA · LAS · TPSK · SER · CER

C. K. Ng (B) · N. K. Noordin · B. M. Ali Department of Computer and Communication Systems Engineering, Faculty of Engineering, Universiti Putra Malaysia, Serdang, Malaysia e-mail: [email protected]; [email protected] N. K. Noordin e-mail: [email protected] B. M. Ali e-mail: [email protected] S. S. Jamuar Department of Electrical Engineering, Faculty of Engineering, Universiti Malaya, Kuala Lumpur, Malaysia e-mail: [email protected]

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1 Introduction Modulation, which is the process of encoding information from message source in a suitable manner for transmission, involves translating a baseband message signal to a bandpass signal at higher frequencies. Modulation may be achieved by varying the amplitude, frequency or phase of a high carrier frequency in accordance with the amplitude of the message signal. Demodulation is the process of extracting the baseband message from the carrier so that it may be processed and interpreted by the intended receiver [1]. In mobile communication systems, occupied spectrum and required transmission power are important criteria for evaluating the modulation schemes. Phase shift keying (PSK) is used for digital modulation because of its excellent error rate characteristics in transmission with power restriction channels. The 16 levels modulation scheme such as 16-quadrature amplitude modulation (QAM) has been proposed to increase spectrum efficiency [2]. Considerable research have resulted in designing spreading sequences that exhibit zero correlation zone (ZCZ) or interference free window (IFW) at some relative zero delay-spread expressed in terms of number of chip intervals [3]. An attractive family of large area synchronous (LAS) sequence, resulting in zero multiple access interference (MAI) within the IFW, is constituted by the combination of large area (LA) and loosely synchronous (LS) codes giving an IFW around the zero delay-spread [4]. The LAS sequence occupies ternary codes having the elements of {+1, 0, −1} [5]. Therefore, it is necessary for the LAS sequence to be modulated by a ternary PSK (TPSK) scheme [6]. Since the LS sequence is seeded into pulse intervals of LA sequence as shown in Fig. 1 to improve its spectrum efficiency, the resultant size of synthesized IFW is much smaller. This is a major drawback to LAS sequence. In addition, the implementation of this LAS sequence is practically more complex. This paper presents a new TPSK modulation scheme for LA sequence in LAS sequence over various channel models that overcome above drawbacks. A modified version of LA sequence called LAS even ternary (LAS-ET) sequence is introduced. The LAS-ET sequence is designed such that the non-zero pulse positions are in even numbers. This rearrangement of LA sequence results in a systematic sequence, which can be synthesized easily. The performances of LAS-ET sequence using TPSK scheme in term of symbol error rate (SER) and chip error rate (CER) are then examined. These performances are evaluated with other PSK schemes over the AWGN, Rayleigh fading and frequency-selective fading channels. In Sect. 2, the system model of CDMA modulation scheme is described. The probabilities of error in TPSK modulation scheme are presented in Sect. 3. The modulation scheme in LAS CDMA system is discussed in Sect. 4. The property of LAS-ET sequence is discussed in Sect. 5. The error rate performance in TPSK scheme over AWGN and multipath fading channel models are discussed and analyzed in Sects. 6 and 7, respectively. This paper is then concluded in Sect. 8.

Fig. 1 LAS sequences construction method

123

A Novel Ternary CDMA Code for TPSK Modulation Scheme d 1 (t )

s1 (t )

c1 (t )

d 2 (t )

d K (t )

c K (t )

h1

2 P1 e j (ω C t +φ1 )

s 2 (t )

c 2 (t )

317

h2

r (t )

∑

2 P2 e j (ω C t +φ2 )

s K (t )

n(t )

z i (t ) e jω C t

ci (t )

hK

2 PK e j (ω C t +φ K )

Fig. 2 A simple CDMA with K users model

2 System Model In this section, a simple CDMA system model with PSK modulation over AWGN channel is described. As shown in Fig. 2 there are K simultaneous users that transmit data, d(t) asynchronously [7]. Both the modulator and demodulator employ complex processing. The kth user’s data signal is expressed as dk (t) =

∞

δ(t − nT )

(1)

n=−∞

where δ(t − nT ) is the binary {+1, −1} pulse of duration T. The complex spreading signal, which is a spreading code, consisting of a number of chips with chip pulse duration TC for the kth user, is expressed as ck (t) =

N −1

δ(t − an TC )

(2)

n=0

where an denotes the nth complex chip value with duration TC and N is The total number of chips in one data pulse also known as spreading factor of the CDMA system given by N=

T TC

(3)

After the modulation with a PSK scheme, the kth transmitted signal is described by (4) sk (t) = 2Pk dk (t)ck (t)e j (ωC t+φk ) √ where 2Pk e j (ωC t+φk ) is the complex carrier signal. Pk represents the kth transmitted signal power, ωC is the common complex carrier frequency, and φk is the carrier phase of the kth carrier. Since the power control is assumed to be perfect, the received signal power of all K users is assumed to be the same. For comparison purposes, each transmitted signal is passed through AWGN channel. In this channel, noise n(t), which is a white Gaussian random process with zero mean and

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power spectral density of N0 /2, is added to the transmitted signal sk (t) prior to reception [8]. Thus, the received signal r (t) is described as r (t) = sk (t) + n(t)

(5)

A simple upper bound for the probability of symbol error or SER in an AWGN channel with a noise density N0 for an arbitrary constellation can be obtained using the union bound [9]. The union bound which provides a representative estimate for SER in a particular modulation signalling can be written as ⎛ ⎞ 2 dmin ⎠ SER = Q ⎝ (6) 2N0 where dmin is the Euclidean distance or minimum distance between signal points in the constellation, and Q(x) is the complementary error function or simply Q-function defined as [10] ∞ Q(x) = x

2 −x 1 dx √ exp 2 2π

(7)

For symmetric constellations, the distance between all constellation points are equivalent, hence (6) gives the average probability of symbol error for a particular constellation set in PSK modulation prior to signal despreading [10]. The received signal is despreaded by its reference complex sequence signal, ci (t). The data signal, z i (t) is then acquired from the despreaded signal.

3 Probability of Error in TPSK Scheme When a spreading sequence has ternary chip elements of {+1, 0, −1}, it is necessary to modulate the sequence using TPSK scheme. The three resulting signals are denoted as si (t) = [s1 (t), s2 (t), s3 (t)]

(8)

The signal constellation of TPSK is shown in Fig. 3. The corresponding √vector space of chips is shown in Table 1. When the signal point of s1 from the origin is E, then the decision threshold, dmin can be obtained as dmin =

√ 3E

√

(9)

Thus, the distance between adjacent points in constellation diagram of TPSK scheme is 3E. Substituting it in (6), the SER in TPSK over AWGN channel is obtained as

3E SER = Q (10) 2N0 In TPSK scheme each symbol corresponds to one chip, so the probability of chip error or CER and SER in TPSK are the same. The CER in TPSK over AWGN channel is given as

3E c CER = Q (11) 2N0

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A Novel Ternary CDMA Code for TPSK Modulation Scheme

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Fig. 3 Constellation diagram of TPSK signals

Table 1 Mapping of TPSK symbols

Transmitted symbol

Vector space

Represented chip

0

π/2

0

1

7π /6

+1

2

11π /6

−1

Fig. 4 The pulse positions of LA(16,38,847)

4 Modulation Schemes in LAS CDMA System The LA sequence in LAS CDMA system is synthesized in such a manner that its N p non-zero ±1 pulses are positioned as shown in Fig. 4. This arrangement of pulse position, Pn with n = 0, 1, 2, . . . forms a configuration of L A(N p , K 0 , L c ) where K 0 is the minimum number of zero padding between pulse interval of non-zero pulses which determine the size of IFW delay-spread in term of chips, while having a total sequence length of L c chips. The arrangement of pulse interval, dn between two adjacent pulses in L A (16, 38, 847) is defined as 0 ≤ n < Np − 1 pn+1 − pn dn = (12) L c − p N p −1 n = N p − 1

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Fig. 5 The synthesization process of LA sequence in LAS-CDMA system Table 2 Mapping of 9-PSK symbols

Transmitted symbol

Vector space

Represented pair of chips

0

π/18

−1, −1

1

5π/18

−1, 0

2

π/2

−1, +1

3

13π /18

0, −1

4

17π /18

0, 0

5

7π /6

0, +1

6

25π/18

+1, −1

7

29π/18

+1, 0

8

11π /6

+1, +1

The limitations imposed on this pulse interval, dn of the LA sequence stated in [3] are as follows: • dn should be even except for d N p −1 • dn = dn+m 0 ≤ n ≤ L c − 1 where m = 1, 2, 3,… The synthesization process of this ternary sequence is shown in Fig. 5. Since the LAS sequence has ternary chip elements of {+1, 0, −1}, it is possible to modulate the sequence with TPSK scheme shown in Fig. 3. If the mapping of two chips into a symbol is possible in LAS sequence, there will be nine possible mappings in complete cycle as shown in Table 2. The I-Q constellation representation signal of this mapping can then be constructed as shown in Fig. 6. √ When the distance of each signal point from the origin is E, then the distance between adjacent points, dmin in the constellation is given as π √ dmin = 2 E sin 9

123

(13)


321

Fig. 6 Constellation diagram of 9-PSK signals

Substituting it in (6), the SER in 9-PSK over AWGN channel is given as

π 2E SER = Q sin N0 9

(14)

Since each symbol corresponds to two chips, thus E = 2E c . Therefore, the CER in 9-PSK over AWGN channel is expressed as

π 4E c sin CER = Q (15) N0 9

5 Large Area Synchronous Even Ternary (LAS-ET) Sequence In this section, we discuss about the properties of proposed LAS-ET sequence. The reason for constructing LAS-ET sequence in even numbers is to allow the sequences to be mapped from two chips to one symbol so that its spectrum efficiency can be improved. When two chips of LAS-ET sequence are mapped into one symbol period, there are only three possible combinations in the mapping instead of nine. This mapped signal set, s(t) is given as s(t) = {[+1, 0], [0, 0], [−1, 0]}

(16)

In order to achieve this mapping, an additional constraint is imposed on the LAS-ET sequence. In this constraint for given length of sequence L c , the pulse interval between two adjacent non-zero pulses dn and the minimum pulse interval K 0 must be both even. The recursive functions for this LAS-ET sequence is obtained as follows: The non-zero pulse positions, Pn (t) of LAS-ET sequence are given as N p −1

Pn (t) =

δ(t − an )

(17)

n=0

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Fig. 7 The pulse positions of LAS–ET(16,38,818)

where an = K 0 n + n(n − 1)

n = 0, . . . , N p − 1

The zero padding, z n in this LAS-ET sequence then become K 0 + 2n − 1 n = 0, . . . , N p − 2 zn = K0 − 1 n = Np − 1

(18)

(19)

Finally, the length of sequence, L c for this LAS-ET sequence therefore become L c = K 0 + K 0 (N p − 1) + (N p − 1)(N p − 2)

(20)

which is an even sequence. For example, in the non-zero pulses for an N p of 16 and the minimum pulse interval, K 0 of 38, the length of sequence, L c is reduced to 818 chips to form LAS − ET (16, 38, 818) compared to original LA sequence of 847 chips. Figure 7 shows the pulse positions of L AS − E T (16, 38, 818) sequence. The duty ratio, D of the spreading sequences is defined as D=

Np Lc

(21)

where N p is the number of non-zero pulses in LAS-ET sequence, and L c is the sequence length. We note that in traditional CDMA sequences, zero pulses do not exist and D is unity. Hence the spectrum efficiency is proportional to the number of available codes to accommodate Nu users. This is shown that there is a large sequence length in L A (16, 38, 847) proposed in [3] which yield a low duty ratio of 0.0189. This low duty ratio is the main drawback of the proposed L A (16, 38, 847) sequence. However, careful investigations of the conditions satisfying the maximum correlation of unity reveal that dn may not always be even. And since the algorithm proposed in [3] does not require Pn be even, the LA sequence length, L of L A (16, 38, 847) can be reduced to 714 chips compared to the original length of 847 chips [11]. The duty ratio of the proposed L A (16, 38, 714) then becomes 0.0224. Therefore, the optimized LA sequence of L A (16, 38, 714) results in an 18.6 % duty ratio increase. Figures 8 and 9 show the cross-correlation properties of the L A (16, 38, 847) and L A (16, 38, 714) sequences respectively. The cross-correlation of L A (16, 38, 714) sequence at zero delay spread is 0.0587 which is significant compared to 2.1088 × 10−17 ≈ 0 of the original L A (16, 38, 847) sequence. This is a major drawback of L A (16, 38, 714) sequence despite exhibiting higher duty ratio. In order to sustain the characteristics of L A (16, 38, 847) sequence without altering the size of IFW and its cross-correlation, a modified sequence, L AS − E T (16, 38, 818) is introduced. Since the spectrum efficiency in CDMA system is proportional to sequence’s duty ratio, D as expressed in (21), the duty ratio of proposed L AS − E T (16, 38, 818) is 0.0196, which is higher than the duty ratio of 0.0189 in the

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Fig. 8 The cross-correlation property in LA(16,38,847) sequences showing no cross-correlation in IFW zone

Fig. 9 The cross-correlation property in LA(16,38,714) sequences showing a significant cross-correlation value at zero delay spread

original L A (16, 38, 847) sequence proposed in [3]. This results in an increase of 3.7 % duty ratio and spectrum efficiency. Figure 10 shows the cross-correlation property of the L AS − E T (16, 38, 818) sequence. Even though the duty ratio of L AS − E T (16, 38, 818) sequence is still low compared to L A (16, 38, 714) sequence proposed in [11]. The cross-correlation property of L AS − E T (16, 38, 818) sequences is almost similar to the original L A (16, 38, 847) sequence, which exhibits a large IFW around the origin. The cross-correlation of L AS − E T (16, 38, 818) sequence at zero delay spread is 4.0332 × 10−17 ≈ 0 compared to 0.0587 as exhibited from L A (16, 38, 714) sequence. This implies that although the L AS − E T (16, 38, 818) sequence has low duty ratio it has

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Fig. 10 The cross-correlation property in LAS-ET(16,38,818) sequences showing no cross-correlation in IFW zone Table 3 The properties of various LAS sequences

LAS sequences

Duty ratio, D

Cross-correlation, Ri,0

L A (16, 38, 847)

0.0189

2.1088 × 10−17 ≈ 0

L A (16, 38, 714)

0.0224

0.0587

L AS − E T (16, 38, 818)

0.0196

4.0332 × 10−17 ≈ 0

a better cross-correlation performance at zero delay spread compared to L A (16, 38, 714) sequence. Table 3 summarizes the properties of the above discussed LAS sequences.

6 Probability of Error for LAS-ET Sequences in TPSK Scheme over AWGN Channel When two chips of LAS-ET sequence are mapped into one symbol, there are only three possible signals in this mapping as denoted in (16). The synthesization process of this mapped LAS-ET sequence is shown in Fig. 11. The I-Q constellation is similar to that shown in Fig. 3. Each mapping of carrier phase in this signalling takes on one of three equal vector spaces shown in Table 4. The SER of this pair of LAS-ET sequence in TPSK scheme is given in (10). Since each symbol corresponds to two chips, therefore the CER for LAS-ET sequence in TPSK scheme is given by

3E c CER = Q (22) N0 Figures 12 and 13 show the CER and SER performance for LAS-ET sequence in TPSK scheme over AWGN channel with BPSK and QPSK schemes as references. In Fig. 12, the curves show that the performance of TPSK signal for ternary CDMA sequence in terms of SER lies between BPSK and QPSK signals. The TPSK signal needs 8 dB of E/N0 to achieve SER of 10−3 . This is based on the distances between adjacent points laid in constellation

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Fig. 11 The synthesization process of LAS-ET sequence Table 4 Mapping of TPSK symbols in LAS-ET sequence

Transmitted symbol

Vector space

Represented pair of chips

0

π/2

0, 0

1

7π /6

+1, 0

2

11π /6

−1, 0

Fig. 12 SER performance of LAS-ET sequence in TPSK scheme over AWGN channel

√ √ √ diagrams of BPSK, TPSK and QPSK which are 2 E, 3E and 2E, respectively. By utilizing the LAS-ET sequence in TPSK scheme, a better performance is achieved compared to both BPSK and QPSK schemes. Figure 13 shows that only 5 dB of E c /N0 is needed to achieve CER of 10−3 in TPSK scheme compared to 7 dB in both BPSK and QPSK schemes, which exhibits a 2 dB gain improvement. Additionally since each symbol in TPSK scheme for LAS-ET sequence corresponds to two chips, E = 2E c , its spectrum efficiency also outperforms BPSK scheme.

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Fig. 13 CER performance of LAS-ET sequence in TPSK scheme over AWGN channel

7 Probability of Error for LAS-ET Sequences in TPSK Scheme over Multipath Fading Channel There are basically two characteristics of radio channels. The first characteristic is that the transmitted signal arrives at the receiver via multiple propagation paths with different time delay and if an extremely short pulse is transmitted, the received signal is spread in time due to the multiple scatterers at different delays. The second characteristic of radio channels is concerned with the time variations in the structure of the medium and if the short pulse transmission is repeated continuously, there are changes in the received signal due to physical changes in the medium. Such changes include variations in the relative delays of signals from the multiple scatterers, since the received signal in a multipath channel consists of a series of attenuations, time delays and phase shifts of the transmitted signal [10]. If there are maximum N multipath components in the channel, the received signal, r (t) may be expressed as r (t) = A

N −1

αn (t) cos [ωc (t − τn (t))]

n=0

= A Re

N −1

αn (t) exp (− jωc τn (t)) exp ( jωc t)

(23)

n=0

where αn (t) is the time-variant attenuation factor associated with the nth propagation path and τn (t) is the corresponding propagation delay. The complex value of multipath channel, h(t, τ ) is expressed as h(t, τ ) =

N −1

αn (t) exp (− jωc τn (t))

n=0

=

N −1 n=0

123

αn (t) exp (− jφn (t, τ ))

(24)


327

Fig. 14 Model for time varying multipath channel

The complex valued h(t, τ ) can be viewed as the sum of vectors of the time-varying amplitude αn (t) and phase φn (t, τ ). The delays τn (t) are associated with the different signal paths that change at different rates and in unpredictable manner. This implies that the complex valued h(t, τ ) can be modelled as a random process. When there are a large number of signal propagation paths, the central limit theorem can be applied. Thus, h(t, τ ) can be modelled as a complex valued Gaussian random process [12]. We assume that n multipath components are resulted from a reflected scattering. Two multipath components are said to be resolvable if their delay difference significantly exceeds the inverse of signal bandwidth (1/W ), where W is the transmitted signal bandwidth. Multipath components that do not satisfy this resolvability criterion cannot be distinguished at the receiver, and thus these components are non-resolvable. These non-resolvable components are combined into a single multipath component resulting in fast variations due to the constructive and destructive interference of amplitude and phase of the combined non-resolvable multipath components. In general, wideband channels have resolvable multipath components while narrowband channels tend to have non-resolvable multipath components [8]. A general model for a time varying multipath channel is illustrated in Fig. 14. The channel model consists of a tapped delay line with uniformly spaced taps. The tap spacing is 1/W , hence, 1/W is the time resolution that can be achieved through the channel. The tap coefficients are denoted as c(t) = αn (t) exp(− jφn t)

(25)

These are usually modelled as complex valued, Gaussian random processes which are usually uncorrelated. The length of the delay line corresponds to maximum delay spread (τmax ) is denoted as τmax =

L W

(26)

where L represents the maximum number of possible multipath signal components.

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In the following sub-sections, we examine the error rate performance of TPSK modulation when the signals are transmitted over Rayleigh fading and frequency selective fading channels. 7.1 Performance of TPSK Modulation over Rayleigh Fading Channel The non-resolvable fading channel also known as frequency non-selective fading channel is Rayleigh fading channel. The signal bandwidth W is assumed to be much smaller than the channel bandwidth. To evaluate the probability of error of TPSK modulation scheme in this slow flat fading channel, the probability of symbol error, PS (γ S ) must be averaged in AWGN channel over the possible ranges of signal strength due to fading. For a fixed attenuation, α the probability of symbol error, PS (γ S ) in term of the received SNR per symbol, γ S is given by

3γ S PS (γ S ) = Q (27) 2 where γS = α2

E N0

(28)

Clearly when α 2 is unity, the probability of symbol error PS (γ S ) in (27) becomes SER as in (10) for TPSK modulation scheme. Hence, the probability of error in AWGN channels is viewed as a conditional error probability. To obtain the average symbol error probabilities, Pe (γ S ) when α is random over the fading channel, the PS (γ S ) must be averaged over the probability density function (pdf) of γ S . Thus ∞ Pe (γ S ) =

PS (γ S ) p (γ S ) dγ S

(29)

0

where p (γ S ) is the pdf of γ S when α is random. Hence, the average probability of symbol error, Pe (γ S ) for TPSK over Rayleigh fading channel is given as ∞ Pe (γ S ) =

PS (γ S ) p (γ S ) dγ S 0 ∞

=

Q

0

3γ S 2

γS 1 dγ S exp − γS γS

(30)

The expression in (30) can be applied for the mapped symbol from two chips of LAS-ET sequence. Since each symbol corresponds to two chips, therefore the average probability of chip error, Pe (γC ) for LAS-ET sequence in TPSK over Rayleigh fading channel is given as ∞ Pe (γC ) =

PC (γC ) p (γC ) dγC 0

∞ = 0

123

1 γC Q 3γC exp − dγC γC γC

(31)


329

Fig. 15 Average probability of symbol error for LAS-ET sequence in TPSK scheme over Rayleigh fading channel

√ where p (γC ) is the pdf of γC when α is random and PC (γC ) = Q 3γC is the probability of chip error when α is fixed in terms of the received SNR per chip γC as equal to γC = α 2

Ec N0

(32)

Figures 15 and 16 show the average probability of chip and symbol errors performance for LAS-ET sequence in TPSK scheme over Rayleigh fading channel with BPSK and QPSK schemes as references. Since the Rayleigh fading channel is based on multipath signal propagation environment, this is considered as the worst case in the error rate performance. As shown in Fig. 15, the TPSK signal needs 25 dB to achieve SER of 10−3 in Rayleigh fading channel compared to 8 dB in AWGN channel resulting performance decrease of 68% in error rate. Figure 15 also shows that the performance of ternary CDMA sequence for TPSK signal in terms of average probability of symbol error Pe (γ S ), which lies between BPSK and QPSK signals. Both BPSK and QPSK signals need 24 and 26 dB respectively to achieve Pe (γ S ) of 10−3 . It is based on the different of decision d√ min that laid in constellation diagram √ threshold, √ of BPSK, TPSK and QPSK which are 2 E, 3E and 2E, respectively. When the LAS-ET sequence is utilized in TPSK scheme, there are only three possible combinations after two of its chips are mapped into one symbol. The average probability of chip error, Pe (γC ) in TPSK scheme outperforms both BPSK and QPSK schemes. Figure 16 shows that TPSK signal needs 22 dB of γC to achieve Pe (γC ) of 10−3 compared to 24 dB in both BPSK and QPSK schemes, which shows a 2 dB gain improvement. However, this error rate performance is still poor when compared to Pe (γC ) in AWGN channel. This is because Rayleigh fading channel is still considered as the worst case in multipath signal propagation environment. 7.2 Performance of TPSK Modulation over Frequency Selective Fading Channel In this sub-section, we consider the case in which the signal bandwidth W exceeds the bandwidth of the channel, hence the channel is said to be frequency selective. The symbol duration

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Fig. 16 Average probability of chip error for LAS-ET sequence in TPSK scheme over Rayleigh fading channel

TS of modulated signal is less than multipath delay spread Tm [12]. The frequency selective channel can be modelled as shown in Fig. 14. The received signal is expressed as r (t) =

L n=1

n + n(t) cn (t)s t − W

(33)

where n(t) represents the AWGN. Therefore, the frequency selective fading channel provides the receiver with up to L replicas of the transmitted signal. Each signal component is multiplied by a corresponding channel tap weight, cn (t) for n = 1, 2, . . . , L. Based on the slow fading assumption, the channel coefficients are considered as constant over the duration of the symbol intervals. Since there are up to L replicas of the transmitted signal s(t), the received signal r (t) is processed in an optimum manner so that its performance of resolvable signal components is achieved equivalent to a communication system with diversity [8]. We assume that there are L diversity fading channels carrying the same signal. Each channel is assumed to be frequency non-selective and slowly fading with Rayleigh pdf. The fading processes among the L diversity channels are assumed to be mutually independent. The signal in each channel is corrupted by AWGN process with zero-mean variance. The noise processes in the L channels are also assumed to be mutually independent. Thus, received SNR per symbol, γ S can be expressed as γS =

L ES 2 αl l = 1, 2, . . . , L N0

(34)

l=1

where γl = αl2 E S /N0 is the instantaneous SNR per symbol on the lth channel. Since the fading on the L channels is mutually independent, the values of γl are also statistically independent. Hence, the pdf, p (γ S ) of L diversity channels in Rayleigh fading channel can be computed as [10]

1 γS L−1 p (γ S ) = γ exp − (35) γl (L − 1)!γl L S

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where γl = γ S /L is the average SNR per symbol per channel. When L = 1, the pdf of (35) corresponds to Rayleigh fading channel system. For L = 2, the performance of the single channel system is identical to the performance obtained in a Rayleigh fading channel with dual diversity. Substituting the pdf p (γ S ) of (35) into (29), the average probability of symbol error, Pe (γ S ) for TPSK over frequency selective fading channel can be obtained as

∞ 3γ S 1 γS L−1 dγ S Pe (γ S ) = Q γ exp − (36) 2 γl (L − 1)!γl L S 0

where the probability of symbol error, PS (γ S ) in term of the received SNR per symbol, γ S is similar to (27). Since the proposed LAS-ET sequence has only three possible combinations after two chips are mapped into one symbol, the expression (36) can be applied to evaluate the average probability of symbol error, Pe (γ S ) for this sequence in TPSK scheme over frequency selective fading channel. To evaluate the average probability of chip error Pe (γC ) for the pair of LAS-ET symbol, it is noted that each symbol corresponds to two chips, E = 2E c and the √ probability of chip error PC (γC ) = Q 3γC in terms of the received SNR per chip γC , which is similar to (32). Finally, the average probability of chip error Pe (γC ) for LAS-ET sequence in TPSK modulation scheme over L diversity order frequency selective channel is given as ∞ Pe (γC ) =

Q 0

3γC

1 γC L−1 dγC γ exp − γl (L − 1)!γl L C

(37)

where γl = γC /L is the average SNR per chip per channel. Figures 17 and 18 show the average probability of chip and symbol errors performance for LAS-ET sequence in TPSK scheme over frequency selective fading channel in various diversity orders L with BPSK and QPSK schemes as references. Since the multiple resolvable signal components from multipath signal propagation environment can be combined for example with RAKE receivers, its error rate performance is likely to improve compared to Rayleigh fading channel. As shown in these figures, when the diversity order L is increased to two, there is a 10 dB gain improvement in order to achieve error rate of 10−3 . Figure 17 shows that the TPSK signal needs 15 dB to achieve average probability of symbol error, Pe (γ S ) of 10−3 in frequency selective fading channel with L = 2, compared to 25 dB in Rayleigh fading channel for frequency selective fading channel with L = 1. It implies that Pe (γ S ) has increased by 66.7%. When the diversity order is increased to ∞, it results in 8 dB of γ S to achieve Pe (γ S ) of 10−3 . This performance is similar to error performance for TPSK scheme over AWGN channel. Therefore, it implies that the multipath fading channel phenomenon can be mitigated by using multiple of RAKE receivers for the example. The two chips of LAS-ET sequence are mapped into one symbol to increase the spectrum efficiency. Then two chips are sent instead of one chip at one time and the error rate in terms of the average probability of chip error, Pe (γC ) in TPSK scheme is seen to outperform both BPSK and QPSK schemes. Figure 18 shows that when L = 2, the TPSK signal needs 12.5 dB of γC to achieve Pe (γC ) of 10−3 compared to 14 dB in both BPSK and QPSK schemes resulting in a 1.5 dB gain improvement. It is also seen from Fig. 18 that the Pe (γC ) of QPSK is identical to BPSK but twice the chips can be sent in the same bandwidth to increase spectrum efficiency. Thus, when compared to BPSK, QPSK provides twice the spectrum efficiency with exactly the same energy efficiency. Similar to QPSK, it also provides twice the spectrum efficiency

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Fig. 17 Average probability of symbol error for LAS-ET sequence in TPSK scheme over frequency selective fading channel for diversity order, L = 1, 2 and ∞

Fig. 18 Average probability of chip error for LAS-ET sequence in TPSK scheme over frequency selective fading channel for diversity order, L = 1, 2 and ∞

but higher (γC ) is needed when compared to BPSK scheme. Hence, the synthesized LAS-ET sequences do not only increase the spectrum efficiency by mapping its two chips into one symbol but also improve the system performance in terms of Pe (γC ).

8 Conclusions This paper presented a novel ternary based LAS-ET sequence to increase spectrum efficiency in TPSK scheme. The two chips of LAS-ET sequence can be mapped in one symbol period yielding only three possible mapped signals. In general, the purpose of mapping process is

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Table 5 Performance for various types of PSK signaling CDMA sequence Walsh–Hadamard

LAS

LAS-ET

Sequence mapping

PSK signalling

CER

{+1, −1}

BPSK

{+1 + 1, −1 +1, −1−1, +1−1}

QPSK

{+1, 0, −1}

TPSK

{−1 − 1, −10, −1 + 1, 0 − 1, 00, 0 + 1, +1 − 1, + 10, +1 + 1}

9-PSK

{+1, 0, −1}

TPSK

{00, +10, −10}

SER 2E N0 Q NE 0

Q

3E 2N0 2E sin π Q 9 N0

Q

Q

Q

Q

TPSK

3E 2N0 3E Q 2N 0

Q

3E c

π 4E c N0 sin 9

Q

2E c N0

2N0

Q

Q

2E c

N0

3E c

2N0 3E c N0

to improve spectral efficiency. It has been shown that a better CER performance in TPSK scheme can be achieved over various channels, which outperforms both BPSK and QPSK schemes at CER of 10−3 . A summary of error probability performances in various PSK signals is shown in Table 5. At the same time, the spectrum efficiency of this LAS-ET sequence is also doubled when a pair of chips is mapped into one symbol. On the other hand, the synthesized L AS − E T (16, 38, 818) sequence is much easier and has better duty ratio than original L A (16, 38, 847) sequence while maintaining the desired cross-correlation property of IFW at some relative zero delay-spread. Since the spectrum efficiency is proportional to the sequence’s duty ratio, the synthesized LAS-ET can increase its spectrum efficiency by two folds.

References 1. Haykin, S. (2001). Communication systems, 4th ed. New York: Wiley. 2. Adachi, F. (1999). BER analysis of 2PSK, 4PSK, and 16QAM with decision feedback channel estimation in frequency-selective slow Rayleigh fading. IEEE Transactions on Vehicular Technology, 48(5), 1563–1572. 3. Li, D. (1999). A high spectrum efficient multiple access code. Fifth Asia-Pacific Conference on Communications and Fourth Optoelectronics and Communications Conference (APCC/OECC ‘99) Proceedings 1 (pp. 598–605). 18–22 Oct. 4. Li, D. (2003). The perspectives of large area synchronous CDMA technology for the fourth-generation mobile radio. IEEE Communications Magazine, 41(3), 114–118. 5. Wei, H., & Hanzo, L. (2006). On the uplink performance of LAS-CDMA. IEEE Transactions on Wireless Communications, 5(5), 1187–1196. 6. Nakamura, M., & Torii, H. (2002). Ternary phase shift keying and its performance. The 5th International Symposium on Wireless Personal Multimedia Communications 3 (pp. 1284–1288). 7. Xie, S., & Rahardja, S. (2005). Performance evaluation for quaternary DS-SSMA communications with complex signature sequences over Rayleigh-fading channels. IEEE Transactions on Wireless Communications, 4(1), 266–277. 8. Goldsmith, A. (2005). Wireless communications. Cambridge: Cambridge University Press. 9. Ziemer, R. E., & Peterson, R. L. (1992). Introduction to digital communication. New York: Macmillan Publishing Company. 10. Proakis, J. G. (2001). Digital Communications, 4th ed. New York: McGraw-Hill. 11. Choi, B.J., & Hanzo, L. (2002) On the design of LAS spreading codes. IEEE 56th Vehicular Technology Conference (VTC 2002-Fall) Proceedings 4, (pp. 2172–2176), 24–28 Sept. 12. Rappaport, T. S. (2002). Wireless comminications, principles and practice, 2nd ed. Upper Saddle River: Prentice Hall Inc.

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Author Biographies Chee Kyun Ng received his Bachelor of Engineering and Master of Science degrees majoring in Computer & Communication Systems from Universiti Putra Malaysia, Serdang, Selangor, Malaysia, in 1999 and 2002, respectively. He has also completed his Ph.D. programme in 2007 majoring in Communications and Network Engineering at the same university. He is currently undertaking his research on wireless multiple access schemes, wireless sensor networks and smart antenna system. His research interests include mobile cellular and satellite communications, digital signal processing, and network security. Along the period of his study programmes, he has published over 30 papers in journals and in conferences.

Nor Kamariah Noordin received her B.Sc. in Electrical Engineering majoring in Telecommunications from University of Alabama, USA, in 1987. She became a tutor at the Department of Computer and Electronics Engineering, Universiti Putra Malaysia, and pursued her Masters Degree at Universiti Teknologi Malaysia and Ph.D. at Universiti Putra Malaysia. She then became a lecturer in 1991 at the same department where she was later appointed as the Head from year 2000 to 2002. She is currently the Deputy Dean (Academic, Student Affairs and Alumni) of the Faculty. During her more than 15 years at the department she has been actively involved in teaching, research and administrative activities. She has supervised a number of undergraduate students as well as postgraduate students in the area of wireless communications, which led to receiving some national and UPM research awards. Her research work also led her to publish more than 100 papers in journals and in conferences.

Borhanuddin Mohd Ali obtained his B.Sc. (Hons) Electrical and Electronics Engineering from Loughborough University in 1979; M.Sc. and Ph.D. from University of Wales, UK, in 1981 and 1985, respectively. He became a lecturer at the Faculty of Engineering UPM in 1985, made a Professor in 2002, and Director of Institute of Multimedia and Software, 2001–2006. In 1997 he co founded the national networking testbed project code named Teman, and became Chairman of the MYREN Research Community in 2002, the successor to Teman. His research interest is in Wireless Communications and Networks where he publishes over 80 journal and 200 conference papers. He is a Senior Member of IEEE and a member of IET and a Chartered Engineer, and the present ComSoc Chapter Chair. He is presently on a 2-year secondment term with Mimos as a Principal Researcher, heading the Wireless Networks and Protocol Research Lab.

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Sudhanshu Shekhar Jamuar received his M.Tech. and Ph.D. in Electrical Engineering from Indian Institute of Technology, Kanpur, India in 1970 and 1977 respectively. He worked as Research Assistant, Senior Research Fellow and Senior Research Assistant from 1969 to 1975 at IIT Kanpur. During 1975–76, he was with Hindustan Aeronautics Ltd., Lucknow. Subsequently he joined the Lasers and Spectroscopy Group in the Physics Department at IIT Kanpur, where he was involved in the design of various types of Laser Systems. He joined as Lecturer Electrical Engineering Department at Indian Institute of Technology Delhi in 1977, where he became Assistant Professor in 1980. He was attached to Bath College of Further Education, Bath (UK), Aalborg University, Aalborg (Denmark) during 1987 and 2000. He was a Professor in the Department of Electrical Engineering at IIT Delhi from 1991 to 2003. He was with University Putra Malaysia during 1996–2008 in the Faculty of Engineering. Presently he is Professor in the Electrical Engineering Department in the Faculty of Engineering, University Malaya (Malaysia) since 2009. He has about 40 papers in the International Journals and has attended several International Conferences and presented papers. He is senior member of IEEE and Fellow of Institution of Electronics and Telecommunications Engineering (India). He is presently the Chapter Chair for IEEE CAS Chapter in Malaysia. He is one of DLP speakers for the term 2008–2009 for the IEEE Circuits and System Society.

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