Kasami Code-Shift-Keying Modulation for Ultra Wideband ...
M3B-2
Kasami Code-Shift-Keying Modulation for Ultra Wideband Communication Systems Yuh-Ren Tsai* and Xiu-Sheng Li Institute of Communications Engineering National Tsing Hua University 101, Sec. 2, Kuang-Fu Rd., Hsinchu 300, Taiwan [email protected] Abstract — Ultra wideband (UWB) communication systems are generally applied to short-range wireless communications. In order to achieve higher rates or to support multiple access capabilities, direct sequence spread spectrum (DSSS) techniques have been introduced to UWB systems. Furthermore, the concept of M-ary code shift keying (M-CSK) was introduced into DSSS systems to achieve higher rates. In this work, we propose an M-CSK modulation technique based on the large set of Kasami sequences since it possesses good code properties. The modulation and demodulation schemes are developed and the system performance is well investigated. It is found that the Kasami M-CSK modulation outperforms other M-CSK modulation techniques or some UWB systems. Furthermore, based on our demodulation scheme, the hardware complexity of receivers can be greatly reduced to O( M 1 3 ) and the implementation for a large M becomes feasible. Index Terms — Modulation; Multiaccess communication; Pseudonoise coded communication.
I.
power efficiency and bandwidth efficiency, in additive white Gaussian noise (AWGN) and UWB channels exposed to multiple user interference is well investigated. Compared with the cases employing other code sets, we found that a better bit error rate (BER) performance can be obtained in most situations, and a higher data rate can be achieved. Furthermore, we introduce two possible demodulation schemes: one is the traditional optimal receiver and the other is our proposed scheme which can greatly reduce the hardware complexity. The rest of this paper is organized as follows. Section II introduces the channel model and the generation of Kasami sequences. Section III describes the modulation and demodulation schemes. Section IV concentrates on the system performance and the comparison with other systems. Finally, this paper is concluded in Section V.
INTRODUCTION
Recently, the investigation activities on the issue of ultra wideband (UWB) systems grow up rapidly [1]-[2]. In general, UWB communication systems are applied to short-range wireless communications for their great advantages, such as coexisting with narrowband systems, low probability of interception, low power consumption, and diversity gain of multipath resolution. Impulse radio (IR) technology is originally proposed for UWB systems [3]. However, in order to diminish the interference from other multiple access users, spread spectrum techniques are introduced and applied to UWB systems. For the purpose of achieving higher rates or supporting multiple access capabilities, direct sequence spread spectrum (DSSS) UWB systems have been proposed in [4]-[5]. In recent years, two different proposals are also proposed for the applications of wireless personal area networks (WPANs). One is the multi-band OFDM (MB-OFDM) system [6], and the other is the direct-sequence UWB (DS-UWB) system [7]. To achieve higher rates, the concept of M-ary code shift keying (M-CSK) was introduced to DSSS systems [8]-[9]. M-CSK uses different codes to represent different symbols. There are many choices of code sets, such as orthogonal sets, bounded correlation sets, and random codes. In this work, we make use of Kasami sequences as the alphabet set. Details of the modulation and demodulation schemes are proposed. The system performance, including the hardware complexity, This work was supported in part by the National Science Council, Taiwan, R.O.C., under Grant NSC 95-2752-E-007-003-PAE and under Grant NSC 94-2219-E-007-006.
II. A.
PRELIMINARIES
Pseudo-Random Sequences
For DSSS, there are many choices of spreading sequences. Particularly, the sequence length, the cross-correlation and the code set size are major properties concerning spreading sequences. The sequence length, denoted by L , is available only for some specific values. For example, the sequence lengths of Walsh sequences are only available for L = 2 n . Furthermore, for any two binary code sequences a = [ a 0 , a1 , , a L −1 ] and b = b0 , b1 , , bL −1 , the discrete periodic cross-correlation function is defined as 1 L −1 (1) θ ab = ∑ (− 1)a (− 1)b . L =0 The code set size is defined as the number of available code sequences in a code set. In practical applications, it is important to distinguish the binary code sequence and the corresponding signal waveform. The relation between a binary code sequence a = [a0 , a1 , , aL−1 ] and the corresponding signal waveform φ (t ) can be presented by L −1
φ (t ) = ∑ (− 1)a p(t − Tc ), =0
0 ≤ t ≤ Ts ,
(2)
where Tc is the chip duration, Ts = LTc is the symbol duration, and p (t ) is the unit pulse shaping function. The signal waveform can be represented as signal vector with elements taking on values of {+ 1, −1} . For convenience sake, we define a function
χ (•) to represent the translation from a binary code sequence to the corresponding signal vector, i.e. χ (a ) = (− 1)a , χ (a) = (− 1) a0 , (− 1) a1 , , (− 1) aL −1 . (3)
[
]
B. Kasami Sequences In this work, we employ the large set of Kasami sequences as the symbol alphabet set since it possesses good code properties, including the large code set size and low cross-correlations. In the construction of the large set Kasami sequences with the length L = 2 n − 1 , the parameter n must be an even integer. In this work, we mainly focus on the case of n = 2 mod 4 , and we briefly introduce how to construct the large set of Kasami sequences below [10]. Let u represent a binary m-sequence vector of the degree n n with the period L = 2 − 1 , D denote the operator which shifts the code phase of a sequence vector cyclically by one unit, and ⊕ denote the exclusive-OR operator. For q = t ( n) ≡ 1 + 2 (n+ 2 ) 2 , u′ = u[t (n) ] forms another m-sequence with the period L , and consequently we have a set of Gold sequences G (u, u ′) ≡ {u, u ′, u ⊕ u ′, u ⊕ Du ′, u ⊕ D 2 u ′,..., u ⊕ D L −1u ′}.(4)
It is noted that G (u, u′) contains SG = L + 2 = 2n + 1 code sequences with the period L . In addition, for q = s( n) ≡ 1 + 2 n 2 , u′′ = u[s(n)] forms another m-sequence with the period L1 = 2 n / 2 − 1 . Now we define a cover sequence c = [c0 , c1 , , cL−1 ] , which is the repetitions of u′′ , i.e. [c0+ jL , c1+ jL , , c( L −1)+ jL ] is equal to u′′ for j = 0, ,2n 2 . We then define a set of cover sequences as L (5) C (u ′′) ≡ 0 L ∪ ∪ 1 D j −1c = {c j , j = 0, , L1 }, 1
1
1
1
[
j =1
]
where ∪ denotes the union of sets, and 0 L is an all-zero sequence with length L . It is noted that there are SC = L1 + 1 = 2 n / 2 elements in C (u′′) . Each element of the large set Kasami sequences, denoted by K (u) , is the exclusive-OR output of an element in G(u, u′) and an element in C (u′′) . It is noted that the code set size of n/2 n K (u) is S K = S C × S G = 2 ( 2 + 1) . The large set of Kasami sequences can also be expressed by generator polynomials. Let
h(x) , h ′( x ) and h ′′(x) denote the generator polynomials of u , u[t (n ) ] and u[s(n ) ] , respectively. Fig. 1 gives an example of the
large set Kasami sequences generator.
C. Channel Models In this work, we consider two channel models, one is additive white Gaussian noise (AWGN) channel model, and the other is ultra-wideband multipath channel model with Tc = 1 n sec [11]. In addition, a RAKE receiver with the finger number 16 is applied for the UWB channel. III.
MODULATION AND DEMODULATION SCHEMES
We propose in this section the modulation and demodulation schemes for M-CSK modulation based on the large set of Kasami sequences.
A. Modulation Scheme In the M-CSK modulation, there are M ( M = 2 k ) different possible symbols, and the transmitter groups the input data stream into k-bit ( k = log2 M ) symbols with a symbol duration Ts . Since the code set size S K = 2 n / 2 ( 2 n + 1) of K (u) is not a power of 2, we select M = 2 3n 2 , and then we have k = 1.5n . We now propose a systematic method for sequences selection and mapping. The input data sequence in a symbol duration is denoted as d = [d 0 , d1 , , d n −1 , d n , , d1.5 n−1 ] , where d i ∈ {0,1} . In Fig. 1, the initial state of h(x) is fixed to an arbitrary nonzero value, such as h = [1, 0, , 0] , and the initial states of h′(x) and h′′(x) are set to be h ′ = [d 0 , d 1 , , d n −1 ] and h ′′ = [d n , d n +1 , , d 1.5 n −1 ] . The output code sequence is consequently a Kasami sequence of K (u ) . The generated Gold sequences set G~(u, u′) in Fig. 1 is the complete set of original Gold sequences with the code sequence u′ excluded, and it follows that S G~ = 2 n = L + 1 . Besides, if we define V (u′) = {0 L , u′, Du′, D 2u′, , D L−1u′}= {v m , m = 0, , L} , (6) the generated Gold sequence set is composed of u and an element from V (u′) , i.e. ~ G (u, u′) = u ⊕ V (u ′)
{
= u, u ⊕ u′, u ⊕ Du′, u ⊕ D 2 u′,
h( x ) = x 6 + x 5 + 1
h = [1,0,...,0] (an arbitrary nonzero vector)
h′ = [d 0 , d1 ,
h′( x) = x 6 + x 5 + x 3 + x 2 + 1 , d n −1 ]
h ′′ = [d n , d n +1 , Fig. 1.
Kasami sequence output
, d 1.5 n −1 ]
h′′( x) = x + x + 1
Large set Kasami sequence generator.
3
We therefore have the set of selected large quences, referred as the reduced set of large quences, being L ~ ~ K ( u ) = G (u , u ′ ) ∪ ∪ j = 1 D j − 1 c ⊕
[
1
{
}
(7)
, u ⊕ D L −1u′ .
Kasami code seKasami code se~ G (u , u ′ )
}] , (8)
j where D j c ⊕ G (u, u' ) denotes the set {D c ⊕ g : g ∈ G (u, u' )} , n/2 n 3n / 2 and S K~ = 2 × 2 = 2 .
B. Demodulation Schemes We propose two demodulation schemes to demodulate the received signal. One is the traditional optimal receiver, and the other is the three-stage demodulation scheme, which can greatly reduce the hardware complexity.
1) Scheme A: Optimal Matched Filters For the M-CSK modulation scheme, the optimum demodulator is a bank of M correlators. The received signal is first correlated with these M correlators and the estimated symbol is determined by picking out the largest correlation value. In this demodulation scheme, we need M correlators and the hardware complexity is O(21.5n ) . This optimum demodulation scheme suffers from the hardware complexity, so the modulation level M is limited. Hence we propose a new demodulation scheme which can greatly reduce the hardware complexity while retaining system performance. 2) Scheme B: Three-Stage Demodulation It is noted that each transmitted Kasami code sequence is the exclusive-OR output of three elementary code sequences, including u , an element of V (u′) , and an element of C (u′′) . In this scheme, we sequentially make use of these three elementary sequences to determine the transmitted symbol. As shown in Fig. 2, we partition the demodulation process into three stages. The effect of code sequence u is eliminated in n/2 Stage 1. In Stage 2, there are a total of S C (= 2 ) α -matched filters, which are used to deal with C (u′′) and V (u′ ) , and the output of each α -matched filter is then sent to Stage 3. Finally, the global maximum is determined from S C ( = 2 n / 2 ) local maximums in Stage 3 and the received symbol can be determined. Let r = [ r0 , r1, , rL−1 ] denote the received signal vector in chip level. Each element ri is the outcome of integrating the received signal in the i -th chip duration, i.e. 1 (i +1)Tc ri = ∫ r (t ) × p (t − iTc ) dt , 0 ≤ i ≤ L − 1 , (9) Tc iTc where r (t ) is the received waveform in the receiver. In Stage 1, r is multiplied by χ (u) “element-by-element”, and the result is denoted as y = [ y0 , y1 , , y L−1 ] , i.e. (10) yi = ri × χ (ui ), i = 0, , L − 1 . n/2 In Stage 2, y is imported into S C (= 2 = L1 + 1) parallel α -matched filters, and the configuration of a α -matched
∑ p (t − iT )
∫ dt
sampling per
χ (u L −1 )
yL −1
i =0
r (t )
In this section, we will compare the system performance among the proposed Kasami code-shift-keying modulation and other systems, in the aspects of power efficiency, bandwidth efficiency, and hardware complexity. These compared systems are M-CSK modulations systems using orthogonal code sequences, m-sequences, Gold sequences, and PPM-UWB systems.
A. Bit Error Probability for AWGN Channel Let Pi , j denote the probability of error decision from symbol i to symbol j , we have (12) Pi , j = Q PTs (1 − ρ i , j ) N 0 = Q (1 − ρ i , j ) × γ , where ρi, j denotes the cross-correlation between φi (t ) and φ j (t ) , and γ = PTs N 0 is the equivalent symbol energy to total noise power density ratio. By applying the union bound, we have the average symbol error probability, conditioning on symbol i
(
) (
)
...
...
χ (c j, 0 )
χ (c j ,1 )
y L −1
y
χ (c j , L−1 )
filter
z j ,0
z j ,1
z j , L −1
∑
α − matched µ1 , m1
r1
r0 χ (u 0 )
χ (u1 )
...
y1
α − matched µ 0 , m0
Tc seconds
Stage 1
Fig. 2.
SYSTEM PERFORMANCE
IV.
y0
c
Tc
r
Subsequently, z j is passed through a matched filter corresponding to u′ to evaluate the correlations between z j and χ ( v m ), m = 1, , L . The matched outputs are denoted as µ j,m . z j is also correlated with χ (0 L ) and the output is denoted as µ j ,0 . Finally, the α -matched filter selects the local maximum µ j among µ j ,m , m = 0, 1, , L , and passes parameters into Stage 3. The parameters are µ j and the related argument m j . In Stage 3, the global maximum among µ j , j = 0, 1, , L1 is picked out and the transmitted Kasami code sequence can also be determined according to the arguments j and the related m j . Finally the estimation of the received symbol is obtained. Notice that the performance of Scheme B is the same as Scheme A, because the received signal in Scheme B is also correlated with all signal waveform alphabets.
L −1
1 Tc
rL−1
filter is shown in Fig. 3. Each α -matched filter is corresponding to an elementary code sequence of C (u′′) . In the j -th α -matched filter ( 0 ≤ j ≤ L1 ), the vector y is first multiplied by χ (c j ) “element-by-element”, and the resultant sequence is denoted as z j = [z j ,0 , z j ,1 , , z j ,L−1 ] , i.e. z j ,i = y i × χ ( c j ,i ), i = 0, , L − 1, j = 0, , L1 . (11)
y1
y0
filter
• • •
Pick the largest one
Decision result
1
Pick the largest one
1
1
χ (u '2 )
χ (u '1 )
zj χ (u 'L −1 )
α − matched µL , mL 1 1
1
µ j ,0
χ (u '1 )
χ (u '0 ) χ (u 'L −1 )
filter
Stage 2
Structure of the three-stage demodulation receiver.
Stage 3
∑ Fig. 3.
Structure of the α -matched filter.
µ j ,m , m = 1, 2,
,L
µ j, mj
modulations, and the improvement in Eb N 0 is about 1 dB. We also found that the M-CSK modulation schemes using orthogo(13) Pi ≤ ∑ Pi , j , for i = 0, 1, , M − 1 , nal sequences, Gold sequences or m-sequences have a nearly j =0 , j ≠i identical performance, since they have almost the same constelIt is noted that the ρi, j of the large set Kasami sequences lation size M . Furthermore, the numerical result obtained from ℜ = {ρ 1 , ρ 2 , ρ 3 , ρ 4 , ρ 5 } = takes values in the set (14) shows good accuracy for a large value of Eb N 0 . More Ac{ [t (n) − 2] L , [s(n) − 2] L , − 1 L , − s(n) L , − t (n) L }. important, our simulation results verify that the performances are cording to (12), (13), and assuming that all symbols are the same for demodulation Scheme A and demodulation Scheme equiprobable, we then have the BER Perror upper bounded by B. 5 1 (14) Perror ≤ β m × Q (1 − ρ m ) × γ , In addition, for PPM-UWB systems, the power efficiency is ∑ 2( M − 1) m=1 the same as that of the M-CSK modulation using orthogonal where β m , 1 ≤ m ≤ 5 denotes the number of signal wave- code sequences. Therefore, according to the results shown in Fig. forms pairs φi (t ), φ j (t ) , for any i ≠ j , 0 ≤ i, j ≤ M − 1 , 4, we can conclude that the power efficiency ηP ,K of Kasami whose cross-correlation is equal to ρ m ∈ ℜ . M-CSK modulation is better than those of other M-CSK modulaTo figure out the upper bound of Perror , the values of β m , tions and the PPM-UWB systems in an AWGN channel, i.e. m = 1, , 5 should be determined. However, there is currently η P ,K > η P ,O = η P , PPM ≈ η P ,G ≈ η P ,m . (15) no solution to solve β m . By exhaustive searching aided by The subscripts O , G , and m represent the M-CSK modulacomputers, the exact values of β m can be determined and the tions using orthogonal sequences, Gold sequences, and bound of Perror can be easily obtained from (14). m-sequences, respectively, and the subscript PPM is used to represent the PPM-UWB systems. B. Power Efficiency To investigate the scalability of the proposed Kasami M-CSK First, we take the case with n = 6 as an example for dis- modulation, we examined four different cases with the polynocussion. There are 6 m-sequences for n = 6 , denoted by u0 , mial degrees being n = 4 , n = 6 , n = 8 and n = 10 . The coru1 , u 2 , u 3 , u 4 and u 5 , and the generator polynomials are responding code lengths are L = 15 , L = 63 , L = 255 and x 6 + x 5 + 1 , x 6 + x + 1 , x 6 + x 5 + x 4 + x + 1 , x 6 + x 5 + x 2 + x + 1 , L = 1023 , respectively, and the generator polynomials are x 6 + x 4 + x 3 + x + 1 , and x 6 + x 5 + x 3 + x 2 + 1 , respectively. h( x) = x 4 + x 3 + 1 , h( x) = x 6 + x 5 + 1 , h( x) = x 8 + x 6 + x 5 + x 4 + 1 Each m-sequence can be used to generate a large set of Ka- and h( x) = x10 + x7 + 1 , respectively. According to the sami sequences, and these 6 different sets of large Kasami above-mentioned results the corresponding Kasami code sets and sequences are denoted by K (u 0 ) , K (u1 ) , K (u 2 ) , K (u3 ) , the implementation of the modulator and demodulator can easily K (u 4 ) , and K (u5 ) . The corresponding reduced sets of large be obtained. Fig. 5 shows the simulation results of the proposed ~ Kasami sequences can be obtained and denoted by K (u 0 ) , scheme for the single-user environment in an AWGN channel ~ ~ ~ ~ ~ K (u 1 ) , K (u 2 ) , K (u 3 ) , K (u 4 ) , and K (u 5 ) . Furthermore, the with different values of n = 4 , 6, 8 and 10.The performance of cross-correlation between any two sequences in one large the M-CSK modulation using orthogonal sequences is also proKasami set takes on values in the set vided as a reference. The results show that the increase of M , ℜ = {15 63 , 7 63 , − 1 63 , − 9 63 , − 17 63 } . Via computer i.e. the increase of n , will effectively enhance the BER perexperiment, we found that the cross-correlation properties are formance, i.e. enhance the power efficiency. It is also shown that ~ the same for all K (u i ), i = 0 ~ 5 , and we have β1 = 0, our proposed scheme is better than M-CSK modulation with β 2 = 96768, β 3 = 129024, β 4 = 3584, and β 5 = 32256 . orthogonal codes for any fixed values of n . Furthermore, for Therefore, we conclude that these 6 reduced sets of large Ka- our proposed scheme, the increase of n from 4 to 6 can sami sequences are equivalent and lead to the same system achieve an improvement of 1.5 dB in E N , and an imb 0 performance. provement of 1 dB in Eb N 0 can be obtained for the increase from n = 6 to n = 8 . 1) AWGN Channel Model being transmitted, upper bounded by M −1
(
[
)
]
Fig. 4 shows the bit error performance Perror versus Eb N 0 with n = 6 for the Kasami M-CSK modulation and
other M-CSK modulations in an AWGN channel. The code length for each modulation scheme is L = 63 , except that the code length is L = 64 for the case using orthogonal sequences. However, the symbol alphabet sizes are different for various modulation schemes. For the proposed Kasami M-CSK modulation we have M = 512 , for the M-CSK modulation using orthogonal or Gold sequences we have M = 64 , and we have M = 63 for M-CSK modulation using m-sequences. The numerical and simulation results show that the Kasami M-CSK modulation outperforms other M-CSK
2) UWB Multipath Channel Model In the simulation for the UWB multipath channel, the chip duration is assumed to be Tc = 1 n sec . To overcome the multipath propagation effect, the M-CSK receiver is assumed to be a RAKE receiver with the finger number being 16, similar to that adopted in the DS-UWB system [7]. Fig. 6 shows the Perror versus Eb N 0 with n = 6 , 8 and 10 for the Kasami M-CSK modulation and other M-CSK modulations in the UWB multipath channel. For the case with n = 6 , the performance of the Kasami M-CSK modulation is slightly worse than that of other schemes. This is mainly due to the fact that the multipath interference degrades the system performance, especially for the
proposed Kasami M-CSK modulation which uses a bounded correlation code set with larger values of cross-correlation. For the case with n = 8 , these three schemes have almost the same performance, whereas the proposed scheme outperforms other schemes for the case with n = 10 . Although the proposed Kasami M-CSK modulation may be inferior to other schemes for the case with n = 6 , it must be noted that the proposed scheme is always superior to other schemes in the bandwidth efficiency with a gain of 50 % as shown next. C. Bandwidth Efficiency
The bandwidth efficiency η B is defined as the ratio of the transmission bit rate to the channel bandwidth. For Kasami M-CSK modulation, we have the bandwidth efficiency log 2 M (L Tc ) 1.5 n . (16) η B,K = = log 2 (2 3 n / 2 ) (2 n − 1) = n 1 / Tc n →∞ 2 For other M-CSK modulations, we have the bandwidth effin n ciency as：η B,O = n 2 (Orthogonal Sequences), η B ,G ≈ n 2 n (Gold Sequences), η B,m ≈ n 2 (m-sequences). In addition, for some PPM-UWB systems, the bandwidth efficiency is n generally limited to η B , PPM = n 2 . Hence, we conclude that the bandwidth efficiency of Kasami M-CSK modulation is 1.5 times better than other M-CSK modulations and PPM-UWB systems. D. Hardware Complexity
The increase of M will effectively enhance the BER performance for M-CSK modulations, but the receiver suffers from the increase of hardware complexity. The optimal receiver structure, like the demodulation Scheme A, uses a bank of M correlators. Therefore, the hardware complexity will be O( M ) = O( 23n 2 ) , and this makes the demodulation Scheme A infeasible for large n . The implementation becomes feasible for large n when the demodulation Scheme B is applied. n/2 Now, only S C = L1 + 1 = 2 α -matched filters are needed, and the complexity is greatly reduced to O( M 1 3 ) = O( 2n 2 ) . When the RAKE receiver is used in UWB multipath chanPerror 10 1
nel, the complexity increases proportionally to the finger number. For a large finger number, a better BER performance can be obtained with the cost of an increase in receiver complexity. In general, there is a trade off between system performance and hardware complexity. E. Multiple Access Environments
It is possible that two or more users are using the same radio channel for UWB applications, and the multiple access interference (MAI) should be taken into consideration. In our discussion, different users in the same radio channel will use different reduced sets of large Kasami sequences. It is assumed that different users are asynchronous, and the power ratio ζ , defined as the signal power of an MAI user to the desired signal power, is considered. The UWB communication systems are generally applied to low-density and short-range transmission, and it is reasonable to consider the power ratio ζ is less than 1. Fig. 7 shows the simulation results for two-user and three-user environments in ~ AWGN and UWB channels. The desired user applies K (u 0 ) as ~ the alphabet set, the other two MAI users apply K (u1 ) and ~ K (u 2 ) . The results show that the BER is inversely proportional to ζ . Furthermore, it is noted that there is no saturation phenomenon for the two-user environment, and the saturation phenomenon due to MAI occurs for the three-user environment in AWGN channel. However, saturation phenomenon occurs in UWB channel, since the multipath interference severely degrades the performance. V.
CONCLUSIONS
In this work, we have proposed an M-CSK modulation technique based on the large set of Kasami sequences for the applications of DSSS UWB communications. The modulation and demodulation schemes were developed, and the system performance, from the aspects of power efficiency, bandwidth efficiency and hardware complexity, was thoroughly investigated. -1
Perror 10
0
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Orthogonal Gold sequence m-sequence
-2
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Bit Error Probability
Bit Error Probability
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SchemeA SchemeB
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Orthogonal (Simulation), L=64, M=64 Gold sequence (Simulation), L=63, M=64 m-sequence (Simulation), L=63, M=63 SchemeA (Simulation), L=63, M=512 SchemeB (Simulation), L=63, M=512 SchemeA&B (Upper bound), L=63, M=512 0
1
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n n n n
= 4 = 6 =8 = 10
Orthogonal Kasami
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Eb / N 0 (dB) Fig. 4. The bit error probability Perror versus Eb N 0 for Kasami M-CSK and other M-CSK modulations with n = 6 in an AWGN channel.
10
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4
5
Eb / N 0 (dB)
6
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8
Fig. 5. The bit error probability Perror versus Eb N 0 for the single-user environment with several values of n in an AWGN channel.
For the bandwidth efficiency, the proposed Kasami sequences M-CSK modulation outperforms other schemes with a gain of 50 %. For the power efficiency, the proposed scheme was found to outperform other schemes with an improvement of 1dB in bit energy-to-noise power spectral density ratio under an AWGN channel; however the power efficiency was found to be almost equivalent under the UWB channel. In addition, the increase of the alphabet size M will enhance the transmission data rate but it increases the hardware complexity dramatically. By using the proposed demodulation Scheme B, the hardware complexity can be greatly reduced to O( M 1 3 ) , and as a result the implementation of receivers for a very large value of M becomes feasible. Furthermore, the proposed Kasami M-CSK modulation can also be applied to asynchronous multiple access environments, and an acceptable bit error rate performance can be obtained if the number of users is small. In conclusion, the proposed Kasami M-CSK modulation substantially enhances power efficiency and bandwidth efficiency, and is therefore quite suitable for the applications of UWB communications or other wireless communications. REFERENCES [1]
K. Siwiak, P. Withington, and S. Phelan, “Ultra-wide band radio: the -1
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Eb / N 0 (dB)
5
6
7
Fig. 6. The bit error probability Perror versus Eb N 0 for the single-user environment with several values of n in the CM1 UWB channel.
-3
-4
-5
ζ ζ ζ ζ
-6
-7
0
2
= 0.5 = 0.3 = 0.1 =0
AWGN, 2 users AWGN, 3 users UWB,CM1,2users UWB,CM1,3users 4
6
8
10
12
14
16
18
Eb / N 0 (dB) Fig. 7. The bit error probability Perror versus Eb N 0 for multiple access with n = 6 and several values of ζ .
Kasami Code-Shift-Keying Modulation for Ultra Wideband Communication Systems Yuh-Ren Tsai* and Xiu-Sheng Li Institute of Communications Engineering National Tsing Hua University 101, Sec. 2, Kuang-Fu Rd., Hsinchu 300, Taiwan [email protected] Abstract — Ultra wideband (UWB) communication systems are generally applied to short-range wireless communications. In order to achieve higher rates or to support multiple access capabilities, direct sequence spread spectrum (DSSS) techniques have been introduced to UWB systems. Furthermore, the concept of M-ary code shift keying (M-CSK) was introduced into DSSS systems to achieve higher rates. In this work, we propose an M-CSK modulation technique based on the large set of Kasami sequences since it possesses good code properties. The modulation and demodulation schemes are developed and the system performance is well investigated. It is found that the Kasami M-CSK modulation outperforms other M-CSK modulation techniques or some UWB systems. Furthermore, based on our demodulation scheme, the hardware complexity of receivers can be greatly reduced to O( M 1 3 ) and the implementation for a large M becomes feasible. Index Terms — Modulation; Multiaccess communication; Pseudonoise coded communication.
I.
power efficiency and bandwidth efficiency, in additive white Gaussian noise (AWGN) and UWB channels exposed to multiple user interference is well investigated. Compared with the cases employing other code sets, we found that a better bit error rate (BER) performance can be obtained in most situations, and a higher data rate can be achieved. Furthermore, we introduce two possible demodulation schemes: one is the traditional optimal receiver and the other is our proposed scheme which can greatly reduce the hardware complexity. The rest of this paper is organized as follows. Section II introduces the channel model and the generation of Kasami sequences. Section III describes the modulation and demodulation schemes. Section IV concentrates on the system performance and the comparison with other systems. Finally, this paper is concluded in Section V.
INTRODUCTION
Recently, the investigation activities on the issue of ultra wideband (UWB) systems grow up rapidly [1]-[2]. In general, UWB communication systems are applied to short-range wireless communications for their great advantages, such as coexisting with narrowband systems, low probability of interception, low power consumption, and diversity gain of multipath resolution. Impulse radio (IR) technology is originally proposed for UWB systems [3]. However, in order to diminish the interference from other multiple access users, spread spectrum techniques are introduced and applied to UWB systems. For the purpose of achieving higher rates or supporting multiple access capabilities, direct sequence spread spectrum (DSSS) UWB systems have been proposed in [4]-[5]. In recent years, two different proposals are also proposed for the applications of wireless personal area networks (WPANs). One is the multi-band OFDM (MB-OFDM) system [6], and the other is the direct-sequence UWB (DS-UWB) system [7]. To achieve higher rates, the concept of M-ary code shift keying (M-CSK) was introduced to DSSS systems [8]-[9]. M-CSK uses different codes to represent different symbols. There are many choices of code sets, such as orthogonal sets, bounded correlation sets, and random codes. In this work, we make use of Kasami sequences as the alphabet set. Details of the modulation and demodulation schemes are proposed. The system performance, including the hardware complexity, This work was supported in part by the National Science Council, Taiwan, R.O.C., under Grant NSC 95-2752-E-007-003-PAE and under Grant NSC 94-2219-E-007-006.
II. A.
PRELIMINARIES
Pseudo-Random Sequences
For DSSS, there are many choices of spreading sequences. Particularly, the sequence length, the cross-correlation and the code set size are major properties concerning spreading sequences. The sequence length, denoted by L , is available only for some specific values. For example, the sequence lengths of Walsh sequences are only available for L = 2 n . Furthermore, for any two binary code sequences a = [ a 0 , a1 , , a L −1 ] and b = b0 , b1 , , bL −1 , the discrete periodic cross-correlation function is defined as 1 L −1 (1) θ ab = ∑ (− 1)a (− 1)b . L =0 The code set size is defined as the number of available code sequences in a code set. In practical applications, it is important to distinguish the binary code sequence and the corresponding signal waveform. The relation between a binary code sequence a = [a0 , a1 , , aL−1 ] and the corresponding signal waveform φ (t ) can be presented by L −1
φ (t ) = ∑ (− 1)a p(t − Tc ), =0
0 ≤ t ≤ Ts ,
(2)
where Tc is the chip duration, Ts = LTc is the symbol duration, and p (t ) is the unit pulse shaping function. The signal waveform can be represented as signal vector with elements taking on values of {+ 1, −1} . For convenience sake, we define a function
χ (•) to represent the translation from a binary code sequence to the corresponding signal vector, i.e. χ (a ) = (− 1)a , χ (a) = (− 1) a0 , (− 1) a1 , , (− 1) aL −1 . (3)
[
]
B. Kasami Sequences In this work, we employ the large set of Kasami sequences as the symbol alphabet set since it possesses good code properties, including the large code set size and low cross-correlations. In the construction of the large set Kasami sequences with the length L = 2 n − 1 , the parameter n must be an even integer. In this work, we mainly focus on the case of n = 2 mod 4 , and we briefly introduce how to construct the large set of Kasami sequences below [10]. Let u represent a binary m-sequence vector of the degree n n with the period L = 2 − 1 , D denote the operator which shifts the code phase of a sequence vector cyclically by one unit, and ⊕ denote the exclusive-OR operator. For q = t ( n) ≡ 1 + 2 (n+ 2 ) 2 , u′ = u[t (n) ] forms another m-sequence with the period L , and consequently we have a set of Gold sequences G (u, u ′) ≡ {u, u ′, u ⊕ u ′, u ⊕ Du ′, u ⊕ D 2 u ′,..., u ⊕ D L −1u ′}.(4)
It is noted that G (u, u′) contains SG = L + 2 = 2n + 1 code sequences with the period L . In addition, for q = s( n) ≡ 1 + 2 n 2 , u′′ = u[s(n)] forms another m-sequence with the period L1 = 2 n / 2 − 1 . Now we define a cover sequence c = [c0 , c1 , , cL−1 ] , which is the repetitions of u′′ , i.e. [c0+ jL , c1+ jL , , c( L −1)+ jL ] is equal to u′′ for j = 0, ,2n 2 . We then define a set of cover sequences as L (5) C (u ′′) ≡ 0 L ∪ ∪ 1 D j −1c = {c j , j = 0, , L1 }, 1
1
1
1
[
j =1
]
where ∪ denotes the union of sets, and 0 L is an all-zero sequence with length L . It is noted that there are SC = L1 + 1 = 2 n / 2 elements in C (u′′) . Each element of the large set Kasami sequences, denoted by K (u) , is the exclusive-OR output of an element in G(u, u′) and an element in C (u′′) . It is noted that the code set size of n/2 n K (u) is S K = S C × S G = 2 ( 2 + 1) . The large set of Kasami sequences can also be expressed by generator polynomials. Let
h(x) , h ′( x ) and h ′′(x) denote the generator polynomials of u , u[t (n ) ] and u[s(n ) ] , respectively. Fig. 1 gives an example of the
large set Kasami sequences generator.
C. Channel Models In this work, we consider two channel models, one is additive white Gaussian noise (AWGN) channel model, and the other is ultra-wideband multipath channel model with Tc = 1 n sec [11]. In addition, a RAKE receiver with the finger number 16 is applied for the UWB channel. III.
MODULATION AND DEMODULATION SCHEMES
We propose in this section the modulation and demodulation schemes for M-CSK modulation based on the large set of Kasami sequences.
A. Modulation Scheme In the M-CSK modulation, there are M ( M = 2 k ) different possible symbols, and the transmitter groups the input data stream into k-bit ( k = log2 M ) symbols with a symbol duration Ts . Since the code set size S K = 2 n / 2 ( 2 n + 1) of K (u) is not a power of 2, we select M = 2 3n 2 , and then we have k = 1.5n . We now propose a systematic method for sequences selection and mapping. The input data sequence in a symbol duration is denoted as d = [d 0 , d1 , , d n −1 , d n , , d1.5 n−1 ] , where d i ∈ {0,1} . In Fig. 1, the initial state of h(x) is fixed to an arbitrary nonzero value, such as h = [1, 0, , 0] , and the initial states of h′(x) and h′′(x) are set to be h ′ = [d 0 , d 1 , , d n −1 ] and h ′′ = [d n , d n +1 , , d 1.5 n −1 ] . The output code sequence is consequently a Kasami sequence of K (u ) . The generated Gold sequences set G~(u, u′) in Fig. 1 is the complete set of original Gold sequences with the code sequence u′ excluded, and it follows that S G~ = 2 n = L + 1 . Besides, if we define V (u′) = {0 L , u′, Du′, D 2u′, , D L−1u′}= {v m , m = 0, , L} , (6) the generated Gold sequence set is composed of u and an element from V (u′) , i.e. ~ G (u, u′) = u ⊕ V (u ′)
{
= u, u ⊕ u′, u ⊕ Du′, u ⊕ D 2 u′,
h( x ) = x 6 + x 5 + 1
h = [1,0,...,0] (an arbitrary nonzero vector)
h′ = [d 0 , d1 ,
h′( x) = x 6 + x 5 + x 3 + x 2 + 1 , d n −1 ]
h ′′ = [d n , d n +1 , Fig. 1.
Kasami sequence output
, d 1.5 n −1 ]
h′′( x) = x + x + 1
Large set Kasami sequence generator.
3
We therefore have the set of selected large quences, referred as the reduced set of large quences, being L ~ ~ K ( u ) = G (u , u ′ ) ∪ ∪ j = 1 D j − 1 c ⊕
[
1
{
}
(7)
, u ⊕ D L −1u′ .
Kasami code seKasami code se~ G (u , u ′ )
}] , (8)
j where D j c ⊕ G (u, u' ) denotes the set {D c ⊕ g : g ∈ G (u, u' )} , n/2 n 3n / 2 and S K~ = 2 × 2 = 2 .
B. Demodulation Schemes We propose two demodulation schemes to demodulate the received signal. One is the traditional optimal receiver, and the other is the three-stage demodulation scheme, which can greatly reduce the hardware complexity.
1) Scheme A: Optimal Matched Filters For the M-CSK modulation scheme, the optimum demodulator is a bank of M correlators. The received signal is first correlated with these M correlators and the estimated symbol is determined by picking out the largest correlation value. In this demodulation scheme, we need M correlators and the hardware complexity is O(21.5n ) . This optimum demodulation scheme suffers from the hardware complexity, so the modulation level M is limited. Hence we propose a new demodulation scheme which can greatly reduce the hardware complexity while retaining system performance. 2) Scheme B: Three-Stage Demodulation It is noted that each transmitted Kasami code sequence is the exclusive-OR output of three elementary code sequences, including u , an element of V (u′) , and an element of C (u′′) . In this scheme, we sequentially make use of these three elementary sequences to determine the transmitted symbol. As shown in Fig. 2, we partition the demodulation process into three stages. The effect of code sequence u is eliminated in n/2 Stage 1. In Stage 2, there are a total of S C (= 2 ) α -matched filters, which are used to deal with C (u′′) and V (u′ ) , and the output of each α -matched filter is then sent to Stage 3. Finally, the global maximum is determined from S C ( = 2 n / 2 ) local maximums in Stage 3 and the received symbol can be determined. Let r = [ r0 , r1, , rL−1 ] denote the received signal vector in chip level. Each element ri is the outcome of integrating the received signal in the i -th chip duration, i.e. 1 (i +1)Tc ri = ∫ r (t ) × p (t − iTc ) dt , 0 ≤ i ≤ L − 1 , (9) Tc iTc where r (t ) is the received waveform in the receiver. In Stage 1, r is multiplied by χ (u) “element-by-element”, and the result is denoted as y = [ y0 , y1 , , y L−1 ] , i.e. (10) yi = ri × χ (ui ), i = 0, , L − 1 . n/2 In Stage 2, y is imported into S C (= 2 = L1 + 1) parallel α -matched filters, and the configuration of a α -matched
∑ p (t − iT )
∫ dt
sampling per
χ (u L −1 )
yL −1
i =0
r (t )
In this section, we will compare the system performance among the proposed Kasami code-shift-keying modulation and other systems, in the aspects of power efficiency, bandwidth efficiency, and hardware complexity. These compared systems are M-CSK modulations systems using orthogonal code sequences, m-sequences, Gold sequences, and PPM-UWB systems.
A. Bit Error Probability for AWGN Channel Let Pi , j denote the probability of error decision from symbol i to symbol j , we have (12) Pi , j = Q PTs (1 − ρ i , j ) N 0 = Q (1 − ρ i , j ) × γ , where ρi, j denotes the cross-correlation between φi (t ) and φ j (t ) , and γ = PTs N 0 is the equivalent symbol energy to total noise power density ratio. By applying the union bound, we have the average symbol error probability, conditioning on symbol i
(
) (
)
...
...
χ (c j, 0 )
χ (c j ,1 )
y L −1
y
χ (c j , L−1 )
filter
z j ,0
z j ,1
z j , L −1
∑
α − matched µ1 , m1
r1
r0 χ (u 0 )
χ (u1 )
...
y1
α − matched µ 0 , m0
Tc seconds
Stage 1
Fig. 2.
SYSTEM PERFORMANCE
IV.
y0
c
Tc
r
Subsequently, z j is passed through a matched filter corresponding to u′ to evaluate the correlations between z j and χ ( v m ), m = 1, , L . The matched outputs are denoted as µ j,m . z j is also correlated with χ (0 L ) and the output is denoted as µ j ,0 . Finally, the α -matched filter selects the local maximum µ j among µ j ,m , m = 0, 1, , L , and passes parameters into Stage 3. The parameters are µ j and the related argument m j . In Stage 3, the global maximum among µ j , j = 0, 1, , L1 is picked out and the transmitted Kasami code sequence can also be determined according to the arguments j and the related m j . Finally the estimation of the received symbol is obtained. Notice that the performance of Scheme B is the same as Scheme A, because the received signal in Scheme B is also correlated with all signal waveform alphabets.
L −1
1 Tc
rL−1
filter is shown in Fig. 3. Each α -matched filter is corresponding to an elementary code sequence of C (u′′) . In the j -th α -matched filter ( 0 ≤ j ≤ L1 ), the vector y is first multiplied by χ (c j ) “element-by-element”, and the resultant sequence is denoted as z j = [z j ,0 , z j ,1 , , z j ,L−1 ] , i.e. z j ,i = y i × χ ( c j ,i ), i = 0, , L − 1, j = 0, , L1 . (11)
y1
y0
filter
• • •
Pick the largest one
Decision result
1
Pick the largest one
1
1
χ (u '2 )
χ (u '1 )
zj χ (u 'L −1 )
α − matched µL , mL 1 1
1
µ j ,0
χ (u '1 )
χ (u '0 ) χ (u 'L −1 )
filter
Stage 2
Structure of the three-stage demodulation receiver.
Stage 3
∑ Fig. 3.
Structure of the α -matched filter.
µ j ,m , m = 1, 2,
,L
µ j, mj
modulations, and the improvement in Eb N 0 is about 1 dB. We also found that the M-CSK modulation schemes using orthogo(13) Pi ≤ ∑ Pi , j , for i = 0, 1, , M − 1 , nal sequences, Gold sequences or m-sequences have a nearly j =0 , j ≠i identical performance, since they have almost the same constelIt is noted that the ρi, j of the large set Kasami sequences lation size M . Furthermore, the numerical result obtained from ℜ = {ρ 1 , ρ 2 , ρ 3 , ρ 4 , ρ 5 } = takes values in the set (14) shows good accuracy for a large value of Eb N 0 . More Ac{ [t (n) − 2] L , [s(n) − 2] L , − 1 L , − s(n) L , − t (n) L }. important, our simulation results verify that the performances are cording to (12), (13), and assuming that all symbols are the same for demodulation Scheme A and demodulation Scheme equiprobable, we then have the BER Perror upper bounded by B. 5 1 (14) Perror ≤ β m × Q (1 − ρ m ) × γ , In addition, for PPM-UWB systems, the power efficiency is ∑ 2( M − 1) m=1 the same as that of the M-CSK modulation using orthogonal where β m , 1 ≤ m ≤ 5 denotes the number of signal wave- code sequences. Therefore, according to the results shown in Fig. forms pairs φi (t ), φ j (t ) , for any i ≠ j , 0 ≤ i, j ≤ M − 1 , 4, we can conclude that the power efficiency ηP ,K of Kasami whose cross-correlation is equal to ρ m ∈ ℜ . M-CSK modulation is better than those of other M-CSK modulaTo figure out the upper bound of Perror , the values of β m , tions and the PPM-UWB systems in an AWGN channel, i.e. m = 1, , 5 should be determined. However, there is currently η P ,K > η P ,O = η P , PPM ≈ η P ,G ≈ η P ,m . (15) no solution to solve β m . By exhaustive searching aided by The subscripts O , G , and m represent the M-CSK modulacomputers, the exact values of β m can be determined and the tions using orthogonal sequences, Gold sequences, and bound of Perror can be easily obtained from (14). m-sequences, respectively, and the subscript PPM is used to represent the PPM-UWB systems. B. Power Efficiency To investigate the scalability of the proposed Kasami M-CSK First, we take the case with n = 6 as an example for dis- modulation, we examined four different cases with the polynocussion. There are 6 m-sequences for n = 6 , denoted by u0 , mial degrees being n = 4 , n = 6 , n = 8 and n = 10 . The coru1 , u 2 , u 3 , u 4 and u 5 , and the generator polynomials are responding code lengths are L = 15 , L = 63 , L = 255 and x 6 + x 5 + 1 , x 6 + x + 1 , x 6 + x 5 + x 4 + x + 1 , x 6 + x 5 + x 2 + x + 1 , L = 1023 , respectively, and the generator polynomials are x 6 + x 4 + x 3 + x + 1 , and x 6 + x 5 + x 3 + x 2 + 1 , respectively. h( x) = x 4 + x 3 + 1 , h( x) = x 6 + x 5 + 1 , h( x) = x 8 + x 6 + x 5 + x 4 + 1 Each m-sequence can be used to generate a large set of Ka- and h( x) = x10 + x7 + 1 , respectively. According to the sami sequences, and these 6 different sets of large Kasami above-mentioned results the corresponding Kasami code sets and sequences are denoted by K (u 0 ) , K (u1 ) , K (u 2 ) , K (u3 ) , the implementation of the modulator and demodulator can easily K (u 4 ) , and K (u5 ) . The corresponding reduced sets of large be obtained. Fig. 5 shows the simulation results of the proposed ~ Kasami sequences can be obtained and denoted by K (u 0 ) , scheme for the single-user environment in an AWGN channel ~ ~ ~ ~ ~ K (u 1 ) , K (u 2 ) , K (u 3 ) , K (u 4 ) , and K (u 5 ) . Furthermore, the with different values of n = 4 , 6, 8 and 10.The performance of cross-correlation between any two sequences in one large the M-CSK modulation using orthogonal sequences is also proKasami set takes on values in the set vided as a reference. The results show that the increase of M , ℜ = {15 63 , 7 63 , − 1 63 , − 9 63 , − 17 63 } . Via computer i.e. the increase of n , will effectively enhance the BER perexperiment, we found that the cross-correlation properties are formance, i.e. enhance the power efficiency. It is also shown that ~ the same for all K (u i ), i = 0 ~ 5 , and we have β1 = 0, our proposed scheme is better than M-CSK modulation with β 2 = 96768, β 3 = 129024, β 4 = 3584, and β 5 = 32256 . orthogonal codes for any fixed values of n . Furthermore, for Therefore, we conclude that these 6 reduced sets of large Ka- our proposed scheme, the increase of n from 4 to 6 can sami sequences are equivalent and lead to the same system achieve an improvement of 1.5 dB in E N , and an imb 0 performance. provement of 1 dB in Eb N 0 can be obtained for the increase from n = 6 to n = 8 . 1) AWGN Channel Model being transmitted, upper bounded by M −1
(
[
)
]
Fig. 4 shows the bit error performance Perror versus Eb N 0 with n = 6 for the Kasami M-CSK modulation and
other M-CSK modulations in an AWGN channel. The code length for each modulation scheme is L = 63 , except that the code length is L = 64 for the case using orthogonal sequences. However, the symbol alphabet sizes are different for various modulation schemes. For the proposed Kasami M-CSK modulation we have M = 512 , for the M-CSK modulation using orthogonal or Gold sequences we have M = 64 , and we have M = 63 for M-CSK modulation using m-sequences. The numerical and simulation results show that the Kasami M-CSK modulation outperforms other M-CSK
2) UWB Multipath Channel Model In the simulation for the UWB multipath channel, the chip duration is assumed to be Tc = 1 n sec . To overcome the multipath propagation effect, the M-CSK receiver is assumed to be a RAKE receiver with the finger number being 16, similar to that adopted in the DS-UWB system [7]. Fig. 6 shows the Perror versus Eb N 0 with n = 6 , 8 and 10 for the Kasami M-CSK modulation and other M-CSK modulations in the UWB multipath channel. For the case with n = 6 , the performance of the Kasami M-CSK modulation is slightly worse than that of other schemes. This is mainly due to the fact that the multipath interference degrades the system performance, especially for the
proposed Kasami M-CSK modulation which uses a bounded correlation code set with larger values of cross-correlation. For the case with n = 8 , these three schemes have almost the same performance, whereas the proposed scheme outperforms other schemes for the case with n = 10 . Although the proposed Kasami M-CSK modulation may be inferior to other schemes for the case with n = 6 , it must be noted that the proposed scheme is always superior to other schemes in the bandwidth efficiency with a gain of 50 % as shown next. C. Bandwidth Efficiency
The bandwidth efficiency η B is defined as the ratio of the transmission bit rate to the channel bandwidth. For Kasami M-CSK modulation, we have the bandwidth efficiency log 2 M (L Tc ) 1.5 n . (16) η B,K = = log 2 (2 3 n / 2 ) (2 n − 1) = n 1 / Tc n →∞ 2 For other M-CSK modulations, we have the bandwidth effin n ciency as：η B,O = n 2 (Orthogonal Sequences), η B ,G ≈ n 2 n (Gold Sequences), η B,m ≈ n 2 (m-sequences). In addition, for some PPM-UWB systems, the bandwidth efficiency is n generally limited to η B , PPM = n 2 . Hence, we conclude that the bandwidth efficiency of Kasami M-CSK modulation is 1.5 times better than other M-CSK modulations and PPM-UWB systems. D. Hardware Complexity
The increase of M will effectively enhance the BER performance for M-CSK modulations, but the receiver suffers from the increase of hardware complexity. The optimal receiver structure, like the demodulation Scheme A, uses a bank of M correlators. Therefore, the hardware complexity will be O( M ) = O( 23n 2 ) , and this makes the demodulation Scheme A infeasible for large n . The implementation becomes feasible for large n when the demodulation Scheme B is applied. n/2 Now, only S C = L1 + 1 = 2 α -matched filters are needed, and the complexity is greatly reduced to O( M 1 3 ) = O( 2n 2 ) . When the RAKE receiver is used in UWB multipath chanPerror 10 1
nel, the complexity increases proportionally to the finger number. For a large finger number, a better BER performance can be obtained with the cost of an increase in receiver complexity. In general, there is a trade off between system performance and hardware complexity. E. Multiple Access Environments
It is possible that two or more users are using the same radio channel for UWB applications, and the multiple access interference (MAI) should be taken into consideration. In our discussion, different users in the same radio channel will use different reduced sets of large Kasami sequences. It is assumed that different users are asynchronous, and the power ratio ζ , defined as the signal power of an MAI user to the desired signal power, is considered. The UWB communication systems are generally applied to low-density and short-range transmission, and it is reasonable to consider the power ratio ζ is less than 1. Fig. 7 shows the simulation results for two-user and three-user environments in ~ AWGN and UWB channels. The desired user applies K (u 0 ) as ~ the alphabet set, the other two MAI users apply K (u1 ) and ~ K (u 2 ) . The results show that the BER is inversely proportional to ζ . Furthermore, it is noted that there is no saturation phenomenon for the two-user environment, and the saturation phenomenon due to MAI occurs for the three-user environment in AWGN channel. However, saturation phenomenon occurs in UWB channel, since the multipath interference severely degrades the performance. V.
CONCLUSIONS
In this work, we have proposed an M-CSK modulation technique based on the large set of Kasami sequences for the applications of DSSS UWB communications. The modulation and demodulation schemes were developed, and the system performance, from the aspects of power efficiency, bandwidth efficiency and hardware complexity, was thoroughly investigated. -1
Perror 10
0
10
-2
10
-1
Orthogonal Gold sequence m-sequence
-2
10
-3
10
Bit Error Probability
Bit Error Probability
10
SchemeA SchemeB
-3
10
-4
10
-5
10
-6
10
-7
10
-1
Orthogonal (Simulation), L=64, M=64 Gold sequence (Simulation), L=63, M=64 m-sequence (Simulation), L=63, M=63 SchemeA (Simulation), L=63, M=512 SchemeB (Simulation), L=63, M=512 SchemeA&B (Upper bound), L=63, M=512 0
1
2
3
-4
10
-5
10
-6
10
n n n n
= 4 = 6 =8 = 10
Orthogonal Kasami
-7
4
5
6
Eb / N 0 (dB) Fig. 4. The bit error probability Perror versus Eb N 0 for Kasami M-CSK and other M-CSK modulations with n = 6 in an AWGN channel.
10
1
2
3
4
5
Eb / N 0 (dB)
6
7
8
Fig. 5. The bit error probability Perror versus Eb N 0 for the single-user environment with several values of n in an AWGN channel.
For the bandwidth efficiency, the proposed Kasami sequences M-CSK modulation outperforms other schemes with a gain of 50 %. For the power efficiency, the proposed scheme was found to outperform other schemes with an improvement of 1dB in bit energy-to-noise power spectral density ratio under an AWGN channel; however the power efficiency was found to be almost equivalent under the UWB channel. In addition, the increase of the alphabet size M will enhance the transmission data rate but it increases the hardware complexity dramatically. By using the proposed demodulation Scheme B, the hardware complexity can be greatly reduced to O( M 1 3 ) , and as a result the implementation of receivers for a very large value of M becomes feasible. Furthermore, the proposed Kasami M-CSK modulation can also be applied to asynchronous multiple access environments, and an acceptable bit error rate performance can be obtained if the number of users is small. In conclusion, the proposed Kasami M-CSK modulation substantially enhances power efficiency and bandwidth efficiency, and is therefore quite suitable for the applications of UWB communications or other wireless communications. REFERENCES [1]
K. Siwiak, P. Withington, and S. Phelan, “Ultra-wide band radio: the -1
Perror 10
emergence of an important new technology,” in Proc. IEEE Vehicular Technology Conf. (VTC), vol. 2, 6-9 May 2001, pp. 1169-1172. [2] D. Porcino and W. Hirt, “Ultra-wideband radio technology-- potential and challenges ahead,” IEEE Commun. Magazine, vol. 41, pp. 66-74, July 2003. [3] M. Z. Win and R. A. Scholtz, “Impulse radio: how it works,” IEEE Commun. Lett., vol. 2, pp. 36-38, Feb. 1998. [4] J. R. Foerster, “The performance of a direct-sequence spread ultra-wideband system in the presence of multipath, narrowband interference, and multiuser interference,” in Proc. IEEE Conf. on Ultra Wideband Systems and Technologies, 21-23 May 2002, pp. 87-91. [5] V. S. Somayazulu,, “Multiple access performance in UWB systems using time hopping vs. direct sequence spreading,” in Proc. IEEE Wireless Commun. and Networking Conf. (WCNC), vol. 2, 17-21 March 2002, pp. 522 -525. [6] “MERGED PROPOSAL #1,” http://grouper.ieee.org/groups/802/15/ pub/TG3a_CFP.html [7] “MERGED PROPOSAL #2,” http://grouper.ieee.org/groups/802/15/ pub/TG3a_CFP.html [8] A. G. Burr, “Capacity improvement of CDMA systems using M-ary code shift keying,” in Proc. IEEE Sixth International Conf. on Mobile Radio and Personal Commun., 9-11 Dec. 1991, pp. 63-67. [9] A. G. Burr, “Spherical codes for M-ary code shift keying,” in Proc. Second IEE National Conf. on Telecommunications, 2-5 Apr. 1989, pp. 67-72. [10] D. V. Sarwate and M. B. Pursley, “Crosscorrelation properties of pseudorandom and related sequences,” Proc. IEEE, vol. 68, pp. 593-619, May 1980. [11] 02490r0P802-15_SG3a-Channel-Modeling-Subcommittee-Report-Final.do c.
Perror10 10
-1
-2
-2
10
Bit Error Probability
Bit Error Probability
10
-3
10
10
n=6
10
n =8 n = 10 Kasami Gold or Goldlike PPM
-4
10
10
-5
10
10
1
2
3
4
Eb / N 0 (dB)
5
6
7
Fig. 6. The bit error probability Perror versus Eb N 0 for the single-user environment with several values of n in the CM1 UWB channel.
-3
-4
-5
ζ ζ ζ ζ
-6
-7
0
2
= 0.5 = 0.3 = 0.1 =0
AWGN, 2 users AWGN, 3 users UWB,CM1,2users UWB,CM1,3users 4
6
8
10
12
14
16
18
Eb / N 0 (dB) Fig. 7. The bit error probability Perror versus Eb N 0 for multiple access with n = 6 and several values of ζ .
