Narrowband interference suppression in wavelet ... - Semantic Scholar
WIRELESS COMMUNICATIONS AND MOBILE COMPUTING Wirel. Commun. Mob. Comput. (2012) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/wcm.2325
RESEARCH ARTICLE
Narrow-band interference suppression in wavelet packets based multicarrier multicode CDMA overlay system Maryam M. Akho-Zahieh and Nasser Abdellatif * Department of Electrical and Computer Engineering, Applied Science Private University, Amman 11931, Jordan
ABSTRACT The wavelet packets based multicarrier (MC) multicode (MCD) code-division multiple-access (CDMA) transceiver consists of the MCD part, which ensures the transmission for high speed and flexible data rate; the MC part contributing to robustness to frequency-selective fading and flexibility for handling multiple data rates; and wavelet packets (WPs) modulation technique, which contributes to the mitigation of the interference problems. As WPs have lower sidelobes compared with sinusoidal carriers, this system is very effective in reducing the problem of inter-carrier interference. Of course, like any CDMA system, the system can suppress a given amount of interference. This paper considers an interference suppression scheme which will enhance the performance of the system. The receiver employs suppression filters to mitigate the effect of narrow-band jammer interference. The framework for the system and the performance evaluation are presented in terms of the bit error rate and the outage probability over a Nakagami fading channel. Also, we investigate how the performance is influenced by various parameters, such as the number of taps of the suppression filter and the ratio of narrow-band interference bandwidth to the spread-spectrum bandwidth. Finally, the performance of the system is compared with the performance of sinusoidal based MC/MCD-CDMA system denoted Sin-MC/MCD-CDMA. Copyright © 2012 John Wiley & Sons, Ltd. KEYWORDS suppression filter; wavelet packet; multicarrier; multicode; CDMA *Correspondence Nasser Abdellatif, Department of Electrical and Computer Engineering, Applied Science Private University, Amman 11931, Jordan. E-mail: [email protected]; [email protected]
1. INTRODUCTION The multicarrier (MC) multicode (MCD) code-division multiple-access (CDMA) system achieves the advantages of both MC-CDMA and MCD-CDMA systems, which are (i) variable data rates without interference scaling and (ii) enhanced robustness to mitigating multipath fading. By combining the properties of MCD and MC techniques, a MC/MCD-CDMA system that uses wavelet packet (WP) as subcarrier is proposed in this paper. The system is denoted as WP-MC/MCD-CDMA. Previous studies [1,2] have shown that the problems posed by sinusoidal carriers can be solved by WPs. Unlike sinusoids, WPs have many attractive properties including: (1) Much lower sidelobes with negligible sidelobe energy leakage compared with sinusoid carriers. Copyright © 2012 John Wiley & Sons, Ltd.
This property is effective in suppressing inter-carrier interference (ICI) and multiple-access interference. (2) Naturally orthogonal and well localized in both time and frequency domains, which relaxes the requirement of frequency or time guard between different user signals. In fact, orthogonality is maintained for overlapped WPs in both time and frequency domains. This is an advantage of using WPs to model communication channels that are characterized not only by frequency selectivity but also by time variation. The WP-MC/MCD-CDMA [1] uses WP as subcarrier instead of sinusoidal function, because of that, it could mitigate the problem of ICI associated with using sinusoidal carrier and suppress interferences caused by multipath fading. Various schemes of MC-CDMA systems have been proposed in recent years, and they can be
Narrow-band suppression in wavelet packets based MC/MCD-CDMA system
If k denotes the kth user, j denotes the j th code substream, and h denotes the hth WPs superstreams, the model of our system for the kth user is shown in Figure 1. The data for the kth user are given by
classified into two groups. The first group is orthogonal MC in which different adjacent subcarriers are orthogonally overlapped. The second group is disjoint MCs in which all subcarriers are not overlapped. The second group contains two subtypes: in the first type, different data streams modulate different carriers, whereas in the second subtype, all carriers are modulated by the same data stream. The system proposed in this paper belongs to the second subtype of the second group. As it is well known, the inherent processing gain of CDMA system will, in many cases, provide the system with a sufficient degree of narrow-band interference rejection capability. But, if the interference signal is powerful enough, the conventional receiver is ineffective in mitigating this problem. Such instances of narrow-band interference may be intentional, as in various military applications, or may be unintentional, as in a commercial CDMA overlay scheme. An interference suppression filter (SF) can be employed to reject the narrow-band interference. A wiener-type filter, described in [3–5], employs a tapped delay line structure to first predict and then subtract out the narrow-band interference. The aim of this paper is to study the effect of narrowband interference binary phase shift keying (BPSK) waveform on the WP-MC/MCD-CDMA system and employs an SF to reject it, then compare the performance of the two systems WP-MC/MCD-CDMA and Sin-MC/MCDCDMA with and without SF.
d kj (t )
∑
bk (t )
a 2 (t ) d kJ (t )
Serial-to-Parallel Converter
Serial-to-Parallel Converter
d k (t )
T t i T =.JH / JH (1)
Y
where …x ./ is a rectangular pulse of duration x. These data have a bit rate of JH T , where T is the bit duration, and Q
dkI .t / and dk .t / represent the inphase .I / and quadrature .Q/ data symbols, respectively. These data are serial-toparallel (S=P) converted into J substreams, each with a lower bit rate H T . The j th data substream corresponding to the kth user signal is given by 1 X
dkj .t / D
i dkj
Y T =H
iD1
.t iT =H /
(2)
After S=P conversion, the substreams are coded by a set of orthogonal signals aj .t /, which has a chip rate of T1c D HNc T ,
and given by
aj .t / D
NX c 1
aji
Y Tc
iD0
.t iTc /
(3)
where Nc is the code length, and Tc is the code pulse i 2 f˙1g with probaduration. Note that aji and dkj bilities P .1/ D P .1/ D 0:5. In order to maintain orthogonality of the coding signals, the maximum number of substreams J is limited to Nc D HTTc . The coded
bk1 (t ) Interference
c k (t )
wp1 (t )
∑
bk1 (t ) c k (t ) wp (t ) h
s (t )
exp(− jω o t )
n(t)
bkH (t )
Channel c k (t )
a J (t )
dki
iD1
The system model under consideration consists of K active users transmitting data simultaneously to the base station.
a1 (t )
1 X
Q
dk .t / D dkI.t /jdk .t / D
2. SYSTEM MODEL AND DESCRIPTION
d k1 (t )
M. M. Akho-Zahieh and N. Abdellatif
wp H (t )
∫ (⋅)dt
x1
0
c k (t ) wp1 (t ) T
∫ (⋅)dt
Suppression filters r (t )
xh
0
rs (t )
exp(− jω o t )
c k (t ) wp h (t ) T
∫ (⋅)dt
0
c k (t ) wp H (t )
xH
T
Parallel-to-Serial Converter
T
H →1
∫ (⋅)dt
z1
T
zj
0
bˆk
a1 (t )
∫ (⋅)dt
0
a j (t ) T
∫ (⋅)dt
0
a J (t )
zJ
Parallel-to-Serial Converter
BPF
dˆ k
J →1
Figure 1. WP-MC/MCD-CDMA system model.
Wirel. Commun. Mob. Comput. (2012) © 2012 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
M. M. Akho-Zahieh and N. Abdellatif
Narrow-band suppression in wavelet packets based MC/MCD-CDMA system
substreams are added, and the resulting signal bk .t / D PJ j D1 aj .t /dkj .t / is again S=P converted into H superstreams bkh .t /. As bk .t / has a bit rate of H T , bkh .t / will 1 have a bit rate of T . The next step is the modulation by the kth user pseudorandom noise (PN) signature sequence
ck .t / D
NX n 1
cki
iD0
Y Tn
.t iTn /
(4)
where Nn is the length of the PN code sequence, cki 2 f˙1g is the i th bit value with probabilities P .1/ D P .1/ D 0:5, and Tn D NTn is the chip duration. Each of the spreading superstreams will be used to modulate a WPs wph .t / given by wph .t / D
X
wh .t iTn /
where the khj l subscripts denote the kth user, j th code, hth wavelet, and lth path, respectively, and dk .t / D Q dkI .t / jdk .t /. The preceding analysis is carried out assuming quadrature phase shift keying (QPSK); if BPSK modulation is used, then the baseband data and the carrier term will be real, such that dk .t / is written as dkI .t / and exp.j !c t / is replaced with cos.!c t /. The receiver shown in Figure 1 is assumed to be synchronous, designed to detect the first substream of the first user’s first WP propagating via the first path. It consists of a bandpass filter (BPF), tap SF, and correlators. The received signal is first passed through the BPF having bandwidth Bs equal to the spread spectrum bandwidth D 2Tn1 , which removes the out-of band noise and lets the desired signal and inferences pass without distortion. It can be shown that the received signal, r.t /, can be written as
(5)
i
r.t / D
Assuming identical power for all users, the transmitted signal sk .t / can be written as sk .t / D
p
hD1 j D1
where dkj h .t / with period T is the data symbol of kth user, rth is the substream of the hth superstream, exp.j !c t / is the carrier signal, and P is the users’ power. The equivalent impulse response for the channel used in this paper can be written as
h.t / D
ˇkl e j kl ı.t kl /
2P
L K X H X J X X
ˇkl aj .t kl /ck .t kl /wph .t kl / h I dkj h .t kl / cos .!o t C kl / i Q Cdkj h .t kl / sin.!o t C kl / C n.t / C =.t /
(9)
(7) =.t / D
where L is the number of propagation paths; ˇkl , kl , and kl are respectively the path gain, the phase delay, and the time delay of lth path of the kth user. The phase kl is assumed to be uniformly distributed over Œ0; 2. The path gain model and distribution function depend on the nature of the channel and the propagation environment. We assume that the channel path gain ˇkl is Nakagami distributed [6]. The output of the channel for the kth user is given by p
2P
where kl D kl !c kl , n.t / is the AWGN, and =.t / is BPSK narrow-band interference jammer given by
lD1
yk .t / D
p
kD1 hD1 j D1 lD1
Re dkj h .t /aj .t /ck .t /wph .t / exp.j !o t / (6)
L X
yk .t / C n.t / C =.t /
kD1
D
H X J X
2P
K X
L H X J X X
p
2=| .t / cosŒ2.fo C /t C
having power =, offset interference carrier with respect to signal carrier , and phase . The information sequence | .t / has a bit rate 1=T| , where T| denotes the duration of one bit. The interference bandwidth is B| D 2=T| , and we assume B| < Bs . An important quantity is the ratio of the interference band to system bandwidth B Tn . The filtered signal is then passed p D B|s D T | through an SF. The impulse response of the filter may be written as
hD1 j D1 lD1
hs .t / D
ˇkl aj .t kl /ck .t kl /wph .t kl / h I dkj h .t kl / cosf!o .t kl / C kl g
(10)
M2 X
˛m ı.t mTn /
(11)
mDM1
i Q C dkj h .t kl / sinf!o .t kl / C kl g (8)
Wirel. Commun. Mob. Comput. (2012) © 2012 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
where ˛0 D 1 and M1 0 and M2 0 represent the number of taps of the filter on the left and right of the center tap [3]. For each tap, the output of the filter is given by
Narrow-band suppression in wavelet packets based MC/MCD-CDMA system
0 rs .t / D @
K X
M. M. Akho-Zahieh and N. Abdellatif
1 yk .t / C n.t / C =.t /A hs .t /
kD1
8 M2 < K X L H X J X X X p 2P ˛m ˇkl aj .t kl mTn /ck .t kl mTn / D : kD1 hD1 j D1 lD1
mDM1
h i Q I m m wph .t kl mTn / dkj h .t kl mTn / cos !o t C kl C dkj h .t kl mTn / sin !o t C kl p
C ˛mb n.t mTn / C ˛m 2=| .t mTn / cosŒ2.fo C /.t mTn / C : m D mT ! and b where kl n.t mTn / is the filtered n o kl AWGN. The output of the filter, rs .t /, is demodulated by a local oscillator, despread by a user specific code sequence, multiplied by the WPs, and correlated over a period T to recover the super bit stream. The resulting data are then parallel-to-serial (P/S) converted again and despread by the MCD to recover the J parallel data substreams. Finally, the correlated outputs from J paths is P/S converted to recover the original data bit. Observe that the output of the first correlator in the WP part denoted as x1 can be written as
Z x1 D
0
1 zDS
r D
h i P Q I ˇ11 Nn T d111 .t / jd111 .t / 2
(16)
and
1 D ˇ11 T zDSI
r
2 P 4 2
1 X
˛m 1
m1 DM1
n
m m cos 111 C j sin 111 ‡1I .m1 /
o m m Q C sin 111 j cos 111 ‡1 .m1 /
rs .t /c1 .t /wp1 .t /Œcos.!o t / j sin.!o t /dt
1 1 1 1 1 C xDSI C xMPI C xMCDI C xWPI D xDS
M2 X
C
(13)
n m m I ˛m2 cos 112 Cj sin 112 ‡1 .m2 /
m2 D1
1 1 where xDS is the desired signal, xDSI is the self1 1 is interference, xMPI is the multipath interference, xMCDI 1 1 the MCD interference, xWPI is the WPs interference, xMUI is the multiuser interference, n1 is suppressed correlated AWGN component, and = .t / is the suppressed narrowband interference. The output of first P=S converter for the first user’s signal may be written as H X
1 1 C xDSI C bO D xDS
(12)
;
After evaluation, it can be shown that
T
1 C xMUI C n1 C =1
9 =
m2
m2
C sin 11 j cos 11
o
3
Q ‡1 .m2 / 5 (17)
where ~ ~ F11 .m1 / C d111;1 F11 .Nn C m1 / ‡1~ .m1 / D d111;0 ~ ~ ~ ‡1 .m2 / D d111;1 F11 .m2 Nn / C d111;0 F11 .m2 /
0
h D1
i h 0 0 0 h h0 h0 h0 xMPI C xMCDI C xWPI C xMUI C nh C =h (14) The output for the first correlator z1 in the MCD part is given by Z z1 D
0
T
~ ~ where d111;1 , d111;0 are the previous data bit and current data bit, respectively, with ~ D I for inphase part and ~ D Q for quadrature part. Fk;i .l/ is the discrete-time aperiodic cross-correlation function defined in [7] and is given by
bO a1 .t /dt Fk;i .l/ D
1 1 1 1 1 C zDSI C zMPI C zMCDI C zWPI D zDS 1 C zMUI
1
e1
Ce n C=
(15)
8 ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ :
NnP 1l j D0 NnP 1Cl j D0
.j / .j C1/
ak ai
;
0 l Nn 1
.j 1/ .j / ai ;
1 Nn l 0
ak
Wirel. Commun. Mob. Comput. (2012) © 2012 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
M. M. Akho-Zahieh and N. Abdellatif
Narrow-band suppression in wavelet packets based MC/MCD-CDMA system
1 , z1 1 The value of the four components zMPI MCDI , zWPI , 1 and zMUI can be written as given in [1, Chapter 4] as follows:
r
1 zMPI
D
M2 X
P 2
n
L X
H X
~ ~1j h D d1j h;1 G1j Œ1l .m/
R11;h0 h Œ1l .m/
0
h D1 ~ O C d1j h;0 G1j Œ1l .m/
H X
RO 11;h0 h Œ1l .m/
0
˛m ˇ1l
h D1
mDM1 lD2
~ ~kj h D dkj h;1 G1j Œkl .m/
m I m C j sin 1l 1 cos 1l
H X
R1k;h0 h Œkl .m/
h0D1
m Q o m j cos 1l 1 C sin 1l
~ O C dkj h;0 G1j Œkl .m/
(18)
H X
RO 1k;h0 h Œkl .m/
h0D1
1 zMCDI
r D
P 2
M2 X L J X X
˛m ˇ1l
mDM1 j D2 lD1
n m I m C j sin 1l 1j cos 1l m Q o m j cos 1l 1j C sin 1l
1 zWPI
r D
P 2
M2 X
H X L J X X
(19)
Z Rhn;xy .kl / D
˛m ˇ1l
mDM1 hD2 j D1 lD1
m Q o m j cos 1l 1j h C sin 1l
1 zMUI D
P 2
M2 X
L K X H X J X X
kl
kl
(20)
.t kl /dt Also, Gxy .:/ and GO xy .:/ are the partial cross-correlation functions defined as in [7] and are given by
˛m ˇkl
mDM1 kD2 hD1 j D1 lD1
Z
n m I m C j sin kl kj h cos kl m Q o m j cos kl kj h C sin kl
ch .t /cn .t kl C T /wpx .t /wpy
0
.t kl C T /dt Z T ch .t /cn .t kl /wpx .t /wpy RO hn;xy .kl / D
n m I m C j sin 1l 1j h cos 1l
r
where Rhn;xy Œkl .m/, RO hn;xy Œkl .m/ are the parI I , dkj are the tial cross-correlation functions; dkj h;1 h;0 inphase part of previous data bit and current data Q Q bit, respectively; dkj h;1; , dkj h;0 are the quadrature component of previous data bit and current data bit, respectively. Following the definition in [7,8], the partial crosscorrelation functions Rhn;xy .kl / and RO hn;xy .kl / can be written as follows:
Gxy .kl / D (21)
GO xy .kl / D
kl
ax .t /ay .t kl C T /dt
0
Z
T
kl
ax .t /ay .t kl /dt
where ~ ~1 D d111;1 G11 Œ1l .m/
H X
The suppressed correlated AWGN component nQ 1 is given by R11;h0 1 Œ1l .m/
0
h D1 ~ GO 11 Œ1l .m/ C d111;0
H X
RO 11;h0 1 Œ1l .m/
0
h D1 ~ ~1j D d1j 1;1 G1j Œ1l .m/
H X
R11;h0 1 Œ1l .m/
h0D1 ~ O C d1j 1;0 G1j Œ1l .m/
H X
RO 11;h0 1 Œ1l .m/
h0D1
Wirel. Commun. Mob. Comput. (2012) © 2012 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
nQ 1 D
Z
M2 X
˛m
mDM1
T
0
a1 .t /
8 < b n.t mTn /Œcos.!c t / j sin.!c t / 0 : 9 3 H NX n1 = X c1i wh0 .t iTn / dt 5 dt (22) ; 0
2 Z 4
T
h D1 iD0
Narrow-band suppression in wavelet packets based MC/MCD-CDMA system
and the suppressed narrow-band interference e =1 is given by e =1 D
r
M2 X
˛m
mDM1
2 Z 4
T 0
= 2
Z
D 2P
0
a1 .t /
k1 D1 h1 D1 j1 D1 l1 D1 k2 D1 h2 D1 j2 D1 L X
c1i wh0 .t
iD0
9 =
E dk1 j1 h1 t k1 l1 dk2 j2 h2 t C vTn k2 l2 E aj1 t k1 l1 aj2 t ClTn k2 l2
H X
h i1 E ck1 wh1 t i1 Tn k1 l1
NX n1NX n1
h D1
3
i1 D0 i2 D0
iTn / dt 5 dt ;
(23)
i2 ck2 wh2 t
3. DETERMINATION OF SUPPRESSION FILTER COEFFICIENTS
˛m ..n m/Tn / C .nTn / D 0 n D M1 ; : : : ; 1; 1; : : : M2
E aj1 .t kl /aj2 .t C vTn kl / D
(24) NX n1
.vTn / is a lowpass autocorrelation function consisting of three components .vTn / D s .vTn / C n .vTn / C j .vTn /
(25)
where s .vTn / is the lowpass version of the desired signal, n .vTn / is due to noise, and j .lTn / is due to narrow-band interference. The s .vTn / is given by
:
2P
L K H J X X X X
1; 0;
vD0 v¤0
and
mDM1 ;m¤0
s .vTn / D E
9 i = C vTn i2 Tn k2 l2 (26) ;
As E aj1 t k1 l1 aj2 t C vTn k2 l2 D 0 when k1 ¤ k2 or l1 ¤ l2 . Furthermore,
It is shown in [3,5] that the coefficients of the SF can be determined using
8 int Œ1=p (29)
Wirel. Commun. Mob. Comput. (2012) © 2012 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
M. M. Akho-Zahieh and N. Abdellatif
Narrow-band suppression in wavelet packets based MC/MCD-CDMA system
where int Œx is defined as the integer part of x. Therefore from (27)–(29), one obtains
multipath components, and Es D 2P T is the mean received symbol energy. Note that in the derivation of the
8 1 K KLHJ ˆ <
C 2Nn ŒEs =No C ==S ; .vTn / D 2P
==SK .1 jvj p/ cos.2vq/; ˆ : 0;
vD0 (30)
jvj int Œ1=p jvj > int Œ1=p
From (25) and (30), we can obtain the coefficients ˛m . Note that the SF can be double sided (DS, M1 ¤ 0 and M2 ¤ 0) or single sided (SS, M1 ¤ 0 and M2 D 0, or M1 D 0 and M2 ¤ 0).
variances, it is assumed that the variation in kl and kl is very small and can be ignored for all K users and L paths, and that the variances are independent of k, h, and j . The 2 is given by variance for the inphase self-interference DSI
4. SIGNAL-TO-NOISE PLUS INTERFERENCE RATIO
i h 2 1 DSI D var zDSI 8 r 2 1 < P 4 X ˛m 1 D var ˇ11 T : 2 m1 DM1 n o m Q m cos 111 ‡1I .m1 / C sin 111 ‡1 .m1 /
To find the signal-to-noise plus interference ratio (SNIR), we need to find the desired signal power and the noise plus interference variances. From (16), the desired inphase signal power S is given by i2 P h I1 D .ˇ11 Nn T /2 (31) S D zDS 2 To calculate the variances, it is assumed that all the interferences and the noise terms are zero mean independent random variables. The process of computing the variances for the different interference terms is quite involved. 2 , 2 2 2 Fortunately, the variances MPI MCDI , WPI , MUI , and 2 nQ 1 , for a system without SF, have been computed in [1, Chapter 4]. Without loss of generality, we invoke the results in [1,3] for the preceding variances with SF, which is given by
C
P .T Nn /2 .MI 1 C NI/ 2
M2 X
MI 1 D 42
2 ˛m
mDM1
2
M2 X
C
C sin 2 D DSI
P T 2 EŒˇ11 2 2 2
C
3
(33) 2 ˛m
˛m1 ˛mO 1
M2 X M2 X
n h i
m m O ˛m2 ˛mQ 2 E cos 112 cos 112
i i h h m m O I I .m2 /‡1 .m Q 2 / CE sin 112 sin 112 E ‡1 3 io h Q Q Q 2/ 5 (35) E ‡1 .m2 /‡1 .m
0
H
Nn .Es =No /
1 X
39 o = 5 ;
m2 D1m Q 2 D1
˛m ˛mC1 5
x m T Nn .JH /2 3 H Z Tn 2 X 2 ^ rh0 1 . / C r h0 1 . / d 5
NI D
m Q 112 ‡1 .m2 /
i h n h i m m O E cos 111 cos 111 E ‡1I .m1 /‡1I .m Q 1/
i h io h Q Q m m O Q 1/ C E sin 111 sin 111 E ‡1 .m1 /‡1 .m
mDM1
M2 X
m1 DM1 m O 1 DM1
(32)
H X H Z Tn X
QJK x m T Nn .JH /2 0 hD1 h0 D1 2 2 ^ rh0 h . / C r h0 h . / d
0
1 X
4
4
h D1
n m I cos 112 ‡1 .m2 /
where 2
˛m 2
m2 D1
2 2 2 2 C MCDI C WPI C MUI C nQ21 T21 D MPI
D
M2 X
(34)
mDM1
with x m D 6 for QPSK and x m D 12 for BPSK, D varŒˇk1 , Q represents the sum of amplitude levels of all Wirel. Commun. Mob. Comput. (2012) © 2012 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
m is a zero means random variable uniAssuming that 11 formly distributed in Œ0; 2 i
h m m Q E sin 111 sin 111
( m R 2 m Q m 1 sin 111 sin 111 d 111 D 12 ; m1 D m Q1 D 2 0 0; m1 ¤ m Q1
Similarly, i
h m m O E cos 111 cos 111 D
(
1 2;
m1 D m Q1
0;
m1 ¤ m Q1
Narrow-band suppression in wavelet packets based MC/MCD-CDMA system
Also, it is evident that
I d111;1
2
= D E 2
2
2
2
Q I I D d111;0 D d111;1 D d111;1 2
2
Q Q D d111;0 D d111;1 D 1
M. M. Akho-Zahieh and N. Abdellatif
"Z
Z
T
0
Z
T
a1 .t /a1 . /
0
T
Z
0
T
0
8 < | .t m1 Tn /| . m2 Tn / : cosŒ2.t m1 Tn / C cosŒ2.t m2 Tn / C
And from [3] ( EŒFk;i .l/Fk;i .m/ D
Nn jlj ;
l D jmj
0;
l ¤ jmj
0 h D1 iD0
P T 2 EŒˇ11 2 D 2 2 1 X
4
PNn T 2
2y m
M2 X
2 ˛m N C 1 n
3
M2 X
˛m
mDM1
"Z
(36)
Z
T 0
0
a1 .t /
| .t mT / cosŒ2.t mTn / C
H NX n1 X
var =Q 1
T
0
o
0 h D1 iD0
c1i wh0 . iTn / dt d dt d
i
(37)
2 Using [3, Eq. 18], also given c1i D 1 and because
E D D ıln , then (37) wh0 t l TNn ; wh0 t n TNn can be written as
2Q 1 D
=H T 2 4
|2 .m1 ; m2 /
M2 X
M2 X
˛m1 ˛m2 |2 .m1 ; m2 /
m1 DM1 m2 DM1
Z1 sign Œ1 p jxNn m1 C m2 j
1
cos Œ2q .xNn m1 C m2 / .1 jxj/ dx (39)
T
0
Z
where |2 .m1 ; m2 / is given in [3, Eq. 19] by
28 r Z M2 < X = T DE4 ˛m a1 .t / : 2 0 mDM1 "Z
0
T
H NX H NX n1 n1 X X c1i wh0 .t iTn /
0 h D1 iD0
The variance can be evaluated as follows: i
a1 .t /a1 . /
(38)
iTn /dt 5 dt
0 h D1 iD0
h
0
Z
T
cosŒ2. m2 Tn / C
=
3 c1i wh0 .t
Z
T
cosŒ2.t m1 Tn / C
T
= 2
˛m 1 ˛m 2
m1 DM1 m2 DM1
0
2 ˛m
3
8 < | .t m1 Tn /| . m2 Tn / :
m1 DM1 ;m¤0
r
M2 X
M2 X
E
where y m D 1 for QPSK and y m D 2 BPSK. The reason that y m and x m in the case of BPSK are double that for QPSK is as follows: as indicated previously, instead of dk .t /, we use dkI .t /, and exp.j !o t / is replaced by cos.!c t / in BPSK modulation. Hence the interference variances are calculated from (17)–(21) by considering only the cosine part of the inphase signal. From (23), the inphase component of narrow jamming signal is given by
e =1 D
M2 X
"Z
2 ˛m N 5 2 n
m2 D1
m1 DM1
D
= 2
D
0 h D1 iD0
9 =
c1i wh0 . iTn / dt d dt d 5 ;
According to this, (35) becomes 2 DSI
H NX H NX n1 n1 X X c1i wh0 .t iTn /
| .t mT / cosŒ2.t mTn /C
H NX n1 X 0 h D1 iD0
92 3 = 7 c1i wh0 .t iTn /dt 5 dt 5 ; 3
given that Nn 1; q D Tn (ratio of the offset of interference carrier frequency to half-spread-spectrum bandwidth) and signŒx D x or zero for x 0 or x < 0, respectively. Accordingly, the total variance for noise and interference Wirel. Commun. Mob. Comput. (2012) © 2012 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
M. M. Akho-Zahieh and N. Abdellatif
Narrow-band suppression in wavelet packets based MC/MCD-CDMA system
is given by 2 2 2 2 2 T2 D MPI C MCDI C WPI C MUI C DSI C nQ21 C 2Q 1
6. OUTAGE PROBABILITY PERFORMANCE
=
P .T Nn /2 .MI 1 C MI 2 C NI C JI/ D 2 P .T Nn /2 D .MI C NI C JI/ 2
(40)
where MI 2 D
JI D
M2 X
2 ˛m y m Nn m1 DM1 ;m¤0
M2 M2 H ==SK X X
Nn2
(41)
m1 DM1 m2 DM1
˛m1 ˛m2 |2 .m1 ; m2 /
(42)
and SK D 2P . Therefore, the output SNIR, , can be written as
D
S T2
D
P 2
.ˇ11 T Nn /2 T2
2 D ˇ11
RESEARCH ARTICLE
Narrow-band interference suppression in wavelet packets based multicarrier multicode CDMA overlay system Maryam M. Akho-Zahieh and Nasser Abdellatif * Department of Electrical and Computer Engineering, Applied Science Private University, Amman 11931, Jordan
ABSTRACT The wavelet packets based multicarrier (MC) multicode (MCD) code-division multiple-access (CDMA) transceiver consists of the MCD part, which ensures the transmission for high speed and flexible data rate; the MC part contributing to robustness to frequency-selective fading and flexibility for handling multiple data rates; and wavelet packets (WPs) modulation technique, which contributes to the mitigation of the interference problems. As WPs have lower sidelobes compared with sinusoidal carriers, this system is very effective in reducing the problem of inter-carrier interference. Of course, like any CDMA system, the system can suppress a given amount of interference. This paper considers an interference suppression scheme which will enhance the performance of the system. The receiver employs suppression filters to mitigate the effect of narrow-band jammer interference. The framework for the system and the performance evaluation are presented in terms of the bit error rate and the outage probability over a Nakagami fading channel. Also, we investigate how the performance is influenced by various parameters, such as the number of taps of the suppression filter and the ratio of narrow-band interference bandwidth to the spread-spectrum bandwidth. Finally, the performance of the system is compared with the performance of sinusoidal based MC/MCD-CDMA system denoted Sin-MC/MCD-CDMA. Copyright © 2012 John Wiley & Sons, Ltd. KEYWORDS suppression filter; wavelet packet; multicarrier; multicode; CDMA *Correspondence Nasser Abdellatif, Department of Electrical and Computer Engineering, Applied Science Private University, Amman 11931, Jordan. E-mail: [email protected]; [email protected]
1. INTRODUCTION The multicarrier (MC) multicode (MCD) code-division multiple-access (CDMA) system achieves the advantages of both MC-CDMA and MCD-CDMA systems, which are (i) variable data rates without interference scaling and (ii) enhanced robustness to mitigating multipath fading. By combining the properties of MCD and MC techniques, a MC/MCD-CDMA system that uses wavelet packet (WP) as subcarrier is proposed in this paper. The system is denoted as WP-MC/MCD-CDMA. Previous studies [1,2] have shown that the problems posed by sinusoidal carriers can be solved by WPs. Unlike sinusoids, WPs have many attractive properties including: (1) Much lower sidelobes with negligible sidelobe energy leakage compared with sinusoid carriers. Copyright © 2012 John Wiley & Sons, Ltd.
This property is effective in suppressing inter-carrier interference (ICI) and multiple-access interference. (2) Naturally orthogonal and well localized in both time and frequency domains, which relaxes the requirement of frequency or time guard between different user signals. In fact, orthogonality is maintained for overlapped WPs in both time and frequency domains. This is an advantage of using WPs to model communication channels that are characterized not only by frequency selectivity but also by time variation. The WP-MC/MCD-CDMA [1] uses WP as subcarrier instead of sinusoidal function, because of that, it could mitigate the problem of ICI associated with using sinusoidal carrier and suppress interferences caused by multipath fading. Various schemes of MC-CDMA systems have been proposed in recent years, and they can be
Narrow-band suppression in wavelet packets based MC/MCD-CDMA system
If k denotes the kth user, j denotes the j th code substream, and h denotes the hth WPs superstreams, the model of our system for the kth user is shown in Figure 1. The data for the kth user are given by
classified into two groups. The first group is orthogonal MC in which different adjacent subcarriers are orthogonally overlapped. The second group is disjoint MCs in which all subcarriers are not overlapped. The second group contains two subtypes: in the first type, different data streams modulate different carriers, whereas in the second subtype, all carriers are modulated by the same data stream. The system proposed in this paper belongs to the second subtype of the second group. As it is well known, the inherent processing gain of CDMA system will, in many cases, provide the system with a sufficient degree of narrow-band interference rejection capability. But, if the interference signal is powerful enough, the conventional receiver is ineffective in mitigating this problem. Such instances of narrow-band interference may be intentional, as in various military applications, or may be unintentional, as in a commercial CDMA overlay scheme. An interference suppression filter (SF) can be employed to reject the narrow-band interference. A wiener-type filter, described in [3–5], employs a tapped delay line structure to first predict and then subtract out the narrow-band interference. The aim of this paper is to study the effect of narrowband interference binary phase shift keying (BPSK) waveform on the WP-MC/MCD-CDMA system and employs an SF to reject it, then compare the performance of the two systems WP-MC/MCD-CDMA and Sin-MC/MCDCDMA with and without SF.
d kj (t )
∑
bk (t )
a 2 (t ) d kJ (t )
Serial-to-Parallel Converter
Serial-to-Parallel Converter
d k (t )
T t i T =.JH / JH (1)
Y
where …x ./ is a rectangular pulse of duration x. These data have a bit rate of JH T , where T is the bit duration, and Q
dkI .t / and dk .t / represent the inphase .I / and quadrature .Q/ data symbols, respectively. These data are serial-toparallel (S=P) converted into J substreams, each with a lower bit rate H T . The j th data substream corresponding to the kth user signal is given by 1 X
dkj .t / D
i dkj
Y T =H
iD1
.t iT =H /
(2)
After S=P conversion, the substreams are coded by a set of orthogonal signals aj .t /, which has a chip rate of T1c D HNc T ,
and given by
aj .t / D
NX c 1
aji
Y Tc
iD0
.t iTc /
(3)
where Nc is the code length, and Tc is the code pulse i 2 f˙1g with probaduration. Note that aji and dkj bilities P .1/ D P .1/ D 0:5. In order to maintain orthogonality of the coding signals, the maximum number of substreams J is limited to Nc D HTTc . The coded
bk1 (t ) Interference
c k (t )
wp1 (t )
∑
bk1 (t ) c k (t ) wp (t ) h
s (t )
exp(− jω o t )
n(t)
bkH (t )
Channel c k (t )
a J (t )
dki
iD1
The system model under consideration consists of K active users transmitting data simultaneously to the base station.
a1 (t )
1 X
Q
dk .t / D dkI.t /jdk .t / D
2. SYSTEM MODEL AND DESCRIPTION
d k1 (t )
M. M. Akho-Zahieh and N. Abdellatif
wp H (t )
∫ (⋅)dt
x1
0
c k (t ) wp1 (t ) T
∫ (⋅)dt
Suppression filters r (t )
xh
0
rs (t )
exp(− jω o t )
c k (t ) wp h (t ) T
∫ (⋅)dt
0
c k (t ) wp H (t )
xH
T
Parallel-to-Serial Converter
T
H →1
∫ (⋅)dt
z1
T
zj
0
bˆk
a1 (t )
∫ (⋅)dt
0
a j (t ) T
∫ (⋅)dt
0
a J (t )
zJ
Parallel-to-Serial Converter
BPF
dˆ k
J →1
Figure 1. WP-MC/MCD-CDMA system model.
Wirel. Commun. Mob. Comput. (2012) © 2012 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
M. M. Akho-Zahieh and N. Abdellatif
Narrow-band suppression in wavelet packets based MC/MCD-CDMA system
substreams are added, and the resulting signal bk .t / D PJ j D1 aj .t /dkj .t / is again S=P converted into H superstreams bkh .t /. As bk .t / has a bit rate of H T , bkh .t / will 1 have a bit rate of T . The next step is the modulation by the kth user pseudorandom noise (PN) signature sequence
ck .t / D
NX n 1
cki
iD0
Y Tn
.t iTn /
(4)
where Nn is the length of the PN code sequence, cki 2 f˙1g is the i th bit value with probabilities P .1/ D P .1/ D 0:5, and Tn D NTn is the chip duration. Each of the spreading superstreams will be used to modulate a WPs wph .t / given by wph .t / D
X
wh .t iTn /
where the khj l subscripts denote the kth user, j th code, hth wavelet, and lth path, respectively, and dk .t / D Q dkI .t / jdk .t /. The preceding analysis is carried out assuming quadrature phase shift keying (QPSK); if BPSK modulation is used, then the baseband data and the carrier term will be real, such that dk .t / is written as dkI .t / and exp.j !c t / is replaced with cos.!c t /. The receiver shown in Figure 1 is assumed to be synchronous, designed to detect the first substream of the first user’s first WP propagating via the first path. It consists of a bandpass filter (BPF), tap SF, and correlators. The received signal is first passed through the BPF having bandwidth Bs equal to the spread spectrum bandwidth D 2Tn1 , which removes the out-of band noise and lets the desired signal and inferences pass without distortion. It can be shown that the received signal, r.t /, can be written as
(5)
i
r.t / D
Assuming identical power for all users, the transmitted signal sk .t / can be written as sk .t / D
p
hD1 j D1
where dkj h .t / with period T is the data symbol of kth user, rth is the substream of the hth superstream, exp.j !c t / is the carrier signal, and P is the users’ power. The equivalent impulse response for the channel used in this paper can be written as
h.t / D
ˇkl e j kl ı.t kl /
2P
L K X H X J X X
ˇkl aj .t kl /ck .t kl /wph .t kl / h I dkj h .t kl / cos .!o t C kl / i Q Cdkj h .t kl / sin.!o t C kl / C n.t / C =.t /
(9)
(7) =.t / D
where L is the number of propagation paths; ˇkl , kl , and kl are respectively the path gain, the phase delay, and the time delay of lth path of the kth user. The phase kl is assumed to be uniformly distributed over Œ0; 2. The path gain model and distribution function depend on the nature of the channel and the propagation environment. We assume that the channel path gain ˇkl is Nakagami distributed [6]. The output of the channel for the kth user is given by p
2P
where kl D kl !c kl , n.t / is the AWGN, and =.t / is BPSK narrow-band interference jammer given by
lD1
yk .t / D
p
kD1 hD1 j D1 lD1
Re dkj h .t /aj .t /ck .t /wph .t / exp.j !o t / (6)
L X
yk .t / C n.t / C =.t /
kD1
D
H X J X
2P
K X
L H X J X X
p
2=| .t / cosŒ2.fo C /t C
having power =, offset interference carrier with respect to signal carrier , and phase . The information sequence | .t / has a bit rate 1=T| , where T| denotes the duration of one bit. The interference bandwidth is B| D 2=T| , and we assume B| < Bs . An important quantity is the ratio of the interference band to system bandwidth B Tn . The filtered signal is then passed p D B|s D T | through an SF. The impulse response of the filter may be written as
hD1 j D1 lD1
hs .t / D
ˇkl aj .t kl /ck .t kl /wph .t kl / h I dkj h .t kl / cosf!o .t kl / C kl g
(10)
M2 X
˛m ı.t mTn /
(11)
mDM1
i Q C dkj h .t kl / sinf!o .t kl / C kl g (8)
Wirel. Commun. Mob. Comput. (2012) © 2012 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
where ˛0 D 1 and M1 0 and M2 0 represent the number of taps of the filter on the left and right of the center tap [3]. For each tap, the output of the filter is given by
Narrow-band suppression in wavelet packets based MC/MCD-CDMA system
0 rs .t / D @
K X
M. M. Akho-Zahieh and N. Abdellatif
1 yk .t / C n.t / C =.t /A hs .t /
kD1
8 M2 < K X L H X J X X X p 2P ˛m ˇkl aj .t kl mTn /ck .t kl mTn / D : kD1 hD1 j D1 lD1
mDM1
h i Q I m m wph .t kl mTn / dkj h .t kl mTn / cos !o t C kl C dkj h .t kl mTn / sin !o t C kl p
C ˛mb n.t mTn / C ˛m 2=| .t mTn / cosŒ2.fo C /.t mTn / C : m D mT ! and b where kl n.t mTn / is the filtered n o kl AWGN. The output of the filter, rs .t /, is demodulated by a local oscillator, despread by a user specific code sequence, multiplied by the WPs, and correlated over a period T to recover the super bit stream. The resulting data are then parallel-to-serial (P/S) converted again and despread by the MCD to recover the J parallel data substreams. Finally, the correlated outputs from J paths is P/S converted to recover the original data bit. Observe that the output of the first correlator in the WP part denoted as x1 can be written as
Z x1 D
0
1 zDS
r D
h i P Q I ˇ11 Nn T d111 .t / jd111 .t / 2
(16)
and
1 D ˇ11 T zDSI
r
2 P 4 2
1 X
˛m 1
m1 DM1
n
m m cos 111 C j sin 111 ‡1I .m1 /
o m m Q C sin 111 j cos 111 ‡1 .m1 /
rs .t /c1 .t /wp1 .t /Œcos.!o t / j sin.!o t /dt
1 1 1 1 1 C xDSI C xMPI C xMCDI C xWPI D xDS
M2 X
C
(13)
n m m I ˛m2 cos 112 Cj sin 112 ‡1 .m2 /
m2 D1
1 1 where xDS is the desired signal, xDSI is the self1 1 is interference, xMPI is the multipath interference, xMCDI 1 1 the MCD interference, xWPI is the WPs interference, xMUI is the multiuser interference, n1 is suppressed correlated AWGN component, and = .t / is the suppressed narrowband interference. The output of first P=S converter for the first user’s signal may be written as H X
1 1 C xDSI C bO D xDS
(12)
;
After evaluation, it can be shown that
T
1 C xMUI C n1 C =1
9 =
m2
m2
C sin 11 j cos 11
o
3
Q ‡1 .m2 / 5 (17)
where ~ ~ F11 .m1 / C d111;1 F11 .Nn C m1 / ‡1~ .m1 / D d111;0 ~ ~ ~ ‡1 .m2 / D d111;1 F11 .m2 Nn / C d111;0 F11 .m2 /
0
h D1
i h 0 0 0 h h0 h0 h0 xMPI C xMCDI C xWPI C xMUI C nh C =h (14) The output for the first correlator z1 in the MCD part is given by Z z1 D
0
T
~ ~ where d111;1 , d111;0 are the previous data bit and current data bit, respectively, with ~ D I for inphase part and ~ D Q for quadrature part. Fk;i .l/ is the discrete-time aperiodic cross-correlation function defined in [7] and is given by
bO a1 .t /dt Fk;i .l/ D
1 1 1 1 1 C zDSI C zMPI C zMCDI C zWPI D zDS 1 C zMUI
1
e1
Ce n C=
(15)
8 ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ :
NnP 1l j D0 NnP 1Cl j D0
.j / .j C1/
ak ai
;
0 l Nn 1
.j 1/ .j / ai ;
1 Nn l 0
ak
Wirel. Commun. Mob. Comput. (2012) © 2012 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
M. M. Akho-Zahieh and N. Abdellatif
Narrow-band suppression in wavelet packets based MC/MCD-CDMA system
1 , z1 1 The value of the four components zMPI MCDI , zWPI , 1 and zMUI can be written as given in [1, Chapter 4] as follows:
r
1 zMPI
D
M2 X
P 2
n
L X
H X
~ ~1j h D d1j h;1 G1j Œ1l .m/
R11;h0 h Œ1l .m/
0
h D1 ~ O C d1j h;0 G1j Œ1l .m/
H X
RO 11;h0 h Œ1l .m/
0
˛m ˇ1l
h D1
mDM1 lD2
~ ~kj h D dkj h;1 G1j Œkl .m/
m I m C j sin 1l 1 cos 1l
H X
R1k;h0 h Œkl .m/
h0D1
m Q o m j cos 1l 1 C sin 1l
~ O C dkj h;0 G1j Œkl .m/
(18)
H X
RO 1k;h0 h Œkl .m/
h0D1
1 zMCDI
r D
P 2
M2 X L J X X
˛m ˇ1l
mDM1 j D2 lD1
n m I m C j sin 1l 1j cos 1l m Q o m j cos 1l 1j C sin 1l
1 zWPI
r D
P 2
M2 X
H X L J X X
(19)
Z Rhn;xy .kl / D
˛m ˇ1l
mDM1 hD2 j D1 lD1
m Q o m j cos 1l 1j h C sin 1l
1 zMUI D
P 2
M2 X
L K X H X J X X
kl
kl
(20)
.t kl /dt Also, Gxy .:/ and GO xy .:/ are the partial cross-correlation functions defined as in [7] and are given by
˛m ˇkl
mDM1 kD2 hD1 j D1 lD1
Z
n m I m C j sin kl kj h cos kl m Q o m j cos kl kj h C sin kl
ch .t /cn .t kl C T /wpx .t /wpy
0
.t kl C T /dt Z T ch .t /cn .t kl /wpx .t /wpy RO hn;xy .kl / D
n m I m C j sin 1l 1j h cos 1l
r
where Rhn;xy Œkl .m/, RO hn;xy Œkl .m/ are the parI I , dkj are the tial cross-correlation functions; dkj h;1 h;0 inphase part of previous data bit and current data Q Q bit, respectively; dkj h;1; , dkj h;0 are the quadrature component of previous data bit and current data bit, respectively. Following the definition in [7,8], the partial crosscorrelation functions Rhn;xy .kl / and RO hn;xy .kl / can be written as follows:
Gxy .kl / D (21)
GO xy .kl / D
kl
ax .t /ay .t kl C T /dt
0
Z
T
kl
ax .t /ay .t kl /dt
where ~ ~1 D d111;1 G11 Œ1l .m/
H X
The suppressed correlated AWGN component nQ 1 is given by R11;h0 1 Œ1l .m/
0
h D1 ~ GO 11 Œ1l .m/ C d111;0
H X
RO 11;h0 1 Œ1l .m/
0
h D1 ~ ~1j D d1j 1;1 G1j Œ1l .m/
H X
R11;h0 1 Œ1l .m/
h0D1 ~ O C d1j 1;0 G1j Œ1l .m/
H X
RO 11;h0 1 Œ1l .m/
h0D1
Wirel. Commun. Mob. Comput. (2012) © 2012 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
nQ 1 D
Z
M2 X
˛m
mDM1
T
0
a1 .t /
8 < b n.t mTn /Œcos.!c t / j sin.!c t / 0 : 9 3 H NX n1 = X c1i wh0 .t iTn / dt 5 dt (22) ; 0
2 Z 4
T
h D1 iD0
Narrow-band suppression in wavelet packets based MC/MCD-CDMA system
and the suppressed narrow-band interference e =1 is given by e =1 D
r
M2 X
˛m
mDM1
2 Z 4
T 0
= 2
Z
D 2P
0
a1 .t /
k1 D1 h1 D1 j1 D1 l1 D1 k2 D1 h2 D1 j2 D1 L X
c1i wh0 .t
iD0
9 =
E dk1 j1 h1 t k1 l1 dk2 j2 h2 t C vTn k2 l2 E aj1 t k1 l1 aj2 t ClTn k2 l2
H X
h i1 E ck1 wh1 t i1 Tn k1 l1
NX n1NX n1
h D1
3
i1 D0 i2 D0
iTn / dt 5 dt ;
(23)
i2 ck2 wh2 t
3. DETERMINATION OF SUPPRESSION FILTER COEFFICIENTS
˛m ..n m/Tn / C .nTn / D 0 n D M1 ; : : : ; 1; 1; : : : M2
E aj1 .t kl /aj2 .t C vTn kl / D
(24) NX n1
.vTn / is a lowpass autocorrelation function consisting of three components .vTn / D s .vTn / C n .vTn / C j .vTn /
(25)
where s .vTn / is the lowpass version of the desired signal, n .vTn / is due to noise, and j .lTn / is due to narrow-band interference. The s .vTn / is given by
:
2P
L K H J X X X X
1; 0;
vD0 v¤0
and
mDM1 ;m¤0
s .vTn / D E
9 i = C vTn i2 Tn k2 l2 (26) ;
As E aj1 t k1 l1 aj2 t C vTn k2 l2 D 0 when k1 ¤ k2 or l1 ¤ l2 . Furthermore,
It is shown in [3,5] that the coefficients of the SF can be determined using
8 int Œ1=p (29)
Wirel. Commun. Mob. Comput. (2012) © 2012 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
M. M. Akho-Zahieh and N. Abdellatif
Narrow-band suppression in wavelet packets based MC/MCD-CDMA system
where int Œx is defined as the integer part of x. Therefore from (27)–(29), one obtains
multipath components, and Es D 2P T is the mean received symbol energy. Note that in the derivation of the
8 1 K KLHJ ˆ <
C 2Nn ŒEs =No C ==S ; .vTn / D 2P
==SK .1 jvj p/ cos.2vq/; ˆ : 0;
vD0 (30)
jvj int Œ1=p jvj > int Œ1=p
From (25) and (30), we can obtain the coefficients ˛m . Note that the SF can be double sided (DS, M1 ¤ 0 and M2 ¤ 0) or single sided (SS, M1 ¤ 0 and M2 D 0, or M1 D 0 and M2 ¤ 0).
variances, it is assumed that the variation in kl and kl is very small and can be ignored for all K users and L paths, and that the variances are independent of k, h, and j . The 2 is given by variance for the inphase self-interference DSI
4. SIGNAL-TO-NOISE PLUS INTERFERENCE RATIO
i h 2 1 DSI D var zDSI 8 r 2 1 < P 4 X ˛m 1 D var ˇ11 T : 2 m1 DM1 n o m Q m cos 111 ‡1I .m1 / C sin 111 ‡1 .m1 /
To find the signal-to-noise plus interference ratio (SNIR), we need to find the desired signal power and the noise plus interference variances. From (16), the desired inphase signal power S is given by i2 P h I1 D .ˇ11 Nn T /2 (31) S D zDS 2 To calculate the variances, it is assumed that all the interferences and the noise terms are zero mean independent random variables. The process of computing the variances for the different interference terms is quite involved. 2 , 2 2 2 Fortunately, the variances MPI MCDI , WPI , MUI , and 2 nQ 1 , for a system without SF, have been computed in [1, Chapter 4]. Without loss of generality, we invoke the results in [1,3] for the preceding variances with SF, which is given by
C
P .T Nn /2 .MI 1 C NI/ 2
M2 X
MI 1 D 42
2 ˛m
mDM1
2
M2 X
C
C sin 2 D DSI
P T 2 EŒˇ11 2 2 2
C
3
(33) 2 ˛m
˛m1 ˛mO 1
M2 X M2 X
n h i
m m O ˛m2 ˛mQ 2 E cos 112 cos 112
i i h h m m O I I .m2 /‡1 .m Q 2 / CE sin 112 sin 112 E ‡1 3 io h Q Q Q 2/ 5 (35) E ‡1 .m2 /‡1 .m
0
H
Nn .Es =No /
1 X
39 o = 5 ;
m2 D1m Q 2 D1
˛m ˛mC1 5
x m T Nn .JH /2 3 H Z Tn 2 X 2 ^ rh0 1 . / C r h0 1 . / d 5
NI D
m Q 112 ‡1 .m2 /
i h n h i m m O E cos 111 cos 111 E ‡1I .m1 /‡1I .m Q 1/
i h io h Q Q m m O Q 1/ C E sin 111 sin 111 E ‡1 .m1 /‡1 .m
mDM1
M2 X
m1 DM1 m O 1 DM1
(32)
H X H Z Tn X
QJK x m T Nn .JH /2 0 hD1 h0 D1 2 2 ^ rh0 h . / C r h0 h . / d
0
1 X
4
4
h D1
n m I cos 112 ‡1 .m2 /
where 2
˛m 2
m2 D1
2 2 2 2 C MCDI C WPI C MUI C nQ21 T21 D MPI
D
M2 X
(34)
mDM1
with x m D 6 for QPSK and x m D 12 for BPSK, D varŒˇk1 , Q represents the sum of amplitude levels of all Wirel. Commun. Mob. Comput. (2012) © 2012 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
m is a zero means random variable uniAssuming that 11 formly distributed in Œ0; 2 i
h m m Q E sin 111 sin 111
( m R 2 m Q m 1 sin 111 sin 111 d 111 D 12 ; m1 D m Q1 D 2 0 0; m1 ¤ m Q1
Similarly, i
h m m O E cos 111 cos 111 D
(
1 2;
m1 D m Q1
0;
m1 ¤ m Q1
Narrow-band suppression in wavelet packets based MC/MCD-CDMA system
Also, it is evident that
I d111;1
2
= D E 2
2
2
2
Q I I D d111;0 D d111;1 D d111;1 2
2
Q Q D d111;0 D d111;1 D 1
M. M. Akho-Zahieh and N. Abdellatif
"Z
Z
T
0
Z
T
a1 .t /a1 . /
0
T
Z
0
T
0
8 < | .t m1 Tn /| . m2 Tn / : cosŒ2.t m1 Tn / C cosŒ2.t m2 Tn / C
And from [3] ( EŒFk;i .l/Fk;i .m/ D
Nn jlj ;
l D jmj
0;
l ¤ jmj
0 h D1 iD0
P T 2 EŒˇ11 2 D 2 2 1 X
4
PNn T 2
2y m
M2 X
2 ˛m N C 1 n
3
M2 X
˛m
mDM1
"Z
(36)
Z
T 0
0
a1 .t /
| .t mT / cosŒ2.t mTn / C
H NX n1 X
var =Q 1
T
0
o
0 h D1 iD0
c1i wh0 . iTn / dt d dt d
i
(37)
2 Using [3, Eq. 18], also given c1i D 1 and because
E D D ıln , then (37) wh0 t l TNn ; wh0 t n TNn can be written as
2Q 1 D
=H T 2 4
|2 .m1 ; m2 /
M2 X
M2 X
˛m1 ˛m2 |2 .m1 ; m2 /
m1 DM1 m2 DM1
Z1 sign Œ1 p jxNn m1 C m2 j
1
cos Œ2q .xNn m1 C m2 / .1 jxj/ dx (39)
T
0
Z
where |2 .m1 ; m2 / is given in [3, Eq. 19] by
28 r Z M2 < X = T DE4 ˛m a1 .t / : 2 0 mDM1 "Z
0
T
H NX H NX n1 n1 X X c1i wh0 .t iTn /
0 h D1 iD0
The variance can be evaluated as follows: i
a1 .t /a1 . /
(38)
iTn /dt 5 dt
0 h D1 iD0
h
0
Z
T
cosŒ2. m2 Tn / C
=
3 c1i wh0 .t
Z
T
cosŒ2.t m1 Tn / C
T
= 2
˛m 1 ˛m 2
m1 DM1 m2 DM1
0
2 ˛m
3
8 < | .t m1 Tn /| . m2 Tn / :
m1 DM1 ;m¤0
r
M2 X
M2 X
E
where y m D 1 for QPSK and y m D 2 BPSK. The reason that y m and x m in the case of BPSK are double that for QPSK is as follows: as indicated previously, instead of dk .t /, we use dkI .t /, and exp.j !o t / is replaced by cos.!c t / in BPSK modulation. Hence the interference variances are calculated from (17)–(21) by considering only the cosine part of the inphase signal. From (23), the inphase component of narrow jamming signal is given by
e =1 D
M2 X
"Z
2 ˛m N 5 2 n
m2 D1
m1 DM1
D
= 2
D
0 h D1 iD0
9 =
c1i wh0 . iTn / dt d dt d 5 ;
According to this, (35) becomes 2 DSI
H NX H NX n1 n1 X X c1i wh0 .t iTn /
| .t mT / cosŒ2.t mTn /C
H NX n1 X 0 h D1 iD0
92 3 = 7 c1i wh0 .t iTn /dt 5 dt 5 ; 3
given that Nn 1; q D Tn (ratio of the offset of interference carrier frequency to half-spread-spectrum bandwidth) and signŒx D x or zero for x 0 or x < 0, respectively. Accordingly, the total variance for noise and interference Wirel. Commun. Mob. Comput. (2012) © 2012 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
M. M. Akho-Zahieh and N. Abdellatif
Narrow-band suppression in wavelet packets based MC/MCD-CDMA system
is given by 2 2 2 2 2 T2 D MPI C MCDI C WPI C MUI C DSI C nQ21 C 2Q 1
6. OUTAGE PROBABILITY PERFORMANCE
=
P .T Nn /2 .MI 1 C MI 2 C NI C JI/ D 2 P .T Nn /2 D .MI C NI C JI/ 2
(40)
where MI 2 D
JI D
M2 X
2 ˛m y m Nn m1 DM1 ;m¤0
M2 M2 H ==SK X X
Nn2
(41)
m1 DM1 m2 DM1
˛m1 ˛m2 |2 .m1 ; m2 /
(42)
and SK D 2P . Therefore, the output SNIR, , can be written as
D
S T2
D
P 2
.ˇ11 T Nn /2 T2
2 D ˇ11
