EFFICIENT IMPLEMENTATION OF DOWNLINK CDMA

EQUALIZATION USING FREQUENCY DOMAIN

APPROXIMATION

F. S. Al-kamali+, M. I. Dessouky, B. M. Sallam, and F. E. El-Samie++

Department of Electrical Communications
Faculty of Electronic Engineering
Menoufia University
Menouf, Egypt
E-mails: +faisalalkamali@yahoo.com, ++fathi_sayed@yahoo.com

ABSTRACT
A signal transimitted through a wireless channel may be severely distorted due to
intersymbol interference (ISI) and multiple access interference (MAI). In this paper,
we propose an efficient CDMA receiver based on a frequency domain equalization
with a regularized zero forcing equalizer and unit clipper decision with parallel
interference cancellation (FDE-RZF-CPIC) to combat both ISI and MAI. This
receiver is suitable for downlink zero padding CDMA (ZP-CDMA) cellular systems.
The effects of the decision function, the channel estimation, the number of cancelled
users, and the user loading on the performance of the proposed receiver are discussed
in the paper. The bit error rate (BER) performance of the proposed receiver is
evaluated by computer simulations. It has been found that the proposed receiver
provides a good BER performance, even at a large number of interfering users. At a
BER of 10-3, the performance gain of the proposed receiver is about 2 dB over the
RAKE receiver with clipper decision and parallel interference cancellation in the
half loaded case (8 users ) and is much larger in the full loaded case (16 users).

Keywords: Downlink CDMA, Decision Functions, PIC, FDE-RZF, Zero Padding,

Channel Estimation.

1 INTRODUCTION suffers from the error propagation phenomena that

contributes to progressively enhanced interference
Recently, single-carrier transmission with [12]. There are several algorithms for interference
frequency domain equalization (FDE) has attracted cancellation in CDMA systems [ 6-13 ]. Most of
much attention for its excellent performance even in these algorithms are designed for the uplink. For
strong frequency selective channels. In practice, the uplink interference cancellation, it is assumed that the
number of fingers in the RAKE receiver is limited base station knows all the spreading codes of the
because of hardware complexity. The use of an FDE active users. For downlink CDMA, the receiver
can alleviate the complexity problem of the RAKE knows only the spreading code of the desired user.
receiver arising from too many paths in a severe As a result, PIC has been assumed to be applicable at
frequency-selective channel. It has been shown that the base station, and not at the mobile terminal where
the FDE can take the place of the conventional only the information stream is to be decoded and the
RAKE receiver with much improved BER spreading codes of the interfering users are unknown.
performance for DS/CDMA signal reception over a However, in last years, many algorithms have been
severe frequency-selective channel [1-5]. This gives proposed for the estimation of the codes of the
the CDMA with FDE the power to compete with interfering users and PIC has been applied for
multi carrier CDMA (MC-CDMA) in fourth downlink CDMA [7]. The main target of this paper is
generation systems. The performance of CDMA is to analyze the performance of PIC for downlink
mainly limited by the interference from other users, CDMA with different decision functions and hence to
which is called the MAI. Therefore, PIC has to be develop an efficient receiver based on FDE and PIC
applied in CDMA receivers [6-13]. that is suitable for downlink CDMA.
PIC has gained a considerable attention for its The remainder of this paper is organized as
potential ability to increase system capacity and its follows: in section 2, the system model of downlink
simplicity. However, the conventional PIC often CDMA is presented. In Section 3, the concept of PIC

Ubiquitous Computing and Communication Journal 1

is discussed. Section 4 deals with the different elements of vectors d m and d m −1 were equal, i.e., if
possible decision functions. FDE for downlink ZP-
a zero padding (or cyclic prefix) process is used. The
CDMA is presented in section 5. In Section 6, the
length of the zero padding must be greater than W. In
proposed FDE-RZF-CPIC algorithm is presented.
this paper we will use a zero padding method as in
Channel estimation is discussed in section 7. The
[1]. So, Eq.(3) can then be written as:
relative performance of the proposed receiver is
compared to some existing approaches in section 8.
r = Hd + n (4)
Finally, Section 9, gives the concluding remarks.
Where H=H0+H1 has now a circulant structure.. H
Notations: The symbols (.)H, (.)T, and (.)-1 designate
can be written as:
complex conjugate transposition of a matrix,
transposition of a matrix, and inverse of a matrix,
respectively. Vectors and matrices are represented in ⎡ h[0] 0 . 0 h[W −1] . . h[0] ⎤
boldface. ⎢ . h[0] . . . . ⎥
⎢ . . . . h[W −1]⎥⎥
2 SYSTEM MODEL ⎢
H = ⎢h[W −1] . . 0 ⎥ (5)
In downlink CDMA, the channel is common ⎢ 0 . . ⎥
with frequency selective fading. We assume that the ⎢ . . . . 0 ⎥
channel parameters vary slowly with time, so that for ⎢ ⎥
sufficiently short intervals the channel is ⎣ 0 . 0 h[W −1] . . h[0] ⎦
approximately a linear time-invariant system. The
baseband channel response can then be expressed by The vector d can be represented as:
Dirac delta functions as follows [9]:
d = CSb (6)
W
h (t ) = ∑h
w
w δ (t − τ w ) (1)
Where C is an (N×L)×(N×L) scrambling code
where hw , and τw are the complex fading and matrix. S is an (N×L)×(K×L) orthogonal code matrix,
propagation delay of the w-th path, and W is the and H 1d m −1 is the inter block interference (IBI)
number of multipath components of the channel term [6]. H0 and H1 can be written as [6]:
impulse response. In this paper, we assume block
fading, where the path gains stay constant over one
block duration. The received signal at the mobile can
be written as[9]: ⎡ h[0] 0 . . . . . 0 ⎤
⎢ . h[0] ⎥
⎢ ⎥
L KW ⎢ . . ⎥
r(t) = ∑∑∑hw AKbk (l)c(t −τ w )sk (t − lTs −τ w ) + n(t) (2) H0 = ⎢
h[W −1] . ⎥
l k w ⎢ ⎥
⎢ . . . ⎥
where Ak is the amplitude, bk(l) ∈ {-1,1} is the lth ⎢ 0 . . h[W −1] . . . h[0]⎥⎦
⎣
bit, sk (t) is the spreading code of user k, and c(t) is
the scrambling code. In matrix form, the received
, and
signal in Eq. (2) can be written as follows [6]:
⎡0 . 0 h[W − 1] . . h[ 0 ] ⎤
⎢. ⎥ (7)
r = H 0 .d m + H 1 . d m − 1 + n m (3) ⎢ . . . ⎥
⎢. . . . ⎥
H 1 = ⎢⎢ . . h[W − 1]⎥
⎥
Where d m is the mth block of the transmitted signal, ⎢. . 0 ⎥
⎢ ⎥
r is the received vector, while nm is the additive ⎢. . ⎥
noise with zero mean and variance of σn2. H0 is the ⎢0
⎣ . . . . . 0 ⎥
⎦
(N×L)×(N×L) matrix describing a multipath channel
having impulse response h(t) of length W. L is the
number of symbols for each user. N=TS/Tc is the The structures of the individual components in
number of chips per bit (spreading factor). TS is the Eq. (6 ) are given bellow [14]:
symbol period. Tc is the chip period. K is the number
of active users. S = diag [ S S ......S ] (8)
We can observe that inter-block interference
(IBI) would disappear from Eq. (3) if the last W-1

Ubiquitous Computing and Communication Journal 2

S = [ s 1 s 2 ........ s K ] in frequency domain can be summarized as follows
(9) [6]:
1- Apply the RAKE receiver on the received signal as
s = [s
k k
(0), s k (1),..........., s k ( N − 1)]
T
(10) follows [14]:
d RAKE = Ψ ( Λ H R Τ ) (16)
T T T
b = [b (1)b ( 2).......b ( L )] (11) Where Λ is a diagonal matrix containing the FFT of
the circulant sequence of H, and RT is the FFT of r.
C = diag [c (1), c ( 2),......, c ( N × L )] (12) 2-Estimate all interferences as follows:

T H
b(l ) = [b1 (l ), b2 (l ),........., bK (l )]
T
(13) bˆ int = U C d RAKE (17)

Where SK is the signature sequence (code), and b is a 3- Discard the detected zero symbols at the end of

vector consisting of the users amplitudes and the each block to produce b int , then take the decision as
transmitted bits, and bK(l) is the lth bit. follows:

In this paper, we suppose a perfect estimation of ~ = f {b } (18)
the downlink interfering codes. So, we can write the b int dec int

received signal as follows:

where fdec(.) is the tentative decision function.
4- Add zeros for padding , then regenerate the MAI
r = HCS d b d + HCUb int + n (14)
as follows:
~
Where Sd is an ((N×L)×L) matrix consisting of the r MAI = H CU b ' int (19)
spreading code of the desired user, U is an ~' ~
(N×L)×((K-1)×L) matrix consisting of the spreading where b int is the zero padded version of b int .
codes of the interfering users, bd is an L×1 vector 5- Use PIC to cancel the effects of interference on the
consisting of the desired symbols, and bint is a ((K- received signal to obtain an interference free signal:
1)×L)×1 vector consisting of the interfering symbols.
In the zero padding method, NZP zeros will be z = r − rMAI (20)
added to the end of NF-NZP symbols to build a block
of NF symbols before transmission. At the receiver,
6- Apply the RAKE receiver on the vector z as
the FFT detection will be performed on the padded
follows:
data block, the detected zeros at the end of this data H −1
block will then be discarded after despreading. The d RAKE = Ψ ( Λ Ψ z) (21)
zero padding process is illustrated in Fig. (1).
7- Descramble and despread the obtained data.
3 PIC FOR DOWNLINK ZP-CDMA 8- Finally, discard the detected zero symbols and
perform hard decision.
Parallel interference cancellation for CDMA Due to error propagation, PIC with hard
systems has attracted much interest, due to its decision may perform worse than PIC with linear or
structured architecture that facilitates easy soft decision functions. On the other hand, hard-
implementation. It was first introduced in 1990 [15]. decision interference cancellation can completely
Such multistage PIC methods attempt to cancel MAI cancel interference when the hard decisions made are
based on tentative decisions. The idea of PIC is to correct [12].
estimate the multiple access and multipath induced
interferences and then to subtract the interference 4 DECISION FUNCTIONS
estimate. The circulant matrix H can be efficiently
diagonalized by the fast Fourier transform (IFFT) The performance of PIC depends on the decision
and inverse fast Fourier transform (FFT). Let Ψ-1 and function used in the interference cancellation
Ψ denote the FFT matrix and the IFFT matrix, iterations, e.g., hard, soft, null zone, unit clipper, and
respectively. hybrid decision functions [12].
A circulant matrix A can be written as (see appendix The general model for the decision function is:
1) :
−1
A = ΨΛΨ (15) y = f dec ( x) (22)
The following, decision functions can be used:
where Λ is the FFT of the circulant sequence of A.
The implementation of the RAKE receiver with PIC ƒ The hard limiter [12]:

Ubiquitous Computing and Communication Journal 3

 16 zeros

1 block (256 chips)

1 slot data (2560 chips )

FFT detect Block- by – block detection

+despreading

FFT detect
+despreading

FFT detect
+despreading

Figure 1: CDMA transmission for FDE detector using zero padding

y
⎧1 x ≥ 0 x
⎪ (23)
y = fdec(x) = ⎨
⎪−1 x < 0 x >1
⎩ ⎧ 1,
⎪ (26)
y = fdec(x) = ⎨ x, x ∈[−1,1]
Hard Limiter
⎪−1, x < −1
⎩
It makes a hard decision for one of the two
possible symbols. It makes a soft bit decision when the soft bit
estimate lies inside the interval [ -1 , 1 ] to avoid
ƒ The null zone function [11]: propagation error , and makes a hard decision when
the soft bit estimate lies outside the interval [-1 , 1] to
y
avoid the noise magnification [10].
1
⎧ 1, x > cn -1 -cn
⎪ (24)
x
y = fdec(x) = ⎨ 0, x ∈[−cn , cn ] cn 1 5 FREQUENCY DOMAIN EQUALIZER FOR
-1
⎪−1, x < −cn DOWNLINK ZP-CDMA.
⎩
Null Zone
The application of FDE techniques makes single
It makes a hard decision when the soft bit carrier modulation a potentially valuable alternative
estimate lies outside the interval [-cn,cn], and sets the to OFDM, especially in regard to its robustness to RF
decision result to zero when the soft bit estimate lies implementation impairments. Linear ZF based chip
inside the interval [-cn , cn]. Where cn is the null level equalization has been one of the most popular
zone decision threshold (0≤ cn ≤1) [11]. equalizers for single user downlink CDMA [16].
Because of the noise enhancement in the ZF
ƒ The linear decision function: equalizers, we propose the application of the
regularized zero forcing equalizer.
y = f dec ( x ) = x (25) The time domain ZF estimation of d is given by [16]:

−1
d ZF = ( H H )
H H
It offers analytical access to the PIC performance, H r (27)
but performs worse than other decision functions.
To encounter the problem of noise enhancement
ƒ The unit clipper decision function [11]: in ZF equalizers, a new regularization term is added
y into Eq. (27) to yields:
1
x
-1
1
-1

Unit Clipper

Ubiquitous Computing and Communication Journal 4

 d RZF = (H H H + αΙ) −1 H H r RAKE receiver with CPIC to estimate, regenerate,
(28) and cancel all the interfering users. Then the FDE-
= M −1H H r = Gr RZF equalizer is used to reduce the ISI effect and to
provide better estimate of desired user's data. In this
section, a specific data detection algorithm for
Where α is a regularization parameter . downlink CDMA is derived which is based on FDE-
The solution of Eq. (28) requires the inversion of RZF equalizer and CPIC. The proposed FDE-RZF-
the matrix M which has dimensions of CPIC system model is depicted in Fig. (2).
(N×L)×(N×L). This inversion process is impractical
in real time. Thus, a simplification is required for this The FDE-RZF-CPIC algorithm can be summarized
inversion process. as follows:
The equalizer matrix G can then be easily calculated
as follows: 1- Estimate all interferences as follows:
−1 H −1
G =Ψ Λ M Λ Ψ (29) bˆ int = U T C H Ψ Λ H .R T (33)
where:
2- Discard the detected zero symbols at the end of

Λ
−1
M
H
= [Λ Λ + αI] −1
(30) each block to produce b int , then take the decision as
follows:

The FDE-RZF algorithm can be summarized as ~
follows: b int = f dec {b int } (34)
1- Apply the FDE-RZF on the received signal as
follows: where fdec(.) is a decision function that transforms
the soft estimate into a unit clipper decision.
Λ RT)
−1 H
d FDE − RZF = Ψ ( Λ M (31) 3- Add zeros for padding , then regenerate the MAI
as follows:
~
3- Then, a better estimation of the symbol of r MAI = H CU b ' int (35)
interest can be obtained as follows:
~' ~
where b int is a zero padded version of b int .
H
bˆ d = S d C d FDE − RZF
T
(32)
4-Use PIC to cancel the effects of interference on
4- Finally, discard the detected zero symbols at the the received signal to get an interference free
end of each block, and then use the decision function. signal:
A major advantage of this equalization method is
its low computational complexity. The price to be Z = r − rMAI (36)
paid is a reduction of the data rate caused by
insertion of zero padding or cyclic prefix. 5- Apply the FDE-RZF to the signal vector Z as
follows:
5.1 Complexity d FDE−CPIC = Ψ ( Λ M .Λ .Ψ
−1 H −1
(Z)) (37)

The complexity of a P-point FFT is of the order of

Plog2P. The FDE provides a complexity of Ο(P log2 6- Then, a better estimation of the symbols of
P ) which shows a significant reduction as compared interest can be obtained as follows:
to that of the direct inversion of a matrix of
dimensions P×P that has a complexity of the order of bˆ d = S dT C H d FDE −CPIC (38)
Ο(P3) [1]. The FDE has also less complexity than
that of the RAKE receiver which has a complexity 7- Finally, discard the detected zero symbols at the
of Ο(P2) [1]. end of each block, and then use the decision function.
The performance of FDE-RZF-CPIC depends
heavily on the channel estimate, not only in the
6 FDE-RZF WITH CLIPPER PIC detection step, but also in the interference regeneration
step. It is more efficient when the system is heavily
This section gives the proposed receiver which is loaded.
used to improve the performance of the PIC for
downlink CDMA. The proposed receiver uses the

Ubiquitous Computing and Communication Journal 5

7 CHANNEL ESTIMATION downlink synchronous ZP-CDMA system, in which
each user transmits BPSK symbols. These symbols
In this section, we consider the channel estimation are spread. After spreading, the resulting sum signal
method which depends on the pilot signal. When the is scrambled using a complex scrambling sequence.
pilot sequence is transmitted, the received signal in The propagation channel is taken to be chip-spaced
Eq. (4) is expressed as: with delay spread of 3Tc. For all simulations, we
r = Dh + n (39) take N=16, and the block size is NF=256 chips with
where the complex channel impulse response h is 16 zeros (NZP=16) at the end of each block as shown
expressed as: in Fig. (1). All users are assigned the same power.
desired
FDE-RZF Descrambling, user's
Implementation
Hard data
of
Despreading,
IFFT And decision
Channel Estimation (Λ Λ+αI)
H -1
remove ZP

Received
signal RAKE
Descrambling 2
Genera Despreading, Unit .
tion of Clipper .
FFT IFFT And .
ΛH remove ZP decision K

Interference
Regeneration

Figure 2: The structure of FDE-RZF-CPIC for downlink ZP-CDMA system.

Figure 3 compares the performance of the FDE-

h = [h1 , h2 ,.........., hW ]T (40) RZF and that of the TDE and the RAKE receiver
for 8 users . It is clear that the equalization in the
time domain is identical to that in the frequency
D is the circulant pilot sequence matrix. domain. The only difference is in the method of
The MMSE channel estimates are found by implementation. Both equalizers have better
minimizing the following squared error quantity: performance than that of the RAKE receiver only.

hˆ = arg min r − Dh
2 SF=16,K=8,α =1
(41) 0
h 10
RAKE Zero padding
Assuming white gaussian noise, the ZF solution is TDE-RZF Zero Padding
given by: FDE-RZF Zero padding

hˆ ZF = (DH D) −1 DH r
-1
(42) 10
AVERAGE BER

However, using zero forcing channel estimation,

the channel estimation accuracy significantly
degrades due to the noise enhancement. -2
10

8 SIMULATION RESULTS
-3
Several simulation experiments are carried out to 10
0 5 10 15
test the performance of the proposed FDE-RZF- SNR
CPIC algorithm and compare it to other algorithms.
The simulation environment is based on the
Figure 3: Performance of FDE-RZF, TDE-RZF and
RAKE receiver Vs the SNR .

Ubiquitous Computing and Communication Journal 6

 single-user RAKE receiver is very poor, even for
high SNR values. Parallel interference cancellation
The performance of the RAKE receiver with PIC improves the performance significantly. Better
and different decision functions is studied. Figures 4, performance can be obtained with the clipper
and 5 illustrate the average BER versus the decision function. Linear decision performance is
threshold of the null zone decision function (cn) at worse than that of the single user RAKE receiver for
different SNR values and different number of users. the heavily loaded case (K=16). This is justified by
the fact that PIC with linear decision is limited by
noise enhancement.
SF=16,K=8 SF=16,K=8
0 0
10 10
NULL ZONE DECISION

-1
10 SNR=[ 6 9 12 15] db 10
-1
AVERAGE BER

AVERAGE BER
-2
10 10
-2

-3 RAKE
10 10
-3
Null zone decision
unit clipper decision
hard decision
-4 soft decision
10 10
-4
0 0.2 0.4 0.6 0.8 1 0 5 10 15
THRESHOLD SNR
Figure 4: Performance of the RAKE receiver with PIC Vs Figure 6: Performance of the RAKE receiver with PIC at
null zone decision threshold (cn) at different SNRs . different decision functions Vs the SNR for the half loaded
case. cn=0.4.
SF=16, K=16 SF=16,K=16
0 0
10 10
NULL ZONE DECISION

-1
10 SNR=[ 6 9 12 15] db
-1
10
AVERAGE BER
AVERAGE BER

-2
10

-2
10 RAKE
-3 Null zone decision
10
unit clipper decision
hard decision

-4 -3
soft decision
10 10
0 0.2 0.4 0.6 0.8 1 0 5 10 15
THRESHOLD SNR
Figure 7: Performance of the RAKE receiver with PIC
Figure 5: Performance of RAKE with PIC Vs null zone at different decision threshold functions Vs the SNR
decision threshold (cn) at different SNR . for a full loaded case. cn=0.4.

The optimal performance is obtained when

cn=0.4. Figures 4, and 5 show that cn is non- The performance of the proposed FDE-RZF-
sensitive to SNR-changes and to system-load CPIC is compared to that of the RAKE receiver,
changes. FDE-RZF equalizer, the RAKE receiver with hard
The effect of the tentative decision function on decision PIC, and the RAKE receiver with unit
the performance of PIC for K=8 (half loaded), and clipper decision PIC.
K=16 (Full loaded ) are studied and shown in Figs. 6, The effect of the regularization parameter on the
and 7. As the number of users increases, the performance of FDE-RZF-CPIC is examined in two
performance deteriorates. The performance of a experiments and shown in Figs. 8 and 9. The optimal

Ubiquitous Computing and Communication Journal 7

 SF=16,K=16
0
10
value of the regularization parameter α is equal to 1.
This value is neither sensitive to SNR-changes nor
to system-load changes. -1
The effect of the choice of the tentative decision 10
function on the performance of the proposed

AVER AG BER
receiver is studied ( Fig. 10 ). The performance with
clipper decision outperforms the performance with -2
10
all other decision functions .
SF=16,K=8 RAKE
0
10 -3
10 FDE-RZF w ith Null zone decision PIC
FDE-RZF-CPIC FDE-RZF w ith CPIC
FDE-RZF w ith hard decision PIC
-1 FDE-RZF w ith linear decision PIC
10 SNR=[ 6 12 ] db -4
10
AVERAGE BER

0 5 10 15
SNR
-2
10 Figure 10: Performance of the proposed receiver
at different decision Functions Vs the SNR.

-3
10

Figures 11, and 12 show the performance of five

-4 reception schemes as a function of SNR of each user
10
10
-2
10
0 for 8 and 16 users, respectively. From Figs. 11, and
Regularization Parameter (α )
12, it can be observed that there is a clear
improvement achieved by FDE-RZF–CPIC scheme
Figure 8: Performance of FDE-RZF-CPIC scheme over other reception schemes. In Fig. 12, BER
Vs regularization parameter (α ) at different SNR performances of all receivers are worse than the
for a half loaded system. . performances in Fig. 11 because of the increment in
the number of users. The FDE-RZF-CPIC scheme
0
SF=16,K=16 improves the performance significantly, without
10 saturation of the performance for high SNRs like the
FDE-RZF-CPIC RAKE receiver. For the heavily loaded case (Fig.
12), the performance of FDE-RZF equalizer is
-1
10 SNR=[ 6 12 ] db greater than that of the RAKE with PIC scheme. This
can be explained by the fact that at heavily loads, the
AVERAGE BER

RAKE receiver sees too much interference, which

10
-2 makes its decisions about interfering users unreliable.

SF=16,K=8,α =1
0
-3 10
10

-1
10
-4
10
AVERAGE BER

-3 -2 -1 0 1
10 10 10 10 10 -2
10
Regularization Parameter (α )

-3
Figure 9: Performance of FDE-RZF-CPIC scheme Vs 10
regularization parameter (α ), at different SNR for a full
RAKE
loaded system..
-4 FDE-RZF
10 hard decision rake+pic
unit clipper decision rake+pic
FDE-RZF-CPIC
0 5 10 15
SNR

Figure 11: Performance of different reception schemes Vs

the SNR for a half loaded system.

Ubiquitous Computing and Communication Journal 8

 users increases.
SF=16,K=16, α =1
0
10 RAKE SF=16
0
FDE-RZF 10
hard decision rake+pic FDE-RZF-CPIC
unit clipper decision rake+pic
FDE-RZF-CPIC -1
-1 10
10
AVERAGE BER

AVERAGE BER
-2
10
-2
10
-3
10

-3
10 -4
0 5 10 15 10
4 6 8 10 12 14
SNR
Number of Cancelled Users
Figure 12: Performance of different reception schemes Vs Figure 14: Performance of FDE-RZF-CPIC scheme Vs
the SNR for a full loaded system.
number of canceled users. α=1, and SNR =12 dB.

The effect of user loading on the performance of The effect of channel estimation accuracy on the
the FDE-RZF-CPIC scheme is studied and presented performance of the FDE-RZF-CPIC scheme for K=8
in Fig. 13. The BER of all receivers degrade with are studied and shown in Figs. 15, and 16. The
increasing the number users. In this case, the BER performance of FDE-RZF-CPIC scheme with ZF
performance of FDE-RZF-CPIC scheme degrades a channel estimation shows a loss of 1 dB at BER of
little bit with increasing the number of users, but it is 10-2 when compared with the case of perfect channel
still better than the other schemes. This observation knowledge. Because the noise enhancement in the
may be due to the MAI. The MAI when the number ZF channel estimation. LMMSE channel estimation
of users is large should be greater than the case when gives better performance.
the number of users is low. Even after interference
SF=16,K=8, LMMSE Channel estimation
cancellation, some residual MAI still exists. 10
0

Therefore, the performance loss may be attributed to rake+pic chann. know n

the residual MAI. rake+pic chann. estimation

FDE-RZF+CPIC chann. know n
SF=16,α =1 FDE-RZF+CPIC chann. estimation
0
10 -1
RAKE 10
AVERAG BER

FDE-RZF
hard decision rake+pic
-1
10 unit clipper decision rake+pic
FDE-RZF w ith CPIC
AVERAG BER

-2
10
-2
10

-3
10 10
-3

0 2 4 6 8 10 12
SNR

10
-4 Figure 15: Performance of Interference Cancellation
4 6 8 10 12 14 Vs the SNR for exact and LMMSE channel estimate.
Number of Users ,K

Figure 13: Performance of different reception schemes Vs

the number of active users. and SNR =12 dB.

Figure 14 depicts the average BER performance

as a function of the number of cancelled users, at a
fixed SNR per user of SNR=12 dB. This graph
shows that the performance of the FDE-RZF-CPIC
scheme improves when the number of cancelled

Ubiquitous Computing and Communication Journal 9

 0
SF=16,K=8, zero padding ,ZF Channel estimation (A.2)
10
where each raw is a circular shift of the raw above
and the first raw is a circular shift of the last raw.
The primary difference between the matrices Q and
-1
10
Q c is that they differ only by elements added in the
upper right and lower left parts to produce the cyclic
AVERAG BER

-2 structure in the raws. If the matrix size S is large

10
and the number of non zero elements on the main
diagonals compared to the number of zero elements
is small (i.e the matrix is sparse), the number of
-3
10
rake+pic chann. known
elements added to the upper right and lower left parts
rake+pic chann. estimation does not affect the matrix, because they are small in
FDE-RZF+CPIC chann. known proportion to the main diagonal elements. It can be
-4
10
FDE-RZF+CPIC chann. estimation
shown from the eigen value distribution of both
0 2 4 6 8 10 12 c
SNR
matrices that Q and Q are asymptotically
equivalent.
Figure 16: Performance of Interference Cancellation Vs c
the SNR for exact and ZF channel estimate. It is known that an SXS circulant matrix Q is
diagonalized by [1]:
9 CONCLUSION −1 c
Λ=Ψ Q Ψ (A.3)
The paper presents an efficient FDE-RZF-CPIC where Λ is an SXS diagonal matrix whose
c
receiver for downlink CDMA. This receiver is elements λ ( s, s ) are the eigenvalues of Q and
implemented using frequency domain approxima- where Ψ is an SXS unitary matrix of eigen vectors
tions rather than the time domain implementation to c
reduce complexity. The comparison studies show that of Q . Thus we have:
*t *t
the proposed receiver outperforms several traditional ΨΨ = Ψ Ψ = I (A.4)
receivers for different loading cases. The sensitivity The elements φ ( s1, s2 ) of Ψ are given by
of the proposed receiver is also studied for different
decision functions and different channel estimation [17,18]:
methods. The obtained results indicate that the ⎡ j 2πs1s2 ⎤
proposed receiver performance is robust for the ψ ( s1, s2 ) = exp ⎢ ⎥ (A.5)
different channel estimation methods. ⎣ S ⎦
2
for s1, s2 = 0,1,........, S − 1 and j = −1
APPENDEX 1 Toeplitz to circulant approximation
The eigen values λ ( s, s ) can be called λ (s ) . For
Let Q be an SXS Toeplitz matrix of the following these eigen values, the following relation holds
form: [17,18]:
⎡ q ( 0) " q ( − l ) 0 ⎤ k ⎡ − j 2πms ⎤
λ ( s ) = q (0) + ∑ q ( m) exp ⎢
⎢ # % % ⎥ m =1 ⎣ S ⎥⎦
⎢
Q = q(k ) % q ( −l )
⎥ (A.1)
(A.6)
⎢ ⎥ −1 ⎡ − j 2πms ⎤
+ ∑ q ( m) exp
⎢ % % # ⎥
m = −l ⎢⎣ S ⎥⎦
⎢⎣ 0 q ( k ) " q (0) ⎥⎦ s = 0,1,........, S − 1
It can be approximated by an SXS circulant matrix c
Because of the cyclic nature of Q ,we define:
Q c defined as [17,18]: q ( S − m) = q ( − m) (A.7)
⎡ q ( 0) " " q ( −l ) 0 " q(k ) " q (1) ⎤ and thus Eq.(A.6) can be written in the form [17,18]:
⎢ # % % % " % ⎥ S −1 ⎡ − j 2πms ⎤
⎢ # q(k ) ⎥ λ ( s ) = ∑ q ( m) exp ⎢ (A.8)
⎣ S ⎥⎦
% % % "
⎢ q(k ) m =0
% % % " ⎥
Q
c
=
⎢ 0 % % % 0
⎥ for s = 0,1,........, S − 1
⎢ ⎥ Thus the circulant matrix can be simply diagonalized
q ( −l )
⎢ # % % %
⎥ by computing the DFT of the cyclic sequence
⎢ q ( −l ) # % % % # ⎥
q (0), q (1),......., q ( S − 1) .
⎢ # % # % % % # ⎥
⎢⎣q ( −1) " q ( −l ) # 0 q(k ) " " q (0) ⎥ ⎦

Ubiquitous Computing and Communication Journal 10

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