Simulation of Adaptive Array Algorithms for CDMA Systems by Zhigang Rong Thesis submitted to the faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Electrical Engineering c Zhigang Rong and VPI & SU 1996 APPROVED: Dr. Theodore S. Rappaport, Chairman Dr. Jeffrey H. Reed Dr. Brian D. Woerner September, 1996 Blacksburg, Virginia Keywords: Adaptive Antennas, Adaptive Algorithm, CDMA
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Simulation of Adaptive Array Algorithms for
CDMA Systems
by
Zhigang Rong
Thesis submitted to the faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Comparing equation (2.34), (2.35) with equation (2.13), (2.14), we see that for a narrow-
band uniform linear array (ULA), there is a correspondence between the normalized element
spacing, dλ , and the sampling period, Ts, in the FIR filter, also the sine of the DOA θi, sin θi,
can be related to the temporal frequency fi of the FIR filter input [7].
Since there is a mapping between the ULA and the FIR filter, a theorem applied to
the FIR filter in the time domain may also be applied to the uniform linear array in space
domain. In time domain, the Nyquist sampling theorem [8] stated that for a bandlimited
signal with highest frequency f , the signal is uniquely determined by its discrete time
samples if the sampling rate is equal to or greater than 2f . If the sampling rate is less than
2f , aliasing will occur. In the space domain, the sampling rate corresponds to the inverse
of the normalized element spacing, and the highest frequency is corresponding to 1 (since
sin θi is always less than 1). From the Nyquist sampling theorem, to avoid spatial aliasing,
we should have1dλ
≥ 2× 1 (2.37)
or equivalently,
d ≤ λ
2. (2.38)
Therefore, the element spacing of an array should always be less than or equal to half of the
carrier wavelength. However, the element spacing cannot be made arbitrarily small since
two closely spaced antenna elements will exhibit mutual coupling effects. It is difficult to
16
CHAPTER 2. FUNDAMENTALS OF ADAPTIVE ANTENNA ARRAYS
generalize these effects since they depend heavily on the type of antenna element and the
array geometry. However, the mutual coupling between two elements typically tends to in-
crease as the distance between elements is reduced [5]. Thus the spacing between elements
must be large enough to avoid significant mutual coupling. In a practical linear arrays, the
element spacing is often kept near a half wavelength so that the spatial aliasing is avoided
and the mutual coupling effect is minimized.
The frequency response of an FIR filter with tap weights w∗i , i = 1, . . . ,M and a sampling
period Ts is given by
H(ej2πf ) =M∑i=1
w∗i e−j2πfTs(i−1), (2.39)
where H(ej2πf ) represents the response of the filter to a complex sinusoid of frequency f .
For the harmonic retrieval problem, if we want to extract the signal with frequency fi,
we need to find a set of complex weights such that the frequency response of the filter
has a higher gain at fi and lower gains (or ideally, nulls) at other frequencies. For the
beamforming problem, since f and Ts are corresponding to sin θ and dλ , respectively, we can
replace f and Ts in equation (2.39) with sin θ and dλ , respectively, to get the beamformer
response,
g(θ) =M∑i=1
w∗i e−j 2π
λ(i−1)d sin θ, (2.40)
where g(θ) represents the response of the array to a signal with DOA equal to θ. So if there
are several signals coming from different directions, and we want to extract the signal with
direction θi, we need to find a set of weights such that the array response has a higher gain
at direction θi and lower gains (or ideally, nulls) at other directions.
The array response g(θ) may also be expressed in vector form as
g(θ) = wHa(θ), (2.41)
where w and a(θ) are defined in equation (2.28) and (2.14), respectively. The beamformer
response may also be viewed as the ratio of the beamformer output to the signal at the
17
CHAPTER 2. FUNDAMENTALS OF ADAPTIVE ANTENNA ARRAYS
reference element when a single plane wave is incident on the array.
The beampattern is defined as the magnitude of g(θ) [7] and is given by
G(θ) = |g(θ)|. (2.42)
Using G(θ), we may define the normalized beamformer response,
gn(θ) =g(θ)
max{G(θ)} (2.43)
where gn(θ) is also known as the normalized radiation pattern or array factor of the array.
The spatial discrimination capability of a beamformer depends on the size of the spatial
aperture of the array; as the aperture increases, discrimination improves. The absolute
aperture size is not important, rather its size in wavelengths is the critical parameter.
To illustrate this point, let us consider a uniform linear array with equal weight for each
element. From equation (2.40), we get the beamformer response
g(θ) =M∑i=1
e−j2πλ
(i−1)d sin θ
=1− e−j 2π
λMd sin θ
1− e−j 2πλd sin θ
(2.44)
=sin(πMd
λ sin θ)
sin(π dλ sin θ)e−jπ
(M−1)dλ
sin θ. (2.45)
The beampattern of this equal-weight beamformer is shown in Figure 2.5. The polar coor-
dinate plot of the beampattern is also shown in Figure 2.6. In Figure 2.5, the normalized
beampattern gain is expressed in dB. From equation (2.45), we see that the null-to-null
beamwidth, θBW , of the array is determined by
πMd
λsin θ = π. (2.46)
The solution of equation (2.46) may be expressed as
θH = arcsin
(λ
Md
)(2.47)
18
CHAPTER 2. FUNDAMENTALS OF ADAPTIVE ANTENNA ARRAYS
−80 −60 −40 −20 0 20 40 60 80−40
−35
−30
−25
−20
−15
−10
−5
0
DOA (degrees)
Bea
mpa
ttern
Gai
n (d
B)
Figure 2.5: Beampattern for an equal-weight beamformer. In this case, the number ofelements is equal to 8, and the element spacing is half of the carrier wavelength. The plotis generated by Programs [P1] and [P2].
0.2
0.4
0.6
0.8
1
60
−120
30
−150
0
180
−30
150
−60
120
−90 90
Figure 2.6: The same beampattern as shown in Figure 2.5 with polar coordinate plot inazimuth. The plot is generated by Programs [P1] and [P2].
19
CHAPTER 2. FUNDAMENTALS OF ADAPTIVE ANTENNA ARRAYS
where θH is the peak-to-null beamwidth, or half of the null-to-null beamwidth. The null-
to-null beamwidth is then obtained by
θBW = 2θH = 2 arcsin
(λ
Md
). (2.48)
From equation (2.48), we see that the beamwidth, which is the width of the main lobe
of the beampattern, is inversely proportional to the term Mdλ . Therefore, if the aperture
size in wavelengths is large, the beamwidth of the array will be small, and the beamformer
will have a high spatial discrimination capability. For a general beamformer with unequal
weights wiue, i = 1, 2, . . . ,M , the weight wiue may be viewed as the product of the equal
CHAPTER 4. ADAPTIVE BEAMFORMING ALGORITHMS USED IN SIMULATION
riCM (l) =
[y(1 + lK)
|y(1 + lK)| ,y(2 + lK)
|y(2 + lK)| , . . . ,y((1 + l)K)
|y((1 + l)K)|
]T(4.49)
ri(l) = aPNriPN(l) + aCMriCM (l) (4.50)
wi(l + 1) =[X(l)XH(l)
]−1X(l)r∗i (l). (4.51)
From the above equations we see that if aCM is set to zero, the LS-DRMTCMA becomes
the LS-DRMTA, therefore the LS-DRMTA can be viewed as a special case of the LS-
DRMTCMA. Also we see that if aPN is set to zero and the GSO procedure is performed
during the adaptation, the algorithm becomes MT-LSCMA. The choice of aPN and aCM
can affect the resulting beampattern and thus the performance of the system. We will
demonstrate this point by simulation in Chapter 5.
We can summarize the LS-DRMTCMA as follows
1. Initialize the p weight vectors w1, . . . ,wp as p identical M × 1 column vectors with
the first element equal to 1 and the other elements equal to 0.
2. Calculate the array output vector using equation (4.46).
3. Despread the ith user’s signal and estimate the nth data bit using equation (4.47).
4. Respread the estimate data bit with the PN code of user i to get a estimate of the
signal waveform of user i over time period [(n− 1)Tb, nTb) using equation (4.48).
5. Calculate the complex-limited output vector of user i using equation (4.49).
6. Calculate the reference signal vector for user i by summing up the weighted respread
signal vector and the weighted complex-limited output vector using equation (4.50).
7. Adapt the weight vector wi of user i using equation (4.51).
8. Repeat step 2 to 7 until the algorithm converges.
71
CHAPTER 4. ADAPTIVE BEAMFORMING ALGORITHMS USED IN SIMULATION
4.6.2 Advantages of LS-DRMTCMA
Since the LS-DRMTCMA utilize both the PN sequence and the constant modulus prop-
erty of the transmitted signal, it possesses all the advantages of LS-DRMTA and has other
advantages that the LS-DRMTA does not possess. The most important advantage is that
it can achieve a much lower BER than the LS-DRMTA. We will discuss this point in Chap-
ter 5. However, since LS-DRMTCMA utilizes the complex-limited output vector of each
user in the adaptation of the weight vectors, this advantage is achieved at the expense of
additional complexity.
4.7 Summary
In this chapter we present four multitarget-type blind adaptive beamformer algorithms,
MT-LSCMA, MT-DD, LS-DRMTA, and LS-DRMTCMA. The LS-DRMTA and the LS-
DRMTCMA are two novel algorithms developed in this research. The derivation and the
advantages of these two novel algorithms are described. In Chapter 5, we will present the
simulation results and compare these four algorithms.
72
Chapter 5
Simulation Results and Discussion
5.1 Introduction
In Chapter 4 we present four multitarget-type adaptive beamformer algorithms for the
base station in a CDMA system. In this chapter, we will compare these four algorithms
under different conditions. The following four cases will be considered in the simulation:
1. AWGN channel
2. System with timing offset
3. System with frequency offset
4. Multipath channel.
Our comparison of the algorithms will focus on the BER performance of different algorithms.
The convergence characteristics of the different algorithms will also be presented.
5.2 Description of System Parameters
The system we consider in the simulation is a direct sequence CDMA (DS-CDMA)
system with a processing gain, N , equal to 15. There are several reasons for choosing such
a small processing gain. The first reason is that we want to reduce the simulation time
and at the same time compare the performance of different adaptive algorithms in a CDMA
system. The second one is that we want to focus on the enhancement of the system capacity
obtained by utilizing the adaptive array, not on the enhancement obtained by using a long
PN sequence for each user. The last one is that N = 15 is the parameter specified by the
73
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
GLOMO program. The modulation scheme used in the system is the binary phase-shift
keying (BPSK), the carrier frequency fc is equal to 2.05 GHz, and the data bit rate Rb
is equal to 128 kbps. An 8-element uniform linear array with half wave-length spacing
between the elements is used in the base station of the system to perform spatial filtering
in the reverse link (from the mobile to the base station). The sampling rate is four times
the chip rate, in other words, there are four data samples per chip. So for the LS-DRMTA
and LS-DRMTCMA, the data block size is equal to 60, which is the number of samples per
bit.
5.3 Simulation Results in AWGN Channel
Let’s first consider one case where there are 8 users in the system transmitting CDMA
signals from different directions. The DOAs of the signals are equally spaced between −70◦
Table 5.1: Signal Parameters of 8 Users Transmitting Signals from Different Directions
User # Power (dBW) DOA (deg)
1 0 -70
2 0 -47
3 0 -24
4 0 -1
5 0 22
6 0 45
7 0 68
8 0 90
Eb/N0 = 20 dB
and 90◦. Figure 5.1 shows the distribution of these 8 users. We assume that there is no
multipath and the radio channel only introduces the additive white Gaussian noise (AWGN).
We also assume perfect power control in the base station, so all the signals impinging on
the array have the same power. The input signal to noise ratio per bit (Eb/N0) is set to 20
dB. Table 5.1 shows the signal parameters of all the user’s signal.
74
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
*
*
User 1
-700
User 2
*User 8
230
*User 3
230
230 230
230
220
230
*User 4
*User 5
*User 6
*User 7
Figure 5.1: Illustration of eight users with DOAs equally spaced between −70◦ and 90◦.
5.3.1 General Result
Figures 5.2 and 5.3 show the beampatterns of all the 8 users generated by using the
LS-DRMTCMA with aPN/aCM = 2. From Figures 5.2 and 5.3 we see that except for
the signals with DOAs near the endfire of the array, most of the signals can be extracted
by nulling out all the other interference. Due to the existence of noise, the nulls are not
constructed perfectly, but the desired user could still have a gain about 20 dB over the
interference as shown in Figures 5.2 and 5.3. For the signals with DOAs near the endfire
of the array (e. g., signals of user 1 and user 8), since the beamwidth of the beam near
the endfire is wider than that of the beam steered to other direction, two or more signals
may fall into one main beam depending on the angle separation of the signals. In such a
case, although the interference is not rejected completely due to the wide beamwidth of the
main beam directed to the endfire, most of the interference coming from other directions is
rejected, thus the overall interference level is reduced. The reason for the wider beamwidth
75
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
−100 −50 0 50 100−60
−50
−40
−30
−20
−10
0
DOA (degrees)
Bea
mpa
ttern
Gai
n (d
B)
User 1
−100 −50 0 50 100−60
−50
−40
−30
−20
−10
0
DOA (degrees)
Bea
mpa
ttern
Gai
n (d
B)
User 2
−100 −50 0 50 100−60
−50
−40
−30
−20
−10
0
DOA (degrees)
Bea
mpa
ttern
Gai
n (d
B)
User 3
−100 −50 0 50 100−60
−50
−40
−30
−20
−10
0
DOA (degrees)B
eam
patte
rn G
ain
(dB
)
User 4
Figure 5.2: Beampatterns corresponding to different users generated by using LS-DRMTCMA. ∗ denotes user in the system. The signal parameters are shown in Table5.1. This plot is generated by Programs [P4] and [P5].
−100 −50 0 50 100−60
−50
−40
−30
−20
−10
0
DOA (degrees)
Bea
mpa
ttern
Gai
n (d
B)
User 5
−100 −50 0 50 100−60
−50
−40
−30
−20
−10
0
DOA (degrees)
Bea
mpa
ttern
Gai
n (d
B)
User 6
−100 −50 0 50 100−60
−50
−40
−30
−20
−10
0
DOA (degrees)
Bea
mpa
ttern
Gai
n (d
B)
User 7
−100 −50 0 50 100−60
−50
−40
−30
−20
−10
0
DOA (degrees)
Bea
mpa
ttern
Gai
n (d
B)
User 8
Figure 5.3: Beampatterns corresponding to different users generated by using LS-DRMTCMA (cntd.). This plot is generated by Programs [P4] and [P5].
76
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
along the endfire of the array can be explained as follows. As shown in Chapter 2, the
sine of the DOA θ corresponds to the temporal frequency f of a FIR filter input, and
the beampattern of an adaptive array can be viewed as the counterpart of the magnitude
response of an FIR filter. So the passband-width ∆f of a FIR filter corresponds to the term
sin θ − sin(θ −∆θ) in the array, where ∆θ is the beamwidth of the main beam. If we want
to keep sin θ − sin(θ −∆θ) constant for all θ (like keeping the passband-width of the FIR
filter constant over all the frequency band), we have
sin θ − sin(θ −∆θ) = ∆f. (5.1)
The term in the left side of equation (5.1) can be expressed as
sin θ − sin(θ −∆θ) = ∆θ cos θ′
(5.2)
where θ′
is a value in [θ −∆θ, θ]. Substituting equation (5.2) into (5.1) we obtain
∆θ =∆f
cos θ′. (5.3)
From equation (5.3) we see that for a fixed ∆f , ∆θ will change with θ. For DOA near the
endfire of the array, θ, and therefore θ′, are close to π/2, so for a fixed passband-width,
∆θ will be large since cos θ′
is close to 0. On the other hand, for DOA near the broadside
of the array, θ′
is close to 0, and ∆θ will be small since cos θ′
is close to 1. Therefore the
beamformer can construct a narrow beam directed to the broadside of the array and a wide
beam directed to the endfire of the array.
The signal constellations of user 5 before and after the beamformer processing are shown
in Figure 5.4 and Figure 5.5, respectively. In Figure 5.5, we have compensated the random
phase effect by multiplying the beamformer output of user 5 by the complex conjugate of
exp{jφ5}, where φ5 is the random phase of user 5. Comparing Figure 5.4 and Figure 5.5, we
see that the interference from different DOAs is indeed rejected and the signal constellation
is reconstructed. Figures 5.6, 5.7, and 5.8 show the original signal waveform, the corrupted
77
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
−8 −6 −4 −2 0 2 4 6 8−8
−6
−4
−2
0
2
4
6
8
Real Part
Imag
Par
t
Figure 5.4: Signal constellation of user 5 before the beamformer processing. The signalparameters are shown in Table 5.1. This plot is generated by Programs [P4] and [P6].
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Real Part
Imag
Par
t
Figure 5.5: Signal constellation of user 5 after the beamformer processing. The signalparameters are shown in Table 5.1. This plot is generated by Programs [P4] and [P6].
78
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
signal waveform, and the reconstructed signal waveform of user 5 over three bit-periods,
respectively. Comparing Figures 5.6, 5.7 and 5.8, we see that the signal is reconstructed in
the beamformer output. Note that the signal shown in Figure 5.8 is the undespread CDMA
signal which will be fed into the matched filter to make a decision on the transmitted data
bit. The rejection of the interference will result in a low BER.
Figure 5.9 shows the convergence curve of the MT-LSDD in port 6 of the beamformer.
The CM criterion shown in Figure 5.9 is defined in equation (3.28). From Figure 5.9 we
see that due to the GSO procedure, after 1000 iterations, the weight vector in port 6 is
replaced by the GSO output and the CM criterion rises sharply but then decreases as the
iteration goes on. For the MT-SDDD, it will take about 8000 iterations (samples) for all
the weight vectors in the 8 ports to converge. The LS-DRMTCMA, on the other hand,
does not need to perform the GSO procedure and therefore can eliminate the sharp rise in
the convergence curve. Figure 5.10 shows the least-squares mean squared error (LS MSE)
defined in equation (4.41) vs. the iteration number in port 6 of the beamformer for the LS-
DRMTCMA which is described in equations (4.46)-(4.51). From Figure 5.10 we see that the
algorithm converges after 4 to 5 iterations. Since in the LS-DRMTCMA and LS-DRMTA,
the data block size in each iteration is equal to the number of data samples per bit, which
is equal to 60 in the simulation, the number of samples required for the LS-DRMTCMA
to converge is 240 to 300, which is far less than the number of samples required for the
MT-SDDD to converge.
5.3.2 BER Performance for AWGN Channel
In this section, we will compare the BER performance of different algorithms under
different conditions. We will consider two Eb/N0 cases, Eb/N0 = 8 dB, and Eb/N0 = 4
dB. For each Eb/N0 case, we will also consider two different DOA distribution cases. One
case is the non-crowded case with all the DOAs of the signals equally spaced between −70◦
and 90◦. Another one is the crowded case with all the DOAs equally spaced between 0◦ to
79
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
20 40 60 80 100 120 140 160 180
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
Sample Number
Am
plitu
de
Figure 5.6: Original signal waveform of user 5. The signal parameters are shown in Table5.1. This plot is generated by Programs [P4] and [P7].
20 40 60 80 100 120 140 160 180
−4
−3
−2
−1
0
1
2
3
4
Sample Number
Am
plitu
de
Figure 5.7: Corrupted signal waveform of user 5. The signal parameters are shown in Table5.1. This plot is generated by Programs [P4] and [P7].
80
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
20 40 60 80 100 120 140 160 180−1.5
−1
−0.5
0
0.5
1
1.5
Sample Number
Am
plitu
de
Figure 5.8: Reconstructed signal waveform of user 5. The signal parameters are shown inTable 5.1. This plot is generated by Programs [P4] and [P7].
0 1000 2000 3000 4000 5000 6000 7000 80000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Sample Number
CM
Crit
erio
n
Figure 5.9: Convergence curve for MT-SDDD in port 6 of the beamformer. The signalparameters are shown in Table 5.1. This plot is generated by Programs [P4] and [P8].
81
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
90◦. The DOA distribution of all the users for both the non-crowded and crowded cases is
illustrated in Figure 5.11. In the following simulation cases, we assume perfect power con-
trol, so all the signals impinging on the array have the same power unless specified otherwise.
Figure 5.12 shows the BER performance of different algorithms for Eb/N0 = 8 dB,
non-crowded DOA case. In Figure 5.12, to generate the BER for the different adaptive
algorithms, we transmit different random bit streams for each user. In the receiver, we use
the beamformer adapted by each algorithm to extract each user’s signal and then feed the
output of the beamformer into the matched filter to despread the signal and estimate the
transmitted data bit. The estimates of the data bits are compared to the original transmit-
ted data bits and the BER of each user is calculated. The BER averaged over all the users
is then calculated and used in the BER performance plot.
From Figure 5.12 we see that the MT-LSCMA cannot work under the low Eb/N0 con-
dition, and its BER performance is very close to that of the conventional receiver. For
the MT-SDDD, when the system is under-loaded (with the number of users less than the
number of antenna elements of the array), the BER increases sharply as the number of
users increases. This is because the MT-SDDD uses the estimate of the data sample as the
desired signal to adapt the weight vectors of the beamformer. When the number of users
increases, the error rate of the data sample estimate becomes larger, and the algorithm
cannot adapt the weight vectors correctly, therefore the improvement due to the spatial fil-
tering becomes smaller and the BER increases. However, comparing the BER performance
of the MT-SDDD with that of the conventional receiver, we see that a large improvement
can still be achieved when the system is under-loaded. When the system is fully-loaded
(with the number of users equal to the number of antenna elements of the array) or over-
loaded (with the number of users greater than the number of antenna elements of the array),
the BER changes smoothly as the number of users increases, and the improvement of the
BER performance over that of the conventional receiver is small. This is because under
82
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
1 1.5 2 2.5 3 3.5 4 4.5 50
0.5
1
1.5
2
2.5
3
3.5
4
Iteration Number
LS
MS
E
Figure 5.10: Convergence curve for LS-DRMTCMA in port 6 of the beamformer. The LS-DRMTCMA is described in equations (4.46)-(4.51). The data block size in each iterationis equal to 60. The signal parameters are shown in Table 5.1. This plot is generated byPrograms [P4] and [P9].
*
*User 1
-700
User 2
*User p
Non-crowded DOA Case
* *User 1 User 2
*User p
Crowded DOA Case
Figure 5.11: DOA distribution of all the users for both the non-crowded and crowded cases.
83
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
4 5 6 7 8 9 10 11 12 13 1410
−6
10−5
10−4
10−3
10−2
10−1
Number of Users
BE
R
Conv. receiver
MT−LSCMA
MT−SDDD
LS−DRMTA
LS−DRMTCMA
Figure 5.12: BER performance of different adaptive algorithms. In this case, Eb/N0 = 8dB, the DOAs of all the users are equally spaced between −70◦ and 90◦. The ratio of thecoefficients aPN/aCM used in the LS-DRMTCMA is set to 2. This plot is generated byPrograms [P10] and [P12].
4 5 6 7 8 9 10 11 12 13 1410
−6
10−5
10−4
10−3
10−2
10−1
Number of Users
BE
R
Conv. receiver
MT−LSCMA
MT−SDDD
LS−DRMTCMA
LS−DRMTA
Figure 5.13: BER performance of different adaptive algorithms. In this case, Eb/N0 = 8dB, the DOAs of all the users are equally spaced between 0◦ and 90◦. The ratio of thecoefficients aPN/aCM used in the LS-DRMTCMA is set to 2. This plot is generated byPrograms [P10] and [P12].
84
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
the fully-loaded and over-loaded situations, the MT-SDDD cannot form deep nulls in the
DOAs of the interference. Also, several signals may fall into a main beam of one output
port, therefore the interference level cannot be reduced to a very low point, and the BER
performance is thus close to that of the conventional receiver.
For LS-DRMTA and LS-DRMTCMA, however, since these two algorithms utilize the
information of the PN sequences of all the users to adapt the weight vectors, they can
constructed deeper nulls in the DOAs of the interference than the MT-SDDD, therefore
the BER of these two algorithms is much lower than that of the MT-SDDD. Figures 5.14
and 5.15 show the beampatterns of user 5 generated by using the LS-DRMTCMA and the
MT-SDDD algorithm, respectively. Comparing Figures 5.14 and 5.15 we see that the LS-
DRMTCMA can generate deeper null in the DOAs of the interference than the MT-SDDD,
therefore can reduce the interference to a lower level. Also, since the LS-DRMTCMA uses
the constant modulus property of the transmitted signal in addition to the PN sequences of
all the users to adapt the weight vectors, it can achieve a lower BER than the LS-DRMTA.
However, the improvement of the LS-DRMTCMA over the LS-DRMTA becomes smaller
when the system is over-loaded. This is because under the over-loaded situation, it is the
interference falling into the main beam of each output port that dominates the overall
interference level. Although the LS-DRMTCMA can form deeper nulls in some DOAs of
the interference than the LS-DRMTA, once there is some interference falling into the main
beam, the overall interference levels of these two algorithms become almost the same, and
thus the BER performance is very close. It is the beamwidth of the main beam and the
DOA distribution of the signals that determines the number of interference falling into
the main beam. For an 8-element uniform linear array with element spacing equal to half
wavelength, from equation (2.47), we obtain the peak-to-null beamwidth of the main beam
directed to the broadside of the array
θH = arcsin
(λ
Md
)= arcsin
(λ
8λ2
)≈ 14.5◦. (5.4)
85
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
−100 −80 −60 −40 −20 0 20 40 60 80 100−50
−45
−40
−35
−30
−25
−20
−15
−10
−5
0
DOA (degrees)
Bea
mpa
ttern
Gai
n (d
B)
Figure 5.14: Beampattern of user 5 generated by using LS-DRMTCMA. In this case,Eb/N0 = 8 dB, the number of users is equal to 8, the DOAs of all the users are equallyspaced between −70◦ and 90◦. The ratio of the coefficients aPN/aCM used in the LS-DRMTCMA is set to 2. This plot is generated by Programs [P4] and [P13].
−100 −80 −60 −40 −20 0 20 40 60 80 100−50
−45
−40
−35
−30
−25
−20
−15
−10
−5
0
DOA (degrees)
Bea
mpa
ttern
Gai
n (d
B)
Figure 5.15: Beampattern of user 5 generated by using MT-SDDD. In this case, Eb/N0 = 8dB, the number of users is equal to 8, the DOAs of all the users are equally spaced between−70◦ and 90◦. The ratio of the coefficients aPN/aCM used in the LS-DRMTCMA is set to2. This plot is generated by Programs [P4] and [P13].
86
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
Since as shown in section 5.3.1, the beamwidth of the main beam directed to the broadside
of the array is always smaller than those of the main beams directed to other directions, if
the angle separation between the DOAs of the received signals is less than 14.5◦, at least
one interference will fall into the main beam of the desired user.
The BER performance of different algorithms for the Eb/N0 = 8 dB, crowded DOA case
is shown in Figure 5.13. From Figure 5.13 we see that the MT-LSCMA still cannot work
under this crowded DOA situation. It was found from experiments that the MT-LSCMA
can only work for high Eb/N0 (e. g., Eb/N0 = 20 dB) and far under-loaded (e. g., number
of users equal to 4) case. Comparing Figure 5.13 with Figure 5.12, we see that the BER
increases as the DOAs of the signals becomes crowded for almost all the test cases. This is
because more and more interference can fall into the main beam of one output port if the
DOAs of the signals becomes crowded. For example, if there exist 8 users in the system, in
the non-crowded DOA case, the angle separation between the DOAs of the received signals
is equal to 23◦, which is larger than θH = 14.5◦, therefore no interference will fall into
the main beam of the desired user, at least for those close to the broadside of the array.
However, in the crowded DOA case, the angle separation becomes 12.86◦, which is smaller
than θH = 14.5◦, thus even for the desired user close to the broadside of the array, two
interference will fall into the main beam of the desired user. Figures 5.16 and 5.17 show the
beampatterns of user 4 generated by using LS-DRMTCMA for both the non-crowded and
crowded case, respectively. Comparing Figures 5.16 and 5.17, we see that in the crowded
case, two more interference fall into the main beam of user 4, thus the interference level
increases and the BER becomes higher. From Figures 5.12 and 5.13, we see that when the
number of users is equal to 8, the BER for LS-DRMTCMA in the non-crowded DOA case
is about 4 × 10−6 while the BER for LS-DRMTCMA in the crowded DOA case becomes
approximately 4 × 10−5, which is 10 times of that in the non-crowded DOA case. In this
situation, however, the LS-DRMTCMA and LS-DRMTA can still achieve a large improve-
ment over the BER performance of the MT-SDDD.
87
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
−100 −80 −60 −40 −20 0 20 40 60 80 100−50
−45
−40
−35
−30
−25
−20
−15
−10
−5
0
DOA (degrees)
Bea
mpa
ttern
Gai
n (d
B)
Figure 5.16: Beampattern of user 4 generated by using LS-DRMTCMA. In this case,Eb/N0 = 8 dB, the number of users is equal to 8, the DOAs of all the users are equallyspaced between −70◦ and 90◦. The ratio of the coefficients aPN/aCM used in the LS-DRMTCMA is set to 2. This plot is generated by Programs [P4] and [P13].
−100 −80 −60 −40 −20 0 20 40 60 80 100−50
−45
−40
−35
−30
−25
−20
−15
−10
−5
0
DOA (degrees)
Bea
mpa
ttern
Gai
n (d
B)
Figure 5.17: Beampattern of user 4 generated by using LS-DRMTCMA. In this case,Eb/N0 = 8 dB, the number of users is equal to 8, the DOAs of all the users are equally spacedbetween 0◦ and 90◦. The ratio of the coefficients aPN/aCM used in the LS-DRMTCMA isset to 2. This plot is generated by Programs [P4] and [P13].
88
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
The BER performance of different algorithms for the Eb/N0 = 4 dB, non-crowded and
crowded DOA cases are shown in Figure 5.18 and Figure 5.19, respectively. Comparing
Figures 5.18 and 5.19 with Figures 5.12 and 5.13, we see that the BER increases as the
Eb/N0 decreases. From Figures 5.18 and 5.19, we see that in the low Eb/N0 case, the LS-
DRMTCMA and LS-DRMTA can still outperform the MT-SDDD for both non-crowded
and crowded DOA cases. Comparing the BER curves of LS-DRMTCMA and LS-DRMTA
in Figures 5.18 and 5.19, we see that the improvement of LS-DRMTCMA over LS-DRMTA
does not degrade very much as the number of users increases, which is different from that in
the Eb/N0 = 8 dB case. This is because for such a low Eb/N0, in addition to the interference,
the noise also plays an important role in determining the BER. Since the difference between
the abilities of the LS-DRMTCMA and LS-DRMTA to reduce the noise changes little as
the number of users increase, the improvement of LS-DRMTCMA over LS-DRMTA does
not degrade very much as the number of users increases. One point to note is that when
the number of users is equal to 4, the BER of the crowded DOA case is less than that of
the non-crowded DOA case. The reason is that when the number of users is equal to 4, the
angle separation between the users is large for both the crowded and non-crowded DOA
cases, so there is no interference falling into the main beam near the broadside of the array,
and only the users near the endfire of the array can affect each other. The beampattern of
user 4 which is near the endfire of the array for both the non-crowded and crowded cases are
illustrated in Figures 5.20 and 5.21, respectively. For the crowded DOA case, the DOAs of
the signals are equally spaced between 0◦ and 90◦, so there is only one user near the endfire
of the array. For the non-crowded case, on the other hand, the DOAs of the signals are
equally spaced between −70◦ and 90◦, thus there are two users near the endfire of the array
and they interference with each other, which results in a higher BER than in the crowded
DOA case.
89
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
4 5 6 7 8 9 10 11 12 13 1410
−6
10−5
10−4
10−3
10−2
10−1
Number of Users
BE
R
Conv. receiver
MT−LSCMA
MT−SDDD
LS−DRMTA
LS−DRMTCMA
Figure 5.18: BER performance of different adaptive algorithms. In this case, Eb/N0 = 4dB, the DOAs of all the users are equally spaced between −70◦ and 90◦. The ratio of thecoefficients aPN/aCM used in the LS-DRMTCMA is set to 2. This plot is generated byPrograms [P10] and [P12].
4 5 6 7 8 9 10 11 12 13 1410
−6
10−5
10−4
10−3
10−2
10−1
Number of Users
BE
R
Conv. receiver
MT−LSCMA
MT−SDDD
LS−DRMTA
LS−DRMTCMA
Figure 5.19: BER performance of different adaptive algorithms. In this case, Eb/N0 = 4dB, the DOAs of all the users are equally spaced between 0◦ and 90◦. The ratio of thecoefficients aPN/aCM used in the LS-DRMTCMA is set to 2. This plot is generated byPrograms [P10] and [P12].
90
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
−100 −80 −60 −40 −20 0 20 40 60 80 100−50
−45
−40
−35
−30
−25
−20
−15
−10
−5
0
DOA (degrees)
Bea
mpa
ttern
Gai
n (d
B)
Figure 5.20: Beampattern of user 4 generated by using LS-DRMTCMA in the non-crowdedDOA case. In this case, Eb/N0 = 4 dB, the number of users is equal to 4, the DOAs of allthe users are equally spaced between −70◦ and 90◦. The ratio of the coefficients aPN/aCMused in the LS-DRMTCMA is set to 2. This plot is generated by Programs [P4] and [P13].
−100 −80 −60 −40 −20 0 20 40 60 80 100−50
−45
−40
−35
−30
−25
−20
−15
−10
−5
0
DOA (degrees)
Bea
mpa
ttern
Gai
n (d
B)
Figure 5.21: Beampattern of user 4 generated by using LS-DRMTCMA in the crowdedDOA case. In this case, Eb/N0 = 4 dB, the number of users is equal to 4, the DOAs of allthe users are equally spaced between 0◦ and 90◦. The ratio of the coefficients aPN/aCMused in the LS-DRMTCMA is set to 2. This plot is generated by Programs [P4] and [P13].
91
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
5.4 BER Performance in Timing Offset Case
In a CDMA system, due to the low SNR and high interference level, the synchronization
of the PN sequence in the receiver cannot be performed perfectly, therefore a timing offset
To exists between the local generated PN sequence and the received PN sequence. This
timing offset will degrade the performance of the system. In this section, we will examine
the BER performance of all the algorithms under different timing offset conditions. We will
consider two timing offset cases, To = 0.25Tc, and To = 0.5Tc, where Tc is the chip period
of the CDMA signal. For each timing offset case, we will also consider two different DOA
distribution cases, the non-crowded DOA case, and the crowded DOA case.
The BER performance of different algorithms for To = 0.25Tc in both non-crowded
and crowded DOA cases are shown in Figures 5.22 and 5.23, respectively. Comparing Fig-
ures 5.22 and 5.23 with Figures 5.12 and 5.13, we see that the BER performance of all the
algorithms in both the non-crowded and crowded DOA cases is degraded due to the timing
offset between the local generated PN sequence and the received PN sequence. However,
comparing the BER curves of LS-DRMTCMA and LS-DRMTA with that of the MT-SDDD,
we see that a big improvement can still be achieved by using the LS-DRMTCMA and LS-
DRMTA in this timing offset case. Also, we see that the LS-DRMTCMA outperforms
the LS-DRMTA. Comparing Figure 5.22 with Figure 5.23, we note that the BER in the
crowded DOA case is higher than that in the non-crowded DOA case for almost all the test
cases, except for the case where the number of users is equal to 4. The reason for this has
been explained in section 5.3.2.
The BER performance of different algorithms for To = 0.5Tc in both non-crowded and
crowded DOA cases is shown in Figures 5.24 and 5.25, respectively. From Figures 5.24 and
5.25 we can see that the improvement of LS-DRMTCMA and LS-DRMTA over the MT-
SDDD becomes very small. The reason is that the BER in this case is now mainly caused
92
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
4 5 6 7 8 9 10 11 12 13 1410
−6
10−5
10−4
10−3
10−2
10−1
100
Number of Users
BE
R
Conv. receiver
MT−LSCMA
MT−SDDD
LS−DRMTA
LS−DRMTCMA
Figure 5.22: BER performance of different adaptive algorithms in timing offset case. Inthis case, Eb/N0 = 8 dB, To = 0.25Tc, the DOAs of all the users are equally spaced between−70◦ and 90◦. The ratio of the coefficients aPN/aCM used in the LS-DRMTCMA is set to2. This plot is generated by Programs [P14] and [P15].
4 5 6 7 8 9 10 11 12 13 1410
−6
10−5
10−4
10−3
10−2
10−1
100
Number of Users
BE
R
Conv. receiver
MT−SDDD
LS−DRMTA
LS−DRMTCMA
MT−LSCMA
Figure 5.23: BER performance of different adaptive algorithms in timing offset case. Inthis case, Eb/N0 = 8 dB, To = 0.25Tc, the DOAs of all the users are equally spaced between0◦ and 90◦. The ratio of the coefficients aPN/aCM used in the LS-DRMTCMA is set to 2.This plot is generated by Programs [P14] and [P15].
93
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
by the timing offset, and the spatial filtering has little effect on the BER performance.
The BER vs. the timing offset for both the non-crowded and crowded DOA cases are
shown in Figures 5.26 and 5.27, respectively. In Figures 5.26 and 5.27, the number of users is
equal to 10, and the Eb/N0 is equal to 8 dB. From Figures 5.26 and 5.27 we see that the BER
of all the algorithms rises as the timing offset increases. When To ≤ 0.3Tc, the improvement
of LS-DRMTCMA and LS-DRMTA over other algorithms is large, but this improvement
decreases as the timing offset increases and the BER is dominated by the timing offset.
Therefore, if we can use the conventional receiver to perform the synchronization of the PN
sequence and reduce the timing offset to 0.3Tc, we can then use the beamformer adapted
by the LS-DRMTCMA to extract the signal, and the output of the beamformer can again
be used in the synchronization. Since the beamformer has rejected most of the interference,
the synchronization can be performed more accurately using the beamformer output, and
the smaller timing offset in turn can result in more improvement due to spatial filtering.
5.5 BER Performance in Frequency Offset Case
In a CDMA system, a frequency offset Fo may exist between the local oscillator output
and the carrier of the signals impinging on the array due to the variation of the electronic
components in the local oscillator. This frequency offset will cause a time-varying phase
shift between the base-band signal in the receiver and that in the transmitter. If this
frequency offset cannot be compensated in the receiver, the BER performance will be de-
graded. In this section, we will examine the BER performance of all the algorithms under
different frequency offset conditions. We will consider two frequency offset cases, Fo = 100
Hz, and Fo = 500 Hz. For each frequency offset case, we will also consider two different
DOA distribution cases, the non-crowded DOA case, and the crowded DOA case.
Since a phase shift in the received signal will cause a rotation of the received signal
94
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
4 5 6 7 8 9 10 11 12 13 1410
−3
10−2
10−1
100
Number of Users
BE
R
Conv. receiver
MT−LSCMA
MT−SDDD
LS−DRMTA
LS−DRMTCMA
Figure 5.24: BER performance of different adaptive algorithms in timing offset case. Inthis case, Eb/N0 = 8 dB, To = 0.5Tc, the DOAs of all the users are equally spaced between−70◦ and 90◦. The ratio of the coefficients aPN/aCM used in the LS-DRMTCMA is set to2. This plot is generated by Programs [P14] and [P16].
4 5 6 7 8 9 10 11 12 13 1410
−3
10−2
10−1
100
Number of Users
BE
R
Conv. receiver
MT−LSCMA
MT−SDDD
LS−DRMTA
LS−DRMTCMA
Figure 5.25: BER performance of different adaptive algorithms in timing offset case. Inthis case, Eb/N0 = 8 dB, To = 0.5Tc, the DOAs of all the users are equally spaced between0◦ and 90◦. The ratio of the coefficients aPN/aCM used in the LS-DRMTCMA is set to 2.This plot is generated by Programs [P14] and [P16].
95
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
−4
10−3
10−2
10−1
100
Timing Offset (Tc)
BE
R
Conv. receiver
MT−LSCMA
MT−SDDD
LS−DRMTA
LS−DRMTCMA
Figure 5.26: BER vs. timing offset for different adaptive algorithms. In this case, Eb/N0 = 8dB, the number of users is equal to 10, the DOAs of all the users are equally spaced between−70◦ and 90◦. The ratio of the coefficients aPN/aCM used in the LS-DRMTCMA is set to2. This plot is generated by Programs [P14] and [P17].
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
−4
10−3
10−2
10−1
100
Timing Offset (Tc)
BE
R
Conv. receiver
MT−LSCMA
LS−DRMTA
LS−DRMTCMA
MT−SDDD
Figure 5.27: BER vs. timing offset for different adaptive algorithms. In this case, Eb/N0 = 8dB, the number of users is equal to 10, the DOAs of all the users are equally spaced between0◦ and 90◦. The ratio of the coefficients aPN/aCM used in the LS-DRMTCMA is set to 2.This plot is generated by Programs [P14] and [P17].
96
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
constellation, the frequency offset will make the received signal constellation rotate along
the unit circle with a speed related to the value of the frequency offset. A large frequency
offset will cause a high rotation speed and a small frequency offset will result in a low
rotation speed. In the receiver, the adaptive algorithm tries to learn the information of
the phase shift and restore the rotated signal constellation to its original one. Actually,
the adaptive algorithm always restores the rotated signal constellation to its closest one.
Therefore, for a BPSK signal, if the algorithm can generate a new weight vector before the
phase shift exceeds π/2, i. e., before the signal constellation passes through the imaginary
axis, the phase shift information can be learned correctly and the signal constellation can be
restored to the right one. Otherwise, the algorithm will make a wrong estimate of the phase
shift and the signal constellation is restored to the wrong one. Let φa denote the maximum
phase shift due to the frequency offset between two consecutive adaptation period then φa
should satisfy
|φa| <π
2. (5.5)
The choice of φa will affect the BER performance of the system and the computational
complexity of the adaptation of the weight vector. We will discuss this point in this section.
The BER performance of different algorithms for Fo = 100 Hz in both the non-crowded
and crowded DOA cases are shown in Figures 5.28 and 5.29, respectively. In Figures 5.28
and 5.29, φa is set to 0.2π, and only the BER performance of the MT-LSCMA, the LS-
DRMTA, and the LS-DRMTCMA is illustrated. The MT-LSCMA and the MT-SDDD,
due to the GSO procedure, lose the phase shift information in their adaptation procedure,
therefore cannot compensate the phase shift caused by the frequency offset. Their BER
performance is similar and thus only the BER performance of the MT-LSCMA is shown
in these figures. From Figures 5.28 and 5.29 we can see that the MT-LSCMA totally fails
in the frequency offset case while the LS-DRMTA and the LS-DRMTCMA can still work
very well, and the LS-DRMTCMA also outperforms the LS-DRMTA in the frequency off-
set case. Comparing Figures 5.28 and 5.29 with Figures 5.12 and 5.13, we note that the
97
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
4 5 6 7 8 9 10 11 12 13 1410
−6
10−5
10−4
10−3
10−2
10−1
100
Number of Users
BE
R
MT−LSCMA
LS−DRMTA
LS−DRMTCMA
Figure 5.28: BER performance of different adaptive algorithms in frequency offset case. Inthis case, Eb/N0 = 8 dB, Fo = 100 Hz, the DOAs of all the users are equally spaced between−70◦ and 90◦. The ratio of the coefficients aPN/aCM used in the LS-DRMTCMA is set to2. The maximum phase shift φa between two consecutive adaptation period is set to 0.2π.This plot is generated by Programs [P18] and [P19].
4 5 6 7 8 9 10 11 12 13 1410
−6
10−5
10−4
10−3
10−2
10−1
100
Number of Users
BE
R
MT−LSCMA
LS−DRMTA
LS−DRMTCMA
Figure 5.29: BER performance of different adaptive algorithms in frequency offset case. Inthis case, Eb/N0 = 8 dB, Fo = 100 Hz, the DOAs of all the users are equally spaced between0◦ and 90◦. The ratio of the coefficients aPN/aCM used in the LS-DRMTCMA is set to2. The maximum phase shift φa between two consecutive adaptation period is set to 0.2π.This plot is generated by Programs [P18] and [P19].
98
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
BER performance of all the algorithms in both the non-crowded and crowded DOA cases is
degraded due to the frequency offset. Comparing Figure 5.28 with Figure 5.29, we can see
that the BER in crowded DOA case is higher than that in the non-crowded DOA case for
almost all the test cases, except for the case where the number of users is equal to 4. This
phenomenon is the same as that shown in the timing offset case.
The BER performance of different algorithms for Fo = 500 Hz in both the non-crowded
and crowded DOA cases is illustrated in Figures 5.30 and 5.31, respectively. Comparing
Figures 5.30 and 5.31 with Figures 5.28 and 5.29, we see that the BERs of both algorithms
change little as the frequency offset increases. Figure 5.32 shows the BER vs. frequency
offset for all the algorithm with the number of users equal to 10. In Figure 5.32, the DOAs
of all the users are equally spaced between −60◦ and 60◦, and φa is set to 0.2π. From
Figure 5.32 we see that for LS-DRMTA and LS-DRMTCMA, the BER performance is al-
most independent of the frequency offset, which indicates that the phase shift caused by
the frequency offset is indeed compensated by the algorithms.
Figure 5.33 shows the BER vs. frequency offset for LS-DRMTA and LS-DRMTCMA
with different φa. From Figure 5.33 we see that by reducing φa, we can get a lower BER.
However, since a smaller φa means a short adaptation period, this small improvement is
paid by a higher computational complexity of the adaptation of the weight vectors. There
is a trade-off between the BER performance and φa (i. e., the computational complexity
of the adaptation). Figure 5.34 shows the BER vs. φa for different frequency offset using
LS-DRMTA. The same plot for LS-DRMTCMA is illustrated in Figure 5.35. From Fig-
ures 5.34 and 5.35 we can see that when φa ≤ 0.2π, the BER curves are flat, and increasing
φa only result in a small degradation of the BER performance. However, once φa > 0.2π,
increasing φa will cause larger degradation of the BER performance. Therefore, 0.2π is the
optimum value for φa.
99
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
4 5 6 7 8 9 10 11 12 13 1410
−6
10−5
10−4
10−3
10−2
10−1
100
Number of Users
BE
R
MT−LSCMA
LS−DRMTA
LS−DRMTCMA
Figure 5.30: BER performance of different adaptive algorithms in frequency offset case. Inthis case, Eb/N0 = 8 dB, Fo = 500 Hz, the DOAs of all the users are equally spaced between−70◦ and 90◦. The ratio of the coefficients aPN/aCM used in the LS-DRMTCMA is set to2. The maximum phase shift φa between two consecutive adaptation period is set to 0.2π.This plot is generated by Programs [P18] and [P19].
4 5 6 7 8 9 10 11 12 13 1410
−6
10−5
10−4
10−3
10−2
10−1
100
Number of Users
BE
R
MT−LSCMA
LS−DRMTA
LS−DRMTCMA
Figure 5.31: BER performance of different adaptive algorithms in frequency offset case. Inthis case, Eb/N0 = 8 dB, Fo = 500 Hz, the DOAs of all the users are equally spaced between0◦ and 90◦. The ratio of the coefficients aPN/aCM used in the LS-DRMTCMA is set to2. The maximum phase shift φa between two consecutive adaptation period is set to 0.2π.This plot is generated by Programs [P18] and [P19].
100
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
0 50 100 150 200 250 300 350 400 450 50010
−6
10−5
10−4
10−3
10−2
10−1
100
Frequency Offset (Hz)
BE
R
MT−LSCMA
MT−SDDD
LS−DRMTA
LS−DRMTCMA
Figure 5.32: BER vs. frequency offset for different adaptive algorithms. In this case,Eb/N0 = 8 dB, the number of users is equal to 10, the DOAs of all the users are equallyspaced between −60◦ and 60◦. The ratio of the coefficients aPN/aCM used in the LS-DRMTCMA is set to 2. The maximum phase shift φa between two consecutive adaptationperiod is set to 0.2π. This plot is generated by Programs [P18] and [P20].
0 50 100 150 200 250 300 350 400 450 50010
−6
10−5
10−4
10−3
10−2
10−1
100
Frequency Offset (Hz)
BE
R
LS−DRMTA, max. phase shift = 0.2*pi
LS−DRMTCMA, max. phase shift = 0.2*pi
LS−DRMTA, max. phase shift = 0.05*pi
LS−DRMTCMA, max. phase shift = 0.05*pi
Figure 5.33: BER vs. frequency offset for different adaptive algorithms with different φa.In this case, Eb/N0 = 8 dB, the number of users is equal to 10, the DOAs of all the usersare equally spaced between −60◦ and 60◦. The ratio of the coefficients aPN/aCM used inthe LS-DRMTCMA is set to 2. This plot is generated by Programs [P18] and [P21].
101
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.4510
−6
10−5
10−4
10−3
10−2
10−1
100
Maximum Phase−shift (pi)
BE
R
Freq. offset = 100 HzFreq. offset = 500 Hz
Figure 5.34: BER vs. φa for different frequency offset using LS-DRMTA. In this case,Eb/N0 = 8 dB, the number of users is equal to 10, the DOAs of all the users are equallyspaced between −60◦ and 60◦. This plot is generated by Programs [P18] and [P22].
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.4510
−6
10−5
10−4
10−3
10−2
10−1
100
Maximum Phase−shift (pi)
BE
R
Freq. offset = 100 HzFreq. offset = 500 Hz
Figure 5.35: BER vs. φa for different frequency offset using LS-DRMTCMA. In this case,Eb/N0 = 8 dB, the number of users is equal to 10, the DOAs of all the users are equallyspaced between −60◦ and 60◦. The ratio of the coefficients aPN/aCM used in the LS-DRMTCMA is set to 2. This plot is generated by Programs [P18] and [P22].
102
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
In all the above simulation cases for LS-DRMTA and LS-DRMTCMA, we assume that
the random phase of each signal impinging on the array is known and is compensated before
the signal is used in the algorithms. However, the LS-DRMTA and the LS-DRMTCMA
can also work without this phase information. Figure 5.36 shows the BER performance
of different adaptive algorithms for Fo = 100 Hz without phase information. Comparing
Figure 5.36 with Figure 5.28, we see that without the phase information, the LS-DRMTA
and LS-DRMTCMA can still achieve almost the same BER performance as that with the
phase information.
4 5 6 7 8 9 10 11 12 13 1410
−6
10−5
10−4
10−3
10−2
10−1
100
Number of Users
BE
R
MT−LSCMA w/o phase inf.
LS−DRMTA w/o phase inf.
LS−DRMTCMA w/o phase inf.
Figure 5.36: BER performance of different adaptive algorithms in frequency offset casewithout phase information. In this case, Eb/N0 = 8 dB, Fo = 100 Hz, the DOAs of all theusers are equally spaced between −70◦ and 90◦. The ratio of the coefficients aPN/aCM usedin the LS-DRMTCMA is set to 2. The maximum phase shift φa between two consecutiveadaptation period is set to 0.2π. This plot is generated by Programs [P18] and [P23].
5.6 BER Performance and Coefficients in LS-DRMTCMA
In the above simulation cases, the ratio of aPN to aCM in the LS-DRMTCMA is always
set to 2. In this section, we will examine the effect of varying aPN/aCM on the BER per-
formance of LS-DRMTCMA.
103
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
Figure 5.37 shows the BER vs. aPN/aCM for the LS-DRMTCMA in different simulation
environments. From Figure 5.37 we see that without timing offset and frequency offset,
the LS-DRMTCMA can always achieve the best BER performance when aPN/aCM = 0.5
no matter what the number of users is and what the Eb/N0 will be. For the case with
timing offset and frequency offset, although the BER performance is not the best when
aPN/aCM = 0.5, the difference between the best BER performance and that obtained at
the point where aPN/aCM = 0.5 is very small. Therefore it is recommended that aPN/aCM
should be set to 0.5 for LS-DRMTCMA to obtain the best BER performance. In Fig-
ure 5.37, the points where aPN/aCM = 100 are actually corresponding to the LS-DRMTA,
and we see that the LS-DRMTCMA with aPN/aCM = 0.5 can achieve a large improvement
over the LS-DRMTA.
Someone may ask if the improvement obtained by varying the term aPN/aCM may have
resulted from the fact that the algorithm may not fully converge when the aPN/aCM is
changed. Figure 5.38 shows the BER vs. aPN/aCM for LS-DRMTCMA with different itera-
tion number. From Figure 5.38 we see that although we have increased the iteration number
from 5 to 8, the BER performance changes little, and the point where aPN/aCM = 0.5 is
still a local minimum of the BER curve. Thus we can conclude that the LS-DRMTCMA
can fully converge with 5 iterations for different aPN/aCM value, and 0.5 is the optimum
value of aPN/aCM for LS-DRMTCMA to achieve the best BER performance.
5.7 BER Performance in Multipath Environment
In a wireless radio channel, the transmitted signal may arrive at the receiver through
different paths with different time delays. These multipath will cause the intersymbol inter-
ference (ISI) and degrade the BER performance of the system. However, if these multipaths
are coming from different DOAs, we can use an adaptive array in the receiver to extract
104
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
10−2
10−1
100
101
102
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
a_PN/a_CM
BE
R
10 users, Eb/No = 8 dB, To = 0, Fo = 0
10 users, Eb/No = 4 dB, To = 0, Fo = 0
8 users, Eb/No = 4 dB, To = 0, Fo = 0
10 users, Eb/No = 8 dB, To = 0.25*Tc, Fo = 0
10 users, Eb/No = 8 dB, To = 0, Fo = 100 Hz
Figure 5.37: BER vs. aPN/aCM for LS-DRMTCMA in different simulation environments.The DOAs of all the users are equally spaced between −70◦ and 90◦. For frequency offsetcase, the maximum phase shift φa between two consecutive adaptation period is set to 0.2π.This plot is generated by Programs [P24] and [P25].
10−2
10−1
100
101
102
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
a_PN/a_CM
BE
R
5 iterations8 iterations
Figure 5.38: BER vs. aPN/aCM for LS-DRMTCMA with different iteration number. Inthis case, Eb/N0 = 8 dB, the number of users is equal to 10, the DOAs of all the users areequally spaced between −70◦ and 90◦. There is no timing offset and frequency offset. Thisplot is generated by Programs [P24] and [P26].
105
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
the path with the strongest power and reject the other ones, therefore reducing the ISI and
improving the BER performance. In this section, we will examine the BER performance of
different algorithms in the multipath environment.
The channel model we use in the simulation is the 2-ray resolvable channel where each
user has two multipaths. The reason for using the 2-ray resolvable channel is that the
bandwidth of the transmitted signal, i. e., the direct sequence spread spectrum signal, is
very high compared to the bandwidth of the wireless radio channel, therefore the multipaths
are resolvable in time. The first issue we need to consider for the channel model is the DOAs
of the multipaths. In the 2-ray channel model, the two multipaths have different DOAs, and
we want to investigate the effect of varying the angle separation between the multipaths
on the performance of different algorithms. In the simulation, we consider two different
multipath angle separation cases. In the first case, the DOA difference between the two
multipaths is set to 10◦ while in the second case, the DOA difference is set to 20◦. As shown
in section 5.3.2, for an 8-element uniform linear array with element spacing equal to half
wavelength, the peak-to-null beamwidth, θH , of the main beam directed to the broadside
of the array is approximately equal to 14.5◦. By setting the multipath angle separation
equal to 10◦ and 20◦, which are less and greater than θH , respectively, we can investigate
the performance of different algorithms under the condition when both multipaths fall into
the main beam and the condition when only one multipath fall into the main beam. The
second issue we need to consider for the channel model is the time delay between the two
multipaths. For different multipath angle separation cases, the time delay between the two
multipaths should be varied . It was shown in [56] and [57] that the time delay between
the multipaths with close DOAs tend to be small and that between the multipaths with
far separated DOAs tend to be large. Since we also want to investigate the performance of
different algorithms under the condition when the time delay between the two multipaths
is less and greater than the chip period Tc, for the first case with small DOA difference, we
set the time delay between the two multipaths to 0.5Tc while for the second case with large
106
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
DOA difference, we set the time delay between the two multipaths to 1.5Tc. The last issue
we need to consider for the channel model is the power ratio between the two multipaths.
As shown in [58], the amplitude of each single multipath varies little over a small range of
distance. Also, since the data bit rate of the system is very high (128kbps), we will consider
the amplitude of each multipath unchanged over the simulation period. In the simulation,
we want to investigate the effect of varying the power ratio between the multipaths on the
performance of different algorithms. So three different power ratios of the first path to the
second path, 0 dB, 6dB, and 10dB, are considered for each multipath angle separation case.
The signal parameters of the multipaths in different simulation cases are shown in Table 5.2.
Table 5.2: Signal Parameters of Multipaths
Case #DOA Difference Time Delay Power Ratio (dB)
Between Multipaths Between Multipaths (First Path/Second Path)
1 10◦ 0.5Tc 0
2 10◦ 0.5Tc 6
3 10◦ 0.5Tc 10
4 20◦ 1.5Tc 0
5 20◦ 1.5Tc 6
6 20◦ 1.5Tc 10
The BER performance of different algorithms for simulation case 1, 2 and 3 are illus-
trated in Figures 5.39, 5.40, and 5.41, respectively. Comparing Figures 5.39, 5.40, and 5.41
with Figure 5.12, we see that the BER performance is indeed degraded by the multipath in
all the cases. Comparing Figures 5.39, 5.40, and 5.41, we note that decreasing the power
ratio between the multipaths will result in a higher BER. This is what we expect since in
these three simulation cases, the angle separation between the multipaths is equal to 10◦,
which is less than θH = 14.5◦, hence both of the multipaths fall into the main beam of
the output port, and a lower power ratio between the multipaths means a higher ISI level,
which will result in a worse BER performance. However, from Figures 5.39, 5.40, and 5.41,
107
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
we see that even for such a small multipath angle separation, using the adaptive array in the
receiver can still reduce the multipath effect and improve the BER performance, although
the improvement becomes smaller when the power ratio between the multipaths decreases.
Also note that the LS-DRMTCMA with aPN/aCM = 0.5 can achieve a large improvement
over that with aPN/aCM = 2 for all the simulation cases. This has confirmed the conclusion
we draw in section 5.6.
The BER performance of different algorithms for simulation case 4, 5 and 6 are illus-
trated in Figures 5.42, 5.43, and 5.44, respectively. From Figures 5.42, 5.43, and 5.44, we
see that although the angle separation between the multipaths is now equal to 20◦, which
is greater than θH = 14.5◦, as in the close multipath DOA case, decreasing the power ratio
between the multipaths will also result in a higher BER. The reason is that the beamformer
cannot ideally form a null in the direction of the second path, so decreasing the power ratio
between the multipaths still will result in a higher ISI level. Also since the angle separation
is now equal to 20◦, the multipaths of one user may fall into the main beam constructed
for another user, hence decreasing the power ratio between the multipaths will increase the
interference level of the desired user. The higher ISI and interference level will then cause
a worse BER performance. Comparing Figures 5.42, 5.43, and 5.44 with Figures 5.39, 5.40,
and 5.41, we see that the BER of the well-separated multipath DOA case is lower than
that of the close multipath DOA case, and the improvement obtained by using the adaptive
array is larger in the well-separated DOA case. The reason is that in the close DOA case
where the DOAs of the multipaths are separated by only 10◦, all the multipaths tend to
fall into the main beam of the output port, and thus the adaptive array cannot rejected
the multipath effectively. However, in the well-separated multipath DOA case, for most of
the users, only one multipath can fall into the main beam of the output port, therefore the
multipath effect can be reduced by the array to a very low level.
108
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
4 5 6 7 8 9 10 11 12 13 1410
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Number of Users
BE
RConv. receiver
MT−SDDD
LS−DRMTA
LS−DRMTCMA, a_PN/a_CM = 2
LS−DRMTCMA, a_PN/a_CM = 0.5
Figure 5.39: BER performance of different adaptive algorithms in multipath environment.In this case, Eb/N0 = 8 dB, the DOAs of the first paths of all the users are equally spacedbetween −70◦ and 90◦. The DOA of the second path is 10◦ less than that of the first path.The power ratio of the first path to the second path is 0 dB, and the time delay betweenthese two paths is 0.5Tc. This plot is generated by Programs [P27] and [P29].
4 5 6 7 8 9 10 11 12 13 1410
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Number of Users
BE
R
Conv. receiver
MT−SDDD
LS−DRMTA
LS−DRMTCMA, a_PN/a_CM = 2
LS−DRMTCMA, a_PN/a_CM = 0.5
Figure 5.40: BER performance of different adaptive algorithms in multipath environment.In this case, Eb/N0 = 8 dB, the DOAs of the first paths of all the users are equally spacedbetween −70◦ and 90◦. The DOA of the second path is 10◦ less than that of the first path.The power ratio of the first path to the second path is 6 dB, and the time delay betweenthese two paths is 0.5Tc. This plot is generated by Programs [P27] and [P29].
109
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
4 5 6 7 8 9 10 11 12 13 1410
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Number of Users
BE
RConv. receiver
MT−SDDD
LS−DRMTA
LS−DRMTCMA, a_PN/a_CM = 2
LS−DRMTCMA, a_PN/a_CM = 0.5
Figure 5.41: BER performance of different adaptive algorithms in multipath environment.In this case, Eb/N0 = 8 dB, the DOAs of the first paths of all the users are equally spacedbetween −70◦ and 90◦. The DOA of the second path is 10◦ less than that of the first path.The power ratio of the first path to the second path is 10 dB, and the time delay betweenthese two paths is 0.5Tc. This plot is generated by Programs [P27] and [P29].
4 5 6 7 8 9 10 11 12 13 1410
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Number of Users
BE
R
Conv. receiver
MT−SDDD
LS−DRMTA
LS−DRMTCMA, a_PN/a_CM = 2
LS−DRMTCMA, a_PN/a_CM = 0.5
Figure 5.42: BER performance of different adaptive algorithms in multipath environment.In this case, Eb/N0 = 8 dB, the DOAs of the first paths of all the users are equally spacedbetween −70◦ and 90◦. The DOA of the second path is 20◦ less than that of the first path.The power ratio of the first path to the second path is 0 dB, and the time delay betweenthese two paths is 1.5Tc. This plot is generated by Programs [P28] and [P29].
110
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
4 5 6 7 8 9 10 11 12 13 1410
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Number of Users
BE
RConv. receiver
MT−SDDD
LS−DRMTA
LS−DRMTCMA, a_PN/a_CM = 2
LS−DRMTCMA, a_PN/a_CM = 0.5
Figure 5.43: BER performance of different adaptive algorithms in multipath environment.In this case, Eb/N0 = 8 dB, the DOAs of the first paths of all the users are equally spacedbetween −70◦ and 90◦. The DOA of the second path is 20◦ less than that of the first path.The power ratio of the first path to the second path is 6 dB, and the time delay betweenthese two paths is 1.5Tc. This plot is generated by Programs [P28] and [P29].
4 5 6 7 8 9 10 11 12 13 1410
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Number of Users
BE
R
Conv. receiver
MT−SDDD
LS−DRMTA
LS−DRMTCMA, a_PN/a_CM = 2
LS−DRMTCMA, a_PN/a_CM = 0.5
Figure 5.44: BER performance of different adaptive algorithms in multipath environment.In this case, Eb/N0 = 8 dB, the DOAs of the first paths of all the users are equally spacedbetween −70◦ and 90◦. The DOA of the second path is 20◦ less than that of the first path.The power ratio of the first path to the second path is 10 dB, and the time delay betweenthese two paths is 1.5Tc. This plot is generated by Programs [P28] and [P29].
111
CHAPTER 5. SIMULATION RESULTS AND DISCUSSION
5.8 Conclusion
In this chapter, we present the simulation results of different adaptive array algorithms
in a CDMA system. We compare the BER performance of different algorithms in various
channel environments (e. g., the AWGN channel, the timing offset case, the frequency
offset case, and the multipath environment). From the comparisons we see that the LS-
DRMTA and the LS-DRMTCMA, the two novel algorithms developed in this research, can
outperform the other algorithms in all the channel environments. The effect of varying the
term aPN/aCM in LS-DRMTCMA on the BER performance is also discussed.
112
Chapter 6
Conclusions and Future Work
6.1 Conclusions
In this thesis, we developed two novel adaptive algorithms for the beamformer used in a
CDMA system. We provide a detailed derivation of these two novel algorithms and create a
MATLABTM simulation testbed to compare the performance of these two novel algorithms
with that of the algorithms presented in the literature. The BER performance of all these
algorithms is compared under different conditions (e. g., the AWGN channel, the timing
offset case, the frequency offset case, and the multipath environment). It was shown from
the simulation results that the two novel algorithms, LS-DRMTA and LS-DRMTCMA,
can outperform the other algorithms in all the test conditions no matter if the system
is over-loaded (i. e., even if the number of users is greater than the number of antenna
elements of the array). It was also shown that the LS-DRMTA and LS-DRMTCMA does
not need to perform the GSO and sorting procedure which are required in the MT-LSCMA
and MT-SDDD, therefore can reduce the system complexity. We also show that unlike
that in the MT-LSCMA and MT-SDDD, the number of output ports is not limited by the
number of antenna elements in LS-DRMTA and LS-DRMTCMA, which can result in a
lower interference level in the beamformer output and make the expansion of the system
easier. We also examine the convergence property of different algorithms and show that the
two novel algorithms can converge faster than the other algorithms. In this thesis, we also
provide a detailed survey of the adaptive beamformer algorithms, which is very useful for
the researchers working in this area.
113
CHAPTER 6. CONCLUSIONS AND FUTURE WORK
6.2 Future Work
A number of possibilities exist for future work based on this thesis. These possibilities
are outlined below.
1. Currently the two novel algorithms are only simulated in the workstation using the
MATLABTM code. It will be useful if these two algorithms can be implemented
in a DSP chip and the 8-element antenna array can be constructed for field trial
measurement.
2. In this thesis, the performance of all the algorithms is evaluated by using a uniform
linear array. In the future, we can use different array geometries, e. g., a circular
array, to examine the performance of the algorithms.
3. The spreading signal used in this research is a short PN sequence (e. g., only 15 chips
per bit). In a realistic CDMA system, a longer PN sequence is always used. To
reduce the computational complexity of the algorithms, we can use only part of the
PN sequence in the LS-DRMTA and LS-DRMTCMA for the adaptation. The effect
of using only one segment of the PN sequence on the performance of the algorithms
should be examined in the future.
4. In this research, we only use a simple channel model to evaluate the performance of
the algorithms. It may be useful if a channel model including the DOA, the time
delay, the power level, and the time-varying property of each multipath can be used
in the simulation.
5. In the simulation of the multipath case, only the path with the strongest power is
extracted by the LS-DRMTA and LS-DRMTCMA. In the future, we can use several
time delayed versions of the respread signal in the LS-DRMTA and LS-DRMTCMA
to extract several multipaths of the transmitted signal, and combine the beamformer
with a RAKE receiver to further enhance the capacity of the system.
114
CHAPTER 6. CONCLUSIONS AND FUTURE WORK
6. In this research, we assume a perfect power control for all the users. In the future,
we can examine the performance of these algorithms under imperfect power control
conditions. It is believed that the algorithms should have the ability to combat the
power variation of the signals.
Finally, in this thesis, all the beamforming algorithms are only used in the base station for
the reverse link. It will be a challenge to use the weight vectors generated by the algorithms
in the reverse link for the beamforming in the forward link, since the reverse link and forward
link are always working at difference frequencies.
115
Appendix A
Differentiation with Respect to a Vector
An issue commonly encountered in the study of optimization theory is that of differ-
entiating a cost function with respect to a parameter vector of interest. The purpose of
Appendix A is to address the more difficult issue of differentiating a cost function with
respect to a complex-valued parameter vector. We begin by introducing some basic defini-
tions [10].
A.1 Basic Definitions
Consider a complex function f(w) that is dependent on a parameter vector w. When
w is complex valued, there are two different mathematical concepts that require individual
attention: (1) the vector nature of w, and (2) the fact that each element of w is a complex
number.
Dealing with the issue of complex numbers first, let xk and yk denote the real and
imaginary parts of the kth element wk of the vector w; that is,
wk = xk + jyk (A.1)
We thus have a function of the real quantities xk and yk. Hence, we may use equation (A.1)
to express the real part xk in terms of the pair of complex conjugate coordinates wk and w∗k
as
xk =1
2(wk + w∗k) (A.2)
116
APPENDIX A. DIFFERENTIATION WITH RESPECT TO A VECTOR
and express the imaginary part yk as
yk =1
2j(wk − w∗k) (A.3)
where ∗ denotes complex conjugation. The real quantities xk and yk are functions of both
wk and w∗k. It is only when we deal with analytic functions f that we are permitted to
abandon the complex-conjugated term w∗k by virtue of the Cauchy–Riemann equations.
However, most functions encountered in physical sciences and engineering are not analytic.
The notion of a derivative must tie in with the concept of a differential. In particular,
the chain rule of changes of variables must be obeyed. With these important points in mind,
we may define certain complex derivatives in terms of real derivatives, as shown by
∂
∂wk=
1
2
(∂
∂xk− j ∂
∂yk
)(A.4)
and∂
∂w∗k=
1
2
(∂
∂xk+ j
∂
∂yk
)(A.5)
The derivatives defined herein satisfy the following two basic requirements:
∂wk∂wk
= 1 (A.6)
∂wk∂w∗k
=∂w∗k∂wk
= 0 (A.7)
(An analytic function f must satisfy ∂f/∂z∗ = 0 everywhere, where z is a complex variable.)
The next issue to be considered is that of differentiation with respect to a vector. Let
w0, . . . , wM−1 denote the elements of an M × 1 complex vector w. We may extend the use
of equation (A.4) and (A.5) to deal with this new situation by writing
∂
∂w=
1
2
∂∂x0− j ∂
∂y0
∂∂x1− j ∂
∂y1
...
∂∂xM−1
− j ∂∂yM−1
(A.8)
117
APPENDIX A. DIFFERENTIATION WITH RESPECT TO A VECTOR
and
∂
∂w∗=
1
2
∂∂x0
+ j ∂∂y0
∂∂x1
+ j ∂∂y1
...
∂∂xM−1
+ j ∂∂yM−1
(A.9)
where we have wk = xk + jyk, for k = 0, 1, . . . ,M − 1. We refer to ∂∂w as a derivative with
respect to the vector w, and to ∂∂w∗ as a conjugate derivative also with respect to the vector
w. These two derivatives must be considered together. They obey the following relations
∂w
∂w= I (A.10)
and∂w
∂w∗=∂w∗
∂w= 0 (A.11)
where I is the identity matrix and 0 is the null matrix. For the subsequent use, we will
adopt the definition of (A.9) as the derivative with respect to a complex-valued vector.
A.2 Examples
In this section, we illustrate some applications of the derivative defined in equation (A.9).
Two examples will be considered.
Example 1 Let x and w denote two complex-valued M × 1 vectors. There are two inner
products, xHw and wHx, to be considered. Let c1 = xHw. The conjugate derivative of c1
with respect to the vector w is
∂c1∂w∗
=∂
∂w∗
(xHw
)= 0 (A.12)
where 0 is the null vector. Consider next c2 = wHx. The conjugate derivative of c2 with
respect to w is∂c2∂w∗
=∂
∂w∗
(wHx
)=
∂
∂w∗
(xTw∗
)= x. (A.13)
118
APPENDIX A. DIFFERENTIATION WITH RESPECT TO A VECTOR
Example 2 Consider next the quadratic form
c = wHRw (A.14)
where R is a Hermitian matrix. The conjugate derivative of c (which is real) with respect
to w is
∂c
∂w∗=
∂
∂w∗
(wHRw
)= Rw. (A.15)
119
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124
Programs
The following programs were used in the preparation of this thesis. All these programs
may be found in ∼zhigang/glomo/simul on the MPRG workstation network.
[P1] postProc/plotBmPattn.m
[P2] postProc/genBmPattn.m
[P3] alg/gso.m
[P4] CMCurve/mtCurveConst.m
[P5] postProc/plotBmPattnNew.m
[P6] postProc/plotConst.m
[P7] postProc/plotWaveform.m
[P8] postProc/plotCMCurve.m
[P9] postProc/plotLSMSECurve.m
[P10] BERprog/AWGN/mtBERNewAWGN8dB.m
[P11] BERprog/AWGN/mtBERNewAWGN4dB.m
[P12] BERprog/BERplotNewA1.m
[P13] postProc/plotBmPattn1by1.m
[P14] BERprog/TimeOffset/mtBERNewTO8dB.m
125
PROGRAMS
[P15] BERprog/BERplotNewT1.m
[P16] BERprog/BERplotNewTBD1.m
[P17] BERprog/BERplotNewT10UAll1.m
[P18] BERprog/FreqOffset/mtBERNewFO8dB.m
[P19] BERprog/BERplotNewFOBC1.m
[P20] BERprog/BERplotNewFOSAAll1.m
[P21] BERprog/BERplotNewFOSAPS1.m
[P22] BERprog/BERplotNewFOSAMrPS1.m
[P23] BERprog/BERplotNewFONPSInf1.m
[P24] BERprog/AllWt/mtBERNewAllWt.m
[P25] BERprog/BERplotNewAllWtTot1.m
[P26] BERprog/BERplotNewAllWtMIt1.m
[P27] mtBERNewAllWtMuS.m
[P28] mtBERNewAllWtMuL.m
[P29] BERprog/BERplotNewAllWtMuT1.m
126
VITA
Zhigang Rong was born on January 25, 1972 in Nanning, China. He received his Bachelor
of Science Degree in Electrical Engineering in July 1993 from University of Science and
Technology of China, Hefei. He joined the Electrical Engineering Department of Virginia
Tech in Fall 1994 to pursue his M. S. In May 1995, he joined the Mobile and Portable Radio
Research Group, where he worked with Dr. T. S. Rappaport in the area of adaptive array