STUDYING THE EFFECT OF DELAY DIVERSITY ON A DS-CDMA DOWNLINK A PROJECT REPORT Submitted in partial fulfillment of the requirements for the award of the Degree of BACHELOR OF TECHNOLOGY in ELECTRICAL ENGINEERING by Sumanth Jagannathan (EE99295) Under the guidance of
Dr. K. Giridhar
DEPARTMENT OF ELECTRICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY MADRAS - 600 036 June 2003
CERTIFICATE
This is to certify that, the report titled Studying the effect of delay diversity on a DS-CDMA downlink is a bonafide record of the work done by Sumanth Jagannathan EE99295, Department of Electrical engineering, Indian Institute of Technology, Madras, under my guidance and supervision, in partial fulfillment of the requirements for the award of the degree of Bachelor of Technology in Electrical Engineering.
Place
Dr. K.Giridhar
Date
Associate Professor Department of Electrical Engineering Indian Institute of Technology, Madras
ii
Acknowledgements I would like to express my deepest and warmest gratitude to my mentor and advisor, Dr. K.Giridhar for the constant guidance and inspiration he has given me during the course of the project. Every time I had a discussion with him, I was filled with more confidence and enthusiasm in my project work. I would like to thank the Department for providing the excellent computing facilities where I carried out my project. I thank Jay Kumar for helping me debug my code and clear my doubts. I also thank Ramanan for the valuable discussions we had, during the course of the project. Last but not the least, I thank my family for their unfailing love, support and encouragement.
Abstract In this project, we investigate the bit error rate on a DS-CDMA downlink as a function of the distance of the mobile from the base station. In a typical DS-CDMA system design, about 50% of the mobiles are in a soft hand-off situation which provides us with macro-cellular diversity. The delay spread in the multipath channel also provides inherent time diversity (micro-cellular diversity). This especially helps at distances farther away from the base station as the delay spread increases with distance. However, the micro-cellular diversity, as opposed to macro-cellular diversity, may not be a true diversity as the various channel taps may be correlated. This is particularly true at distances close to the base station. Even when the channel taps are independent, the fact that the channel taps might be arbitrarily spaced can affect the performance of a receiver which limits itself to chip-spaced sampling. In this project we simulate the DS-CDMA downlink assuming a chip rate of 1.2288M cps and the IS-95 base station specific scrambling code. An exponential power delay profile, a distance specific choice for delay spread and Rician ‘K’ factor are assumed. We compare the RAKE error rate performance under various situations including no handoff versus soft handoff, independent fading versus correlated fading,
chip-spaced sampling versus sub-chip sampling, and with and without pilot channel cancellation at the mobile receiver. Interesting insights into the DS-CDMA downlink performance are obtained, and some of these results, to the best of our knowledge, have not appeared so far in open literature.
v
Contents Acknowledgements
iii
Abstract
iv
List of Figures
ix
List of Tables
xii
1 Introduction
1
1.1
The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2 The System Model 2.1
2.2
The Transmitter
5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.1.1
Power distribution . . . . . . . . . . . . . . . . . . . . . . . .
8
2.1.2
The PN Sequence generator . . . . . . . . . . . . . . . . . . .
9
The Discrete Multipath Channel Model . . . . . . . . . . . . . . . . .
9
2.2.1
Delay-spread model . . . . . . . . . . . . . . . . . . . . . . . .
12
2.2.2
Fading models . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.3
2.2.3
Correlated Channel Taps . . . . . . . . . . . . . . . . . . . . .
14
2.2.4
Channel normalization . . . . . . . . . . . . . . . . . . . . . .
16
The Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.3.1
Pulse shaping . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.3.2
Noise addition at the Receiver . . . . . . . . . . . . . . . . . .
20
2.3.3
Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . .
20
2.3.4
Pilot Cancellation . . . . . . . . . . . . . . . . . . . . . . . . .
22
3 Simulation Model and Results 3.1
3.2
25
Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.1.1
The Cell site and delay spread . . . . . . . . . . . . . . . . . .
25
3.1.2
The Model for Soft handoff . . . . . . . . . . . . . . . . . . .
26
3.1.3
Pulse shaping and the channel . . . . . . . . . . . . . . . . . .
27
3.1.4
Channel Normalization . . . . . . . . . . . . . . . . . . . . . .
29
3.1.5
The Chip-matched Receive Filter . . . . . . . . . . . . . . . .
30
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
3.2.1
Simulation Parameters . . . . . . . . . . . . . . . . . . . . . .
31
3.2.2
The Effect of the Line of Sight path . . . . . . . . . . . . . . .
32
3.2.3
The Effect of Soft handoff . . . . . . . . . . . . . . . . . . . .
33
vii
3.2.4
The Effect of correlated channel taps and sub-chip sampling at the receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.5
35
The Effect of Pilot cancellation and the Rake finger acceptance threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
4 Conclusion
40
Appendix A
43
A.1 Walsh codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B
43 44
B.1 PN sequences and the Code Circle . . . . . . . . . . . . . . . . . . . . Appendix C
44 47
C.1 Correlated Rayleigh sequence generation with single IDFT operation Appendix D
47 49
D.1 The RAKE Combiner . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography
49 51
viii
List of Figures 2.1
Complex baseband system model for the downlink of a DS-CDMA system
5
2.2
The transmitter block on the DS-CDMA downlink . . . . . . . . . . .
7
2.3
Time Varying Channel . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.4
The Exponential Power Delay Profile for the Discrete Multipath Channel Impulse Response . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.5
Exponential model for correlated channel taps . . . . . . . . . . . . .
15
2.6
The receiver block on the DS-CDMA downlink . . . . . . . . . . . . .
18
2.7
Implementation of pulse shaping and noise addition . . . . . . . . . .
19
2.8
Conventional Channel Estimation by correlation of the received signal with the complex spreading sequence of the Pilot channel . . . . . . .
21
2.9
The Pilot cancellation block . . . . . . . . . . . . . . . . . . . . . . .
23
3.1
Cell arrangement in a two-way soft handoff. The value in % refers to the amount of power received by the mobile from base station B,
3.2
keeping 100% power received from base station A as reference . . . .
27
The optimized pulse shaping and channel block . . . . . . . . . . . .
29
3.3
The Chip Matched Filter . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
No handoff situation. Channel taps are independent, Rake acceptance threshold = -3dB, Eb/No = 6dB, Tx SIR = -6dB. . . . . . . . . . . .
3.5
34
Soft handoff situation. Channel taps are independent, Rake acceptance threshold = -3dB, Eb/No = 6dB, Tx SIR = -6dB. . . . . . . . . . . .
3.6
30
34
Comparison of BER performance with correlated/independent channel taps & T c/T c/2 sampling at the receiver. Rake acceptance threshold = -3dB, Eb/No = 6dB, Tx SIR = -6dB. . . . . . . . . . . . . . . . .
3.7
The improvement in BER with pilot cancellation. Rake acceptance threshold = -3dB . Soft handoff, Independent channel taps &
Tc 2
sam-
pling at the receiver. Eb/No = 6dB, Tx SIR = -9.5dB. . . . . . . . . 3.8
35
37
The improvement in BER with pilot cancellation. Rake acceptance threshold = -6dB . Soft handoff, Independent channel taps &
Tc 2
sam-
pling at the receiver. Eb/No = 6dB, Tx SIR = -9.5dB. . . . . . . . .
38
B.1 Fibonacci implementation of LFSR . . . . . . . . . . . . . . . . . . .
44
B.2 Galois implementation of LFSR . . . . . . . . . . . . . . . . . . . . .
44
B.3 The PN sequence Code Circle . . . . . . . . . . . . . . . . . . . . . .
45
B.4 Cross correlation for the in-phase PN sequence . . . . . . . . . . . . .
46
B.5 Cross correlation for the quadrature-phase PN sequence . . . . . . . .
46
x
C.6 Modified Smith’s Simulator . . . . . . . . . . . . . . . . . . . . . . .
47
D.7 A RAKE combiner with M fingers . . . . . . . . . . . . . . . . . . . .
49
xi
List of Tables 2.1
Transmit power distribution among the various channels . . . . . . .
3.2
Distance from BTS Vs Delay spread
. . . . . . . . . . . . . . . . . .
26
3.3
Power received by a mobile in 2-way handoff . . . . . . . . . . . . . .
28
3.4
Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . .
31
3.5
BER Vs Distance of the mobile from the base station for different values of K
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
8
32
Chapter 1 Introduction 1.1
The Problem
Traditionally, researchers have investigated the BER performance for different SNR’s in a DS-CDMA system. This helps in understanding the performance of the DSCDMA system and in comparing various types of transmit/receive schemes. However, such a study is limited in the sense that a particular power delay profile is assumed for the channel irrespective of the distance of the mobile from the base station. In a real DS-CDMA system, as a mobile moves towards or away from the base station, it experiences a variation in the channel. This variation is not due to fading alone but also due to the delay spread of the multipath channel. The delay spread of a multipath channel increases as the mobile moves away from the base station and therefore the channel taps vary in position as the mobile moves to or from the base station. Therefore, for a fixed SNR, the BER performance may vary depending on the distance of the mobile from the base station. It would be of interest to model the delay spread and investigate the BER performance of the DS-CDMA system as a function of distance of the mobile from the base station. 1
The delay spread in the channel may cause interference at the receiver, but there is also a some diversity available as the same signal appears at the receiver at different delays. When combined properly using the RAKE receiver, this can help in averaging out the interference and strengthening the signal. This time diversity available due to the inherent characteristics of the multipath channel is termed as Micro-cellular diversity. However this kind of diversity may not be easy to exploit. First of all, the various paths of the multipath channel have to be chip time resolvable (this is satisfied at distances away from the base station) and secondly, since the channel may be arbitrarily spaced, the receiver working at chip rate (or sub-chip rate) must be able to estimate to the channel to a reasonable extent. Also, the diversity due to the delay spread of the channel can be exploited only if the channel taps are independent. Suppose the different channel taps are correlated, there may not be any diversity in the channel that the receiver could make use of. In a typical DS-CDMA system design, about 50% of the mobiles are in soft handoff. When a mobile is in soft handoff, it receives the same signal from more than one base station. In such a case, the mobile will receive multiple copies of the signal not only due to the delay spread of the channel but also from the other base stations that it is talking to. This is another form of diversity that the RAKE receiver can exploit called Macro-cellular diversity. Since a mobile in soft handoff will receive independent copies of the signal from different base stations, the macro-cellular diversity obtained
2
is a true diversity and will definitely lead to an improvement in performance. It is of interest to investigate the performance improvement that is available due to micro-cellular diversity and due to macro-cellular diveristy in terms of the BER versus distance of the mobile from the base station. In this project we simulate the DS-CDMA downlink assuming a chip rate of 1.2288M cps and the IS-95 base station specific scrambling code. An exponential power delay profile, a distance specific choice for delay spread and Rician ‘K’ factor are assumed. We compare various situations such as no handoff versus soft handoff, independent fading versus correlated fading and chip-spaced sampling versus sub-chip sampling at the receiver. We also investigate the effect of pilot cancellation at the receiver for different pilot strengths. Interesting insights into the DS-CDMA downlink performance are obtained through our simulations. We find that the line of sight component of the multipath channel is the dominant factor in determining the BER at distances near the base station. Near the edge of the cell site, micro-cellular diversity is available but it is not adequate to compensate for the absence of a strong LOS component. In such a situation the macro-cellular diversity available due to soft handoff becomes an important factor in reducing the BER near the edge of the cell site. Further, when the channel taps are themselves correlated, we find that macro-cellular diversity is the most important factor in maintaining a reasonable error rate performance at the edge of the cell site.
3
1.2
Outline of Thesis
Chapter one gives an introduction to the problem and the motivation behind it. The models used to represent the different blocks of a DS-CDMA system are described in detail in Chapter two. The models assumed for the delay spread of the multipath channel, correlated channel taps and pilot cancellation are also discussed. Chapter three describes the simulation model which has been primarily optimized for time. BER simulation results for the DS-CDMA system as a function of the distance of the mobile from the base station are also shown in Chapter three. Chapter four concludes the thesis by providing more insight into the simulation results.
4
Chapter 2 The System Model In this chapter, the models that have been used to represent the sub-blocks of the DS-CDMA downlink system will be discussed in detail. We follow the problem formulation and symbol convention as in [8]. The complex baseband system model for the DS-CDMA downlink is illustrated in Fig. 2.1.
s0 (i)
Transmitter
s1 (i)
r(t) Channel h(t)
Rake Receiver
sˆk (n)
Pilot Channel
n(t) Channel
sN −1 (i)
Estimation
Multi-user Interference
Fig.2.1: Complex baseband system model for the downlink of a DS-CDMA system
5
2.1
The Transmitter
The downlink of a DS-CDMA system uses Walsh codes (Orthogonal codes) for channelization and spreading, and a pseudo-orthogonal sequence or a Pseudo-Noise sequence (PN sequence) for complex scrambling. The IS-95 or CDMA2000 standards as such use only one PN sequence generator, but each base station is assigned a unique shift on the code circle of the PN sequence. This helps in differentiating between the various base stations. In our model, we use the short PN sequence for this purpose. Further details about the PN sequence generation and the code circle can be found in Appendix B.1. In order to differentiate between the different channels, each channel is assigned a unique Walsh code. This helps in demultiplexing the channels at the receiver. Walsh codes are discussed in Appendix A.1. Fig. 2.2 shows the spreading and modulation for a downlink user. Quaternary Phase Shift Keying (QPSK) is applied for data modulation in the downlink. Each pair of two bits are serial-to-parallel converted and mapped to the I and Q branches respectively. The data in the I and Q branches of channel k are spread to the chip rate by the same channelization (Walsh) code wck . This spread signal is then scrambled by a cell specific scrambling code pni where i refers to the ith bit. The scrambling code is updated every bit period based on the code circle of the PN sequence while the Walsh code is fixed for all the bits. Additional channels can be multiplexed by using
6
different Walsh codes while retaining the same scrambling code. Square-Root Raised Cosine (SRRC) filters with roll-off factor of 0.22 are employed for pulse shaping. The pulse shaped signal is subsequently upconverted and transmitted.
sk (i)
Serial to Parallel
Channelization
Complex Scrambling
SRRC Pulse Shaping
Walsh Code (wck )
xk (t)
j
PN Sequence(pni )
Fig.2.2: The transmitter block on the DS-CDMA downlink Let there be N channels (including N − 3 user traffic channels and the pilot, sync and paging channels). Therefore, the desired traffic channel has N − 1 interfering channels. The ith data symbol for traffic channel k, sk (i) is spread using using the symbol-period-dependent spreading waveform wk,i (t). Hence, the transmitted signal of traffic channel k can be expressed as
xk (t) =
∞
sk (i)wk,i (t − iT )
(2.1)
i=−∞
where T is the symbol duration. Each symbol is assumed to have unit amplitude; i.e., |sk (i)|2 = 1
7
(2.2)
The spreading waveform for the k th user and the ith bit consists of a complex chip sequence, ck,i (j) convolved with the chip pulse shape (SRRC) g(t), giving
SF −1 1 wk,i (t) = √ ck,i (j)g(t − jTc ) SF j=0
(2.3)
where ck,i (j) = wck (j) ∗ pni (j), ∀ j = 0 : SF − 1 and SF is the spreading factor 2.1.1
Power distribution
The pilot, paging and synchronization channels are common to all the mobile stations and therefore a fixed amount of power is assigned to these channels so that they can be received without much error even at the edge of the cell site. The pilot channel is the most important in this regard as the channel estimation depends critically on the received pilot signal. Table 2.1 shows the typical distribution of the transmit power among the various channels on a DS-CDMA downlink.
Name of the channel Percentage of transmit power Pilot channel
20%
Paging channel
14%
Sync channel
2%
All Traffic channels
64%
Table 2.1: Transmit power distribution among the various channels
8
2.1.2
The PN Sequence generator
The short PN sequence of IS-95 and CDMA2000 [7, 9, 10] has a period of 215 − 1 and an all zero code is inserted to make the period 215 . The generator polynomial for the I-arm and the Q-arm short PN sequences are given respectively as: PI (x) = x15 + x13 + x9 + x8 + x7 + x5 + 1
and
(2.4)
PQ (x) = x15 + x12 + x11 + x10 + x6 + x5 + x4 + x3 + 1
(2.5)
Based on the above polynomials, the PN sequences for the I-arm (CshortI ) and the Q-arm (CshortQ ) can be generated by the following linearrecursions: CshortI : i(n) = i(n − 15) ⊕ i(n − 10) ⊕ i(n − 8) ⊕ i(n − 7) ⊕ i(n − 6) (2.6) ⊕ i(n − 2) CshortQ : q(n) = q(n − 15) ⊕ q(n − 12) ⊕ q(n − 11) ⊕ q(n − 10) ⊕ q(n − 9) (2.7) ⊕ q(n − 5) ⊕ q(n − 4) ⊕ q(n − 3) 2.2
The Discrete Multipath Channel Model
An L-tap fading multipath channel as shown in Fig.2.3 is considered. The channel is characterized by the following equivalent baseband impulse response h(t) =
L−1
αi δ(t − τi )
(2.8)
i=0
where τi is the delay of the ith tap of the multipath channel w.r.t the first tap (τ0 = 0) and αi is the complex gain of that tap. 9
Rician fade variable (α0 )
Rayleigh fade variable (α1 ) Delay of the
2nd path τ1 Transmitted signal
Rayleigh fade variable (αL−1 ) Delay of the
Lth path τL−1
Fig.2.3: Time Varying Channel We assume that the channel follows an exponential power delay profile given by the following equation. p(τ ) = exp(−β0 τ )
(2.9)
The power delay profile is limited to the −10dB level w.r.t to the maximum. If the maximum is 1, then the limit will be at 0.1. The delay at which a signal level of −10dB is received w.r.t the first path is denoted as ∆τ . The channel consists of 5 taps equally spaced from 0 to ∆τ where time 0 corresponds to the first path that the 10
mobile station encounters. The exponential power delay profile is shown in Fig. 2.4. The interval ∆τ is split into 4 parts in order to obtain a 5-tap discrete time multipath channel. The first path is assumed to be a line of sight Rician fading channel whereas the remaining 4 paths are taken as non-line of Sight Rayleigh fading channels. 1
0.9
0.8
Relative power distribution
0.7
0.6
e−β0τ
0.5
0.4
0.3
0.2
0.1
0 0
∆τ/4
∆τ/2
3∆τ/4
∆τ
Delay spread (τ)
Fig.2.4: The Exponential Power Delay Profile for the Discrete Multipath Channel Impulse Response
11
2.2.1
Delay-spread model
As the mobile station goes further away from the base station, the transmitted wave undergoes more scattering in space and finally reaches the mobile by means of multiple reflections. Therefore, one can expect a mobile that is farther away from the base station to see paths that are much more spread in time than one that is closer to the base station. This heuristic is captured in our delay spread model. ∆τ is a measure of the delay-spread of the channel and is given by Equation 2.10 where d is the distance of the mobile from the base station and c is the speed of light.
∆τ = 20% d/c
(2.10)
Therefore, for a given distance from the base station we can obtain the 5-tap channel with the power delay profile parameter β0 calculated from : β0 = (ln 10)/∆τ 2.2.2
(2.11)
Fading models
The multiple taps present in the channel are assumed to undergo slow, frequenceselective fading. This assumption is reasonable as for a mobile speed of 100 Km/hr, a 2GHz carrier and a chip rate (1/T c) of 1.2288 Mchips/sec, the doppler frequency (fd ) is 185Hz. Therefore fd << 1/T c which implies a slow fading environment as the Doppler spread is very small compared to the bandwidth of the baseband signal. 12
The narrow pulse width (=Tc) means that the multipaths will be resolved most of the time and the transmitted signal will undergo frequency selective fading. 2.2.2.1
Rayleigh fading model
When all the paths in the channel are Non Line of Sight (NLOS), then a Rayleigh fading model can be assumed for each path where the Power Spectral Density (PSD) of the Rayleigh variable follows Jake’s spectrum. The Rayleigh fading simulation model can be found in Appendix C.1. Now, depending on whether we assume uncorrelated scattering or correlated scattering, the various taps of the multipath channel may be independent or correlated. The generation of correlated channel taps is discussed in section 2.2.3. 2.2.2.2
Rician fading model
When a Line of Sight (LOS) path is present, then a Rician fading model is adopted for that path with the mean of the fade variable being specified by a K parameter. The K parameter indicates that the power present in the LOS component of the Rician fading path is K times the power in the NLOS component. The Rician fade variable with a particular K parameter can be generated from the Rayleigh fade variable as shown in Equation 2.12 where is the R Rician variable and J is the Rayleigh variable. √ R = ( K + 1) × J
13
(2.12)
In our simulation, we assume the first path to be a LOS Rician fading channel with the other 4 channels being Rayleigh. For any given value of K say K0 , the line of sight component is assumed to decay exponentially from K = K0 at 300m from the base station to K = 0.1 at 3km from the base station i.e the edge of the cell. This decay of the LOS component as a function of distance is given by K(d) = K0 · e−B·(d−300)
(2.13)
where d is the distance of the mobile from the base station and B= 2.2.3
log(K0 /0.1) 2700
(2.14)
Correlated Channel Taps
Let the channel have L taps and let Xi , ∀ 1 ≤ i ≤ L be L independent Rayleigh fade variables with unit energy. The channel taps positions are fixed at delays τi , ∀ 1 ≤ i ≤ L with the fade variable Xi corresponding to the delay τi . The L correlated channel taps Yj , ∀ 1 ≤ j ≤ L are generated from Xi as follows. The fade variable Yj corresponds to the delay τj and hence should be more correlated with Xj . Also, the correlation between Yj and Xi must reduce as the time difference |τj − τi | increases. Therefore we assume an exponential model for the magnitude correlation, |ρ| = E[|Yj ||Xi∗ |] as a function of |τj −τi | as shown in Fig. 2.5 while the phase correlation ∠ρ is taken to be linear with (τj − τi ). ∠ρ equals zero when τj = τi . The model also assumes a cut-off point of 10T c at which we have |ρ| = 0.01. 14
For a time difference more than 10T c, we assume that the paths are uncorrrelated with each other. 1
0.9
0.8
Magnitude Correlation |ρ|
0.7
0.6
0.5
0.4
−γ|ρ|
e 0.3
0.2
0.1 0.01 0
Tc
2Tc
3Tc
4Tc
5Tc
6Tc
7Tc
8Tc
9Tc
10Tc
Delay in mutiples of chip time (Tc)
Fig.2.5: Exponential model for correlated channel taps
Let τ = (τj − τi ). Then we can define the magnitude correlation |ρ| and the phase correlation ∠ρ as follows. |ρ| = exp(−γ|τ |)
(2.15)
π (τ ) 10T c
(2.16)
∠ρ =
15
where ρ = E[Yj Xi∗ ] ∀ 1 ≤ i, j ≤ L
(2.17)
Thus the correlated taps Yj are obtained from Xi by the following linear combination
Y1 Y2 = .. . YL
1
ρ12
ρ21
1
.. . ρL1 ρL2
ρ13 . . . ρ1L ρ23 . . . ρ2L .. ... . ρL3 . . . 1
X1 X2 .. . XL
(2.18)
where ρij = E[Yj Xi∗ ] Finally, each variable Yi , ∀ 1 ≤ i ≤ L is normalized by the
(2.19) L j=1
|ρij |2 to ensure
that the new fade variables are also of unit energy. For generating correlated Rician fading channel taps, we need to first generate correlated Rayleigh fading variables and then add on a DC component to each variable depending on the corresponding K parameters. 2.2.4
Channel normalization
The channel is normalized such that the expected energy of the signal does not change when passed through it. This helps in simplifying calculations while adding noise as the power of noise to be added can be made independent of the channel. Also in the
16
multi-cell case, the received signal power can be defined without bothering about the individual channels. When there is no fading i.e the gains of the channel taps are fixed, then one can normalize the channel as follows: L−1
|αi |2 = 1;
(2.20)
i=0
In the presence of independent fading, the gains of the channel are time varying and therefore we must ensure that the expected energy in the channel is unity. This can be done as follows: L−1
E[αi αi∗ ] = 1;
(2.21)
i=0
For the case of correlated channel taps, the procedure for forming the fade variables ensures that the expected energy in the channel is unity.
2.3
The Receiver
The receiver structure after the signal is brought down to baseband consists of an SRRC (Square Root of Raised Cosine) filter followed by a chip-time or higher rate sampler. Noise is added to the signal before the SRRC filter (refer section 2.3.2). This is followed by the de-spreader and de-scrambler. The channel estimation module runs parallely and estimates the mutli-path channel parameters using the incoming pilot signal. It is assumed that the mobile receiver is able to time synchronize with 17
the base station and hence the propagation delay is not considered in our model. The different channels are demultiplexed in the receiver corresponding to each tap of the estimated multipath channel and also from the neigbouring base station in the case of soft hand-off. These signals along with the channel estimates are then sent to the RAKE combiner which is explained in appendix D.1. The typical receiver structure is illustrated in Fig. 2.6.
Received baseband signal
SRRC filter
Correlator
for user k
RAKE Combiner
Sgn(.)
sˆk (i)
Fig.2.6: The receiver block on the DS-CDMA downlink
2.3.1
Pulse shaping
Data is pulse shaped through a Square Root Raised Cosine (SRRC) filter at the transmitter. The receiver at the base station has a filter matched to the pulse shaping filter at the transmitter. Both these filters have a roll-off of 0.22. In the simulator the transmitter and the receiver filters are cascaded and they constitute a Raised Cosine (RC) filter. This rearrangement is shown in Fig. 2.7. In our simulations, the noise was directly added to the received signal without any filtering. Strictly speaking, the noise must have been SRRC filtered as shown in Fig. 2.7. However, we do not expect any significant change in the results due to this. 18
Square Root Raised Cosine Filter
Square Root Raised Cosine Filter
Channel Gaussian Noise Generator
equivalent to
Square Root Raised Cosine Filter
Square Root Raised Cosine Filter
Gaussian Noise Generator
Channel
Square Root Raised Cosine Filter
Raised Cosine Filter
Gaussian Noise Generator
equivalent to
Channel
Square Root Raised Cosine Filter
Fig.2.7: Implementation of pulse shaping and noise addition Two side lobes on either side of the raised cosine filter were considered for implementing the filters and all the filters were implemented as symmetric FIR filters. Symmetric FIR filters have a delay equal to half the order of the filter. In order to take this delay into account we padded each transmitted data block by an additional data segment whose size is equal to the delay. Therefore, if the delay of the filter is ND samples we repeated the first ND samples of the data at the end of the data
19
block. After the filtering, we removed the first ND output samples of the filter. We also ignored the transient of the filter. 2.3.2
Noise addition at the Receiver
AWGN noise is added to the front end of the reciever i.e before the SRRC filter. The noise to be added is calculated directly based on the required SNR value and the transmit signal power as the channel has been normalized to unit energy. Let the required SNR (Eb/No) be XdB, SF be the Spreading Factor, and T be the power of the traffic channel. Then, the noise variance is given by
σ2 = 2.3.3
SF · T X
2 · 10 10
(2.22)
Channel Estimation
The conventional channel estimator as mentioned in [4] is adopted. The channel is estimated by correlating the received pilot signal with the complex conjugate of the PN sequence. This is depicted in Fig. 2.8 where M refers to the number of bits over which the correlation is performed, ci,k is the complex conjugate PN code and bi (m) are the pilot bits which in our case is -1 ∀m. In this regard, it is important to note that in our work we have assumed that the receiver is able to synchronize with the base station in order to obtain the correct PN offset.
20
Fig.2.8: Conventional Channel Estimation by correlation of the received signal with the complex spreading sequence of the Pilot channel After performing the correlation, inorder to ensure that we do not include many weak taps into our estimated channel, we set a threshold of say 3dB or 6dB i.e no tap that is more than 3dB/6dB weaker than the strongest tap will be included in the estimated channel. This is to prevent the weak taps (which may be spurious taps) from affecting the BER performance. At the same time, the threshold must be small enough to include genuine taps, which if ignored may once again deteriorate the performance. Let r(n) be the received signal after sampling, where n refers to the nth chip. Let ci,0 = pni ∗ wc0 be the complex spreading sequence for the pilot channel where i refers
21
to the ith bit. c¯i,0 will be a vector consisting of SF (spreading factor) elements i.e [c1,i,0 c2,i,0 . . . cj,i,0 . . . cSF,i,0 ]. Let us assume that we estimate the channel till L×Tc and that we perform the correlation over M bits. ˆ i , we form the Let c¯M ci,0 c¯i,0 . . . M times]. Then to estimate the channel h i,0 = [¯ correlation matrix Ci as follows
c¯M 1,i,0
0 c¯M c¯M 2,i,0 1,i,0 .. . c¯M 2,i,0 .. C¯i = . c¯M SF ·M,i,0 0 c¯M SF ·M,i,0 .. . 0 0 0
...
0 .. .
...
0 c¯M SF ·M −L,i,0 c¯M SF ·M −L+1,i,0
...
.. .
...
c¯M SF ·M,i,0
(2.23)
We then form the received vector, r¯ = [r{i · SF + 1} r{(i + 1) · SF + 1} · · · r{(i + M ) · SF + 1}]T and perform the correlation to estimate the channel as shown in Eq. 2.24. ˆi = √ h
1 (C¯iH · r¯) P · (−1 − j1) · 2M · SF
(2.24)
where P is the transmit power of the Pilot channel. 2.3.4
Pilot Cancellation
One way of reducing interference at the receiver is to estimate the interference that comes from the Pilot channel and then cancel it from the demultiplexed traffic chan22
nel. Since the pilot bits are known and the channel has been estimated, the receiver can reconstruct the channel and estimate what amount of interference is caused due to the pilot channel and this interference can be removed. The pilot cancellation block is shown in Fig. 2.9.
received baseband signal
Channel Estimator
Complex Pilot sequence Raised Cosine after spreading Filter and scrambling
Chip Spaced Estimated channel
To the Despreader and the Rake receiver
Pilot signal is cancelled here
Fig.2.9: The Pilot cancellation block
The channel is estimated at the mobile receiver in steps of T c or
Tc 2
from the
received signal as described in section 2.3.3. The mobile then reconstructs the transmitted pilot signal as the pilot bits are known. This signal is then modulated with a Raised Cosine pulse to take care of both the transmit and receiver SRRC’s and is passed through the estimated discrete time channel to estimate the received pilot channel. The estimated pilot channel is then cancelled from the received signal before sending it to the despreader and the RAKE receiver. However, pilot cancellation mainly helps only when the pilot signal is much
23
stronger the traffic channel, in which case there is considerable amount of interference from the pilot channel. Cancelling this interference can improve performance. However, when the pilot strength is comparable to the strength of the traffic channel, the interference caused will not be very high and cancelling the pilot may even degarde the BER performance as the cancellation may occur at the wrong taps.
24
Chapter 3 Simulation Model and Results This chapter describes the optimized simulation model designed to evaluate the Bit Error Rate (BER) on the downlink of a DS-CDMA system as a function of the distance of the distance of the mobile from the base station. The simulation model is essentially the same as the system model described in Chapter 2 but for the optimization of some blocks in order to make the simulation faster. The simulation results are also discussed in this chapter.
3.1
Simulation Model
3.1.1
The Cell site and delay spread
A circular cell site of radius 3km has been assumed. The BER is evaluated starting at a distance of 300m and continuing in steps of 300m till we reach the edge of the cell, i.e 3km. Table 3.2 shows the value of delay spread (∆τ ) for the various distances from the base station. This shows that more chip-time resolvable paths are available as we move farther away from the base station. There is no delay-diversity available at a distance of 25
300m as all the taps are crowded within a quarter of a chip time whereas at the edge of the cell, we have 2 to 3 chip resolvable paths and this may lead to some diversity. Distance of mobile from the BTS Delay spread 300m
0.2458T c
600m
0.4915T c
900m
0.7373T c
1200m
0.9830T c
1500m
1.2288T c
1800m
1.4746T c
2100m
1.7203T c
2400m
1.9661T c
2700m
2.2118T c
3000m
2.4576T c
Table 3.2: Distance from BTS Vs Delay spread
3.1.2
The Model for Soft handoff
In order to simulate soft handoff, we assume two circular cell sites A and B of radius 3km each which are tangentially aligned as shown in Fig. 3.1. Let us consider a mobile in cell site A at different distances from the base station A. We assume that till a distance of 1800m, the signal from base station B is not strong enough and therefore the mobile is not in handoff. From a distance(d) of 2100m onwards, the mobile is in 2-way handoff. For 300m ≤ d < 2100m, the power received 26
Base station A
Base station B 20% 70% 2100m 2700m
Cell A
Cell B 3Km
3Km
3000m 2400m 40%
100%
Fig.3.1: Cell arrangement in a two-way soft handoff. The value in % refers to the amount of power received by the mobile from base station B, keeping 100% power received from base station A as reference by the mobile from base station A is P and from base station B is 0. Table 3.3 shows the power recieved by the mobile from base stations A and B for 2100m ≤ d ≤ 3000m. 3.1.3
Pulse shaping and the channel
The channel model as described in section 2.2 is simulated. The multipath delays are discretized to steps of 0.01T c in order to save simulation time. This is done in order to handle arbitrary delays in the channel (fractions of T c) as we need to store the raised cosine at a sampling rate equal to the minimum time step in the channel. The minimum time step (Tstep ) of the channel is defined to be the maximum value such that each delay of the multipath channel can be expressed as an integral multiple of 27
Distance of mobile Power from BTS A Power from BTS B
Total power received
from the BTS 2100m
P
0.2P
1.2P
240m
P
0.4P
1.4P
2700m
P
0.7P
1.7P
3000m
P
P
2P
Table 3.3: Power received by a mobile in 2-way handoff Tstep . This minimum time step will vary for different distances from the base station due to the different delay spreads. Also, in some cases the sampling rate can be very high leading to unnecessary delay in the simulation. Hence the Tstep of the channel is fixed at 0.01T c. This value of Tstep leads to a good trade-off of accuracy versus simulation time. Although the raised cosine samples are stored in steps of Tstep , performing the modulation operation at the rate of Tstep will require a lot of memory and time. But this is unnecessary as finally at the receiver, the signal will be brought down to chip rate (T c) or a rate of T c/2. Hence we can define an effective pulse shaping filter ˜ (h(t)) at chip time or T c/2 by incorporating the channel into the pulse shaping filter as shown in Fig. 3.2. Let p(n·Tstep ) be the raised cosine filter sampled at Tstep . Let us define the following filters sampled at T c but with different offsets corresponding to τi ∀ 0 ≤ i ≤ 4 where
28
α0
Raised Cosine Filter
Receiver
i · Tstep
k · Tc
α1 α2 α3 α4
Delay τ1 Delay τ2 Delay τ3 Delay τ4
Effective pulse shape filter and channel ˜ h(t)
Receiver
k · Tc
Fig.3.2: The optimized pulse shaping and channel block τ0 = 0. pi (k · T c) = αi × p(
k · Tc τi − )∀0≤i≤4 Tstep Tstep
(3.1)
Then, the effective pulse shape filter combined with the channel is ˜ · T c) = h(k
4
pi (k · T c)
(3.2)
i=0
3.1.4
Channel Normalization
We have a 5-tap channel with a LOS Rician fading tap (with a specified K parameter) and 4 Rayleigh fading taps. The variance (σi2 , ∀ 0 ≤ i ≤ 4) of the Rayleigh variables is obtained from the exponential power delay profile as σi2 = e−β0 ·τi
29
(3.3)
Now for a Rayleigh variable α, E[αα∗ ] = 2 × σ 2 . Therefore in order to ensure a unit energy channel, we normalize all the fade variables by the following factor. N ormalizingf actor = K · σ02 + 2
4
σi2
(3.4)
i=0
3.1.5
The Chip-matched Receive Filter
For the purpose of simulation, the receiver can be optimized by defining an equivalent chip-matched filter with a single finger rather than having a conventional RAKE receiver with L fingers as shown in Fig. 3.3. 64
τL−1 − τ0
1
c∗ (n)
64
r(n)
α0∗ τL−1 − τ1
1
c∗ (n −
α1∗
τ1 ) Tstep
64
0
1
c∗ (n −
τL−1 ) Tstep
∗ αL−1
64
Chip Matched Filter
1
c∗ (n)
Fig.3.3: The Chip Matched Filter
30
3.2
Results
In this section, we present the error rate simulation results for the DS-CDMA downlink system. We present the variation of the BER as a function of the distance of the mobile from the base station under conditions of no handoff and soft handoff, correlated and independent fading and chip-rate and sub-chip correlation at the receiver. Also, the effect of pilot cancellation for different pilot strengths is observed. 3.2.1
Simulation Parameters
The various parameters used in the simulation are given in Table 3.4. Parameter
Value
Number of bits used in the simulation
105
Speed of the mobile (Jake’s spectrum)
100 Km/hr
Carrier Frequency
2 GHz
Data rate on Traffic channel
19.2 Kbps
Chip rate (IS-95)
1.2288 M cps
Spreading factor
64
Transmit Signal to Interference ratio(SIR)
−6dB/ − 9.5dB
Number of channel taps
5
Eb Received SNR ( N ) o
6dB
Channel estimation window
256 chips
Threshold for accepting RAKE fingers
−3dB/ − 6dB w.r.t. strongest tap
Table 3.4: Simulation parameters
31
The power distribution as mentioned in section 2.1.1 is used for our simulations. Three traffic channels are simulated along with the pilot, sync and and paging channels. Walsh codes 0, 1 and 32 are used for channelizing the Pilot, Paging and Sync channels, respectively. The mobile speed of 100Km/hr corresponds to a Doppler spread of 185Hz. 3.2.2
The Effect of the Line of Sight path
Table 3.5 gives the BER tabulated for different values of the line of sight component K for the case of independent channel taps with a Tc-spaced correlator at the receiver. The value K refers to the LOS component at 300m from the base station. Irrespective of this initial value of K, the LOS component is assumed to decay to 0.1 at a distance of 3km as described in section 2.2.2.2. Distance of the mobile
Line of Sight Component K
from the base station
K=0
K=6
K = 20
K = 100
300m
1.30 × 10−1
4.78 × 10−2
1.63 × 10−2
6.01 × 10−3
900m
1.32 × 10−1
8.07 × 10−2
4.47 × 10−2
1.36 × 10−2
1800m
1.24 × 10−1
1.09 × 10−1
9.77 × 10−2
7.79 × 10−2
2400m
1.23 × 10−1
1.16 × 10−1
1.13 × 10−1
1.12 × 10−1
3000m
1.24 × 10−1
1.20 × 10−1
1.20 × 10−1
1.21 × 10−1
Table 3.5: BER Vs Distance of the mobile from the base station for different values of K We can see an improvement in the BER performance as the line of sight path 32
increases in strength. However at the edge of the cell (3km), we have assumed the line of sight component to be almost negligible. Therefore the BER converges to the case when all the channel taps are undergoing independent Rayleigh fading. Depending on the strength of the LOS component and the distance from the base station, the BER may even be an order better than the NLOS case. Thus we can conclude that the presence of a line of sight component definitely improves the BER. 3.2.3
The Effect of Soft handoff
The soft handoff model as described in section 3.1.2 is implemented. A mobile in a soft handoff situation receives more interference than a mobile which is not in handoff. However, it also gains macro-cellular diversity from another base station. The signal received from the other base station is definitely fading independently from the signal coming from the closer base station. Therefore, the diversity obtained is a true diversity which leads to a considerable improvement in the BER performance. Figures 3.4 and 3.5 show the BER versus Distance curves for the no handoff and soft handoff cases respectively for different values of the LOS component. The received SNR is fixed at 6dB, the transmit SIR is −6dB and the Rake acceptance threshold is −3dB. We can see that macro-cellular diversity helps in improving the BER performance at distances which are farther away from the base station and closer to the edge of the cell.
33
−1
BER
10
−2
10
K=0 K=6 K=20 K=100 −3
10
300
600
900
1200 1500 1800 2100 2400 Distance of mobile from base station in metres
2700
3000
Fig.3.4: No handoff situation. Channel taps are independent, Rake acceptance threshold = -3dB, Eb/No = 6dB, Tx SIR = -6dB.
−1
BER
10
−2
10
K=0 K=6 K=20 K=100 −3
10
300
600
900
1200 1500 1800 2100 2400 Distance of mobile from base station in metres
2700
3000
Fig.3.5: Soft handoff situation. Channel taps are independent, Rake acceptance threshold = -3dB, Eb/No = 6dB, Tx SIR = -6dB. 34
Further results in this thesis are shown under the following conditions:
1. The mobile receiver is in soft handoff. 2. The line of sight component(K) is 20. 3.2.4
The Effect of correlated channel taps and sub-chip sampling at the receiver 0
10
Independent taps, Tc sampling Independent taps, Tc/2 sampling Correlated taps, Tc sampling Correlated taps, Tc/2 sampling
−1
BER
10
−2
10
−3
10
300
600
900
1200 1500 1800 2100 2400 Distance of mobile from base station in metres
2700
3000
Fig.3.6: Comparison of BER performance with correlated/independent channel taps & T c/T c/2 sampling at the receiver. Rake acceptance threshold = -3dB, Eb/No = 6dB, Tx SIR = -6dB. 35
From Fig. 3.6, we can see that the BER performance in the case of correlated channel taps is worse than when the channel taps are independent. This is especially visible at distances farther away from the base station where the delay spread is higher and more chip resolvable paths exist. The reason for this is that when the channel taps are independent, the receiver is able to exploit the micro-cellular time diversity that is available due to the multipath channel. However, when the channel taps are correlated, this diversity is lost. Although the channel may have many chip resolvable paths, there is not much diversity that the receiver can exploit from them as they are correlated. This is the reason for the poorer BER performance as compared to the independent channel taps case. Fig. 3.6 also gives us the comparison between chip spaced sampling at the receiver and sub-chip sampling at the receiver. We can see that the performance of a receiver working at T c/2 is better than a receiver working at T c. There are two main reasons for this. In the case of independent channel taps, the receiver working at T c/2 is able to pick more diversity from the channel as it can detect strong taps even if they are spaced less than a chip. The second reason is common to both independent as well as correlated channel taps. In our channel model, we have assumed the channel to be discretized in steps of 0.01T c. Therefore, a receiver working at T c/2 can have a channel estimate which is closer to the original channel than the estimate from a receiver working at T c. This helps greatly in improving the BER performance.
36
3.2.5
The Effect of Pilot cancellation and the Rake finger acceptance threshold
The pilot cancellation block as described in section 2.3.4 was implemented. It was observed that when the strength of the pilot channel equalled that of the strength of the traffic channel, the effect of pilot cancellation could not be observed. In fact, in some cases the performance was slightly worse with pilot cancellation. −1
10
BER
No pilot cancellation Pilot cancellation
−2
10
300
600
900
1200
1500
1800
2100
2400
2700
3000
Distance of mobile from base station in metres
Fig.3.7: The improvement in BER with pilot cancellation. Rake acceptance threshold = -3dB . Soft handoff, Independent channel taps &
Tc 2
sampling at the receiver. Eb/No = 6dB, Tx SIR = -9.5dB. When the pilot strength was increased to 10dB above the traffic channel strength, 37
the effect of pilot cancellation was pronounced. In IS-95, the pilot channel usually has 30% of the total downlink power, and it defines the coverage region of the cell. Therefore, this asssumption of 10dB more power in the pilot is reasonable. Since the pilot channel contributed to considerable interference, the BER performance improved greatly when pilot cancellation was performed. This can be observed in Fig 3.7 where the BER curves have been plotted with and without pilot cancellation. The threshold for accepting Rake fingers during channel estimation has been fixed at -3dB. −1
10
BER
No pilot cancellation Pilot cancellation
−2
10
300
600
900
1200
1500
1800
2100
2400
2700
3000
Distance of mobile from base station in metres
Fig.3.8: The improvement in BER with pilot cancellation. Rake acceptance threshold = -6dB . Soft handoff, Independent channel taps & sampling at the receiver. Eb/No = 6dB, Tx SIR = -9.5dB.
38
Tc 2
When the threshold for accepting Rake fingers during channel estimation is decreased to -6dB, the BER performance improves as shown in Fig. 3.8. The reason for this is as follows. During the Rake combining, combining strong paths will help in improving performance. At the same time, including weaker paths, which may be spurious might hamper the performance. Therefore, a threshold of -3dB was set for accepting Rake fingers during channel estimation. Such a sharp threshold might help in improving the performance near the base station where there is not much time diversity and there is a higher chance of spurious paths. However, the same threshold might cause genuine taps to be excluded at distances farther away from the base station. Therefore, decreasing the threshold for accepting Rake fingers to -6dB helps in such a situation. Also, the improvement in the BER performance due to pilot cancellation is higher when the Rake threshold is -6dB than when it is -3dB. In our simulations we also increased the channel estimation window from 256 chips to 1024 chips but the improvement in BER was marginal. Hence, we have shown all the results for a channel estimation window which is 256 chips in length.
39
Chapter 4 Conclusion We implemented a simulator for the downlink of a DS-CDMA system and investigated the BER performance of the RAKE receiver as a function of the distance of the mobile from the base station. Various models were proposed to model the delay spread in the channel, correlated channel taps and the line of sight component of the channel. Based on the simulation results, we can make the following conclusions.
1. There are a combination of effects which come into play while determining the performance of a DS-CDMA system. The presence of a line of sight component in the multipath channel helps tremendously in improving the BER performance especially at distances close to the base station where the LOS component has sufficient strength as compared to the NLOS components. In fact this LOS component can overcome the lack of time diversity due to correlated fading, at distances close to the base station. 2. We find that the macro-cellular diversity obtained due to soft handoff plays an important role in reducing the BER near the edge of the cell site. In the 40
presence of uncorrelated channel taps, there is some micro-cellular diversity that the RAKE reciever can make use of. But this diversity seems to be inadequate as the BER performance becomes poorer with increasing distance when the mobile is not in hand off. This is due to the inability of the receiver (working at chip-rate or twice chip-rate) to synchronize to a channel which is arbitrarily spaced. This not only causes loss of orthogonality but also introduces some correlation among the various estimated taps. The soft handoff situation, on the other hand, leads to improved performance as an independent copy of the signal is received from another base station. 3. However, the micro-cellular diversity is not to be ignored. We find that when the channel is modelled to have correlated taps, the performance is much worse when compared to a channel having independent taps. Therefore, there is some amount of diversity advantage that is obtained due to the multipath channel, although it may not be very prominent. 4. It was also observed that when the receiver worked at twice the chip rate, the BER performance was better than when the receiver worked at chip rate. The main reason for this is that the receiver is better “matched” to the channel, and can pick some strong taps in the channel which were previously missed out due to chip rate sampling.
41
5. The effect of pilot cancellation was investigated and it was observed that pilot cancellation at the receiver helped in improving the performance when the pilot channel was much stronger than the traffic channels. Also, it was observed that the a 6dB threshold for RAKE finger acceptance led to better performance than a 3dB threshold. This is especially true at distances farther away from the base station.
To our knowledge, many of these results are new, and have not appeared in open literature. It will be interesting to extend this work to Transmit diversity DS-CDMA downlink connections, and also incorporate the effect of Orthogonal Variable Spreading Factor (OVSF) codes.
42
Appendix A A.1
Walsh codes
Walsh codes are orthogonal functions which are generated by mapping codeword rows of special square matrices called Hadamard matrices. These matrices contain one row of all zeros, and the remaining rows each have equal numbers of ones and zeros. Walsh functions can be constructed for block length N = 2n . The Hadamard matrix of desired length can be generated by the following recursive procedure: 0 0 0 0 1 0 0 0 , H4 = H1 = [0], H2 = 0 0 1 0 1 0 1 1
0 1 HN HN , H2N = ¯N 1 HN H 0
(A.1)
where N is a power of 2 and the overscore denotes the binary complement of the bits in the matrix. Each row of H2N presents a Walsh function. These functions have zero correlation between each other. In the transmitter, each bit is spread by a Walsh function; therefore, the spreading factor is equal to N. These codes can be used if all the users of the are synchronized in time to a very high accuracy, because the cross-correlation between different shifts of Walsh functions is not zero.
Appendix B B.1
PN sequences and the Code Circle
PN sequences can be generated by means of an LFSR (Linear Feedback Shift Register) the feedback taps for which are specified by a generator polynomial. Let g(x) =
m i=0
gi xi be the generator polynomial for the PN Sequence. The PN
sequence can then be generated by a LFSR as shown in Figures B.1 and B.2.
Fig.B.1: Fibonacci implementation of LFSR
Fig.B.2: Galois implementation of LFSR
The PN sequence is periodic and the periodicity depends on the number of taps (N ) the shift register. A maximal PN sequence will have a periodicity of 2N − 1. We
can view one period of the PN code as a circle; any sub-sequence is then an arc of the circle. Fig. B.3 depicts the code circle for a PN sequence. Vector a
Vector b
Fig.B.3: The PN sequence Code Circle Consider two vectors a and b of size M × 1 where 0 < M < 2N − 1 at different offsets on the code circle. Let the value aT b be denoted as 6. PN sequences have the property that as M becomes larger, 6 will tend to zero. In other words, the PN sequences are pseudo orthogonal and the longer the correlation, the better the orthogonality. The correlation properties of the In-phase and Quadrature-phase PN sequences used in our DS-CDMA downlink model are shown in Fig. B.4 and B.5.The cross correlation has been plotted on the y axis versus the offset between the PN sequences on the x axis for different lengths of the PN sequences namely L=4, 16, 128 and 1024. We can see that for a length of 128, the PN sequences show reasonable amount of pseudo-orthogonality. 45
L=4
L=16
1.1
1.1
0.9
0.9
0.7
0.7
0.5
0.5
0.3
0.3
0.1
0.1
−0.1
−0.1
−0.3 −0.5 −2
−0.3 2
6
10
14
18
−0.5 −2
22
2
6
L=128 1.1
0.8
0.9
0.6
0.7
0.4
0.5
0.2
0.3
0
0.1
2
6
10
14
18
22
14
18
22
L=1024
1.0
−0.2 −2
10
14
18
−0.1 −2
22
2
6
10
Fig.B.4: Cross correlation for the in-phase PN sequence L=4
L=16
1.0
1.0
0.8
0.8
0.6 0.4
0.6
0.2
0.4
0 0.2
−0.2 −0.4
0
−0.6 −0.2
−0.8 −1.0 −2
2
6
10
14
18
−0.4 −2
22
2
6
L=128 1.1
0.8
0.9
0.6
0.7
0.4
0.5
0.2
0.3
0
0.1
2
6
10
14
18
22
14
18
22
L=1024
1.0
−0.2 −2
10
14
18
−0.1 −2
22
2
6
10
Fig.B.5: Cross correlation for the quadrature-phase PN sequence 46
Appendix C C.1
Correlated Rayleigh sequence generation with single IDFT operation
N i.i.d zero mean complex Gaussian variates
Multiplication by Filter sequence
{FM [k]}
N−point Complex IDFT
N Correlated Rayleigh Samples
k = 0, 1, . . . , N − 1
Fig.C.6: Modified Smith’s Simulator
The filter sequence FM [k] is defined as follows
FM [k] =
0 k=0
1 k = 1, 2, . . . , km − 1 k 2· 1−( )2 N ·fm
km π km −1 √ k = km − arctan 2 2 2km −1 0 k = km + 1, . . . , N − km − 1
km π km −1 √ k = N − km − arctan 2 2 2km −1
1 k = N − km + 1, . . . , N − 2, N − 1 2· 1−( N −k )2 N ·fm
(C.1)
Where,
N = Number of correlated Rayleigh samples to be generated fm =
fd fs
= Normalized Doppler frequency
fd = Maximum Doppler frequency fs = Sampling frquency km = fm · N Fig. C.6 shows the simulator for generating rayleigh variables whose PSD follows Jake’s spectrum. The model used here is a modification of Smith’s rayleigh fading simulator of Clarke’s model, given by D.J. Young and N.C. Beaulieu [5] which requires just one IDFT operation as opposed to the two IDFT operations in Smith’s original simulator. Thus the modified simulator consists of only one branch. The modified filter sequence, FM [k], ensures that the output of a single IDFT operation will have independent real and imaginary parts and hence can be used directly for generating the required Rayleigh sequence without the need of second IDFT operation. Thus, we can generate Rayleigh samples having same spectral properties as that of generated by Smiths simulator by using only one IDFT operation. This results in significant reduction in the computational complexity of the algorithm. Also, this method requires less memory compared to that of Smith’s algorithm because of only one IDFT operation.
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Appendix D D.1
The RAKE Combiner
The Rake receiver is based on the principle of Maximal Ratio Combining [2]. It is optimal for a multipath channel where the receiver encounters time-shifted versions of the original signal. If these multipath components are delayed in time by more than a chip duration, they appear like uncorrelated noise at a CDMA receiver. However since there is useful information in the multipath components, the Rake receiver attempts to collect the time-shifted versions of the original signal by providing a separate correlation receiver for each of the multipath components. Fig. D.7 shows a typical Rake receiver implementation. Correlator 1 α1∗
r(t) received baseband signal
Correlator 2
T 0
(·)dt
α2∗
Correlator M ∗ αM
Fig.D.7: A RAKE combiner with M fingers
sgn(·)
The channel estimates (α1 , . . . , αM & τ1 , . . . , τM ) are used for combining the different multipath signals. Hence channel estimation forms a critical component of a system using the Rake receiver. The Rake receiver can also pick up diversity from other base stations in the case of soft hand-off by providing separate correlators for those paths. Although the Rake receiver attempts to gain diversity by combining signals from various paths, it also combines more interference. However, in the presence of good spreading codes, this interference will behave like noise while the desired signal is strengthened. Hence, the RAKE receiver performs better for a multipath channel or for a channel with diversity than any other receiver.
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Bibliography [1] Theodore S.Rappaport, Wireless Communications: Principles and Practice. Pearson Education Asia, 2002. [2] John G.Proakis, Digital Communications - 4th edition. McGraw-Hill, 2001. [3] Fakhrul Alam, “Simulation of Third Generation CDMA Systems”, MS Thesis, Virginia Polytechnic Institute & State University, 1999. [4] Ansgar Scherb, Volker Kuehn and Karl-Dirk Kammeyer, “Pilot Aided Channel Estimation for Short-Code DS-CDMA”, IEEE Seventh International Symposium on Spread Spectrum Techniques and Applications , 2002 [5] Young, D. J., Beaulieu, N. C., “The Generation of Correlated Rayleigh Random Variates by Inverse Discrete Fourier Transform,” IEEE Transactions on Communications, vol. 48, pp. 1114-1127, July 2000. [6] Esmael H. Dinan and Bijan Jabbari, “Spreading Codes for Direct Sequence CDMA and Wideband CDMA Cellular Networks”, IEEE Communications Magazine, September 1998. [7] Joseph C.Liberti, Jr., Theodre S.Rappaport, Smart antennas for wireless communications: IS-95 and third generation CDMA applications, Prentice Hall, 1999. [8] Gregory E. Bottomley, Tony Ottosson, Yi-Pin Eric Wang, “A Generalized RAKE
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Receiver for Interference Suppression”, IEEE Journal on selected areas in communications, Vol. 18, No. 8, AUGUST 2000 [9] 3GPP2
Technical
specification,
3GPP2
C.S0002-B,
available
at
http://www.3gpp2.org [10] TIA/EIA/IS-2000-2: Physical Layer Standard for cdma2000 Spread Spectrum Systems, TIA/EIA Interim Standard (URL: http://www.tiaonline.org) [11] Scott Baxter, http://www.howcdmaworks.com
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