1. On the Design of RAKE Receivers with Non-uniform Tap SpacingBy
K. B. Baltzis and J. N. Sahalos
RadioCommunications Lab., Department of Physics,
Aristotle University of Thessaloniki, Greece.
July 2006
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki

2. CONTENTS 1. Abstract. 2. Introduction.
3. Transmitter and Channel Model.
4. Proposed Receiver Model.
5. The Maximum Power Minimum Correlation (MPMC) Criterion.
6. Numerical Examples.
7. Conclusions.
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloni

3. ABSTRACT
The effect of Non-Uniform tap spacing on the performance of a RAKE receiver is studied.
A new RAKE receiver, the MPMC RAKE, is suggested.
Taps positions optimization is done according to the MPMC criterion.
MPMC criterion considers the total received signal autocorrelation properties at the correlators outputs of the receiver.
Numerical results, comparisons and discussions are provided.
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki

4. DS/SS (WCDMA) Used in 3G Communication Systems.
INTRODUCTION
DS/SS (WCDMA) Used in 3G Communication Systems.
RAKE diversity Used to combat Multipath Fading.
Maximal Ratio Combining, MRC
Equal Gain Combining, EGC
(Generalized) Selection Combing, (G)SC
Maximum Likelihood criterion, ML
Implementation
Strategies
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloni

5. xi, ith-correlator output Z, decision variable RAKE receiver model
INTRODUCTION
r(t), received signal
W, signal bandwidth
L, number of branches
xi, ith-correlator output
Z, decision variable
RAKE receiver model
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki

6. Usually taken equal to chip period.
INTRODUCTION
TAP SPACING:
Usually taken equal to chip period.
MRC is optimum under the assumption of independent
branch signals, Dong and Beaulieu, [2002].
ML criterion is optimal when tap spacing is set less
than chip duration, Kim et al., [2000].
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki

7. ASSUMPTIONS The modulation scheme is a BPSK one.
TRANSMITTER AND CHANNEL MODEL
ASSUMPTIONS
The modulation scheme is a BPSK one.
Signal energy per bit Eb is assumed equal for all users.
Channel is modeled as a WSSUS frequency-selective Rayleigh fading one.
Transmitted pulses are time-limited rectangular.
Power Delay Profile (PDP) is uniform or exponential.
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki

8. EQUIVALENT LOWPASS TRANSMITTED SIGNAL OF THE kth–USER:
TRANSMITTER AND CHANNEL MODEL
EQUIVALENT LOWPASS TRANSMITTED SIGNAL OF THE kth–USER:
bit energy
processing gain
binary data sequence of the kth user
PN signature sequence of the kth user
the largest integer not greater than
chip duration
normalized chip waveform
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloni

9. TOTAL RECEIVED EQUIVALENT LOWPASS SIGNAL
TRANSMITTER AND CHANNEL MODEL
TOTAL RECEIVED EQUIVALENT LOWPASS SIGNAL
AT THE RECEIVER FROND END (K ACTIVE USERS):
channel impulse response of the kth user’s link at delay τ and time instant t
time of arrival of the kth user’s signal
low pass equivalent process of AWGN
Complex zero-mean Gaussian random process
It is related to the PDP function g(t) with the expression:
Delay depends on the time instant t:
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki

10. ASSUMPTIONS Desired user channel impulse response can be estimated.
PROPOSED RECEIVER MODEL
ASSUMPTIONS
Desired user channel impulse response can be estimated.
Amplitude, phase and timing of the desired user’s signal are known.
Chip waveform shaping filters in transmitter and receiver are known.
Average received signal energy is the same for all users (power control).
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki

11. CORRELATOR OUTPUT: DESIRED SIGNAL COMPONENT: ISI COMPONENT:
PROPOSED RECEIVER MODEL
CORRELATOR OUTPUT:
DESIRED SIGNAL COMPONENT:
ISI COMPONENT:
MAI DUE TO THE kth USER
COMPONENENT:
AWGN COMPONENT:
Symbol denotes the convolutional operator
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki

12. the first bit of the desired user data sequence
PROPOSED RECEIVER MODEL
the first bit of the desired user data sequence
the discrete crosscorrelation function between the desired and the kth user
the autocorrelation function of the chip waveform.
DESIRED USER CHANNEL IMPULSE RESPONSE:
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki

13. PROPOSED RECEIVER MODEL
ITS MAIN CHARACTERISTIC
IS THE NON-UNIFORM TAP
SPACING
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki

14. Block diagram of Correlation Coefficients Estimator (CCE) Unit
PROPOSED RECEIVER MODEL
Block diagram of Correlation Coefficients Estimator (CCE) Unit
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki

15. Block diagram of SUM1 and SUM2 Units
PROPOSED RECEIVER MODEL
Block diagram of SUM1 and SUM2 Units
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki

16. Multi-objective optimization problem
THE MAXIMUM POWER MINIMUM CORRELATION (MPMC) CRITERION
DEFINITION OF MPMC CRITERION
Optimum receiver performance is gained when a simultaneous maximization of the sum of squares of average received signal power in each branch and minimization of the sum of squares of autocorrelation between each pair of branches takes place
Multi-objective optimization problem
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki

17. the taps settings vector
THE MAXIMUM POWER MINIMUM CORRELATION (MPMC) CRITERION
DEFINITIONS:
the taps settings vector
the total signal average power coefficients vector
the total signal autocorrelation coefficients matrix
It is:
the autocorrelation function of X(t) given b1,0
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki

18. PROBLEM: Euclidean norm of Applied in the Decision Unit
THE MAXIMUM POWER MINIMUM CORRELATION (MPMC) CRITERION
PROBLEM:
Euclidean norm of
Applied in the Decision Unit
Hilbert-Schmidt norm of
Created in CCE Unit
Calculated in SUM1 and SUM2 Units
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki

19. LEXICOGRAPHIC method has been adopted for the optimization
THE MAXIMUM POWER MINIMUM CORRELATION (MPMC) CRITERION
Finally MPMC criterion is defined as:
the antiderivative function of g(t)
the inverse function of g(t)
LEXICOGRAPHIC method has been adopted for the optimization
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki

20. SIMULATION CHARACTERISTICS:
NUMERICAL RESULTS
SIMULATION CHARACTERISTICS:
Processing gain N = 256
Constant tap spacing in MRC RAKE Tr = Tc
Constant tap spacing in ML RAKE, (Kim et al.), Tr = 0.7Tc
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloni
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki

21. tmax = 2Tc K = 10 Uniform PDP NUMERICAL RESULTS INTERFERENCE LIMITED
SYSTEM
TAPS SETTINGS:
MPMC 3RAKE:
0.28, 0.98, 1.69 (Tc )
MPMC 4RAKE:
0.28, 0.65, 1.3, 1.7 (Tc )
MPMC 4RAKE: 40 – 50% smaller Pe compared to ML 4RAKE at Eb/N0 = 15dB
60 – 80% smaller Pe compared to ML 4RAKE at Eb/N0 = 30dB
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloni

22. Pe<10-3 tmax = 2Tc NUMERICAL RESULTS Uniform PDP
INTERFERENCE LIMITED
SYSTEM
MPMC RAKE: 40 – 50% increase in the number of users compared to MRC RAKE
20 – 30% increase in the number of users compared to ML RAKE
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloni
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki

23. Uniform PDP NUMERICAL RESULTS 3rd tap 2nd tap 1st tap
OPTIMUM TAPS POSISTIONS
MPMC 3RAKE
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloni

24. ARE NOT AFFECTED FROM THE NUMBER OF USERS
NUMERICAL RESULTS
ARE NOT AFFECTED FROM THE NUMBER OF USERS
OR THE VALUE OF SIGNAL TO NOISE RATIO.
EXAMPLE (MPMC 3RAKE, tmax = 2Tc):
OPTIMUM TAPS POSITIONS
ONLY CHANNEL CHARACTERISTICS
HAVE AN IMPACT ON THEM.
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloni
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki

25. MORE SIGNIFICANT IMPOVEMENT IN PERFORMANCE FOR THE CASES OF:
NUMERICAL RESULTS
Exponential PDP, tspr = Tc
Exponential PDP, tspr = 2Tc
MORE SIGNIFICANT IMPOVEMENT IN
PERFORMANCE FOR THE CASES OF:
UNIFORM PDP
LARGER CHANNEL SPREAD
Uniform PDP, tmax = Tc
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloni

26. MORE SIGNIFICANT IMPOVEMENT IN PERFORMANCE FOR THE CASES OF:
NUMERICAL RESULTS
Exponential PDP, tspr = Tc
Exponential PDP, tspr = 2Tc
MORE SIGNIFICANT IMPOVEMENT IN
PERFORMANCE FOR THE CASES OF:
UNIFORM PDP
LARGER CHANNEL SPREAD
Uniform PDP, tmax = Tc
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloni

27. NUMERICAL RESULTS
IMPERFECT CHANNEL IMPULSE RESPONSE ESTIMATION OR / AND PARTIAL KNOWLEDGE OF CHANNEL PDP DEGRADES RECEIVER PERFORMANCE
TWO CASES ARE STUDIED:
THE “OPTIMIZED TAPS”
1. IMPERFECT DESIRED USER CHANNEL IMPULSE RESPONSE
ESTIMATION
THE “NON-OPTIMIZED TAPS”
ESTIMATION.
2. TAPS OPTIMIZATION IS DONE ACCORDING TO THE AVERAGE AND
NOT THE INSTANTANEOUS PDP VALUE.
TAPS POSITIONS DO NOT CHANGE.
THIS IS ALSO THE CASE BEFORE THE TAPS ACQUIRE THEIR OPTIMIZED VALUES (TRAINING PERIOD)
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloni

28. K = 10 MPMC 3RAKE Uniform PDP
NUMERICAL RESULTS
MPMC 3RAKE
Uniform PDP
K = 10
OPTIMIZED TAPS: ITS PERFORMANCE COMPENSATES FOR THE COMPLEXITY
NON-OPTIMIZED TAPS : SIMPLE – LOW COMPUTATIONAL COST
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloni
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki

29. CONCLUSIONS
A RAKE receiver with non-uniform taps distribution has been proposed.
Determination of the optimum taps positions is based on the correlation properties of the signal components in each branch.
The Maximum Power Minimum Correlation (MPMC) criterion has been proposed for the optimization of taps distribution (multi-objective optimization problem).
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloni
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki

30. CONCLUSIONS
Comparisons with other implementations have exhibited the improved performance of the proposed receiver especially at higher values of signal to noise ratio.
Optimum taps settings depend only on channel characteristics.
Channel estimation errors does not affect significantly receiver performance.
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloni

31. On the Design of RAKE Receivers with Non-uniform Tap SpacingBy
K. B. Baltzis and J. N. Sahalos
RadioCommunications Lab., Department of Physics,
Aristotle University of Thessaloniki, Greece.
July 2006
 RadioCommunications Laboratory - Dept. of Physics - Aristotle University of Thessaloniki