 Open Access
 Total Downloads : 613
 Authors : Vaishali Patil, Prof Jaikaran Singh, Prof Mukeshtiwari
 Paper ID : IJERTV1IS3083
 Volume & Issue : Volume 01, Issue 03 (May 2012)
 Published (First Online): 30052012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Optimization of Generalized Discrete Fourier Transform for CDMA
Vaishali Patil
Department of Electronics and Communication
Sri SatyaSai Institute of Science and Technology, Sehore M.P.
Prof Jaikaran Singh
Department of Electronics and Communication
Sri SatyaSai Institute of Science and Technology, Sehore M.P.
Prof MukeshTiwari Department of Electronics and Communication
Sri SatyaSai Institute of Science and Technology, Sehore M.P.
ABSTRACT
Generalized Discrete Fourier Transform (GDFT) with nonlinear phase is a complex valued, constant modulus orthogonal function set. GDFT can be effectively used in several engineering applications, including discrete multitone (DMT), orthogonal frequency division multiplexing (OFDM) and code division multiple access (CDMA) communication systems. The constant modulus transforms lik e discrete Fourier transform (DFT), Walsh transform, and Gold codes have been successfully used in above mentioned applications over several decades. However, these transforms are suffering from low cross correlation features. This problem can be addressed by using GDFT transform. This paper describes the optimization of Generalized Discrete Fourier Transform with NonLinear Phase for code division multiple access (CDMA). We have also implemented Gold, Walsh and DFT codes. Their performance is analyzed and compared on the basis of various parameters such as Maximum Value of OutofPhase AutoCorrelation, Maximum Value of OutofPhase CrossCorrelation, MeanSquare Value of AutoCorrelation, MeanSquare Value of Cross Correlation and merit factor. The result of simulation in the form o f co mparison of Biterrorrate (BER) and Signaltonoise (SNR) ratio for various spreading codes is presented.
Keywords
CDMA, Simu lation, Orthogonal, BER, SNR, corre lation

INTRODUCTION
CDMA uses unique spreading codes to spread the baseband data before transmission. The signal is transmitted in a channel, which is below noise level. The receiver then uses a correlator to dispread the wanted
signal, which is passed through a narrow band pass filter. Unwanted signals will not be dispread and will not pass through the filter. Codes take the form of a carefu lly designed one/zero sequence produced at a much higher rate than that of the baseband data. The rate of a spreading code is referred to as chip rate rather than bit rate. The concept of CDMA is based around the fact that a data sequence is mult iplied by a spreading code or sequence which increases the bandwidth of the signal. Then within the receiver the same spreading code or sequence is used to extract the required data. Only when the required code is used, does the required data appear fro m the signal. The process of e xtracting the data is called corre lation. When a code exactly the same as that used in the transmitter is used, then it is said to have a correlation of one and data is e xtracted. When a spreading code that does not correlate is used, then the data will not be extracted and a different set of data will appear. This means that it is necessary for the same spreading code to be used within the transmitter and receiver for the data to be extracted.
1.1 CDMA CODE TYPES
There are several types of codes that can be used within a CDMA system for providing the spreading function:
PN codes: Pseudorandom number codes
(pseudonoise or PN code) can be generated very easily. These codes will sum to zero over a period of time. A lthough the sequence is deterministic because of the limited length of the linear shift register used to generate the
sequence, they provide a PN code that can be used within a CDMA system to provide the
spreading code required. They are used within many systems as there is a very la rge number that can be used.
1
A feature of PN codes is that if the same versions of the PN code are time shifted, then they become almost orthogonal, and can be used as virtually orthogonal codes within a CDMA
system.
Truly orthogonal codes: Two codes are said to be orthogonal if when they are mu ltip lied together the result is added over a period of time they sum to ze ro. For e xa mp le a codes 1 1 1 1 and 1 1 1 1 when multip lied together give 1 1 – 1 1 which gives the sum ze ro. An exa mple of an orthogonal code set is the Walsh codes used within the IS95 / CDMA2000 system.
Let {er(n)} be a periodic, constant modulus, comple x
sequence which can be e xpressed as the rth power of the first primitive Nth root of unity z1 raised to the nth power as [1]
(1)
n = 0,1,2,……….., N 1 and r = 0,1,2,………………., N1
The comple x sequence (1) over a finite d iscretetime interval in a geo metric series is e xpressed as follows [10],[11]:
Among known binary spreading code families, Gold
codes have been successfully used for asynchronous communicat ions in direct sequence CDMA (DS CDMA)
systems due to their lo w cross correlation features.
Walsh, Go ld, Wa lshlike [3] and several other binary spreading code sets are designed to optimize so called
even correlation functions. However, the odd correlations
(2)
are as important as even correlations. Therefore, Fuku masa, Kohno and Ima i proposed a new set of comple x pseudorandom noise (PN) sequences, called equal odd and even (EOE) sequences, with good odd and even correlations [4]. EOE sequences are generated by using one of the binary code sets like Go ld or Walsh.
More recently, research has refocused on constant modulus spreading codes for radio communicat ions applications due to the efficiency limitations of nonlinear
Then, (2) is rewritten as the definition of the discrete Fourier transform (DFT) set {ek(n)} satisfying the orthonorma lity conditions
gain characteristics of co mmonly used power a mp lifiers in transceivers. Hence, the comple x roots of unity are
wide ly proposed as comple x spreading codes by several authors in the literature. All codes of such a set are placed on the unit circ le of the comp le x p lane. FrankZadoff, Chu and Oppermann have forwarded a variety of co mp le x spreading codes [5][8]. Moreover, Oppermann has shown that FrankZadoff and Chu sequences are special
cases of his fa mily of spreading sequences [9]. This paper introduces generalized d iscrete Fourier
transform (GDFT) with nonlinear phase. GDFT provides a unified theoretical fra me work where popular constant
modulus orthogonal function sets including DFT and others are shown to be special solutions. Therefore,
(3)
GDFT provides a foundation to explo it the phase space in its entirety in order to improve correlat ion properties of constant modulus orthogonal codes. We present GDFT and demonstrate its improved correlations over the popular DFT, Go ld, Walsh and Oppermann fa milies leading to superior communicat ions performance for the scenarios considered in the paper.

DISCRETE FOURIER TRANS FORM (DFT)
The notation (*) represents the comple x conjugate
function of a function. One might rewrite the first primitive Nth root of unity as where , and it is called the fundamental frequency defined in radians. We are going to extend the phase functions in (3) in order to define the nonlinear phase GDFT in the fo llo wing section.

GEN ERALIZED DISCRET E FOURIER TRANS FORM (GDFT)
(10)
Let us generalize (2) by re writ ing the phase as the diffe rence of two functions
and expressing a constant modulus orthogonal set as follows,
The otation indicates that conjugate and transpose operations applied to the matrix, and is the identity matrix. Therefore, the transform kernel generating A GDFT matrix through this methodology is expressed as follows:
(11)
Therefore, by inspection
(4)
(5)
4.1 ALGORITHM
S tep 1) Find N X N DFT
k,n=0,1,,N1
S tep 2) Find G1 & G2
Hence, the basis functions of the new set are defined as
(6)
We call th is orthogonal function set as the Generalized Discrete Fourier Transform (GDFT).

GDFT DES IGN MET HODS
Step 3) Calculate GDFT
(7)
The square GDFT matrix as a product of the three orthogonal matrices as follows
,
,
(8)
Where G1 and G2 are constant modulus diagonal matrices and written as follows:
(9)
and
Step 4) Find dam, dcm, dmax, RAC, RCC, F

Maximum value of outofphase Auto correlation:

Maximum value of outofphase Cross correlation:

Mean squa re Value of AutoCorrela tion,RAC.

Mean squa re Value of CrossCorrela tion,RCC

The Meri t Fa ctor.
S tep 5) S imulate using AWGN S tep 6) Plot BER vs S NR


Optimal Design of GDFT
The optima l design of phase shaping function (n), based on a performance metric is presented in this section . The phase function {k(n) n}, of (8) is now deco mposed into two functions in the time variab le as follows:
The linear term of phase function kn is highlighted due to its significance in the orthogonality require ments . The GDFT fra mewo rk offers us the fle xib ility to define the phase shaping function according to the design require ments. Note that any function will give us an orthogonal GDFT.
The crosscorrelation sequence of a GDFT basis function pair (k ,l) with length N is defined as
The autocorrelation function of a GDFT basis function as

RES ULT & DISCUSS ION
The GDFT and other codes are implemented in M atlab and the results are shown below
Fig 1 : The Periodic constant modulus sequence er(n)
Fig 1 shows er(n) which is a periodic constant modulus sequence and plays an important role in Generalized Discrete Fourier Transform. It is a complex valued sequence as the rth power of the first primitive Nth root of unity. Hence it has zeros on the unit circle in zplane as shown in fig. 2
Fig 2 : Pole/zero plot for primitive Nth root of unity
As spreading codes are to be dispread after transmission, the correlator function is required for it. It has been found that the GDFT provides good Autocorrelation and Crosscorrelation values. Following figures depicts the results of simulation of different spreading codes such as Walsh, Gold, DFT and GDFT. Their Correlation is displayed along with Bit error rate and SignaltoNoise ratio.
Fig 3 : Performance comparison of various codes
Fig 4 : Optimum Design of (n) using RAC
Fig 5 : Optimum Design of (n) using RCC
The ma in advantage of the proposed method is the ability to design a wide collection of constant modulus orthogonal code sets based on the desired correlation performance mimic king the specs of interest. Moreover, the proposed GDFT technique can also be considered as a natural enhancement to DFT to obtain improved performance. Note that the autocorrelation magnitude functions of the codes in any GDFT set are the same. displays autocorrelation function of a size 16 GDFT code optimized based on along with size 16 DFT set for comparison purposes.
Similarly, crosscorrelation functions (CCF) of the first and second codes of optima l GDFT design based on metric and DFT set for a re displayed in Fig. 6. These figures highlight the me rit of the proposed GDFT fra me work over the traditional DFT.
In order to compare performance of code families, several objective performance metrics are used. All the metrics used in this study depend on aperiodic correlat ion functions (ACF) of the spreading code set
Table 1 : Performance comparison of various codes
Code
dam
Dcm
Dmax
RAC
RCC
F
Walsh
1.000
0.031
1.000
1.960
0.002
2.213
Gold
1.000
0.306
1.000
0.247
0.007
0.786
8 x 8
DFT
0.875
0.326
0.875
1.637
0.006
0.002
16 x 16
DFT
0.937
0.088
0.937
7.318
0.002
0.001
GDFT
0.342
0.454
0.454
0.335
0.014
0.006

CONCLUS ION
In this paper, we have discussed the MATLAB implementation of Walsh, Gold, DFT and GDFT codes. These codes are simulated and their results are presented. It has been observed from the results that GDFT provides better and effective correlation functions which can be exploited in optimum way in asynchronous CDM A communication systems. However, there is further scope for optimization.

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