Performance of Quadratic and Exponential Multiuser Chirp Spread ...

Performance of Quadratic and Exponential Multiuser Chirp Spread Spectrum Communication Systems Muhammad Ajmal Khan, Raveendra K. Rao, Xianbin Wang Department of Electrical and Computer Engineering Western University, London, Ontario, Canada, N6A 5B9. Abstract-A novel non-linear chirp spread spectrum modu­

lation (CSSM) is introduced for binary data transmission in a multi-user (MU) environment. Two subclasses of non-linear signals namely quadratic (Q-CSSM) and exponential (E-CSSM) modulations are described and their properties are given. The chirp rates in these modulations are varied as a function of user in an MU environment using the orthogonal structure inherent in non-linear chirp signals. A generic MU communication system model that employs non-linear chirp signals is presented and its bit error rate (BER) performance is analyzed in additive white Gaussian noise (AWGN) channel, and Rayleigh and Nakagami­ m

fading environments as a function of the number of users

in the system, signal-to-noise ratio (SNR), and multiple access interference (MAl). An investigation of the trade off between bandwidth and the number of users in the system is provided for both Q- and E-CSSM. Numerical results demonstrate that these proposed modulations with proper chirp rate assignment are very effective in reducing MAl.

I.

INTRODUCTION

The number of users for wireless and mobile communi­ cation networks is growing at a rapid pace. Thus, demand­ ing advanced multiple-access techniques which can accom­ modate more number of users in the available bandwidth while reducing multiple access interference (MAl), multipath channel distortions and Doppler spreading. Various multiple­ access techniques such as frequency division multiple access (FDMA), time division multiple access (TDMA), code division multiple access (CDMA), etc. are available to provide wireless services to a large number of users [1]. Chirp spread spectrum (CSS) is a multiple-access technique that is robust to multipath interference, can realize higher processing gains and have been used in wireless communications to improve the performance due to its anti-jamming characteristic [2] [3]. The conventional chirp signals are linear frequency-modulated signals and have been widely used in radar and sonar applications. Chirp signals have also been used in various other applications such as combating multipath interference [4], spread spectrum techniques [5], modulation in multiple-access schemes [6]-[8], equalization [9], and channel estimation [10]. In chirp modulation, the frequency is varied linearly or nonlinearly as a function of time. Linear chirp signals have been widely analyzed for applications in spread spectrum multiple-access systems such as in [11] [12], where authors have successfully applied linear chirp signals with unique chirp rate as well as multi-chirp rate in multiuser environment. Linear chirp modulation in fading channels as well as in the maximum ratio combining (MRC) diversity system has been investigated in [13] [14], and closed-form error probability

expressions have been derived. A multiuser MIMO commu­ nication system using linear chirp spread spectrum has been recently proposed in [15] [16] and analyzed using computer simulations. In the literature, nonlinear chirp modulation is sparsely treated. Takeuchi and Yamanouchi [17] developed a modulator and demodulator for DPSK modulated nonlinear chirp signals in a single user environment. Machineni et. a!. [18] have proposed multiuser schemes using nonlinear logarithmic chirp modulation. In this paper, we propose new multiuser schemes using nonlinear chirp modulations. Two subclasses of modulations referred to as Q-CSSM and E-CSSM are proposed for ap­ plications in MU environments. In particular, we exploit the signaling structure in these modulations to design the multiuser communication system for minimum MAl among the users. Mathematical conditions that guarantee orthogonality among the set of chirp signaling rates are derived for minimum MAL Also, a comparison of bandwidth requirements for Q-CSSM and E-CSSM as a function of number of users in the system is provided. The error performance of the proposed multiuser schemes is obtained over AWGN channel, and Rayleigh and Nakagami-m fading channels through Monte Carlo simula­ tions. The main contributions of the paper are: (i) derivation of conditions to achieve unique chirp rate for each user, (ii) comparison of proposed nonlinear chirp modulations, and (iii) performance analysis of the proposed chirp modulations in multiuser environment through Monte Carlo simulations. The paper is organized as follows: Section II describes the multiuser chirp spread spectrum modulation (CSSM), followed by derivations associated with conditions for unique Q-CSSM and E-CSSM that minimize MAl in Sections III and IV, respectively. In Section V, a comparison of these two systems is given. Performance analysis of MU communication system for the two modulations is presented in Section VI. The paper is concluded in Section VII. II.

NONLINEAR CHIRP SPREAD SPECTRUM SY STEM

A chirp (also known as frequency modulated) signal is a sinusoidal signal whose frequency varies with time. Mathe­ matically, chirp signal is represented by [19] (1) where a is the chirp rate (or frequency modulated rate), 'f/(t) and v(t) are phase and frequency functions, respectively, and they are related as v(t) -!ft'f/(t). The instantaneous 2a'f/(t) and frequency of the chirp signal is given by f(t) the bandwidth is f(T) 2a'f/(T). =

B

=

=

=

In binary chirp modulation, a positive chirp rate (i-e. up­ chirp signal) is used to represent bit 1 and a negative of the same rate is used (i-e. down-chirp signal) to represent bit -1. Thus, the transmitted baseband signal with the information bit b E ±1 can be represented as [19]

s(t;b)

=

y!v(t)ej27rbQ1](t)

(2)

A multiuser chirp spread spectrum (CSS) communication system with K synchronous users for coherent detection is shown in Figure 1. The input data of each user bk is assumed to be a sequence of ±1 from an equally likely and statistically independent data source.

proposed multiuser Q-CSSM scheme can be viewd in time­ frequency (TF) plane, as shown in Figure 2. In this Figure, unique chirp rates (CR) for a 5-user system are shown for information bits b E ±1. BKmax

"--'---'--'-;::=�=�=:':::==:;'----'---''--A

>u C

..::> �

u.

Chirp Despreading & Demodulation with Chirp Rate a,

User 2 Data

-- Information bit = + 1

,

,,

a'

Chirp Despreading & Demodulation with Chirp Rate a

,

- - - Information bit = -1

,

,User 5 , , User 4 ,

,

,

,,

,,

User.

,,

Use� 3 ,User 2

,,

User

,,

User 2 User 1

_User 1

6,

- -

0

Chirp Despreading & Demodulation with Chirp Rate aK

User 5

Time

-=-����

-

T

Fig. 2: Unique CR for Multiuser Q-CSSM in TF Plane.

Fig. 1: Multiuser CSSM System Model.

A unique chirp rate is assigned to each user to transmit syn­ chronously in the system. For a K -user system, the baseband signal for the kth user is defined by

Sk(t;bk) y!v(t)ej27rbkQk1](t), where CXk is the chirp rate assigned

k

=

=

to user

1,2, . . . ,K. (3) k and bk E ±1 is

In the multiuser Q-CSSM system, chirp rates should be assigned to the users in such a way as to minimize the cross correlation between their signals. In this section, we develop conditions on the chirp rates in order to minimize the cross correlation and thus, minimizing multiple access interference (MAl) among users. Specifically, consider User k and User m with the Q-CSSM signals Sk(t;bk) and sm(t;bm), respectively.

sk(t;bk) sm(t;bm)

the information bit of user k. After spreading, each user's data is transmitted over an independent Rayleigh fading channel and each fading channel is assumed to be uncorrelated from each other. Then, data is perturbed by the additive white Gaussian noise (AWGN) with zero-mean and power spectral density (PSD) No/2. We assume AWGN to be statistically independent from channel as well as independent of the fading amplitude. The received signal is de­ spreaded with the assigned chirp rate and decorrelator detector is employed to decide each user's data. The perfect knowledge of the channel state information (CSI) is also assumed to be available at the receiver but transmitter has no information of the channel. III.

Q UADRATIC

CHIRP SPREAD SPECTRUM SY STEM

If the instantaneous frequency of the signal varies quadrat­ ically with time, then it is known as quadratic chirp spread spectrum modulation (Q-CSSM). It has non-linear frequency variation and can be defined by phase function 1](t) It31 and frequency function v (t) 3t2 , thus the quadratic chirp spread spectrum (CSS) signal for the kth user can be represented by =

=

k

=

1, 2, . . . ,K.

(4)

=

=

v3 tej27rbkQkt3 v3 t ej27rbmQmt3

(5)

The signal energy is given by [20]:

Es

=

l T Is(t;b)12 dt

=

T3

(6)

The cross correlation between the two signals is given by [20] [21]:

T 1 r Sk(t;bk)S�(t;bm)dt JESkEsm Jo -bmQm)T3 sinc [(bkCXk - bmcxm) T3] j7r(bkQk e

Pk,m(bk,bm) =

=

(7)

where * denotes conjugate of the signal. We notice that corre­ lation in (7) contains sinc term, which plays an important role to minimize the correlation. Thus, we consider the following three scenarios:

Scenario

1: When signals of users are orthogonal, MAl is minimum and thus, the signals have zero cross correlation. Therefore, setting (7) to zero, we get

ej7r(bkQk-bmQm)T3 sinc [(bkCXk - bmcxm) T3]

=

0

(8)

which gives The instantaneous frequency in Q-CSSM is given by f(t) 6cxt2 and the bandwidth is f(T) 6cxT2. The =

B

=

=

(9)

Scenario

2: For the same user transmission, it is required to distinguish between two different data bits that is between up-chirp and down-chirp of the same user. Thus, up-chirp and down-chirp of the same user should have zero cross correlation.

Pk,k(bk, -bk)

=

ej7r(2bkG:k)T3 sinc [(2bko:k) T3]

=

0

BKmax

O: k bk

=

2T3'

C2

=

±1, ±2,...

(10)

>­ () c , CI> ,

l!!

u.

(11)

,, ,

,

,

,

,

,

User User 2

' .!lser 2

=

In order to reduce MAl in the proposed Q-CSSM system, unique chirp rates to each user is assigned using (9) and (11). Therefore, we assign the unique chirp rate to kth user in a K-User system as follows k

=

1,2, . . . ,K.

(12)

The total bandwidth required for the multiuser system from (12) is obtained as BK 20:KT (2K - 1)jT2, which gives the maximum number of users, the system can support, as =

Kmax

_

-

=

l

BK=ax T2 +

2

1

J

(13)

o

0:k

=

2k - 1 2T3

l�J BK

=

l BT2 J

2k - 1 2T3 2K - 1

(14)

EXPONENTIAL CHIRP SPREAD SPECTRUM SY STEM

In exponential chirp spread spectrum modulation (E­ et CSSM), the signal is defined by the phase function T)(t) et , where instantaneous and the frequency function v(t) frequency of the signal varies exponentially with time. Thus, the signal for the kth user is defined by =

=

k

=

1,2, . . . ,K.

(15)

The instantaneous frequency in E-CSSM is given by f(t) and the bandwidth is B f(T) 20:eT. Figure 3 demonstrates the proposed multiuser E-CSSM system in time­ frequency (TF) plane, where unique chirp rates (CR) of 5-user system is shown for information bits b E ± 1. =

20:et

=

Fig, 3: Unique CR for Multiuser E-CSSM in TF Plane,

and

O:k

=

x

ej7r(bkG:k-b=G:=)(eT +1) sinc

[(bkO:k - bmo:m) (eT - 1)]

(16)

Similarly, we obtain the following conditions that reduce MAL Cl (17) Cl ±1, ±2, . . . O:k ± O: m ---' =

eT - 1

=

2(eT - 1)

, C2

=

±1, ±2, , . .

(18)

Hence, the maximum number of users the system can support is given by Kmax

_

-

l

BK=ax (eT

- 1) + T 2T

J

(19)

From (17) and (18), we assign the chirp rates in multiuser exponential CSS system as

O:k

=

l J

2k - 1 2(eT 1)

B

BK

_

V.

=

l

2k - 1 B (eT - 1) 2(eT - 1) ( 2K 1)T _

J

(20)

COMPARISON BETW EEN QUADRATIC AND EXPONENTI AL

CSSM

In order to find the relationship between quadratic and exponential CSSMs, we compare the required bandwidth for a fixed number of users in both the systems. To accommodate K users, if BQ and BEare the required bandwidths for quadratic and exponential CSSMs, respectively, then equating (13) and (19), we get

=

Following similar approach as explained in Section III for Q-CSSM, we derive the cross correlation between exponential chirp signals as well as conditions to minimize MAL Thus, the cross correlation is given by [20] [21]:

Pk,m(bk,bm)

T

Time

In order to reduce MAl as well as to utilize the whole available bandwidth B, we assign the chirp rates as gIven below

IV.

,

US'e(3

g.

When same user transmits the same information bit (i-e. bm), the autocorrelation becomes unity, Pk,k(bk,bk) 1.

=

- Information bit = + 1

- - - Information bit = -1

which gives

C2

r,--.----.---r;==:::':===::c:==�==:;-r-----,---.---_____::1

=

(21) Typically, symbol duration T is of the order of 10 -3, which makes the right-hand side of (21) greater than unity, Therefore, it is evident from (21) that quadratic CSSM requires higher amount of bandwidth than exponential CSSM to provide optimum performance without MAL VI.

NUMERICAL RESULTS

In this section, we present the error performance of the proposed multiuser Q-CSSM and E-CSSM systems through computer simulations, The success of proposed multiuser schemes using nonlinear CSSM to reduce MAl is demonstrated by using the conditions derived in (14) and (20) to assign

unique chirp rate to each user. Numerical results are obtained by Monte Carlo simulations using over 107 samples for information bits and the generation of the fading envelopes at each signal-to-noise ratio (SNR). All users are assumed to transfer data synchronously and their data is detected at the receiver coherently.

--1 - * -2 - � -3 · · · D · · 5

Figures 4 and 5 depict the error performance of multiuser Q-CSSM scheme over AWGN and Rayleigh fading channel, respectively. These performance results are obtained for differ­ ent number of users (K 1,2,3, and 5). Chirp rates to each user is assigned using the conditions derived in (14) for Q­ CSSM system. It is evident from these figures that 2-, 3-, and 5-User Q-CSSM system have the same performance as that of I-User system, which verifies that MAl has been completely removed by using chirp rate conditions.

User Users Users Users

=

10° r-----�----�--�==�====� - e -1 User · · + · · 2 Users · - * - . 3 Users - "If -5 Users

10� '----�---'--'---�--' -10 -5 o

Fig. 6: Multiuser E-CSSM over Nakagami-m (m Channel.

>-

:::

�

01

t

c: :5

10

1

-

I- -- . .. � ...

....... .. "".

1�

2

t:: 10-3

5

10

Fig. 4: Multiuser Q-CSSM over AWGN Channel.

- + -1 User - * - 2 Users . -8- · 3 Users · · · D · · 5 Users

�. 'fl.

��

�

\

8, 10-4

� �

2) Fading

'� ,,� . '"

�

-5

.... .

=

'