Secure Communication in Asynchronous Noise Phase Shift Keying ...

Secure Communication in Asynchronous Noise Phase Shift Keying CDMA Systems Ramin Vali

Stevan M. Berber

Department of Electrical and Computer Engineering The University of Auckland Auckland, New Zealand

Department of Electrical and Computer Engineering The University of Auckland Auckland, New Zealand

Abstract—Gaussian distributed random signals are analyzed for encoding and spreading users’ message sequences in code division multiple access systems (CDMA). Using theoretical analysis and practical implementation in DSP technology it has been shown that the random sequences generated from the Gaussian distribution can be efficiently used in this CDMA system. In this way the security of the message transmission is enhanced due to the random nature of all signals generated and used for users’ message signal spreading and coding in this communication system. The signal processing blocks of the system, including the transmitter, receiver and the channel, are theoretically described. The expressions for the probability of error are derived for the case when the additive white Gaussian noise and fading are present in the channel. The system, including the transceiver and the channel, is implemented in DSP technology and the theoretical results are confirmed by measurements on the designed system. Moreover the acquisition phase of a synchronization block using Gaussian distributed random signals is also theoretically analyzed and implemented in DSP technology. Keywords Noise Phase Shift Acquisition; Slow Flat Fading;

Keying;

According to the knowledge of the authors, spread spectrum signal acquisition has not been attempted using a Gaussian distributed random signal as the spreading sequence. In order to confirm feasibility of the noise based CDMA system a prototype of the system has been designed in DSP technology using TIGERSharc Analog Devices Processor. The prototype incorporated the CDMA transceiver, the noise and fading generators. The system is investigated and BER curves were generated for various scenarios, including the influence of spreading factor, number of users in the system as well as the influence of noise and fading on signal transmission. In Section 2 the theoretical model of the system in the presence of noise is presented. The influence of fading is analyzed in Section 3. In Section 4 the process of signal acquisition using NPSK is presented. This section also includes the relevant results from the synchronization block and the assessment criteria for NPSK as a synchronization sequence. Section 5 presents the conclusions and discussion.

Synchronization;

II.

THEORETICAL MODEL OF THE SYSTEM IN THE PRESENCE OF NOISE

I. INTRODUCTION Classic direct sequence CDMA systems have been using pseudo random sequences with good orthogonal characteristics, Walsh functions or wavelets for signal spreading and user information encoding [1],[2]. In principle, these sequences allow spread spectrum characteristics of the transmitted signals and enhance the security of the communication system as well as anti-jamming protection of the system due to their spread spectrum characteristics. Further enhancement of the security of the system has been achieved by using chaotic sequences for signal spreading and coding [3], [4]. In order to increase the security of users’ information transmission in CDMA system, a system that uses the noise signals as carriers of users’ information, called noise phase shift keying (NPSK CDMA) was analyzed and proposed in [5] for the case when transmission channel is represented by the additive white Gaussian noise. In this paper we present further results obtained in the case when fading is present in the system and the results obtained on the prototype system implemented in DSP technology. The problem of sequence synchronization using noise signals is also addressed [6],[7].

The NPSK CDMA system is shown in fig 1. Users’ information bits are generated in the form of a binary sequence having values Ȗig that are taken from the alphabet {-1, +1}. N users’ sequences are spread by N spreading sequences xtg generated from N Gaussian random generators. These random sequences are orthogonal and uncorrelated, Gaussian and, they are also independent. The modulated signal at the output of the transmitter is

§N · rt (t) = ¨¨¦γig xtg ¸¸ 2 cosωct . © g=1 ¹

(1)

and the received signal is

rr (t) = rt (t) + ξt (t) §N · = ¨¨¦γ ig xtg ¸¸ 2 cosωct + 2ξtI (t) cosωct − 2ξtQ (t)sinωct. © g=1 ¹

(2)

where ȟt(t) is a bandpass noise and ȟtI(t) and ȟtQt) are Gaussian random processes having zero mean and the same variance

978-1-4244-2204-3/08/$25.00 ©2008 IEEE

ı2= N0/2. After coherent demodulation in the receiver the single user information bits are detected using correlation procedure. Because the variance of the channel noise is σ2=N0/2 and the energy of a bit is Eb = 2ȕPc = 2ȕσc2, the probability of error in this CDMA system is [5] § N +1 § E ·−1 · 1 p = erfc ¨ +¨ b ¸ ¸ ¨ β ¨© N0 ¸¹ ¸ 2 © ¹

−1/ 2

g

(3)

.

This probability of error depends on three values: the signal to noise ratio in the system, the number of users in the

2 iβ

st(1)

(1) i

. . x(g) t .

γ

It was also shown that this probability of error decreases if the spreading factor increases, as shown in fig 3, and can be reduced by increasing the spreading factor 2ȕ of the system. However, an increase of the spreading factor causes an increase in the required bandwidth of the spread signal. Thus, a

xt(1)

xt(1)

γ

system N and the spreading factor 2ȕ. If the number of users increases then the probability of error will increase due to the residual inter-user interference in the system, as shown in fig 2. This increase in probability is more pronounced at the higher signal to noise ratio. Namely, in this case the inter-user interference becomes the dominant generator of the errors in the system.

(g ) i

¦β

zi(1)

(.)

γ~i(1)

t = 2 ( i −1) +1

2 cos(ωc t )

2 cos(ωct ) FIR

. . (N ) . xt

¦

FIR

αt Fading

γ i( N ) s

x

2 iβ

st( g )

. . .

(g) t

zi( g )

(.)

t =2 ( i −1) β +1

ξt (t )

. . .

xt( N )

AWGN

z i( N )

2 iβ

¦β

(N ) t

γ~i( g )

γ~i( N )

(.)

t =2 ( i −1) +1

Noise-like Sequence Generator

Synchronization Block

Figure 1. DS-CDMA CPSK system

The NPSK system, Theory and Implementation Results SF = 10 0

The NPSK system, Theory and Implementation Results SF = 10 0

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10

10

10

-1

10

Bit Error Rate

Bit Error Rate

10

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-2

8 Users Theoretical 8 Users DSP 1 User DSP 1 User Theoretical 4 Users DSP 4 Users Theoretical BPSK Theory Single User

-3

-1

0

1

-2

8 Users Theoretical 8 Users DSP 1 User DSP 1 User Theoretical 4 Users DSP 4 Users Theoretical BPSK Theory Single User

8 Users 4 Users 10

1 User

-4

-2

10

-1

2 3 Eb/No (dB)

4

5

6

Figure 2. Dependence of probability of error on the number of users

7

10

-3

8 Users 4 Users 1 User

-4

-2

-1

0

1

2 3 Eb/No (dB)

4

5

6

7

Figure 3. Dependence of probability of error on the system spreading factor 2ȕ

compromise must be found between the value of the spreading factor, which is equivalent to the processing gain, and the width of the transmitted signal spectrum in the communication system. It is worth to note that in the case when the spreading factor is 2000 the BER curves for the 8 user system and BPSK system nearly overlap, i.e., the system which uses noise sequences with high spreading factors becomes equivalent to the BPSK system, as can be seen from Fig 3. These results were also confirmed by simulation using simulator of this communication system [5],[8],[9]. The CDMA signal generated in the transmitter is composed of the signals that have Gaussian distribution of the amplitudes. Because these signals are added to each other the probability density function of the composite signal is also Gaussian. Therefore, in the case of communication interception the signal intercepted will have characteristics of the Gaussian noise signal and the procedure of its demodulation and decoding will be extremely difficult. Also, the reception of the signal depends on the goodness of the synchronization system used and the ability of the receiver to acquire this synchronization. Due to complexity of random sequences synchronizations the difficulties for signal interception increases. The possible number of users in the system is of a particular interest for the system designer. Namely, this number specifies the capacity of a base station in, for example, cellular mobile phones system. Using relation (3) we can derive the expression for the number of users in this form β

THEORETICAL MODEL OF THE SYSTEM IN THE PRESENCE OF FADING

In the case when Rayleigh flat fading is present in the system the characteristics of the system will deteriorate significantly [10]. The influence of fading is represented by a multiplier shown in fig. 1, where multiplication factor at affects all the chips inside a bit interval. Thus, having in mind (2), the received signal now is

§N · rr (t) = at ¨¨¦γ ig xtg ¸¸ 2 cosωc t + 2ξtI (t) cosωc t − 2ξtQ (t) sinωct . (5) © g=1 ¹ The signal at the output of the g-th user correlator for the ith message bit, can be expressed as [11] 2 βi

¦rt xtg =

t =2 β (i −1) +1

t =2 β ( i −1) +1

¦ai γ ig (xtg ) +

· § N g g ¨ ai ¦γ i xt + ξt ¸ xtg ¦ ¸ ¨ g=1 t =2 β ( i −1) +1 © ¹

2

t =2β (i−1)+1

2βi

2βi N · g § n n ¨ ¸ a γ x x + xtgξt ¦ ¦ ¨ i ¦i t¸t t =2β (i−1)+1 © n=1,n≠g t=2β (i−1)+1 ¹ .(6)

In the case when fading is flat, the fading parameter has the same value in a bit interval, i.e., ai = a. Then, for γ 1g = +1 the sampled value of the random function z1g is 2β

( )

z1g = a¦ xtg t =1

2

+a

N

2β

¦ ¦x

n=1, g ≠n t =1

2β

x + ¦ xtgξt

g n t t

t =1

(7)

and the probability of error for the g-th user is equal to [11] 1 p g = erfc 2

βa 2 Eb

.

(8)

( N + 1)a 2 Eb + βN 0

Because the fading parameter at is a random variable the probability of error in each bit interval have different probability of error depending on this random variable, i.e., this probability of error can be treated as a function of Rayleigh random variable a, i.e. pg = p(a), which has the Rayleigh density function. Therefore, the probability of error for the g-th user, denoted by p, assuming the AWGN and flat fading are present in the channel, will be equal to the mean value of the probability of error function for AWGN case found in respect to fading parameter a as a Rayleigh random variable, which is expressed by this integral ∞

β a 2 Eb 1 2 a −a 2 / b . p = ³ erfc e da 2 + + β 2 ( N 1 ) a E N 0 b b 0

(4)

The dependence of N on the probability of error for the signal to noise ratio as a parameter shows that when this number increases the probability of error increases in any user’s channel, which is a consequence of inter-user interference. Also this number rises when the signal to noise ratio is higher.

zig =

2βi

−1

§E · N= −1 − β ¨¨ b ¸¸ . [erfc−1(2 pg )]2 © N0 ¹

III.

=

(9)

This integral can be solved using numerical integration. The lower noiseless bound can be found for the case when the noise level is reduced to zero. In this case we may have ∞

p LB =

β β . 1 2a −a2 / b 1 erfc e da = erfc ³ 2 ( N + 1) 0 b 2 ( N + 1)

(10)

This bound is equal to the bound obtained for the case when only AWGN is present in the channel. In fig. 4 the dependence of the probability of error p on the signal to noise ratio is presented for 1, 4 and 8 users. In contrast to the case when only AWGN is present in the channel, in this case there is not any improvement in when the number of users decreases, because the influence of the fading dominates over the influence of the interuser interference and the noise power in the system. IV. TIMING ACQUSITION USING NPSK In order to correctly decode the message bits, a spreadspectrum receiver must generate a spreading sequence that is synchronized to the received signal. Sequence synchronization is typically implemented in two stages for all serial and parallel search algorithms. In this paper the assumption is that the acquisition phase uses a serial search algorithm. The first, acquisition, or coarse alignment, reduces the relative delay to within one chip interval. Once the incoming sequence has been

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Bit Error Rate

8 Users 10

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10 10 10

-6

-8

-10

4 Users 1 User 2 β = 100 Theory 1 User 2 β = 100 4 Users 2 β = 100 Theory 4 User 2 β = 100 8 Users 2 β = 100 Theory 8 User 2 β = 100 BPSK Theory Single User no Fade

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Bit Error Rate

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0

1 User

10 10

-3

-2

-1

0

1

2

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5

6

7

SNR (Eb/N0) in dB

Figure 4. Fading: Dependence of probability of error on SNR for the number of users N as a parameter

acquired, the second stage, called tracking, provides and maintains fine synchronization [11]-[13]. A relative delay exists between the received and the locally generated sequences, caused by propagation delays, clock drift and other non-linear behavior in the wireless channel. Any sequence misalignment causes the demodulator output to fall, in accordance with the auto-correlation function, and increases the probability of symbol error. The presence of data impedes sequence synchronization, so a separate channel is used to transmit an unmodulated pilot signal. The pilot signal has a fixed period and is repeated, which allows the system multiple attempts to gain synchronization. The period of the pilot sequence has two impacts on the system, the range of possible delays that can be synchronized and, the mean time to reacquire the signal in case of synchronization loss. Walsh or PN pilot sequence periods are typically a power of two. These sequences have been extensively used for acquisition and tracking before [11]-[13]; however, using Gaussian distributed random signal for the purpose of acquisition has not been attempted before. It is imperative to mask the signals (including the synchronization channel ones) in order to preserve the security of the system. In a CDMA system, users begin transmission at the start of the pilot sequence. Therefore, once the pilot is synchronized, and the relative offset estimated, user sequences can be aligned. Non-coherent code acquisition must precede carrier synchronization as the signal energy is spread over a wide spectral band. The receiver and transmitter pilot sequences are offset by the channel delay IJ. Suppose the receiver pilot is xt0 (t − τ ) ; a time shifted replica of the transmitter pilot xt0 (t ) . Synchronization is attained by advancing the receiver generated pilot one chip and computing the correlation with the transmitter pilot. The result is processed using an appropriate decision-rule and search strategy. A serial-search acquisition system attempts to align the pilots by shifting the receiver pilot

10

-12

2 β = 100 2 β = 1000 2 β = 5000 2 β = 10000 BPSK

-14

-16

-5

0

5

10

15

SNR (E /N ) b

0

Figure 5. Fading: Dependence of probability of error on SNR for the spreading factor as a parameter

in discrete steps. Once the two sequences are aligned a control signal Cinitial is generated at the output of the Synchronization block. This signal initializes the gth user Noise Sequence generator at the time instant that the transmitted sequence of the gth user is supposed to start. Thus, when the transmitter starts to generate message bits for the gth user, the gth user noise spreading sequence generator at the receiver side will be synchronized to the incoming sequence. Other strategies, such as parallel or windowed serial searches, are generally more computationally expensive. In order to gauge the performance of a particular spreading sequence for synchronization purposes, we have to define some performance criteria. Since the synchronization block is correlator based, a peak in the correlator output indicates a potentially correct timing estimate. However, channel impairments affect the correctness of this timing estimation. The probability that synchronization is truly achieved is called probability of detection and is denoted by Pd. The probability of a correlator peak at the wrong timing is called probability of false alarm and is denoted by Pf. These two as well as the threshold used to detect the correlator peaks (denoted by Z) will be the performance criteria for synchronization properties of the spreading sequence [11]-[13]. Fig. 6 shows what will happen to Pd and Pf if we use a range of thresholds. As can be seen when Z is extremely low both probabilities are at their maximum of 1. However, the most desirable part of the curve is when Pd is close to 1 while Pf is close to zero. This is called the optimum threshold. If we run a set of statistically significant experiments to get the threshold for different values of signal to noise ratio and a multitude of users, then we can use the values of threshold in our practical implementation. The results for NPSK are presented in fig. 7 below. As can be seen, the threshold will increase as the number of users increase. This means that inter user interference is

1

6

x 10

4

0.9 5

0.8

SNR = 0

Optimum Threshold

Threshold Value

Probability

0.7 0.6

Detection

0.5 0.4 0.3 0.2

SNR = 4 SNR = 8 SNR = 12

4

3

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1

Failure

0.1 0 0

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1

1.5

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2.5

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Z-Value

3.5 x 10

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0 0

2

4 Users

6

8

Figure 7. Change in the threshold value for different values of noise and IUI with Pf = 0.01%

Figure 6. Pd and Pf for different values of threshold Z

affecting the performance and the threshold has to rise in order to maintain the desired probability of detection [11]-[13]. Lastly, we can gain insight into the synchronization performance of the system by looking at the Receiver Operating Characteristics (ROC) plot. This is basically values of Pd and Pf plotted against each other. The higher the curve is the better. Inter-user interference and higher noise values as well as lowering the spreading factor will result in an increased probability of failure and decreased probability of detection. This translates into a lower synchronization quality and higher acquisition time. The receiver operating characteristics for the NPSK system discussed is given in fig. 8 below.

system, we include a visual result in the form of an image (fig. 9) which is sent across the channel. The random patches indicate where the synchronization has been lost due to the false alarm.

Figure 9. The result of image transmission for 2 users and Signal to noise ratio of 15 dB 1 0.9

2 Users SNR = 4 8 Users SNR = 4 2Users SNR = 12 8Users SNR = 12

0.8 0.7

Pd

0.6 Increasing SNR and decreasing IUI

0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

Pf

Figure 8. The Receiver Operating Characteristics for 2 and 8 users with different values of noise and spreading factor of 100

The effect of NPSK synchronization on the bit error rate curve can be derived; however this requires a paper on its own. Derivation of the exact bit error rate performance for NPSK is a potential future direction worth exploring. Nevertheless to show the importance of synchronization in a spread spectrum

V. CONCLUSIONS AND DISCUSSION A mathematical model for a CDMA system, which uses random noise signals as carriers of users’ information, is developed and presented. The system characteristics are analyzed assuming that the AWGN and slow flat fading are present in the channel. This bit error rate expression shows that the system characteristics are deteriorated when the number of users increases, and improved when the spreading factor increases. In particular the number of users is calculated as a function of probability of error, which could give the evidence about the capacity of the CDMA system. It was also confirmed that the noise sequences can be used as spreading sequences in CDMA systems. This is done by both using noise sequences as code spreaders for the message, as well as employing the noise sequences (for the first time) as the synchronization pilot. The latter confirmation is important since in a secure communication system, it is essential to protect the pilot signal as well as the data in order to make the system intercept proof. Due to the random nature of noise sequences their application substantially increases the security of these kinds of CDMA systems, because the spread signal characteristics of AWGN and can hardly be distinguished from the channel noise both in time and frequency domain as well as the probability

density function.. The theoretical findings related to both the message modulation and synchronization block are confirmed by measurements on the prototype of the system analyzed.

[7]

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