A Chaotic Interleaving Scheme for the Continuous

Wireless Pers Commun DOI 10.1007/s11277-010-0047-z

A Chaotic Interleaving Scheme for the Continuous Phase Modulation Based Single-Carrier Frequency-Domain Equalization System Emad S. Hassan · Xu Zhu · Said E. El-Khamy · Moawad I. Dessouky · Sami A. El-Dolil · Fathi E. Abd El-Samie

© Springer Science+Business Media, LLC. 2010

Abstract In this paper, we propose a chaotic interleaving scheme for the continuous phase modulation based single-carrier frequency-domain equalization (CPM-SCFDE) system. Chaotic interleaving is used in this scheme to generate permuted versions from the sample sequences to be transmitted, with low correlation among their samples, and hence a better bit error rate (BER) performance can be obtained. The proposed CPM-SC-FDE system with chaotic interleaving combines the advantages of the frequency diversity, the low complexity, and the high power efficiency of the CPMSC-FDE system and the performance improvements due to chaotic interleaving. The BER performance of the CPM-SC-FDE system with and without chaotic interleaving is evaluated by computer simulations. Also, a comparison between the proposed chaotic interleaving and the conventional block interleaving is performed. Simulation results show that, the proposed chaotic interleaving scheme can greatly improve the performance of the CPM-SC-FDE system. Furthermore, the results show that this scheme outperforms the conventional block E. S. Hassan · M. I. Dessouky · S. A. El-Dolil · F. E. A. El-Samie (B) Department of Electronics and Electrical Communications, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt e-mail: [email protected] E. S. Hassan e-mail: [email protected] M. I. Dessouky e-mail: [email protected] S. A. El-Dolil e-mail: [email protected] X. Zhu Department of Electrical Engineering and Electronics, University of Liverpool, Liverpool L69 3GJ, UK e-mail: [email protected] S. E. El-Khamy Department of Electrical Engineering, Faculty of Engineering, Alexandria University, Alexandria 21544, Egypt e-mail: [email protected]

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interleaving scheme in the CPM-SC-FDE system. The results also show that, the proposed CPM-SC-FDE system with chaotic interleaving provides a good trade-off between system performance and bandwidth efficiency. Keywords

SC-FDE · CPM · Chaotic interleaving · Frequency-domain equalization

1 Introduction Future wireless communications are required to support high-speed and high-quality multimedia transmission. However, there exist challenges including frequency selective channels, due to the existence of multipaths in communication. The single-carrier frequencydomain equalization (SC-FDE) system is one of the most promising systems that can be used in a severe frequency-selective environment due to its effectiveness and low complexity [1–5]. Compared to the orthogonal frequency-division multiplexing (OFDM) system [6,7], the SC-FDE system has a lower peak-to-average power ratio (PAPR), a less sensitivity to frequency synchronization errors, and a higher frequency diversity gain [3]. Continuous phase modulation (CPM) is widely used in wireless communication systems, because of the constant envelope of the transmitted signals, which is required for efficient power transmission, and its ability to exploit the diversity of the multipath channel [8–12]. In [10] and [11], CPM was used to solve the problems associated with the PAPR of the OFDM system and the SC-FDE system, respectively. In [13], a new low-complexity linear FDE approach for CPM signals was developed. Some other novel equalization algorithms in frequency-domain for CPM signals are given in [14]. In [15] and [16], a CPM-SC-FDE structure for broadband wireless communication systems was proposed. Although the use of CPM in SC-FDE systems gives a good performance and an acceptable receiver complexity, typical CPM signals do not usually provide high bandwidth efficiency when compared with varyingenvelope modulation schemes such as pulse amplitude modulation (PAM) and quadrature amplitude modulation (QAM) [16,17]. Strong mechanisms for error reduction such as powerful error correction codes [18] and efficient interleaving schemes [19] are required to reduce the channel effects on the data transmitted with the CPM-SC-FDE system. Since the channel errors caused by the mobile wireless channels are bursty in nature, interleaving is a must in mobile communication systems. Several interleaver schemes have been proposed. The simplest and most popular of such schemes is the block interleaver [19,20]. In spite of the success of this scheme to achieve a good performance in wireless communication systems, there is a need for a much powerful scheme for severe channel degradation cases. Chaotic maps have been proposed for a wide range of applications in communications [21], and cryptography [22–25]. Due to the inherent strong randomization ability of these maps, they can be efficiently used for data interleaving. In [26], a chaotic interleaving scheme has been proposed to improve the performance of the CPM-OFDM system. In this paper, we propose a chaotic interleaving scheme for the CPM-SC-FDE system [16]. This scheme is based on the 2-D chaotic Baker map presented in [24,25]. The idea of chaotic interleaving is to generate permuted sequences with low correlation between their samples from the sample sequences before transmission over the channel, thus a better BER performance can be achieved. Moreover, it increases the security of the communication system. The rest of this paper is organized as follows. Section 2 presents the proposed CPM-SCFDE system model. The block interleaving and the proposed chaotic interleaving schemes are

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explained in Sect. 3. The equalizer design and the phase demodulator structure are explained in Sects. 4 and 5, respectively. The bandwidth efficiency and the multipath diversity of CPM signals are explained in Sect. 6. Section 7 introduces the simulation results. Finally, section 8 gives the concluding remarks.

2 The Proposed CPM-SC-FDE System Model The block diagram of the proposed CPM-SC-FDE system is illustrated in Fig. 1. As will be shown in the next sections, the proposed modifications will be in the interleaver and the equalizer blocks. A block length of K symbols is assumed with x(n) (n = 0, 1,…, K-1) representing the data sequence after symbol mapping. During each T-second symbol interval, x(n) is passed through a phase modulator (PM) to get the constant envelope sequence, s(n). This sequence is then interleaved to get s I (n) with the subscript I referring to the interleaving process. Then, each data block is pre-appended with a cyclic prefix (CP) to mitigate the inter-block interference (IBI). The CP length must be longer than the channel impulse response. Finally, the continuous-time CPM-SC-FDE signal, s I (t) is generated at the output of the digital-to-analog (D/A) converter. According to [16], s I (t) can be written as: s(t) = Ae jφ(t) = Ae j[2π hx(t)+θ ]

(1)

where A is the signal amplitude, h is the modulation index, θ is an arbitrary phase offset used to achieve CPM [10], and x(t) is the real-valued message signal given by: x(t) = Cn

K 

Ik qk (t)

(2)

k=1

where Ik are the M-ary real-valued data symbols, M is the number of constellation points, qk (t) are the orthogonal subcarriers, and Cn is a normalization constant. The real-valued data symbols, Ik , can be written as [16]: ⎧ ⎨ {X (k)}, k ≤ K /2 Ik = (3) ⎩ −{X (k − K /2)}, k > K /2 where {X(k)}, {X(k)} are the real and the imaginary part of {X(k)}, respectively.

(a) s(n) sI (n) CP QAM x(n) Phase Chaotic + Mapping Modulator Interleaving

I/P Data

PA

To multipath channel

Transmitter

(b) rI(t)

sI (t) D/A

A/D

rI (n)

CP

-

sI (n) FDE

x(n) Phase Chaotic De- s(n) QAM DeDemodulator interleaving mapping

O/P Data

Receiver

Fig. 1 The CPM-SC-FDE system model with chaotic interleaving

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The transmitted signal s I (t) passes through the multipath channel. The channel impulse response is modelled as a wide-sense stationary uncorrelated scattering (WSSUS) process consisting of L discrete paths: h(t) =

L−1 

h(l)δ(t − τl )

(4)

l=0

where h(l) and τl are the channel gain and delay of the lth path, respectively. The continuous-time received signal r I (t) can be expressed as: r I (t) =

L−1 

h(l)s I (t − τl ) + n(t)

(5)

l=0

where n(t) is a complex additive white Gaussian noise (AWGN) with single-sided power spectral density N0 . The output of the analog-to-digital (A/D) converter is sampled at t = iT/JK, where J is the oversampling factor, then the CP is discarded. The ith (i = 0, 1, . . . , JK − 1) sample of the received signal r I (t) is given by: r I (i) =

L J −1

h(l)s I (i − l) + n(i)

(6)

l=0

Defining NDFT = JK, the received signal r I (i) is transformed into the frequency domain by using the NDFT -point discrete Fourier transform (DFT). The received signal on the mth (m = 0, 1,…, NDFT -1) subcarrier is given by: R I (m) = H (m)S I (m) + N (m)

(7)

where R I (m), H(m), S I (m), and N(m) are the NDFT -points DFT of r I (i), h(i), s I (i), and n(i), respectively.

3 Interleaving Mechanisms Error correction codes are usually used to protect signals through transmission over wireless channels. Most of the error correction codes are designed to correct random channel errors. However, channel errors caused by mobile wireless channels are bursty in nature. Interleaving is a process to rearrange the samples of the transmitted signal so as to spread error bursts over multiple code words. The simplest and most popular of such interleavers is the block interleaver. We first review the basics of the conventional block interleaving [19]. Then we present the proposed chaotic interleaving mechanism. 3.1 The Block Interleaving Mechanism The idea of block interleaving can be explained with the aid of Fig. 2. After PM, block interleaving is applied to the signal samples. The samples are first arranged to a matrix in a row-by-row manner and then read from this matrix in a column-by-column manner. Let us take a look at how the block interleaving mechanism can correct error bursts. Assume that an error burst affects four consecutive samples (1-D error burst) as shown in Fig. 2b with shades. After de-interleaving, the error burst is effectively spread among four different rows

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Fig. 2 Block interleaving of an 8 × 8 matrix

as shown in Fig. 2c, resulting in a small effect for the 1-D error burst. With a single error correction capability, it is obvious that no decoding error will result from the presence of such 1-D error burst. This simple example demonstrates the effectiveness of the block interleaving mechanism in combating 1-D error bursts. Let us examine the performance of the block interleaving mechanism when a 2-D (2×2) error burst occurs [19], as shown in Fig. 2b with shades. Figure 2c indicates that this 2×2 error burst does not spread, effectively, so that there are adjacent samples in error in the first and the second rows. As a result, this error burst can not be corrected using a single error correction mechanism. That is, the conventional block interleaving mechanism can not combat the 2 × 2 error bursts. 3.2 The Proposed Chaotic Interleaving Mechanism As mentioned in the previous subsection, the block interleaver is not efficient with 2-D error bursts. As a result, there is a need for advanced interleavers for this task. The 2-D chaotic Baker map in its discretized version is a good candidate for this purpose. After PM, the signal samples can be arranged into a 2-D format then randomized using the chaotic Baker map. The chaotic interleaver generates permuted sequences with lower correlation between their samples and adds a degree of encryption to the transmitted signal. The discretized Baker map is an efficient tool to randomize the items in a square matrix. Let B(n 1 , . . ., n k ), denote the discretized map, where the vector, [n 1 , . . ., n k ], represents the secret key, Skey . Defining N as the number of data items in one row, the secret key is chosen such that each integer n i divides N, and n 1 + · · · + n k = N. Let Ni = n 1 + · · · + n i−1 . The data item at the indices (q, z), is moved to the indices [24,26]:  B(n 1 ,cdots,n k ) (q, z) =

N (q − Ni ) + z mod ni



N ni

 ,

ni N



 z − z mod

N ni



 + Ni

(8)

where Ni ≤ q < Ni + n i , 0 ≤ z < N , and N1 =0. In steps, the chaotic permutation is performed as follows: (1)

An N × N square matrix is divided into N rectangles of width n i and number of elements N.

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Fig. 3 Chaotic interleaving of an 8 × 8 matrix

(2)

The elements in each rectangle are rearranged to a row in the permuted rectangle. Rectangles are taken from left to right beginning with upper rectangles then lower ones.

(3)

Inside each rectangle, the scan begins from the bottom left corner towards upper elements.

Figure 3 shows an example for the chaotic interleaving of an (8 × 8) square matrix (i.e. N = 8). The secret key, Skey = [n 1 , n 2 , n 3 ] = 2, 4, 2. Note that, the chaotic interleaving mechanism has a better treatment to both 1- and 2-D error bursts than the block interleaving mechanism. Errors are better distributed to samples after de-interleaving in the proposed chaotic interleaving mechanism. As a result, a better BER performance can be achieved with this proposed mechanism. Moreover, it adds a degree of security to the communication system. At the receiver of the proposed system with chaotic interleaving, the received signal is then passed through an A/D converter, then the CP is discarded and the remaining samples are equalized as will be discussed in the next section.

4 Equalizer Design In this section, the design of the frequency domain equalizer is discussed. As shown in Fig. 4, the received signal is equalized in the frequency domain. The equalized signal is then transformed back into the time domain by an inverse DFT (IDFT). Let W(m), (m = 0, 1, . . . , NDFT -1), denote the equalizer coefficients for the mth subcarrier, the time-domain equalized signal s I (n), which is the soft estimate of s I (n), can be expressed as follows: s˜ I (n) =

1

NDFT −1

NDFT

m=0

RI (m)

rI(n) DFT

W(m)

Fig. 4 The frequency domain equalizer (FDE)

123

W (m)R I (m) e j2π mn/NDFT

sI (n) = IDFT{W(m)RI(m)} IDFT

(9)

A Chaotic Interleaving Scheme for The Continuous Phase Modulation Based Single-Carrier

The equalizer coefficients W(m) are selected to minimize the mean squared error between the equalized signal s˜ I (n) and the original signal s I (n). These coefficients are computed according to a certain optimization rule leading to several types of equalizers such as: • The zero-forcing (ZF) equalizer: 1 H (m)

W (m) =

(10)

• The minimum mean square error (MMSE) equalizer: W (m) =

H ∗ (m)

(11)

|H (m)|2 + (E b /N0 )−1

• The regularized zero forcing (RZF) equalizer: W (m) =

H ∗ (m)

(12)

|H (m)|2 + β

where (.)∗ denotes the complex conjugate and β is the regularization parameter. The RZF equalizer described in (12) avoids the problems associated with the MMSE equalizer, such as the measurement of the signal power and the noise power, which are not available prior to equalization. Moreover, the RZF equalizer avoids the noise enhancement caused by the ZF equalizer by introducing the regularization parameter β into the equalization process. Considering the MMSE equalizer, the equalized signal can be expressed as: s˜ I (n) =

1 NDFT 

NDFT −1

|H (m)|2 S(m)

m=0

|H (m)|2 + (E b /N0 )−1

e j2π mn/NDFT 

signal

+

1 NDFT 

NDFT −1

|H (m)|∗ N (m)

m=0

|H (m)|2 + (E b /N0 )−1

e j2π mn/NDFT

(13)



noise

The de-interleaving is then applied to the equalized samples. Afterwards, a phase demodulation step is applied to recover the data as explained in the next section.

5 Phase Demodulator In this section, the design of the phase demodulator is discussed. Its block diagram is illustrated in Fig 5. It starts with a finite impulse response (FIR) filter to remove the out-of-band noise. The filter is designed using the windowing technique [27]. The filter impulse response with length L f and normalized cut-off frequency f nor (0 < f nor ≤ 1), can be expressed as follows [16]:

L −1 sin 2π f nor n − f2

0≤n ≤ Lf −1 g(n) = (14) , L −1 π n − f2 In Eq. (14), if n = (L f −1)/2, g(n) = 2πfnor /π.

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s(i)

f(i) FIR Filter {g(n)},n=0,1,..Lf-1

(i) arg(.)

Phase Unwrapper

{q0(m)} m=0,1,..NDFT -1 .

x0

x0

{qk(m)} m=0,1,..NDFT -1 .

xk

xk

{qK-1(m)} m=0,1,..NDFT -1

xK-1

xK-1

Matched filters

Fig. 5 Phase demodulator

The output of FIR filter can be expressed as: L f −1

f (i) =



g(n)˜s (i − n)

(15)

i=0

Afterwards, the phase of the filtered signal f(i) is obtained: ϕ(i) = arg ( f (i)) = φ(i) + δ(i)

(16)

where φ (i) denotes the phase of the desired signal, and δ(i) denotes the phase noise. Then, a phase unwrapper is used to minimize the effect of any phase ambiguities and to make the receiver insensitive to phase offsets caused by the channel nonlinearities. Finally, the obtained signal is passed through a bank of K matched filters to get soft estimates of the data symbols x(n) (n = 0, …, K − 1).

6 Bandwidth Efficiency and Multipath Diversity of CPM Signals The bandwidth efficiency is an important quality metric for a modulation scheme, since it quantifies how many information bits per second can be loaded per unity of the available bandwidth. To evaluate the bandwidth efficiency of a signal, its bandwidth needs to be estimated. Using Taylor expansion, the CPM signal described in Eq. (1), when θ = 0, can be rewritten as:  ∞   ( j2π h)n n s(t) = Ae j2π hx(t) = A x (t) n! n=0   (2π h)2 2 (2π h)3 3 = A 1 + j2π hx(t) − (17) x (t) − j x (t) + · · · 2! 3! The subcarriers are centered at the frequencies ±i/T Hz, i = 1, 2,…, K/2. The effective double-side bandwidth of the message signal, x(t), is defined as W = K/T Hz. According to Eq. (17), the bandwidth of s(t) is at least W, if the first two terms only of the summation are considered. Depending on the modulation index value, the effective bandwidth can be greater than W. A useful bandwidth expression for the CPM signal is the root-mean-square (RMS) bandwidth [28]: BW = max(2π h, 1)W

Hz

(18)

As shown in Eq. (18), the signal bandwidth grows with 2π h, which in turn reduces the bandwidth efficiency. Since the bit rate is R = K(log2 M)/T bps, the bandwidth efficiency of the CPM signal, η, can be expressed as:

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η=

log2 M R bps/Hz = BW max(2π h, 1)

(19)

The bandwidth efficiency of a CPM signal is controlled by two parameters, M and 2π h. On the other hand, the bandwidth efficiency of an OFDM signal is log2 M, which depends only on M. The Taylor expansion given in Eq. (17) reveals how a CPM signal exploits the frequency diversity in the channel for a large modulation index. This is not necessary the case, however. For a small modulation index, only the first two terms in Eq. (17) contribute and the other terms can be ignored, Eq. (17) can be rewritten as: s(t) ≈ A [1 + j2π hm(t)]

(20)

In this case, the CPM signal does not have the frequency spreading given by the higher-order terms. Therefore, the CPM signal does not have the ability to exploit the frequency diversity of the channel [16].

7 Numerical Results and Discussion In this section, simulation experiments are performed to compare between the CPM-SC-FDE system and the CPM-OFDM system described in [10]. Another comparison study between the effect of using the proposed chaotic interleaving scheme and the traditional block interleaving scheme in the CPM-SC-FDE system is presented. 4-ary (M = 4) pulse amplitude modulation (PAM) data symbols are used in the simulations. Each block contains K = 64 symbols and each symbol is sampled 8 times (J = 8). A channel model following the exponential delay profile in [16] with a root mean square (RMS) delay spread τrms = 2 µs is adopted except in Fig. 13. The channel is assumed to be perfectly known at the receiver. The SNR is defined as the average ratio between the received signal power and the noise power, which is given by SNR = A2 /N0 . The FIR filter has an impulse response length of L f = 11 and a normalized cut-off frequency of f nor = 0.2 [10]. Figure 6 shows a performance comparison between the CPM-SC-FDE system without interleaving and the CPM-OFDM system described in [10] using the MMSE equalizer in both systems. It is clear that, the CPM-SC-FDE system significantly outperforms the CPMOFDM system. For example, at a BER = 10−3 , the CPM-SC-FDE system provides an SNR gain of about 5 dB over the CPM-OFDM system. Figure 7 demonstrates the effect of the choice of the regularization parameter on the proposed CPM-SC-FDE system with chaotic interleaving. The variation of the BER with the regularization parameter is clear in this figure at different SNR values. The objective of this figure is to choose an optimum value for the regularization parameter if an RZF equalizer is to be used in the proposed system. According to this figure, the best choice of β is 10−2 . Figure 8 shows the BER performance of the proposed CPM-SC-FDE system with chaotic interleaving using the ZF equalizer, the RZF equalizer (with β = 10−2 ), and the MMSE equalizer. The results show that, the MMSE equalizer outperforms both the ZF equalizer and the RZF equalizer. For example, at a BER = 10−3 , the MMSE equalizer outperforms the RZF equalizer and the ZF equalizer by about 0.3 and 9.5 dB, respectively. Figure 9 shows a performance comparison between the conventional SC-FDE system [1], the CPM-SC-FDE system without interleaving [16], the CPM-SC-FDE system with block interleaving, and the proposed CPM-SC-FDE system with chaotic interleaving using the MMSE equalizer in all systems. It is clear that the proposed CPM-SC-FDE system with

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10

CPM-OFDM, No interleaving [10] CPM-SC-FDE, No Interleaving

-1

Bit Error Rate

10

-2

10

-3

10

-4

10

0

5

10

15

20

25

30

Eb/No [dB] Fig. 6 BER performance of the CPM-SC-FDE system and the CPM-OFDM system -1

10

-2

10

-3

BER

10

-4

10

-5

10

SNR = 10 dB SNR = 15 dB SNR = 20 dB SNR = 25 dB

-6

10

-4

10

-3

10

-2

10

-1

10

0

10

β Fig. 7 BER vs. the regularization parameter at different SNRs in the proposed CPM-SC-FDE system with chaotic interleaving

chaotic interleaving outperforms all the other systems. For example, at a BER=10−3 , the proposed CPM-SC-FDE system with chaotic interleaving provides SNR gains of 2 and 1 dB over the CPM-SC-FDE system [16] and the CPM-SC-FDE system with block interleaving, respectively. Figures 10 and 11 show the effect of the modulation index on the performance of the CPM-SC-FDE system, at a fixed SNR = 20 dB for both the single path and the multipath cases with and without chaotic interleaving, respectively. In both cases, the performance of the CPM-SC-FDE system over multipath channels, outperforms its performance over a single path channel for large modulation index values, which verifies the analysis in Sect. 6. In multipath channels, the performance of the CPM-SC-FDE system with and without chaotic interleaving is better than its performance in a single path channel for 2 π h > 0.2 and 2 π h > 0.4, respectively. Based on the performance shown in Fig. 10 and the bandwidth

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10

ZF-FDE RZF-FDE MMSE-FDE

-1

Bit Error Rate

10

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-4

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35

Eb/No [dB] Fig. 8 BER performance of the proposed CPM-SC-FDE system with chaotic interleaving using the ZF equalizer, the RZF equalizer, and the MMSE equalizer 10

0 SC-FDE [1] CPM-SC-FDE, No Interleaving [16] CPM-SC-FDE, Block Interleaving CPM-SC-FDE, Chaotic Interleaving

BER

10

10

10

10

-1

-2

-3

-4

0

5

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15

20

25

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Eb/No [dB] Fig. 9 BER performance of the SC-FDE system, the CPM-SC-FDE system without interleaving, the CPM-SC-FDE system with block interleaving, and the proposed CPM-SC-FDE system with chaotic interleaving

efficiency given by Eq. (19), it can be deduced that a moderate value of the modulation index achieves a significant utilization of the frequency diversity, while maintaining a high bandwidth efficiency. Figure 12 shows a performance comparison between the CPM-SC-FDE system with and without chaotic interleaving in terms of the modulation index in the multipath channel case. It is clear that, the proposed CPM-SC-FDE system with chaotic interleaving outperforms the CPM-SC-FDE system without interleaving [16], at relatively small modulation index values. This means that the proposed chaotic interleaving scheme improves the bandwidth efficiency in the CPM-SC-FDE system. Table 1 shows the percentage of improvement in the bandwidth efficiency in the proposed CPM-SC-FDE system with chaotic interleaving over the CPM-SC-FDE system without inter-

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10

Single-path Multipath

-1

10

-2

BER

10

-3

10

-4

10

-5

10

-6

10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2π h Fig. 10 Impact of the modulation index on the performance of the CPM-SC-FDE system with chaotic interleaving for both the single path and multipath cases using an MMSE equalizer at an SNR = 20 dB 0

10

Multipath Single-path -1

Bit Error Rate

10

-2

10

-3

10

-4

10

-5

10

-6

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0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2πh Fig. 11 Impact of the modulation index on the performance of the CPM-SC-FDE system without interleaving for both the single path and multipath cases using an MMSE equalizer at an SNR = 20 dB

leaving when M=4. For example, at a BER of 8 × 10−6 , the CPM-SC-FDE system without interleaving needs 2 π h = 1.4, and hence the bandwidth efficiency will be η = 1.43 bps. On the other hand, the proposed CPM-SC-FDE system with chaotic interleaving needs 2 π h = 1.2, i.e. η = 1.67 bps, which means that the chaotic interleaving achieves an improvement in the bandwidth efficiency of about 16%, when it used with the CPM-SC-FDE system. We can go to a conclusion that the proposed CPM-SC-FDE system with chaotic interleaving achieves a trade-off between the performance and the bandwidth efficiency, which is one of the most serious problems in CPM based systems. Figure 13 shows the performance of the conventional SC-FDE system [1], the CPM-SCFDE system without interleaving [16], and the proposed CPM-SC-FDE system with chaotic

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10

CPM-SC-FDE Without Chaotic Interleaving CPM-SC-FDE With Chaotic Interleaving

-1

10

-2

BER

10

-3

10

-4

10

-5

10

-6

10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2π h Fig. 12 BER performance of the CPM-SC-FDE system with and without chaotic interleaving vs. the modulation index at an SNR = 20 dB

Table 1 Percentage of improvement in bandwidth efficiency (G) in the CPM-SC-FDE system with and without chaotic interleaving BER

CPM-SC-FDE (without interleaving)

CPM-SC-FDE (with chaotic interleaving)

G (%)

1 × 10−4

η = 1.818 bps (2π h = 1.1)

η = 2 bps (2π h = 0.9)

10

8 × 10−6

η = 1.43 bps (2π h = 1.4)

η = 1.67 bps (2π h = 1.2)

16.8

0

10

SC-FDE [1] CPM-SC-FDE, No Interleaving [16] CPM-SC-FDE, Chaotic Interleaving

-1

BER

10

-2

10

-3

10

-4

10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Normalized RMS Delay Fig. 13 Effect of the RMS delay on the performance of the conventional SC-FDE system, the CPM-SC-FDE system and the proposed CPM-SC-FDE system at an SNR = 20 dB

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interleaving, in terms of the BER versus the RMS delay spread, which is normalized to the symbol period. The MMSE equalizer is used with all systems at a fixed SNR = 20 dB. As shown in this figure, in flat fading (i.e., τrms = 0), the performance of the CPM-SC-FDE system converges with a small performance loss to the conventional SC-FDE system, due to the effect of the phase demodulator threshold. In frequency selective channels (τrms >0), however, the proposed CPM-SC-FDE system with chaotic interleaving achieves a significant performance gain over the other systems. It also provides a better performance than that in the flat fading case by exploiting the channel frequency diversity efficiently, especially at high delay spreads.

8 Conclusion An efficient chaotic interleaving scheme has been proposed for the CPM-SC-FDE system. The proposed scheme improves the BER performance of the CPM-SC-FDE system more than the traditional block interleaving scheme, where it generates permuted sequences from the samples to be transmitted with lower correlation. The performance of the proposed CPM-SC-FDE system with chaotic interleaving was studied over a multipath fading channel with MMSE equalization. The obtained results show a noticeable performance improvement achieved by the proposed system over the conventional SC-FDE system and the CPM-SCFDE system without interleaving, especially at high RMS delay spreads. Simulation results have shown that, the proposed CPM-SC-FDE system with chaotic interleaving makes a good trade-off between the performance and the bandwidth efficiency, where it achieves an efficient utilization of the frequency diversity and maintains the high bandwidth efficiency.

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A Chaotic Interleaving Scheme for The Continuous Phase Modulation Based Single-Carrier 11. Buzid, T., & Huemer, M. (2009). Single carrier transmission with frequency domain equalization (SC/FDE) system with a PAPR of unity. In Proceedings of ICACT-09 (Vol. 1, pp. 459–462). Feb. 2009 12. Tsai, Y., Zhang, G., & Pan, J.-L. (2005). “Orthogonal frequency division multiplexing with phase modulation and constant envelope design”. In IEEE Milcom, 4, 2658–2664. 13. Thillo, W., Horlin, F., Nsenga, J., Ramon, V., Bourdoux, A., & Lauwereins, R. (2009). Low-complexity linear frequency domain equalization for continuous phase modulation. IEEE Transactions on Wireless Communications, 8(3), 1435–1441. 14. Pancaldi, F., & Vitetta, G. M. (2006). Equalization algorithms in the frequency domain for continuous phase modulations. IEEE Transactions on Communications, 54(4), 648–658. 15. Hassan, E. S., Zhu, X., El-Khamy, S. E., Dessouky, M. I., El-Dolil, S. A., & Abd El-Samie, F. E. (2009). A continuous phase modulation single-carrier wireless system with frequency domain equalization. In Proceedings of ICCES-09, Cairo, Egypt, 14–16 Dec. 2009. 16. Hassan, E. S., Zhu, X., El-Khamy, S. E., Dessouky, M. I., El-Dolil, S. A., & Abd El-Samie, F. E. (2010). Performance evaluation of OFDM and single-carrier systems using frequency domain equalization and phase modulation. International Journal of Communication Systems (in press). 17. Barbieri, A., Fertonani, D., & Colavolpe, G. (2009). Spectrally efficient continuous phase modulations. IEEE Transactions on Wireless Communications, 8(3), 1564–1572. 18. Castello, D.J., Hagenauer, J., Imai, H., & Wicker, S. (1998). Applications of error-control coding. IEEE Transactions on Information Theory, 44, 2531–2560. 19. Shi, Y. Q., Zhang, X. M., Ni, Z.-C., & Ansari, N. (2004). Interleaving for combating error bursts. IEEE Circuts and systems magazine, 4, 29–42 (First Quarter 2004). 20. Nguyen, V. D., & Kuchenbecker, H. (2001). Block interleaving for soft decision viterbi decoding in ofdm systems. In IEEE VTC (Vol. 1, pp. 470–474). 2001. 21. Jovic, B., & Unsworth, C. (2007). Chaos-based multi-user time division multiplexing communication system. IET Communications, 1(4), 1751–8628. 22. Matthews, R. (1998). On the derivation of a chaotic encryption algorithm. Cryptologia XIII, 1, 29–41. 23. Deffeyes, K. S. (1991). Encryption system and method. US Patent, no. 5001754, March 1991. 24. Fridrich, J. (1998). Symmetric ciphers based on two-dimensional chaotic maps. International Journal of Bifurcation and Chaos, 8, 1259–1284. 25. Han, F., Yu, X., & Han, S. (2006). Improved baker map for image encryption,” in ISSCAA, 2006, pp. 1273–1276. 26. Hassan, E. S., El-Khamy, S. E., Dessouky, M. I., El-Dolil, S. A., & Abd El-Samie, F. E. (2009). New interleaving scheme for continuous phase modulation based OFDM systems using chaotic maps. In Proceedings of WOCN-09, Cairo, Egypt, 28–30 April 2009. 27. Proakis, J. G., & Manolakis, D. G. (1996). Digital signal processing: Principles, algorithms, and applications (3rd edn). NJ: Prentice Hall. 28. Proakis, J. G., & Salehi, M. (1994). Communication Systems Engineering. New Jersey: Prentice Hall.

Author Biographies Emad S. Hassan received the B.Sc. and M.Sc. degrees in Electrical Engineering from Menoufia University, Egypt in 2003 and 2006, respectively. He is currently an Assistant Lecturer in the Dept. of Electronics and Electrical Communications, Faculty of Electronic Engineering, Menoufia University. In 2008, he joined the Communications Research Group at Liverpool University, Liverpool, UK, as a Visitor Research Student doing research on wireless communication. He is currently working towards the Ph.D. degree in Communications Engineering from the Menoufia University. His areas of interests are CDMA, OFDM, SC-FDE, MIMO and CPM based systems.

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E. S. Hassan et al. Xu Zhu received the B.Eng. degree (with first class honors) from the Huazhong University of Science and Technology, Wuhan, China, in 1999, and the Ph.D. degree from the Hong Kong University of Science and Technology, Hong Kong, in 2003, both in Electrical and Electronic Engineering. Since May 2003, she has been with the Department of Electrical Engineering and Electronics, the University of Liverpool, Liverpool, U.K., where she is currently a lecturer. Dr. Zhu was the vice chair of the 2006 and 2008 ICA Research Network International Workshops, which were held in Liverpool, U.K. She has served as a session chair and a technical program committee member for various conferences, such as IEEE GLOBECOM 2009 and IEEE VTC Spring-2009. Her research interests include MIMO, OFDM, equalization, blind source separation, cooperative communications and crosslayer optimization, etc.

Said E. El-Khamy received the B.Sc. (Honors) and M.Sc. degrees from Alexandria University, Alexandria, Egypt, in 1965 and 1967 respectively, and the Ph.D. degree from the University of Massachusetts, Amherst, USA, in 1971. He joined the teaching staff of the Department of Electrical Engineering, Faculty of Engineering, Alexandria University, Alexandria, Egypt, since 1972 and was appointed as a Full-time Professor in 1982 and as the Chairman of the Electrical Engineering Department from September 2000 to September 2003. He is currently an Emeritus Professor. Prof. El-Khamy has published more than three hundreds scientific papers in national and international conferences and journals and took part in the organization of many local and international conferences. His Current research areas of interest include Spread-Spectrum Techniques, Mobile and Personal Communications, Wave Propagation in different media, Smart Antenna Arrays, Space-Time Coding, Modern Signal Processing Techniques and their applications in Image Processing, Communication Systems, Antenna design and Wave Propagation problems. Prof. El-Khamy is a Fellow member of the IEEE since 1999. He received many prestigious national and international prizes and awards including the State Appreciation Award (Al-Takderia) of Engineering Sciences for 2004, the most cited paper award from Digital Signal Processing journal for 2008, the IEEE R.W.P. King best paper award of the Antennas and Propagation Society of IEEE, in 1980, the the A. Schuman’s-Jordan’s award for Engineering Research in 1982. He is also a Fellow of the Electromagnetics Academy and a member of Tau Beta Pi, Eta Kappa Nu and Sigma Xi.

Moawad I. Dessouky received the B.Sc. (Honors) and M.Sc. degrees from the Faculty of Electronic Engineering, Menoufia University, Menouf, Egypt, in 1976 and 1981, respectively, and the Ph.D. from McMaster University, Canada, in 1986. He joined the teaching staff of the Department of Electronics and Electrical Communications, Faculty of Electronic Engineering, Menoufia University, Menouf, Egypt, in 1986. He has published more than 140 scientific papers in national and international conference proceedings and journals. He is currently the head of the Dept. Electronics and Electrical Communications, Faculty of Electronic Engineering, Menoufia University. He has received the most cited paper award from Digital Signal Processing journal for 2008. His current research areas of interest include spectral estimation techniques, image enhancement, image restoration, super resolution reconstruction of images, satellite communications, and spread spectrum techniques.

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A Chaotic Interleaving Scheme for The Continuous Phase Modulation Based Single-Carrier Sami A. El-Dolil received his B.Sc. and M.Sc. degrees in Electronic Engineering from Menoufia University, Menouf, Egypt, in 1977 and 1981, respectively. In 1986 he joined the Communications Research Group at Southampton University, Southampton, England, as a Research Student doing research on teletraffic analysis for mobile radio communication. He received his Ph.D. degree from Menoufia University, Menouf, Egypt, in 1989. He was a Post Doctor Research Fellow at the Department of Electronics and Computer Science, University of Southampton, 1991–1993. He is working as a Professor at the Department of Electronics and Electrical Communications, Faculty of Electronic Engineering, Menoufia University, Menouf, Egypt. His current research interests are in high-capacity digital mobile systems and multimedia networks.

Fathi E. Abd El-Samie received the B.Sc. (Honors), M.Sc., and Ph.D. from the Faculty of Electronic Engineering, Menoufia University, Menouf, Egypt, in 1998, 2001, and 2005, respectively. He joined the teaching staff of the Department of Electronics and Electrical Communications, Faculty of Electronic Engineering, Menoufia University, Menouf, Egypt, in 2005. He is a co-author of about 100 papers in national and international conference proceedings and journals. He has received the most cited paper award from Digital Signal Processing journal for 2008. His current research areas of interest include image enhancement, image restoration, image interpolation, super resolution reconstruction of images, data hiding, multimedia communications, medical image processing, optical signal processing, and digital communications.

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