A chaotic interleaving scheme for continuous-phase modulation-based orthogonal frequency-division...

Wireless Pers CommunDOI 10.1007/s11277-010-0047-z

A Chaotic Interleaving Scheme for the Continuous PhaseModulation Based Single-Carrier Frequency-DomainEqualization 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 contin-uous phase modulation based single-carrier frequency-domain equalization (CPM-SC-FDE) system. Chaotic interleaving is used in this scheme to generate permuted versionsfrom the sample sequences to be transmitted, with low correlation among their sam-ples, and hence a better bit error rate (BER) performance can be obtained. The pro-posed CPM-SC-FDE system with chaotic interleaving combines the advantages of thefrequency diversity, the low complexity, and the high power efficiency of the CPM-SC-FDE system and the performance improvements due to chaotic interleaving. The BERperformance of the CPM-SC-FDE system with and without chaotic interleaving is evaluatedby computer simulations. Also, a comparison between the proposed chaotic interleaving andthe conventional block interleaving is performed. Simulation results show that, the proposedchaotic interleaving scheme can greatly improve the performance of the CPM-SC-FDE sys-tem. 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, Egypte-mail: [email protected]

E. S. Hassane-mail: [email protected]

M. I. Dessoukye-mail: [email protected]

S. A. El-Dolile-mail: [email protected]

X. ZhuDepartment of Electrical Engineering and Electronics, University of Liverpool, Liverpool L69 3GJ, UKe-mail: [email protected]

S. E. El-KhamyDepartment of Electrical Engineering, Faculty of Engineering, Alexandria University,Alexandria 21544, Egypte-mail: [email protected]

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interleaving scheme in the CPM-SC-FDE system. The results also show that, the proposedCPM-SC-FDE system with chaotic interleaving provides a good trade-off between systemperformance 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-qualitymultimedia transmission. However, there exist challenges including frequency selective chan-nels, due to the existence of multipaths in communication. The single-carrier frequency-domain equalization (SC-FDE) system is one of the most promising systems that can beused 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 tofrequency 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 efficientpower 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 OFDMsystem and the SC-FDE system, respectively. In [13], a new low-complexity linear FDEapproach for CPM signals was developed. Some other novel equalization algorithms in fre-quency-domain for CPM signals are given in [14]. In [15] and [16], a CPM-SC-FDE structurefor broadband wireless communication systems was proposed. Although the use of CPM inSC-FDE systems gives a good performance and an acceptable receiver complexity, typicalCPM signals do not usually provide high bandwidth efficiency when compared with varying-envelope modulation schemes such as pulse amplitude modulation (PAM) and quadratureamplitude modulation (QAM) [16,17].

Strong mechanisms for error reduction such as powerful error correction codes [18] andefficient interleaving schemes [19] are required to reduce the channel effects on the datatransmitted with the CPM-SC-FDE system. Since the channel errors caused by the mobilewireless channels are bursty in nature, interleaving is a must in mobile communication sys-tems. Several interleaver schemes have been proposed. The simplest and most popular ofsuch schemes is the block interleaver [19,20]. In spite of the success of this scheme to achievea good performance in wireless communication systems, there is a need for a much powerfulscheme for severe channel degradation cases. Chaotic maps have been proposed for a widerange of applications in communications [21], and cryptography [22–25]. Due to the inherentstrong 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 theCPM-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 ofchaotic interleaving is to generate permuted sequences with low correlation between theirsamples from the sample sequences before transmission over the channel, thus a better BERperformance can be achieved. Moreover, it increases the security of the communicationsystem.

The rest of this paper is organized as follows. Section 2 presents the proposed CPM-SC-FDE system model. The block interleaving and the proposed chaotic interleaving schemes are

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

explained in Sect. 3. The equalizer design and the phase demodulator structure are explainedin Sects. 4 and 5, respectively. The bandwidth efficiency and the multipath diversity of CPMsignals are explained in Sect. 6. Section 7 introduces the simulation results. Finally, section8 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 willbe shown in the next sections, the proposed modifications will be in the interleaver and theequalizer blocks. A block length of K symbols is assumed with x(n) (n = 0, 1,…, K-1) rep-resenting 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 sI (n) with the subscript I referring to the interleav-ing process. Then, each data block is pre-appended with a cyclic prefix (CP) to mitigatethe inter-block interference (IBI). The CP length must be longer than the channel impulseresponse. Finally, the continuous-time CPM-SC-FDE signal, sI (t) is generated at the outputof the digital-to-analog (D/A) converter.

According to [16], sI (t) can be written as:

s(t) = Ae jφ(t) = Ae j[2πhx(t)+θ ] (1)

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

x(t) = Cn

K∑

k=1

Ikqk(t) (2)

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 datasymbols, Ik , can be written as [16]:

Ik =⎧⎨

⎩

�{X (k)}, k ≤ K/2

−�{X (k − K/2)}, k > K/2(3)

where �{X(k)}, �{X(k)} are the real and the imaginary part of {X(k)}, respectively.

Receiver

FDE A/D CP -

Phase Demodulator

QAM De-mapping

O/P Data

rI(t) rI(n) I(n) (n)

I/P Data

QAM Mapping

Phase Modulator

CP + D/A

x(n) s(n)

PA To

multipath channel

Transmitter

Chaotic Interleaving

sI(n) sI(t)

Chaotic De-interleaving

(n)

(a)

(b)

xss

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

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The transmitted signal sI (t) passes through the multipath channel. The channel impulseresponse is modelled as a wide-sense stationary uncorrelated scattering (WSSUS) processconsisting of L discrete paths:

h(t) =L−1∑

l=0

h(l)δ(t − τl) (4)

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

rI (t) =L−1∑

l=0

h(l)sI (t − τl) + n(t) (5)

where n(t) is a complex additive white Gaussian noise (AWGN) with single-sided powerspectral density N0.

The output of the analog-to-digital (A/D) converter is sampled at t = iT/JK, where J isthe oversampling factor, then the CP is discarded. The ith (i = 0, 1, . . . , JK − 1) sample ofthe received signal rI (t) is given by:

rI (i) =L J−1∑

l=0

h(l)sI (i − l) + n(i) (6)

Defining NDFT = JK, the received signal rI (i) is transformed into the frequency domainby using the NDFT-point discrete Fourier transform (DFT). The received signal on the mth(m = 0, 1,…, NDFT-1) subcarrier is given by:

RI (m) = H(m)SI (m) + N (m) (7)

where RI (m), H(m), SI (m), and N(m) are the NDFT-points DFT of rI (i), h(i), sI (i), and n(i),respectively.

3 Interleaving Mechanisms

Error correction codes are usually used to protect signals through transmission over wirelesschannels. 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. Interleavingis a process to rearrange the samples of the transmitted signal so as to spread error burstsover multiple code words. The simplest and most popular of such interleavers is the blockinterleaver. We first review the basics of the conventional block interleaving [19]. Then wepresent 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, blockinterleaving is applied to the signal samples. The samples are first arranged to a matrix in arow-by-row manner and then read from this matrix in a column-by-column manner. Let ustake a look at how the block interleaving mechanism can correct error bursts. Assume thatan error burst affects four consecutive samples (1-D error burst) as shown in Fig. 2b withshades. 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 errorcorrection capability, it is obvious that no decoding error will result from the presence of such1-D error burst. This simple example demonstrates the effectiveness of the block interleavingmechanism 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 2×2error burst does not spread, effectively, so that there are adjacent samples in error in thefirst and the second rows. As a result, this error burst can not be corrected using a singleerror correction mechanism. That is, the conventional block interleaving mechanism can notcombat 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 errorbursts. As a result, there is a need for advanced interleavers for this task. The 2-D chaoticBaker map in its discretized version is a good candidate for this purpose. After PM, the signalsamples 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 theirsamples 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(n1, . . ., nk), denote the discretized map, where the vector, [n1, . . ., nk], represents thesecret key, Skey . Defining N as the number of data items in one row, the secret key is chosensuch that each integer ni divides N, and n1 + · · · + nk = N .

Let Ni = n1 + · · · + ni−1. The data item at the indices (q, z), is moved to the indices[24,26]:

B(n1,cdots,nk )(q, z) =(

N

ni(q − Ni ) + z mod

(N

ni

),

ni

N

(z − z mod

(N

ni

))+ Ni

)(8)

where Ni ≤ q < Ni + ni , 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 ni and number ofelements 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. Rect-angles 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 upperelements.

Figure 3 shows an example for the chaotic interleaving of an (8 × 8) square matrix (i.e.N = 8). The secret key, Skey = [n1, n2, n3] = 2, 4, 2. Note that, the chaotic interleavingmechanism has a better treatment to both 1- and 2-D error bursts than the block interleavingmechanism. Errors are better distributed to samples after de-interleaving in the proposed cha-otic interleaving mechanism. As a result, a better BER performance can be achieved with thisproposed 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 thenpassed through an A/D converter, then the CP is discarded and the remaining samples areequalized 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 inFig. 4, the received signal is equalized in the frequency domain. The equalized signal is thentransformed 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 sub-carrier, the time-domain equalized signal sI (n), which is the soft estimate of sI (n), can beexpressed as follows:

s̃I (n) = 1

NDFT

NDFT−1∑

m=0

W (m)RI (m) e j2πmn/NDFT (9)

DFT IDFT rI(n) RI(m)

W(m) I(n) = IDFT{W(m)RI(m)} s

Fig. 4 The frequency domain equalizer (FDE)

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The equalizer coefficients W(m) are selected to minimize the mean squared error betweenthe equalized signal s̃I (n) and the original signal sI (n). These coefficients are computedaccording to a certain optimization rule leading to several types of equalizers such as:

• The zero-forcing (ZF) equalizer:

W (m) = 1

H(m)(10)

• The minimum mean square error (MMSE) equalizer:

W (m) = H∗(m)

|H(m)|2 + (Eb/N0)−1(11)

• The regularized zero forcing (RZF) equalizer:

W (m) = H∗(m)

|H(m)|2 + β(12)

where (.)∗ denotes the complex conjugate and β is the regularization parameter. The RZFequalizer described in (12) avoids the problems associated with the MMSE equalizer, suchas the measurement of the signal power and the noise power, which are not available priorto equalization. Moreover, the RZF equalizer avoids the noise enhancement caused by theZF equalizer by introducing the regularization parameter β into the equalization process.Considering the MMSE equalizer, the equalized signal can be expressed as:

s̃I (n) = 1

NDFT

NDFT−1∑

m=0

|H(m)|2 S(m)

|H(m)|2 + (Eb/N0)−1e j2πmn/NDFT

︸︷︷︸signal

+ 1

NDFT

NDFT−1∑

m=0

|H(m)|∗ N (m)

|H(m)|2 + (Eb/N0)−1e j2πmn/NDFT

︸︷︷︸noise

(13)

The de-interleaving is then applied to the equalized samples. Afterwards, a phase demodu-lation 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 illus-trated in Fig 5. It starts with a finite impulse response (FIR) filter to remove the out-of-bandnoise. The filter is designed using the windowing technique [27]. The filter impulse responsewith length L f and normalized cut-off frequency fnor (0 < fnor ≤ 1), can be expressed asfollows [16]:

g(n) =sin

(2π fnor

(n − L f −1

2

))

π(

n − L f −12

) , 0 ≤ n ≤ L f − 1 (14)

In Eq. (14), if n = (L f −1)/2, g(n) = 2π fnor/π .

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(i) f(i)

0

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

(i) Phase Unwrapper

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

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

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

0

k

K-1

k

K-1

Matched filters

.

.

s

x x

xx

x x

Fig. 5 Phase demodulator

The output of FIR filter can be expressed as:

f (i) =L f −1∑

i=0

g(n)s̃(i − n) (15)

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 thereceiver insensitive to phase offsets caused by the channel nonlinearities.

Finally, the obtained signal is passed through a bank of K matched filters to get softestimates 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 itquantifies how many information bits per second can be loaded per unity of the availablebandwidth. To evaluate the bandwidth efficiency of a signal, its bandwidth needs to be esti-mated. Using Taylor expansion, the CPM signal described in Eq. (1), when θ = 0, can berewritten as:

s(t) = Ae j2πhx(t) = A∞∑

n=0

[( j2πh)n

n!]

xn(t)

= A

[1 + j2πhx(t) − (2πh)2

2! x2(t) − j(2πh)3

3! x3(t) + · · ·]

(17)

The subcarriers are centered at the frequencies ±i/T Hz, i = 1, 2,…, K/2. The effectivedouble-side bandwidth of the message signal, x(t), is defined as W = K/T Hz. According toEq. (17), the bandwidth of s(t) is at least W, if the first two terms only of the summationare considered. Depending on the modulation index value, the effective bandwidth can begreater than W. A useful bandwidth expression for the CPM signal is the root-mean-square(RMS) bandwidth [28]:

BW = max(2πh, 1)W Hz (18)

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

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η = R

BW= log2 M

max(2πh, 1)bps/Hz (19)

The bandwidth efficiency of a CPM signal is controlled by two parameters, M and 2πh. Onthe other hand, the bandwidth efficiency of an OFDM signal is log2M, which depends onlyon M.

The Taylor expansion given in Eq. (17) reveals how a CPM signal exploits the frequencydiversity 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 otherterms can be ignored, Eq. (17) can be rewritten as:

s(t) ≈ A [1 + j2πhm(t)] (20)

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

7 Numerical Results and Discussion

In this section, simulation experiments are performed to compare between the CPM-SC-FDEsystem and the CPM-OFDM system described in [10]. Another comparison study betweenthe effect of using the proposed chaotic interleaving scheme and the traditional block inter-leaving scheme in the CPM-SC-FDE system is presented. 4-ary (M = 4) pulse amplitudemodulation (PAM) data symbols are used in the simulations. Each block contains K = 64symbols and each symbol is sampled 8 times (J = 8). A channel model following the expo-nential delay profile in [16] with a root mean square (RMS) delay spread τrms = 2 µs isadopted except in Fig. 13. The channel is assumed to be perfectly known at the receiver. TheSNR 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 = 11and a normalized cut-off frequency of fnor = 0.2 [10].

Figure 6 shows a performance comparison between the CPM-SC-FDE system withoutinterleaving and the CPM-OFDM system described in [10] using the MMSE equalizer inboth systems. It is clear that, the CPM-SC-FDE system significantly outperforms the CPM-OFDM system. For example, at a BER = 10−3, the CPM-SC-FDE system provides an SNRgain of about 5 dB over the CPM-OFDM system.

Figure 7 demonstrates the effect of the choice of the regularization parameter on the pro-posed CPM-SC-FDE system with chaotic interleaving. The variation of the BER with theregularization parameter is clear in this figure at different SNR values. The objective of thisfigure is to choose an optimum value for the regularization parameter if an RZF equalizer isto 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 chaoticinterleaving using the ZF equalizer, the RZF equalizer (with β = 10−2), and the MMSEequalizer. The results show that, the MMSE equalizer outperforms both the ZF equalizerand the RZF equalizer. For example, at a BER = 10−3, the MMSE equalizer outperforms theRZF 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 blockinterleaving, and the proposed CPM-SC-FDE system with chaotic interleaving using theMMSE equalizer in all systems. It is clear that the proposed CPM-SC-FDE system with

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0 5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

Eb/No [dB]

Bit

Err

or R

ate

CPM-OFDM, No interleaving [10]

CPM-SC-FDE, No Interleaving

Fig. 6 BER performance of the CPM-SC-FDE system and the CPM-OFDM system

10-4

10-3

10-2

10-1

100

10-6

10-5

10-4

10-3

10-2

10-1

β

BE

R

SNR = 10 dB

SNR = 15 dBSNR = 20 dB

SNR = 25 dB

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

chaotic interleaving outperforms all the other systems. For example, at a BER=10−3, theproposed CPM-SC-FDE system with chaotic interleaving provides SNR gains of 2 and 1 dBover 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 theCPM-SC-FDE system, at a fixed SNR = 20 dB for both the single path and the multipathcases with and without chaotic interleaving, respectively. In both cases, the performanceof the CPM-SC-FDE system over multipath channels, outperforms its performance over asingle 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 withoutchaotic interleaving is better than its performance in a single path channel for 2 πh > 0.2and 2 πh > 0.4, respectively. Based on the performance shown in Fig. 10 and the bandwidth

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0 5 10 15 20 25 30 3510

-4

10-3

10-2

10-1

100

Eb/No [dB]

Bit

Err

or R

ate

ZF-FDE

RZF-FDEMMSE-FDE

Fig. 8 BER performance of the proposed CPM-SC-FDE system with chaotic interleaving using the ZFequalizer, the RZF equalizer, and the MMSE equalizer

0 5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

Eb/No [dB]

BE

R

SC-FDE [1]

CPM-SC-FDE, No Interleaving [16]CPM-SC-FDE, Block Interleaving

CPM-SC-FDE, Chaotic Interleaving

Fig. 9 BER performance of the SC-FDE system, the CPM-SC-FDE system without interleaving, theCPM-SC-FDE system with block interleaving, and the proposed CPM-SC-FDE system with chaotic inter-leaving

efficiency given by Eq. (19), it can be deduced that a moderate value of the modulationindex achieves a significant utilization of the frequency diversity, while maintaining a highbandwidth efficiency.

Figure 12 shows a performance comparison between the CPM-SC-FDE system with andwithout 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 theCPM-SC-FDE system without interleaving [16], at relatively small modulation index values.This means that the proposed chaotic interleaving scheme improves the bandwidth efficiencyin the CPM-SC-FDE system.

Table 1 shows the percentage of improvement in the bandwidth efficiency in the proposedCPM-SC-FDE system with chaotic interleaving over the CPM-SC-FDE system without inter-

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210

-6

10-5

10-4

10-3

10-2

10-1

100

2π h

BE

R

Single-path

Multipath

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

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210

-6

10-5

10-4

10-3

10-2

10-1

100

2πh

Bit

Err

or R

ate

Multipath

Single-path

Fig. 11 Impact of the modulation index on the performance of the CPM-SC-FDE system without interleavingfor 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 withoutinterleaving needs 2 πh = 1.4, and hence the bandwidth efficiency will be η = 1.43 bps. On theother hand, the proposed CPM-SC-FDE system with chaotic interleaving needs 2 πh = 1.2,i.e. η = 1.67 bps, which means that the chaotic interleaving achieves an improvement in thebandwidth efficiency of about 16%, when it used with the CPM-SC-FDE system. We can goto a conclusion that the proposed CPM-SC-FDE system with chaotic interleaving achievesa trade-off between the performance and the bandwidth efficiency, which is one of the mostserious problems in CPM based systems.

Figure 13 shows the performance of the conventional SC-FDE system [1], the CPM-SC-FDE system without interleaving [16], and the proposed CPM-SC-FDE system with chaotic

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210

-6

10-5

10-4

10-3

10-2

10-1

100

2π h

BE

R

CPM-SC-FDE Without Chaotic Interleaving

CPM-SC-FDE With Chaotic Interleaving

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

Table 1 Percentage of improvement in bandwidth efficiency (G) in the CPM-SC-FDE system with andwithout chaotic interleaving

BER CPM-SC-FDE(without interleaving)

CPM-SC-FDE (withchaotic interleaving)

G (%)

1 × 10−4 η = 1.818 bps (2πh = 1.1) η = 2 bps (2πh = 0.9) 10

8 × 10−6 η = 1.43 bps (2πh = 1.4) η = 1.67 bps (2πh = 1.2) 16.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

10-4

10-3

10-2

10-1

100

Normalized RMS Delay

BE

R

SC-FDE [1]

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

Fig. 13 Effect of the RMS delay on the performance of the conventional SC-FDE system, the CPM-SC-FDEsystem 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 thesymbol period. The MMSE equalizer is used with all systems at a fixed SNR = 20 dB. Asshown in this figure, in flat fading (i.e., τrms = 0), the performance of the CPM-SC-FDEsystem converges with a small performance loss to the conventional SC-FDE system, dueto 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 significantperformance gain over the other systems. It also provides a better performance than that inthe flat fading case by exploiting the channel frequency diversity efficiently, especially athigh delay spreads.

8 Conclusion

An efficient chaotic interleaving scheme has been proposed for the CPM-SC-FDE sys-tem. The proposed scheme improves the BER performance of the CPM-SC-FDE systemmore than the traditional block interleaving scheme, where it generates permuted sequencesfrom the samples to be transmitted with lower correlation. The performance of the proposedCPM-SC-FDE system with chaotic interleaving was studied over a multipath fading channelwith MMSE equalization. The obtained results show a noticeable performance improvementachieved by the proposed system over the conventional SC-FDE system and the CPM-SC-FDE system without interleaving, especially at high RMS delay spreads. Simulation resultshave shown that, the proposed CPM-SC-FDE system with chaotic interleaving makes agood trade-off between the performance and the bandwidth efficiency, where it achieves anefficient utilization of the frequency diversity and maintains the high bandwidth efficiency.

References

1. Falconer, D., Ariyavisitakul, S., Benyamin-Seeyar, A., & Eidson, B. (2002). Frequency domainequalization for single-carrier broadband wireless systems. IEEE Communications and Magagement,40(4), 58–66.

2. Gusmo, A., Dinis, R., & Esteves, N. (2003). On frequency-domain equalization and diversity combiningfor broadband wireless communications. IEEE Communication Letters, 51(7), 1029–1033.

3. Pancaldi, F., Vitetta, G., Kalbasi, G. R., Al-Dhahir, N., Uysal, M., & Mheidat, H. (2008). Single-carrierfrequency domain equalization. IEEE Signal Processing Magazine, 25(5), 37–56.

4. Sari, H., Karam, G., & Jeanclaude, I. (1995). Transmission techniques for digital terrestrial TVbroadcasting. IEEE Communications Magazine, 33(2), 100–109.

5. Zhu, X., & Murch, R. (2004). Layered space-frequency equalization in a single-carrier MIMO systemfor frequency-selective channels. IEEE Transaction on Wireless Communicaions, 3, 701–708.

6. Nee, R. V., & Prasad, R. (2000). OFDM for wireless multimedia communications. Norwood: ArtechHouse.

7. Schulze, H., & Luders, C. (2005). Theory and application of OFDM and CDMA wideband wirelesscommunication. New York: John Wiley.

8. Anderson, J., Aulin, T., & Sundeberg, C. (1986). Digital phase modulation. New York: PlennumPress.

9. Kiviranta, M., Mammela, A., Cabric, D., Sobel, D. A., & Brodersen, R. W. (2005). Constantenvelope multicarrier modulation: Performance evaluation in AWGN and fading channels. IEEEMilcom, 2, 807–813.

10. Thompson, S. C., & Ahmed, A. U. (2008). Constant-envelope OFDM. IEEE Transaction on Com-munications, 56(8), 1300–1312.

123


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 phasemodulation 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-complexitylinear frequency domain equalization for continuous phase modulation. IEEE Transactions on WirelessCommunications, 8(3), 1435–1441.

14. Pancaldi, F., & Vitetta, G. M. (2006). Equalization algorithms in the frequency domain for continuousphase 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 domainequalization. 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 domainequalization 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 inofdm systems. In IEEE VTC (Vol. 1, pp. 470–474). 2001.

21. Jovic, B., & Unsworth, C. (2007). Chaos-based multi-user time division multiplexing communicationsystem. 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. InProceedings of WOCN-09, Cairo, Egypt, 28–30 April 2009.

27. Proakis, J. G., & Manolakis, D. G. (1996). Digital signal processing: Principles, algorithms, andapplications (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 Elec-trical Engineering from Menoufia University, Egypt in 2003 and2006, respectively. He is currently an Assistant Lecturer in the Dept.of Electronics and Electrical Communications, Faculty of ElectronicEngineering, Menoufia University. In 2008, he joined the Commu-nications Research Group at Liverpool University, Liverpool, UK,as a Visitor Research Student doing research on wireless communi-cation. He is currently working towards the Ph.D. degree in Com-munications Engineering from the Menoufia University. His areasof interests are CDMA, OFDM, SC-FDE, MIMO and CPM basedsystems.

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Xu Zhu received the B.Eng. degree (with first class honors) from theHuazhong University of Science and Technology, Wuhan, China, in1999, and the Ph.D. degree from the Hong Kong University of Sci-ence and Technology, Hong Kong, in 2003, both in Electrical andElectronic Engineering. Since May 2003, she has been with the Depart-ment of Electrical Engineering and Electronics, the University of Liver-pool, Liverpool, U.K., where she is currently a lecturer. Dr. Zhu was thevice chair of the 2006 and 2008 ICA Research Network InternationalWorkshops, which were held in Liverpool, U.K. She has served as asession chair and a technical program committee member for vari-ous conferences, such as IEEE GLOBECOM 2009 and IEEE VTCSpring-2009. Her research interests include MIMO, OFDM, equaliza-tion, blind source separation, cooperative communications and cross-layer optimization, etc.

Said E. El-Khamy received the B.Sc. (Honors) and M.Sc. degreesfrom Alexandria University, Alexandria, Egypt, in 1965 and 1967respectively, and the Ph.D. degree from the University of Massachu-setts, Amherst, USA, in 1971. He joined the teaching staff of theDepartment of Electrical Engineering, Faculty of Engineering, Alexan-dria University, Alexandria, Egypt, since 1972 and was appointed as aFull-time Professor in 1982 and as the Chairman of the Electrical Engi-neering Department from September 2000 to September 2003. He iscurrently an Emeritus Professor. Prof. El-Khamy has published morethan three hundreds scientific papers in national and international con-ferences and journals and took part in the organization of many localand international conferences. His Current research areas of interestinclude Spread-Spectrum Techniques, Mobile and Personal Communi-cations, Wave Propagation in different media, Smart Antenna Arrays,Space-Time Coding, Modern Signal Processing Techniques and theirapplications in Image Processing, Communication Systems, Antenna

design and Wave Propagation problems. Prof. El-Khamy is a Fellow member of the IEEE since 1999. Hereceived many prestigious national and international prizes and awards including the State AppreciationAward (Al-Takderia) of Engineering Sciences for 2004, the most cited paper award from Digital Signal Pro-cessing journal for 2008, the IEEE R.W.P. King best paper award of the Antennas and Propagation Soci-ety of IEEE, in 1980, the the A. Schuman’s-Jordan’s award for Engineering Research in 1982. He is also aFellow 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. degreesfrom the Faculty of Electronic Engineering, Menoufia University,Menouf, Egypt, in 1976 and 1981, respectively, and the Ph.D. fromMcMaster University, Canada, in 1986. He joined the teaching staffof the Department of Electronics and Electrical Communications, Fac-ulty of Electronic Engineering, Menoufia University, Menouf, Egypt,in 1986. He has published more than 140 scientific papers in nationaland international conference proceedings and journals. He is currentlythe head of the Dept. Electronics and Electrical Communications, Fac-ulty of Electronic Engineering, Menoufia University. He has receivedthe most cited paper award from Digital Signal Processing journal for2008. His current research areas of interest include spectral estima-tion techniques, image enhancement, image restoration, super resolu-tion reconstruction of images, satellite communications, and spreadspectrum techniques.

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Sami A. El-Dolil received his B.Sc. and M.Sc. degrees in ElectronicEngineering from Menoufia University, Menouf, Egypt, in 1977 and1981, respectively. In 1986 he joined the Communications ResearchGroup at Southampton University, Southampton, England, as aResearch Student doing research on teletraffic analysis for mobile radiocommunication. He received his Ph.D. degree from Menoufia Univer-sity, Menouf, Egypt, in 1989. He was a Post Doctor Research Fellowat the Department of Electronics and Computer Science, University ofSouthampton, 1991–1993. He is working as a Professor at the Depart-ment of Electronics and Electrical Communications, Faculty of Elec-tronic Engineering, Menoufia University, Menouf, Egypt. His currentresearch interests are in high-capacity digital mobile systems and mul-timedia networks.

Fathi E. Abd El-Samie received the B.Sc. (Honors), M.Sc., andPh.D. from the Faculty of Electronic Engineering, Menoufia Univer-sity, Menouf, Egypt, in 1998, 2001, and 2005, respectively. He joinedthe teaching staff of the Department of Electronics and Electrical Com-munications, Faculty of Electronic Engineering, Menoufia University,Menouf, Egypt, in 2005. He is a co-author of about 100 papers innational and international conference proceedings and journals. He hasreceived the most cited paper award from Digital Signal Processingjournal for 2008. His current research areas of interest include imageenhancement, image restoration, image interpolation, super resolutionreconstruction of images, data hiding, multimedia communications,medical image processing, optical signal processing, and digital com-munications.

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