A Study on Non-contiguous Carrier Interferometry Code for Cognitive Radio ∗ Department
Mithun Mukherjee∗ , Prashant Kumar† , Rakesh Matam‡
of Electronics and Communication Engineering, National Institute of Technology Hamirpur, India of Electronics and Communication Engineering, Birla Institute of Technology Patna, India ‡ Department of Computer Science Engineering, Indian Institute of Information Technology Guwahati, India Email:
[email protected],
[email protected],
[email protected] † Department
Abstract—Non-contiguous Carrier Interferometry (NC-CI) signature waveforms have been proposed for Dynamic Spectrum Access (DSA) and Cognitive Radio (CR) system in the literature. Walsh-Hadamard (WH) codes cannot ensure orthogonality for non-contiguous carrier allocation, when the number of noncontiguous carrier is not a multiple of 4. In literature, NC-CI and binary orthogonal codes of any code length are proposed for non-contiguous spectrum allocation. In this paper the crosscorrelation properties of NC-CI signature waveforms for noncontiguous CI orthogonal frequency division multiplexing (NCCI/OFDM) and NC-CI multicarrier code division multiple access (NC-CI/MC-CDMA) have been evaluated through simulation and analysis over AWGN channel. Time domain NC-CI waveforms have also been examined for non-contiguous allocation. Crosscorrelation analysis and time domain waveforms show that NCCI codes maintain orthogonality over non-contiguous subcarrier allocation in single user and multiuser environments. For WH code, it has been shown that orthogonality can be maintained with a loss of data rate. Index Terms—Non-contiguous Carrier Interferometry (NCCI), CR, DSA, NC-CI/MC-CDMA, NC-SOFDM, NC-OFDM, Cognitive NCI-OSDM.
I. I NTRODUCTION NE of the major obstacles in obtaining high data rate on wireless channel is the lack of a contiguous wide bandwidth. Most part of the frequency spectrum has been already allocated in small blocks for different wireless applications. So a contiguous wideband of the frequency spectrum is not easily available [1]. According to a report by the FCC [2], spectrum congestion is mainly due to inefficient spectrum allocation rather than spectrum scarcity. New FCC policies signified improved spectrally efficient wireless systems via sharing of the available spectrum by Dynamic Spectrum Allocation (DSA). As a result, innovative techniques like cognitive radio (CR) [3] seems to be a promising solution to spectrum congestion problem through the introduction of the opportunistic usage of frequency bands that are not heavily occupied by Licensed Users (LUs) [4]. In DSA, CR transreceiver can detect and harvest unused frequency bands to transmit information while avoiding to LUs. For better utilization of spectrum, CR need to transmit over multiple non-contiguous frequency holes. To achieve this objective, the Physical Layer architecture (PHY) must be highly flexible and adaptable [5]. Currently, multicarrier based multiple access schemes are
O
actively considered in designing DSA system because of increased spectrum sharing adaptability provided by wide transmission bands [6]. In IEEE 802.22, which is the first world-wide cognitive radio standard, Non-Continuous OFDM (NC-OFDM) [7], [8] has been adopted in the CR environment. NC-OFDM uses non-contiguous blocks of subcarriers to transmit data symbols of Secondary Users (SU) by deactivating the subcarriers used by the Licensed Users (LU) to avoid interference. Non-contiguous Orthogonal Frequency Division Multiplexing (NC-OFDM) [9] and Non-Orthogonal Frequency Division Multiplexing (NOFDM) [10], along with their noncontiguous variant (NC-NOFDM) have been discussed in literature to support CR. NC-OFDM decreases data rate by its transmission over (N − M ) active subcarriers when M of the total N subcarriers are deactivated to avoid interference to LUs. BER performance of NC-OFDM system remains same with the decrease in data rate. However in NC-multicarrier code division multiple access (NC-MC/CDMA) system, the deactivation of M subcarriers introduces loss of orthogonality among different users spreading codes [11]. In MC-CDMA, well known orthogonal Walsh-Hadamard (WH) [12] codes are used to assign multiple codes. But WH codes exist for certain code length (2p , p ∈ I+ ). Specifically, orthogonal WH codes are possible when N = 4m, where N is the code length and m is a natural number. Hence, if the number of active subcarrier in a NC-MC-CDMA doesn’t happen to be a multiple of 4, then orthogonal WH code set does not exist to maintain orthogonality among spreading codes. This results in degradation of BER performance. The performance can be improved with adaptive spreading code selection [11] to minimize the loss of orthogonality. A novel orthogonal complex spreading code referred as Carrier Interferometry (CI) [13]–[16] code has been well studied in OFDM and MC-CDMA system. It has been stated in recent literature that CI based OFDM (CI-OFDM) and CI based MC-CDMA (CI/MC-CDMA) provide better performance and better Peak to Average Power Ratio (PAPR) property as compared to conventional OFDM and MC-CDMA systems. CI codes [13], [14] have unique features which allow MC-CDMA to: 1) have orthogonal spreading sequence of any length N, (N ∈ I+ ), 2) support additional (N − 1) pulse π phase shifted) to accommodate extra users or shape (with N
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information using Pseudo-Orthogonal CI (PO-CI) codes [13] with no expansion in bandwidth. So, CI codes having good spectral sharing characteristics and provide flexible system capacity [17]. A new overloaded multiple access concept, i.e., Walsh Hadamard-spread CI multicarrier Code Division Multiple Access (WH-spread CI/MC-CDMA) [18]–[20] has been discussed which combines the spreading gain diversity of WH-code (CDMA) and phase characteristic property of CI signal (CI/MC-CDMA) to support more users than the orthogonal dimension N . Non-contiguous spectrum access schemes based on NCCI codes have been considered strong candidates for CR and DSA [21], [22]. It has been pointed out in [5] that the performance of adaptive NC-CI/MC-CDMA outperforms adaptive NC-multicarrier code division multiple access (NCMC/CDMA) using Hadamard-Walsh codes. Also, by applying non-contiguous spread orthogonal frequency division multiplexing (NC-SOFDM) with CI code, the loss of orthogonality among spreading codes caused by deactivating subcarriers can be eliminated [21], [22] in DSA. Also, Non-Contiguous CI (NC-CI) based orthogonal frequency division multiplexing (NC-OFDM) improves the performance in terms of BER and PAPR reduction [23]. Cognitive NC-CI Orthogonal Signal Division Multiplexing (Cognitive NCI-OSDM) [24] transmission system has been proposed with high transmission capacity and good adaptability to variable mobile speeds. This identifies possibilities for its future application in autonomous vehicular communications. Variable rate transmission scheme based on orthogonal property of NC-CI codes of any length has been proposed in [25]. Orthogonal property of non-contiguous CI (NC-CI) codes ensures time and frequency diversity gain in non-contiguous allocation compared to contiguous allocation. In recent literature various applications of CI and NC-CI have been discussed [26]–[30]. The rest of the manuscript is organized as follows: related work based on NC-CI codes are presented in Section II. Simulation study on NC-CI codes in non-contiguous spectrum has been presented in Section III. Orthogonal property of NC-CI codes has been highlighted in section IV. Finally, we present our concluding remarks in Section V. II. R ELATED WORK ON N ON - CONTIGUOUS CI CODES NC-OFDM operated over non-contiguous spectrum inherits the shortcomings of OFDM, such as higher PAPR and low transmission rate due to the disabled subcarriers within spectrum holes [9]. In [23], Zhang proposed a method to construct NC-CI codes for the non-contiguous multicarrier system which results in NCI-OFDM system and maintains the advantages of CI codes in NC-OFDM. It has been observed that NCI-OFDM has a higher transmission rate compared with the traditional NC-OFDM. Also, the transmission rate of non-contiguous pseudo-orthogonal CI (NPCI)-OFDM is approximately two fold of NCI-OFDM. Also, PAPR values are lower in NCIOFDM and NPCI-OFDM compared to NC-OFDM system [23]. In the cognitive radio, the activated and deactivated subcarriers are changeable, i.e. the available number of the sub-
carriers is dynamic. Therefore, in the real application, the selfadaptive method [5] can be employed to construct the NCI and NPCI codes with variable length, which further yields the adaptive NCI-OFDM and NPCI-OFDM systems. Different non-contiguous systems based on NC-CI have been discussed in following subsections. A. Orthogonal Signal Division Multiplexing (OSDM) with NCI codes Besides the traditional OFDM transmission, another scheme named Orthogonal Signal Division Multiplexing (OSDM), also known as SD-OFDM, has been proposed to improve BER performance and spectrum efficiency over the frequency selective fading channel [31]. By benefiting from spreading gain in the time domain, OSDM outperforms the traditional OFDM. As in OFDM, OSDM has the same problems of PAPR and Inter-Carrier Interference (ICI). Carrier Interferometry OSDM (CI-OSDM) scheme combining CI codes and OSDM for adaptable transmission has been designed and evaluated in [32]. A cognitive non-continuous Carrier Interferometry OSDM (Cognitive NCI-OSDM) transmission system has been proposed by Zhang [24] to provide autonomous broadband communications among mobile vehicles, especially in an environment without fixed terrestrial vehicular base stations. Similar to an NCI-OFDM system, the NCI codes can be employed in a non-continuous OSDM system to form an NCI-OSDM system. With NCI codes, a non-continuous OSDM system will benefit from features such as higher transmission capacity, low PAPR and ICI [23]. In [24], it has been observed that NCI-OSDM outperforms NCIOFDM, non-contiguous OSDM (NC-OSDM) and NC-OFDM systems. B. Adaptive Non-Contiguous spread orthogonal frequency division multiplexing (SOFDM) for DSA In DSA, it is required to choose a multicarrier based technology such as non-contiguous OFDM (NC-OFDM) [7], [11] for physical layer. Recently, Spread Orthogonal Frequency Division Multiplexing (SOFDM) has demonstrated better performance compare to OFDM in multipath fading channel [21], [33]. Traditionally, WH codes are used in SOFDM system. In non-contiguous spectrum allocation, the loss of orthogonality among spreading codes degrades the performance. To overcome the problem, it has been pointed in [11] the performance of Non-Contiguous SOFDM (NC-SOFDM) can be improved by using adaptive spreading code selection which minimize (or even eliminate) the loss of orthogonality. On the other hand, in many cases NC-SOFDM based on WH code sets needs to deactivate more subcarriers which results in degradation of data rate. Recently, Li et al. [21], [22] proposed NC-SOFDM based on CI codes. Also, a novel code called Binary Code based on WH code has been developed for any code length. It has been shown in [21], that NC-SOFDM based on binary code or CI code sets outperforms the adaptive NC-SOFDM with adaptive WH code sets and traditional NC-OFDM. It is
−1
10
WH code−Case 1 Simulated WH code−Case 1 Analytical WH code−Case 2 Simulated WH code−Case 2 Analytical WH code−Case 3 Binary code−Case4 NC−CI code−Case 5
−2
10
BER
interesting to note that in some cases, NC-SOFDM with CI code performs better than the NC-SOFDM with binary code [21]. NC-CI signature waveforms maintain orthogonality of any code length. So, NC-SOFDM based on NC-CI codes [21], [22] maintains same data rate as that of traditional OFDM and improves the performance by exploiting the diversity gain and eliminating the orthogonality loss.
−3
III. S IMULATION STUDY OF NC-CI CODES In DSA, for Secondary Users (SUs) it is required to deactivate the subcarriers which are used by Primary User (PU)s to avoid interference. For example, it has been assumed that total N narrowband subcarriers are available. PU uses M subcarriers. These M subcarriers are deactivated for the SUs to avoid interference. So in total (N − M ) subcarriers are activated for SUs in NC-MC-CDMA system operating in a DSA network. Theoretically, a (N − M ) subcarrier based NC-MC/CDMA supports (N − M ) users with (N − M ) orthogonal spreading code. However, if (N − M ) ̸= 4m, where m is any natural number, then such a group of (N − M ) orthogonal WH codes does not exist. Different spreading codes and their orthogonality are discussed in Table I for non-contiguous spectrum allocation. BER performances of NC-MC/CDMA with WH code (case 1, case 2 and case 3), NC-CI code and Binary code [21] over AWGN channel have been shown in Figure 1. NC-MC/CDMA with WH code (case 1, case 2) cannot eliminate the loss of orthogonality. So, over AWGN channel, BER performance of NC-CI/MC-CDMA does not maintain single user bound ) (√ 2Eb 2 (i.e., Q N0 ) due to non zero value of E[ρk,l ], where th ρk,l is the cross-correlation between k user and lth user signature waveforms. Here, Eb represents bit energy, which is
−4
10
5
10
SNR (dB)
15
20
25
Fig. 1. BER performance of NC-MC/CDMA with different spreading codes discussed in Table III with N = 16, M = 5 and K = 11 users.
assumed to be same for all users. ( BER performance of)case1 √ 2 and case2 corresponds to Q and Eb −1 M 2(K−1) N 2 +( N0 ) ( ) √ 2 Q respectively over AWGN Eb −1 N ′ −N +M 2(K−1)
N ′2
+( N ) 0
channel. It is also observed from figure that the simulation results match the theoretical results, confirming the performance analysis. From the figure, it is also clear that (√ ) WH code (case 2Eb 3) maintains single user bound (Q N0 ) over AWGN. But the overall data rate of secondary users/data stream has been decreased. Performance of both NC-CI code and Binary code [21] are equal to single user performance. For the generation of CI pulse shape in time domain over non-contiguous bands, Weighted Inverse Fast Fourier Transform (IFFT) has been used. 20 15 10 5 0 −5
0
0.2
0.4
Time
0.6
0.8
1
(a)
4∆f 0
2∆f ∆f
4∆f 8∆f
3∆f
10∆f 9∆f 11∆f
(b)
4∆f 16∆f 18∆f 17∆f 19∆f
24∆f 26∆f 25∆f 27∆f
Frequency
It has been discussed in literature that two fundamental paradigm shifts must emerge if 4G is to succeed. First, new bandwidth sharing/allocation strategy and the second refers to the transformation of handhelds, presenting the vision of transition from hardware defined to software defined radio. In [26], a multicarrier multiplatform transceiver has been discussed to operate over wide bands while mitigating interference. Also, this system model enables a single piece of hardware to operate across the many emerging platforms, including WLAN, PAN, fixed broadband, 4G, 3G, and 2G cellular. A general form of transmitter and receiver with CI signalling has been proposed that support Direct Sequence CDMA (DSCDMA), Multicarrier CDMA (MC-CDMA), Time Division Multiple Access (TDMA), and OFDM modes over multiple non-contiguous bands. OFDM, MC-CDMA, and TDMA transmitter can be constructed using appropriately weighted IFFTs. Using the multicarrier CI signal, the weighting applied to the IFFT depends on incoming data stream, multiple access techniques and available number of subcarriers per bands used in particular multiple access technique.
10
Magnitude
C. Multi-mode Transmission with Weighted Inverse Fast Fourier Transform (IFFT)
Fig. 2. (a) Time domain representation and (b) Frequency domain representation of NC-CI signature waveform over Non-Contiguous spectrum
An example of the multi-carrier signal with NC-CI codes has been illustrated in Figure 2. Figure 2 (a) represents one period of a multi-carrier signal operating over four non-contiguous bands [26], and Figure 2 (b) represents the corresponding frequency domain representation. The NC-CI waveform shown in Fig. 2 is also obtained using Tektronix
TABLE I O RTHOGONALITY AMONG DIFFERENT SPREADING CODES FOR NON - CONTIGUOUS SPECTRUM ALLOCATION WITH (N − M ) ACTIVE SUBCARRIERS USED IN NC-SOFDM AND NC-MC/CDMA
Case
Code
Code length
Case 1
WH
N , (N − M ) ̸= 4m
Case 2
WH
Case 3
WH
N ′ , N ′ = 4m and N ′ > (N − M ), where, N ′ is the smallest multiple of 4 which is larger than (N − M ) N ′ , N ′ = 4m and N ′ < (N − M ) where, N ′ is the largest multiple of 4 which is smaller than (N − M )
Case 4
Binary [21]
Case 5
NC-CI [5], [21], [23]
Orthogonality
Variance of Partial cross-correlation ( 2 ) E[ρk,l ]
Summary
Non-orthogonal
M N2
Non-orthogonal
N ′ −(N −M ) N ′2
Loss of orthogonality between spreading codes. Each user/data stream spreads data with N length WH codes and send over (N − M ) subcarriers.
Orthogonal
0
(N − M )
Orthogonal
0
(N − M )
Orthogonal
0
ArbExpress-AXW100 and MATLAB. The generated waveform is observed in YOKOGAWA DLM 2.5GS/s 200 MHz MSO and it is noteworthy that four distinct frequency bands are obtained for SUs. In the following section we will discuss the orthogonality property of NC-CI codes in multiuser noncontiguous allocation. IV. O RTHOGONALITY A NALYSIS OF NC-CI SIGNATURE WAVEFORMS
As discussed in section III, there are basically two methods to generate NC-CI codes; a) method by Zhang and b) method by Wu et al.. Although different approaches have been taken for NC-CI codes, both methods generate orthogonal NC-CI signature waveform. CI codes discussed in CI/MC-CDMA [13], [14], maintain orthogonality over contiguous allocation. It has been shown in [13], that N orthogonal signature waveform can be placed at ( N k∆f ), k = 1, 2, . . . , N − 1 positions over contiguous band. For simplicity, we have taken total N = 16 available subcarriers i.e., (fi = fc + (i − 1)∆f, i = 1, 2, 3 . . . 16). fi is the frequency of ith narrow band subcarrier with center frequency fc . ∆f is selected such that orthogonality between carrier frequencies can be maintained. Typically, ∆f = 1/Tb where Tb is bit duration. In a particular instant, we have assumed
Loss of orthogonality can be reduced. Each user/data stream will only use first (N − M ) elements of its spreading code with length N ′ . Each user/data stream spread information over available (N − M ) subcarriers and ignore the last [N ′ − (N − M )] elements of its information Need to deactivate more [(N −M )−N ′ ] subcarriers. So, all available spectrum for secondary user/data stream cannot be fully utilized. Each user/data stream will use first N ′ (N ′ = 4m) elements of spreading code and spread information over available N ′ subcarriers. Also overall data rate of number of secondary users/data stream has been decreased. With this spreading code matrix, 4m user/data stream are spread onto 4m subcarriers to increase the diversity gain. Meanwhile, the rest of the user/data stream is also spread onto different subcarriers. When only one data symbol is left, there is no spreading any more. Notice that each subsystem also helps to increase the frequency distance to other subsystems Adaptively allocate orthogonal CI spreading codes, eliminating the loss of orthogonality entirely among spreading codes, regardless of the number of deactivated subcarriers.
that A = [1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1], where Ai = 0 if the ith subcarrier is deactivated and Ai = 0 if the ith subcarrier is active [5], [21], [22]. Such a case is shown in Figure 4. The signature waveform of k th user in NC-CI/MCCDMA with NC-CI codes corresponds to b ck (t)
=
2π
2π
2π
2π
2π
2π
ej2π0∆f t .ej 11 0k + ej2π1∆f t .ej 11 1k + ej2π3∆f t .ej 11 2k + ej2π4∆f t .ej 11 3k + ej2π5∆f t .ej 11 4k + ej2π6∆f t .ej 11 5k + 2π 2π ej2π7∆f t .ej 11 6k + ej2π10∆f t .ej 11 7k + 2π 2π ej2π12∆f t .ej 11 8k + ej2π13∆f t .ej 11 9k + 2π
ej2π15∆f t .ej 11 10k
(1)
Figure 5 (a) and Figure 5 (b) illustrate the signature waveform of 6th and 7th user in contiguous spectrum allocation with N = 16 subcarriers and signature waveform of 6th and 7th user in NC-CI/MC-CDMA by superposition of all (N − M ) = 11 active non-contiguous subcarriers. It is clear from Figure 5, that the power variation is more in NCCI signature envelope compared to contiguous CI signature envelope. This leads to a higher PAPR in NC-CI compared to contiguous CI based system. From Figure 5 (a), it is clear that in contiguous CI signature
15
th
6 user
10
Magnitude
7th user
orthogonal positions
5 0
0
0.5
1 Time
1.5
2
(a)
10 Magnitude
6th user
Fig. 3. Generation of NC-CI signature waveform and frequency plot over non-contiguous spectrum (Fig. 2) using Tektronix ArbExpress-AXW100 and MATLAB. Instrument used: YOKOGAWA DLM 2.5GS/s 200 MHz Mixed Signal Oscilloscope (specification: FFT Points 1.25K, Window: Rectangular, 72MHz center, 182MHz Span), and Tektronix AFG 3021B Arbitary function generator 250MS/s, 25 MHz (specification 1 MHz, 2 V (p-p)max ). N subcarrier
th
7 user
5
0
−5
0
0.5
1 Time
1.5
2
(b) Fig. 5. (a) Envelope of 6th user and 7th user signature waveform with N = 16 contiguous subcarriers; (b) Envelope of 6th user and 7th user signature waveform (eq. 1) with NC-CI codes with (N − M ) = 11 active subcarriers;
f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16
conjugate of cl (t). fi
Deactivated subcarrier
fi
Activated subcarrier
Fig. 4. Non-contiguous subcarrier with deactivated subcarrier; N = 16 total subcarrier and M = 5 deactivated subcarrier
waveform superposition of one user’s all carriers creates a main lobe in time and makes components of all other user’s signature waveforms zero at {( N k∆f ), k = 1, 2, . . . , N − 1} orthogonal position. From Figure 5 one may conclude that no orthogonal positions are possible in NC-CI signature waveforms as found in contiguous CI signature waveform. To examine the orthogonality of NC-CI signature waveforms in time, we shall analyse orthogonality among NC-CI signature waveforms mathematically. The “orthogonality” of signature waveforms is defined as zero cross-correlation between two signature It ( waveforms. 2π is clear from Eq. (2) that, when k = l, 1 + ej 11 1(k−l) + 2π
2π
2π
2π
2π
2π
2π
2π
2π
2π
ej 11 2(k−l) +ej 11 3(k−l) +ej 11 4(k−l) +ej 11 5(k−l) +ej 11)6(k−l) + 2π 2π 2π 2π ej 11 7(k−l) + ej 11 8(k−l) + ej 11 9(k−l) + ej 11 10(k−l) = 11, ( 2π 2π 2π and when k ̸= l, 1 + ej 11 1(k−l) + ej 11 2(k−l) + ej 11 3(k−l) + ej 11 4(k−l) +ej 11 5(k−l) +ej)11 6(k−l) +ej 11 7(k−l) +ej 11 8(k−l) + 2π 2π ej 11 9(k−l) + ej 11 10(k−l) = 0. Hence, it is proved that signature waveforms using NC-CI code set are orthogonal, i.e. [ ∫T ] { 1 ∗ 1 k=l T 0 ck (t)cl (t) dt Re = (3) ∫ 1 T ∗ 0 k ̸= l c (t)c (t) dt k k T 0 where ck (t) and cl (t) represents the k th and lth user’s signature waveform respectively, and c∗l (t) denotes the complex
V. C ONCLUSION This paper discusses the different methods of generating NC-CI signature waveforms used in non-contiguous multicarrier communications for Dynamic Spectrum Access (DSA). DSA enables the secondary users to transmit over unoccupied frequency spectrum of the primary user. WH codes are suboptimal for non-contiguous allocation of subcarriers to the secondary users as orthogonal codes are not available for all values of subcarriers. Non-contiguous communications based on OFDM and MC-CDMA using NC-CI and WH codes have been evaluated extensively. The results are validated for single and multiuser environments. It has been shown that orthogonality between NC-CI signature waveforms is maintained even over multiple sub-bands of aggregated noncontiguous spectrum making it suitable for Cognitive Radio. R EFERENCES [1] Q. Zhao and B. Sadler, “A survey of dynamic spectrum access,” IEEE Signal Processing Magazine, vol. 24, no. 3, pp. 79 –89, May 2007. [2] FCC, “Notice of proposed rulemaking and order, facilitating opportunities for flexible, efficient and reliable spectrum use employing cognitive radio technologies,” Tech. Rep. ET Docet No. 03-108, 2003. [3] S. Haykin, “Cognitive radio: brain-empowered wireless communications.” IEEE Journal on Selected Areas in Communications, vol. 23, no. 2, pp. 201–220, 2005. [4] I. Mitola, J. and J. Maguire, G.Q., “Cognitive radio: making software radios more personal,” Personal Communications, IEEE, vol. 6, no. 4, pp. 13 –18, aug 1999. [5] Z. Wu, P. Ratazzi, V. Chakravarthy, and L. Hong, “Performance evaluation of adaptive non-contiguous MC-CDMA and non-contiguous CI/MC-CDMA for dynamic spectrum access,” in 3rd Int. Conf. Cognitive Radio Oriented Wireless Netw. Commun., May 2008, pp. 1 –6. [6] T. Luo, F. Lin, T. Jiang, M. Guizani, and W. Chen, “Multicarrier modulation and cooperative communication in multihop cognitive radio networks,” Wireless Communications, IEEE, vol. 18, no. 1, pp. 38 –45, february 2011.
1 T
∫
T 0
1 = T
ck (t)c∗l (t) dt
∫
T(
2π
2π
2π
2π
2π
ej2π0∆f t .ej 11 0k + ej2π1∆f t .ej 11 1k + ej2π3∆f t .ej 11 2k + ej2π4∆f t .ej 11 3k + ej2π5∆f t .ej 11 4k +
0 2π
2π
2π
2π
2π
t j 11 6k ej2π6∆f t .ej 11 5k +)ej2π7∆f .e + ej2π10∆f t .ej 11 7k + ej2π12∆f t .ej 11 8k + ej2π13∆f t .ej 11 9k + (
ej2π15∆f t .ej 11 10k . e−j2π0∆f t .e−j 11 0l + e−j2π1∆f t .e−j 11 1l + e−j2π3∆f t .e−j 11 2l + e−j2π4∆f t .e−j 11 3l + 2π
2π
2π
2π
2π
e−j2π5∆f t .e−j 11 4l + e−j2π6∆f t .e−j 11 5l + e)−j2π7∆f t .e−j 11 6l + e−j2π10∆f t .e−j 11 7l + e−j2π12∆f t .e−j 11 8l + 2π
2π
2π
2π
2π
e−j2π13∆f t .e−j 11 9l + e−j2π15∆f t .e−j 11 10l dt ∫ 2π 2π 2π 2π 2π 2π 2π 1 T( 1 + ej 11 1(k−l) + ej 11 2(k−l) + ej 11 3(k−l) + ej 11 4(k−l) + ej 11 5(k−l) + ej 11 6(k−l) + ej 11 7(k−l) + = T 0 ) 2π 2π j 2π 11 e 8(k−l) + ej 11 9(k−l) + ej 11 10(k−l) + high frequency term dt ( 2π 2π 2π 2π 2π 2π 2π 2π = 1 + ej 11 1(k−l) + ej 11 2(k−l) + ej 11 3(k−l) + ej 11 4(k−l) + ej 11 5(k−l) + ej 11 6(k−l) + ej 11 7(k−l) + ej 11 8(k−l) + ) 2π 2π ej 11 9(k−l) + ej 11 10(k−l) (2) 2π
2π
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