Concatenative Complete Complementary Code Division Multiple Access and Its Fast Transform

Over multipath channels, complete complementary code division multiple access and convolutional spreading code division multiple access provide inter-channel interference free transmission with an enhanced spectral efficiency. However, the convolutional spreading (CS) operation of the systems is computationally complex and involves a high peak-to-average power ratio. To address such issues, we propose the concatenative complete complementary code (CCCC) division multiple access, named (CCC-CDMA). Since the CCCCs can be generated from the rows of the Walsh-Hadamard or discrete Fourier transform matrices, the CS operation can be implemented using corresponding fast transforms to reduce computational complexity. Simulation results show that the enlargement of the spreading factor strengthens the robustness against clipping noise. The binary CCCC generated by Walsh-Hadamard matrix exhibited excellent robustness against Doppler frequency shifts.


Concatenative Complete Complementary Code Division Multiple Access and Its Fast Transform Hikaru Mizuyoshi and Chenggao Han
Abstract-Over multipath channels, complete complementary code division multiple access and convolutional spreading code division multiple access provide inter-channel interference free transmission with an enhanced spectral efficiency.However, the convolutional spreading (CS) operation of the systems is computationally complex and involves a high peak-to-average power ratio.To address such issues, we propose the concatenative complete complementary code (CCCC) division multiple access, named (CCC-CDMA).Since the CCCCs can be generated from the rows of the Walsh-Hadamard or discrete Fourier transform matrices, the CS operation can be implemented using corresponding fast transforms to reduce computational complexity.Simulation results show that the enlargement of the spreading factor strengthens the robustness against clipping noise.The binary CCCC generated by Walsh-Hadamard matrix exhibited excellent robustness against Doppler frequency shifts.

I. INTRODUCTION
A. Background C ODE division multiple access (CDMA) and orthogonal frequency division multiple access (OFDMA) are two major multiplexing schemes of current digital communication system.When comparing direct spread (DS)-CDMA with OFDMA, DS-CDMA distinguishes users based on the previously assigned signals, called spreading sequences, while the indexes of frequency sub-carrier are used for user identification in OFDMA.More specifically, CDMA multiplies each user's modulated symbols by the corresponding spreading sequence while those are multiplied by several one-to-one sinusoidal waves.Thus, if we treat the sampled sinusoidal waves/signals as spreading sequences in a unified manner, the essential differences between DS-CDMA and OFDMA are as follows: Hikaru Mizuyoshi was with the Graduate School of Informatics and Engineering, The University of Electro-Communications, Chofu-shi, Tokyo 182-8585, Japan.He is now with Honda Motor Company Ltd., Haga-machi, Tochigi-ken 321-3321, Japan (e-mail: mizuyoshi@uec.ac.jp).
Chenggao Han is with the Graduate School of Informatics and Engineering, The University of Electro-Communications, Chofu-shi, Tokyo 182-8585, Japan (e-mail: han.ic@uec.ac.jp).
Digital Object Identifier 10.1109/TWC.2023.3250659 1) Number of spreading sequences Since each user transmits modulated symbols over multiple sub-carriers, OFDMA assigns multiple sequences while a single sequence is assigned to each user in DS-CDMA.

2) Synchronization
Each OFDMA packet is composed of synchronously summarized spread signals, and the transmitted symbols are detected synchronously at the receiver.Meanwhile, DS-CDMA transmits the spread signals asynchronously.Hence, synchronization is not required at both sides.
3) The cyclic prefix (CP) scheme An OFDM-based system utilizes the CP scheme to convert the aperiodic convolution between the transmitted packet and channel impulse response (CIR) into periodic convolution.Therefore, we may regard OFDMA as a special CDMA that assigns multiple sinusoidal signals for each user, transmits and receives signals in the synchronous manner, and employs CP scheme.
In terms of signal design, the performance of CDMA over multipath channel primarily depends on the correlation properties of the employed spreading sequences.The maximum spectral efficiency (SE) [1] can be achieved by a sequence set (SS) with the ideal correlation properties.In other words, the auto-correlation function of each sequence is zero except at zero shift, and the cross-correlation functions of any distinct sequence pair are zero at all shifts.Unfortunately, such an SS is non-existent, and instead, the pseudo-random sequences which have small side-lobes, e.g., Gold sequences, Kasami sequences, and maximum-length sequences (M-sequences) [2], were widely employed for DS-CDMA.
Even the side-lobes of the pseudo-random sequence are designed as a small value, the non-ideal cross-correlation causes inter-channel interference (ICI) and involves a near-far problem.Thus, DS-CDMA can be classified as an ICI limited system since the overall cell capacity cannot be increased by increasing the transmission powers.Therefore, a complex power control unit is essential to combat the near-far problem.On the contrary, OFDMA utilizes the sinusoidal waves that have the ideal periodic cross-correlation property and is an ICI-free system.
While the cross-correlation property of the employed spreading sequences is associated with ICI, the autocorrelation property affects the accuracy of detected symbols over multipath channels.Accordingly, DS-CDMA spreads the modulated symbols using the sequences that have sharp autocorrelation and it generally attains full path diversity over a multipath fading channel, i.e., the order is equal to the number of independent paths.Meanwhile, since the periodic autocorrelations of the sinusoidal waves are constant in amplitude across all shifts the achievable diversity order of the naive OFDM/OFDMA is only one.Therefore, in the practical uses of OFDM/OFDMA, codings over sub-carriers is usually applied to improve the attainable diversity order [3], [4].Nevertheless, obtaining the full path diversity for OFDM/OFDMA is not a easy task [5], [6].

B. Related Works
The signals with the ideal correlation properties have been investigated by numerous researchers to realize ICI-free communication systems.Such approaches can be classified into two classes: complete complementary code (CCC) [7], [8], [9], [10] and zero-correlation zone (ZCZ) sequences [11], [12], [13].The former utilizes multiple sequences to realize an ideal correlation sum while the ideal correlation properties are designed to achieve on ZCZ in periodic manner for the latter.
The CCC-based ICI-free system, called complete complementary coded CDMA (CC-CDMA), was proposed by Suehiro et al. [14], [15].In CC-CDMA, each sub-packet spread by a different sequence should be passed through an individual matched filter.Since as the separation of sub-packets is completed on time or frequency domain, it can be further classified into two categories: time division multiplex CC-CDMA [15] and frequency division multiplex CC-CDMA [16].The former requires zero padding (ZP) or CP schemes for each sub-packet to prevent inter-sub-packet interferences cause by multipath propagation, and hence, it is inferior to latter in terms of SE but is superior in system performance and implementation complexity [17], [18], [19].
The concept of ZCZ sequences first appeared in [11] and originally applied to realize ICI-free quasi-synchronous CDMA to maintain the orthogonality between channels if user's time delays occur within a few chips [20], [21].Subsequently, the ZCZ sequence based CDMA with convolutional spreading (CS) scheme [15] was investigated by Weerashinghe et al. termed as CS-CDMA [22], [23].The authors show that associate with multiple-input single-output, CS-CDMA provides complete transmit and path diversities [24].Later, Yue et al. realized an SE higher than DS-CDMA and chipinterleaved block spread CDMA proposed in [25] by utilizing iterative partial multiuser detection to CS-CDMA [26].
Both CC-and CS-CDMAs utilize the CS scheme to achieve a high SE, which involves high peak-to-average power ratio (PAPR).Accordingly, Weerasinghe et al. investigated the robustness against clipping noise for various ZCZ sequences and indicated that the M-sequence based ZCZ (M-ZCZ) sequence as the most robust [27].Moreover, the sequence selection also affects the performance of CS-CDMA over fast fading channels and the inferior-to-superior association is related to the use of a channel estimation at the receiver.
A key difference between CC-and CS-CDMA is that, in CC-CDMA, each user utilizes multiple spreading sequences as well as OFDMA while a single sequence is used in CS-CDMA.Consequently, CC-CDMA requires multiple CPs/ZPs to prevent inter-sub-packet interferences over multipach channels while single CP is required for each packet in CS-CDMA, and generally, CS-CDMA achieves higher SE than CC-CDMA.Moreover, since a larger ZCZ of the employed SS translates to a higher SE of CS-CDMA, we desire a simple construction of ZCZ sequences with the largest ZCZ.
On the other hand, Han et al. have proposed a special class of CCC, named concatenative CCC (CCCC), and constructed the binary and polyphase CCCCs using Walsh Hadamard (WH) and discrete Fourier transform (DFT) matrices, respectively [32].In given CCCC, ZCZ sequences can be constructed by concatenating the sequences in each complementary code set and elongating ZCZ by padding zeros for each sequence before concatenation.Thus, in association with the CS-CDMA scheme, the concatenative complete complementary code division multiple access (CCC-CDMA) provides a simple ZP scheme to enhance SE.Furthermore, Han et al. proposed an OFDM like fast Fourier transform based implementation structure for the transmitter of CCC-CDMA to reduce the computational complexity of the CS operation, and showed a tradeoff relationship between clipping resistance and computational complexity [33], [34].

C. Contributions
In this study, we present a comprehensive fast transform (FT) based transceiver structures of both binary and polyphase CCC-CDMAs by introducing interleaver/deinterleaver components.We prove that the outputs of the proposed structure are equivalent to that of the CS-CDMA employing the ZCZ sequences constructed from CCCC and provide the performance and complexity analysis under assumption that the maximum likelihood (ML) detection is employed at the receiver.The numerical results indicate that CCC-CDMA enhances the resistance against clipping noise by simply incrementing of the employed sequence length i.e., increment of spreading factor (SF), and the binary CCC-CDMA has excellent robustness against Doppler frequency shifts.Compared with OFDMA, the proposed CCC-CDMA is also a synchronous multiple access transmission that can be implemented by FTs, however, it is superior to OFDMA at the achievable diversity order over multipath fading channel, and serves as a simple countermeasure against clipping noise while being robust against frequency shifts.

D. Paper Organization
The remainder of this paper is organized as follows.In Section II, after defining the correlation and correlation sum, we introduce some sequence set/family with ideal correlation (sum).In Section III, we briefly review the conventional CC-CDMA and CS-CDMA.In Section IV, we present the novel FT-based tranceiver structures of CCC-CDMA.The performance of CCC-CDMA over Reighy fading multipath channel is analyzed in Section V.The numerical results are shown in Section VI and, finally, we conclude this study in Section VII.

E. Notations
A vector is denoted by a bold lowercase letter and is also represented with its entries as v = (v n ) N −1 n=0 .0 N denotes an all zero vector of length N .For given vector v, let v(a : b) = (v n ) b n=a be the length-(b − a + 1) partial vector of v.For simplicity, we identify the vector v and a sequence v(n) A matrix is denoted by a bold uppercase letter and an M × N matrix A with its . The mth row and the nth column vectors of A are denoted by a m and a n , respectively.Moreover, A * and A H denote the complex conjugate and Hermitian transpose of A, respectively.The determinant and rank of A are denoted by det(A) and rank(A), respectively.Consider f i N to be the ith row of the N -dimensional DFT matrix and h i N be the ith row of the N -dimensional WH matrix H N , whose recursive generation is given by . An indexed set resembles a set of numbered elements and is denoted by outline S = {s n } N −1 n=0 while a family S stands for a collection of sets.δ(τ ) represents Kronecker's delta function and [x] L denotes a non-negative integer that is less than L and satisfies (x − [x] L ) mod L = 0.For an integer m, m 2 denotes the binary extension of m and m 2 ⊕ n 2 represents the bit-wise exclusive OR logic of two binary vectors m 2 and n 2 .The expectation operation of a random variable x is denoted as E{x}.

II. PRELIMINARY
Here, a set For two sequences s of length L and s ′ of length L ′ , their corresponding aperiodic convolution and aperiodic correlation [2] are defined as follows: 2), then it is called the auto-correlation of s and is denoted as ϕ A (s; τ ).Otherwise, it is called the cross-correlation between s and s ′ .
Since ϕ A (s, s ′ ; τ ) takes non-zero values on the interval −L ′ < τ < L, to coordinate with the convolution given in (1), and we define two length-(L + L ′ − 1) vectors as follows: Notice that the τ th entry of ϕ A (s, s ′ ) is given by ) and we proved the following equalities in Appendix A.
For two sequences s and s ′ of length-L, the periodic convolution and periodic correlation are defined as follows: For the case τ ≥ 0, the periodic convolution and correlation are related with the aperiodic convolution and correlation, respectively, as Similar to the aperiodic case, if we define length-L vectors as it is not difficult to observe that the following equalities hold for sequences s, s ′ , and s ′′ of the same length.For a length L sequence s, let S e [s] := (s([e + ℓ] L )) L−1 ℓ=0 be the e-shifted sequence of s.Then, we obtain the following form: For two SSs, (N, L)-S and (N, L ′ )-S ′ , the aperiodic correlation sum can be defined as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
If S = S ′ , it is called the aperiodic auto-correlation sum and is denoted by Φ A (S; τ ).Otherwise, ( 14) is called the aperiodic cross-correlation sum.
A. ZCZ-SS Definition 1: A sequence s is called the perfect sequence (PS) if the periodic auto-correlation of s is zero except for zero-shift, i.e., ϕ P (s; τ ) = E s δ(τ ), where E s := ss H . From (13), each shifted version of a PS S e [s], 0 ≤ e < L, is also PS.
Chu-sequence [28] generated by is a well-known polyphase PS, where Q resembles an integer relatively prime to L and, in this study, we let Q = 1 for simplicity.
Meanwhile, a is considered as the M-sequence of length-L, that is, a binary sequence with the periodic auto-correlation function Subsequently, the two real-valued sequence generated by is the PS called the modified maximum-length sequence [35].Definition 2: An (M, L)-S is called a ZCZ-SS, and it is denoted by (M, L; Z)-ZCZ, if the periodic auto-correlation of each sequence is zero for 0 < |τ | ≤ Z and the periodic cross-correlation between any distinct sequence pair is zero for |τ | ≤ Z, i.e., Then, each (M, L; Z)-ZCZ satisfies Z ≤ ⌊L/M ⌋ − 1 [36].However, the equality may not be achieved for a ZCZ-SS with a small alphabet size and the bound is considered as Z ≤ ⌊L/2M ⌋ for binary ZCZ-SS.For a PS s of length-L, let Z = ⌊ L M ⌋ − 1, then the SS constructed by is an (M, L; Z)-ZCZ.
Meanwhile, from the length-7 M-sequence a = (− − + + + − +), the PS generated using (16) m=0 is called a complete complementary code (CCC), denoted by (M, N, L)-CCC, if the sequences in each row are complementary SS and the cross-correlation sum between any two distinct complementary SSs is zero for all shifts, i.e., For a given is (M, N L; Z)-ZCZ, then we call C Concatenative CCC (CCCC) and denote it as (M, N, L; Z)-CCCC.
To describe the constructions proposed in [32] in a unified manner, we let Ω = H N for binary CCCC construction while for polyphase case Ω = F N and ω k be the kth row of Ω.
with the interleavring rule k = π (m) (n) be the (N, N, N )-SQF comprising the rows of Ω. Subsequently, if we use an unexacting expression k = k 2 for binary case and specify the interleaving rule as follows: then the resultant SQFs are the binary (N, N, N ; N/2)-CCCC and polyphase (N, N, N ; N − 1)-CCCC.Notice the interleaving rule given in ( 23) can be deinterleaved by and the proposed constructions are optimal in the sense that the qualities on the (conjectured) bounds, Z = N − 1 and N/2 respectively, hold for both cases.In the practical applications of (N, N, L; Z)-CCCC, a large merit figure η = (Z + 1)/L ≤ 1 is expected to achieve a high SE and we may improve it by ZP scheme [32].
while the polyphase CCCC , can be constructed with the natural addition for indexes.
Although the resultant CCCCs are of the same length L = N = 4, the ZCZ of the binary CCCC is Z = 2 while it is increased to Z = 3 for the polyphase case.Accordingly, the SS constructed by ( 22) is a (4,16;2)-ZCZ with merit figure η = 2/3.However, if we terminate 0 2 for each sequence before concatenation as then the resultant SS is a (4,24;4)-ZCZ with a improved merit figure η = 4/5.

III. BRIEF REVIEW OF CC-CDMA AND CS-CDMA
We consider the down-link of M users in MA systems equipped with single transmitting/receiving antenna.For the sake of establishing a unifying description, let u m , 0 ≤ m < M , be the mth user's modulated symbol vector1 of length-K selected from constellation S K , and we assume that the antenna transmits the elements of ⃗ x serially.We consider the general case that users are distributed on the distinct positions and the mth user receives the transmitted signal over length-(P (m) + 1) quasi-static multipath channel h m P = h (m) (p) Thus, the received signal can be expressed as follows: where ⃗ ξ m denotes the mutually independent circularly symmetric additive white Gaussian noise (AWGN) with zero mean and variance E ⃗ ξ In the present analysis, we let max 0≤m<M P (m) ≤ G and assume the mth user's CIR h m P can be perfectly recovered at the receiver of the mth user.Owing to the guard-part 0 G , from the received signal, we can obtain N inter-sub-packet interference free vectors of length-K + L + P (m) − 1 as

A. Review of CC-CDMA [14]
for 0 ≤ n < N , where we let x n := . Each y m n is then inputed into the matched filter of the corresponding spreading sequence c m n and the outputs of matched filter can be summarized as follows: We proved in Appendix B that the CC-CDMA provides ICI-free relationship as where r m = r (m) (τ ) , E = LN , and η m denotes a length-(K + P (m) ) complex Gasussian random vector with zero mean and variance Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Let y m be the mth user's CP removed signal of length L. As despreading process of CS, we then calculate the periodic cross-correlation between the received signal y and the mth user's spreading sequence s m as r m = ϕ P (y m , s m ) and detect the transmitted symbols from the first K + P (m) output r m = r (m) (τ ) . In Appendix C, we proved that if we let E = E s m , then CS-CDMA provides the same relationship given in (29).

C. The Comparison of CC-and CS-CDMAs
The CC-and CS-CDMAs provide the same ICI-free input-output relationship and achieves higher SE than the DS-CDMAs employing Gold sequence, M-sequence, and Walsh sequence [18], [19], [37], [38].When comparing the SEs of these two systems, CC-CDMA transmits KM symbols using packet of length N (K + L + G − 1), where the number of users is bounded by M ≤ N .Accordingly, the SE of CC-CDMA is bounded by where L C denotes the length of each sequence.Meanwhile, for CS-CDMA, to transmit K symbols for each user, it is necessary to employ a ZCZ-SS of length L ≥ M (Z + 1), where Z ≥ K + G − 1. Accordingly, the length of packet is L + G ≥ M (K + G) + G and the SE of CS-CDMA is bounded by When comparing (30) with (31), CS-CDMA achieves higher efficiency than CC-CDMA in the case M (L C − 1) > G and vise versa.Additionally, a large L allows K ≫ G and the SE of CC-and CS-CDMAs are close to that of OFDMA given by η OFDMA = K/(K + G).

IV. CCC-CDMA WITH ITS IMPLEMENTATIONS
In this section, we introduce interleaver/deinterleaver components and propose a novel comprehensive FT-based implementation structure for CCC-CDMA.

A. FT-Based Tranceiver Structures of CCC-CDMA
The FT-based transmitter structure of CCC-CDMA is illustrated in Fig. 3. µ m (τ ) denotes the state of the mth user's shiftregister (SR) of length N at time τ and we assume that the initial states are set to 0 N for all users.Subsequently, the modulated symbols are serially fed into SR and, after each shift, the N -points FT is performed over SR to yield signal µ m (τ )Ω until all modulated symbols are shifted out from the SR.Consequently, the transmitter performs totally K +N −1 FTs, that begins and terminates at the states µ m (0) = u (m) (0), 0 N −1 and µ m (K + N − 1) = 0 N −1 , u (m) (K − 1) , respectively.
Let p .With the mth user's interleaver with length-G is copied to the head of x, as illustrated in Fig. 4. Notice that the former overlapped part convert aperiodic convolution to periodic one according to (9) while the later CP part acts as guard against multipath.
At the mth user's receiver, after the removal of CP, the length-N D vector y m is arranged into a cyclic shift-register (CSR) whose tth value at time τ can be expressed by ν Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

C. The Proof of Equivalence
While the ZCZ-SS generated from CCCC is employed for CS-CDMA, we proved in Appendix D that the outputs at the transmitter and receiver of the mth user are represented by respectively.
In the FT-based transmitter structure, since the value of the ℓth SR at time τ is given by µ (m) ℓ (τ ) = u (m) (τ − ℓ), the kth output of the FT can be expressed as while the mth user's signal at time τ is represented by Since the nth input of the delay is connected with the kth output of FT by interleaving rule k = π (m) (n), substituting (38) into (39), we obtain where the last equality is caused by the symmetry of Ω.Thus, if the interleaving rule is specified using (23), the mth user's output can be expressed as Let τ = n ′ D + ℓ ′ for 0 ≤ n ′ < N and 0 ≤ ℓ ′ < D. Resultantly, since u (m) (τ ) = 0 if τ < 0 or τ ≥ K, (40) can be rewritten for the case n ′ > 0 as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
and for the case n ′ = 0 as Thus, it is not difficult to observe that in Fig. 4, the summation of the overlapped part realize (43), while (42) guarantees that the copied part comprises the CP of CS-CDMA.Moreover, since it is common for all users, the OCP can be processed after the summation of all user's signals.
In the proposed receiver structure, since the kth output of the interleaver at time τ can be expressed as v where we let n = [−n ′ ] N for the last equation.
or F −1 N for binary or polyphase cases, respectively.Then, we have and since all elements of the binary CCCC are real values, the output of (45) coincides with (37).

V. PERFORMANCE ANALYSIS OF CCC-CDMA
In this section, we evaluate bit error rate (BER) for systems having the input-output relationship defined in (29) and evaluate the computational complexities required for CCC-CDMA.We assume that ML detection is employed at the receiver and, for the sake of notational simplicity, in the following, we omit the user index m.

A. Upper-Bound on BER Over Rayleigh Fading Channels
We first consider the pair-wise error probability (PEP) that the transmitted symbol vector u is erroneously detected into û.Accordingly, the conditional PEP is upper bounded as where e := u − û and H denotes the Teoplitz matrix of size K × (K + P ) given by Since η in ( 29) is composed of the Gaussian random variables with zero mean and variance N 0 , Hη H is also a Gaussian vector with zero mean and covariance matrix E{Hη H ηH H } = N 0 I K .Consequently, ℜ eHη H contains the Gaussian distribution with mean zero and variance D ℜ eHη H = N0 2 ∥eH∥ 2 and the conditional PEP is upper-bounded as follows: where Q(x) denotes the Gaussian Q-function and E represents the Teoplitz matrix of size (P + 1) × (K + P ) can be expressed using ē = [e 0 P ] as follows: Notice that xE = 0 K+P has solution if and only if x = 0 P +1 and hence, rank(E) = P + 1.From Craig's representation [39] (also see [40]), the bound described in (48) is written as Since the channel is assumed to subject quasi-static uncorrelated multipath Rayleigh fading, h P consists of mutually independent Gaussian random variable with zero mean and variance , where we assume σ p > 0, p = 0, 1, • • • , P , for simplicity.
Subsequently, since h P has the probability density function given by the (unconditional) PEP is bounded by where B = ∥E∥ 2 + 4N 0 sin 2 α Σ −1 P .Since Σ P is a nonnegative diagonal matrix and ∥E∥ 2 = [ϕ A (e, i − j)] P,P i=0,j=0 is Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
the auto-correlation matrix of e, which is a non-negative definite Hermitian matrix.Thus, we have det (B) ≥ det ∥E∥ 2 , and PEP is bounded as where for a positive integer x, we let for the binomial coefficient n C r .The right-hand side of bound ( 53) is finite if and only if det ∥E∥ 2 ̸ = 0, that is, rank ∥E∥ 2 = P + 1. Accordingly, if the input-output relationship is given by ( 29) over the length-(P + 1) multipath Reyleigh fading channel, then such systems provide the full path diversity of order P + 1.
) and E := ∪ u∈S E(u), where E(u) := {e = u − û} û∈S for each u ∈ S. (see details in [41].)Then, BER is given by where n b (e) is the number of bit errors involved by e and it can be simplified to for BPSK and QPSK modulations, where w(e) := ∥e∥ 2 /4 denotes normalized weight of e.

B. Complexity
We first consider the computational complexity in terms of the complex multiplications required for the CS operation.While an (N, N, N ; Z)-CCCC is employed in the CCC-CDMA, the conventional CC-and CS-CDMAs require complex multiplications of order O(N 2 ) to spread a modulated symbol.In the proposed FT-based structure, the number of multiplications required for spreading a modulated symbol is order of O (K+N )N K log 2 N , primarily because the size-N FFT is performed as (K +N −1) ≈ K +N times.Thus, for the case K ≫ N , the polyphase CCC-CDMA has a complexity of order O(N log 2 N ), less than that of OFDM(A) O(log 2 K).Moreover, the binary CCC-CDMA completely eliminates the multiplications in CS operation at the expense of SE.
Considering the channel equalization complexity, OFMD(A) provides the ML detection with a single tap equalization.For CCC-CDMA, on the other hand, we may recover the transmitted symbols under the ML criterion using the Viterbi algorithm.To detect a transmitted symbol, we need to compare metrics for at least |S| P states, each of which requires a complex multiplication [42].Thus, overall complexity of CCC-CDMA O(N log 2 N ) + O(|S| P ) is much higher than that of OFDM(A) O(log 2 K) especially for a high order modulation and under a large multipath environment.Although we may achieve a certain diversity order by employing a low complexity equalizer such as the frequency domain linear equalizer proposed in [43], in this paper, we limit our discussions and performance evaluations for the receiver under the ML criterion.

VI. SIMULATION RESULTS
We have shown in the previous section that, the performance of systems that provide the input-output relationship shown in (29) over Rayleigh fading multipath channel does not depend on the sequence selection.Hence, the binary and polyphase CCCCs, Chu-ZCZs, and M-ZCZs with the same SF achieve the same performance.Thus, in this study, we evaluate the resistances against clipping noise and Doppler frequency shifts.For CCCCs, the resistance against clipping noise is considered to monotonically weaken with the increment of padding zeros [33] and, in this study, we only test the crude CCCCs.
The evaluations are executed by computer simulations over four paths (P = 3) Rayleigh fading channel with uniformly distributed PDP for the cases of CCCC, Chu-ZCZ, and OFDMA of lengths L = 256/1024 while L = 255/1023 for M-ZCZ.For each case, we assumed M = N users transmit QPSK modulated symbols of the same length-(N/2 − P + 1) per packet, which is the maximum length of the binary CCCC that provides the lowest SE.Thus, for the CCCCs, Chu-ZCZ, and OFDMA, M = N = √ L = 16 and 32 for L = 256 and 1024, respectively, while the lengths of the corresponding QPSK symbol vectors are 6 and 14, respectively, and these are 1 symbol short for M-ZCZ.We also normalized the power delay profile (PDP) to P p=0 σ 2 p = 1.Therefore, at both the transmitter and receiver, the signal to noise ratio (SNR) can be expressed by Ēx /σ 2 , where Ēx = E x /L denotes the average transmission energy.At the receiver, the ML with Viterbi algorithm is utilized to recover the transmitted symbols, and we plotted each BER point after accumulating over 5, 000 bit errors.

A. Clipping Resistances
In Fig. 6, we plotted the SNR vs. BER over the quasi-static Rayleigh fading channel for two clipping levels: PAPR 0 = 2 dB and 4 dB.Under the assumption that polar clipping occurs at the transmitters, we replaced the signal x by x ′ as , where E th = Ēx ×PAPR 0 denotes the clipping threshold and detected the transmitted symbols based on the ML criterion with the perfect CIR at all receivers.
We can observe from Fig. 6 that, as indicated in [27], the best performances are achieved by M-ZCZ for all cases and their BER curves exhibit full path diversity without any error floors.Meanwhile, OFDMA exhibits the worst performance for all cases owing to a lack in diversity order while error floors are appeared in the BER curves of CCCCs and Chu-ZCZ.Evidently, Fig. 6 also indicates that the lengthening employed Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.sequence enhances the robustness against clipping noise with a higher SE, it is a simple but efficient countermeasure to mitigate the influence of clipping noise.
To explain the effect of SF, in Fig. 7, we plotted Pr{PAPR > PAPR 0 }, the complementary cumulative probability function of PAPR := max L−1 ℓ=0 |x(ℓ)| 2 / Ēx after testing 10 8 packets.It is evident from the figure that, the curves of the polyphase CCCC-ZCZ, Chu-ZCZ, and M-ZCZ with the same length overlap each other and hence the occurrence probabilities of them are almost identical.Thus, the difference in distribution of the despread clipping noise causes different tolerances and it has the minimum energy for the white Gaussian distribution.Thus, M-ZCZ constructed from a PR sequence has the best resistance, and despite the fact that the enlargement of SF increases the probability of clipping, the resistance against clipping noise is enhanced with a large SF because the distribution of the despread clipping noise for these cases is close to the white Gaussian and we may benefit from processing gain.When comparing the Chu-ZCZ of the same length, although CCCC has higher error floor level, the low complexity implementation of CCC-CDMA enables easy enlargement of SF to combat clipping noise.

B. Doppler Resistance
Under the assumption of a carrier frequency of 2 GHz and symbol duration T s = 100 µs, in Fig. 8, we compared BER performances over the fast fading channels with normalized maximum Doppler frequencies f max T s = 0.04, 0.01, corresponding to moving velocities f max T s are 216 km/h, 54 km/h, respectively.The time varying channel was generated based on Jakes model [44] and we used the Rayleigh fading simulation model given in [45].At the receiver, the ML detection was utilized with imperfect CIR information, i.e., the CIR at each begin of packet was used for ML detection.
When comparing the curves of the case f max T s = 0.04, the binary CCCC appears to be more resistant than M-ZCZ against fast fading and the polyphase sequences exhibit a similar weakness.In fact, over the fast fading channel, the M-ZCZ and binary CCCC disperse ICIs over all users with similar strengths while the polyphase sequences concentrate them on few neighboring channels.Thus, for the polyphase sequences, a sophisticatedly designed channel estimation and equalization can be employed to mitigate the ICI caused over fast fading channels.

VII. CONCLUSION
This study presented CCC-CDMA with its comprehensive FT-based implementation structure of the transceiver to reduce the computational complexity of CS operation.Simulation results show that the enlargement of SF strengthens the clipping resistance and the binary CCC-CDMA exhibits excellent robustness against Doppler frequency shifts.

APPENDIX
A. The Proofs of ( 4) and (5) The τ th entry of the left-hand side of (4) can be calculated as and it coincides with that of the right-hand side given by ψ A (ψ A (s, s ′ ), s ′′ ; τ ) Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Manuscript received 23
March 2022; revised 15 August 2022 and 19 November 2022; accepted 18 February 2023.Date of publication 8 March 2023; date of current version 12 December 2023.This work was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI under Grant JP16K06339.The associate editor coordinating the review of this article and approving it for publication was M. C. Gursoy.(Corresponding author: Chenggao Han.)

Fig. 1
Fig. 1 illustrates the tranceiver structure of CC-CDMA.Let C = {C m } M −1 m=0 be an (M, N, L)-CCC.In CC-CDMA, the mth CS-SS C m = {c m n } N −1 n=0 is assigned

k
(τ ) represent the kth output of FT at time τ and let p m k := p

n=0
and the delayed signals are summarized to yield the mth user's signal vector x m = (x m n ) N −1 n=0 .The resultant signal is then added with other signals as x = M −1 m=0 x m .Finally, the length-(K +G−T −1) vector x(N D − G : N D + K − T − 2), called the overlapped CP (OCP), is appended to the head of signal x to yield the signal ⃗ x, where the length-(K − T − 1) signal x(N D : N D + K−T −2) is removed from the tail and summarized with x(0 :

.
nD (τ ) is inputed into the kth input of the FT v (m) k (τ ) with the deinterleaving rule k = π (m) (n) specified by (24).Let v m (τ ) := v The N points transform is then performed over the outputs of the deinterleaver as w m (τ ) = v m (τ ) Ω and the resultant signal is inputed to delay ζ −[−n−1] N to yield r m n = (0 n , w m n , 0 N −n−1 ) N −1 n=0 .Lastly, we detect the transmitted data based on the summarized vector r m = N −1 n=0 r m n .