1 Introduction

A duplex communication system [1] is a point-to-point system composed of two connected parties or devices that can communicate with one another in both directions, normally termed as uplink and downlink individually. To circumvent the mutual interference between the uplink and downlink, the Frequency Division Duplex (FDD) [1] and Time Division Duplex (TDD) [1] apparatuses are commonly deployed in communication systems. In the FDD system, the uplink and downlink signals are gapped in different frequency bands, while in the TDD system, the uplink and downlink transmissions are identified in different time slots. In this way, the uplink and downlink can be treated as being orthogonal in the FDD or TDD apparatus since they utilize different frequency or time resources.

Recently, No Division Duplex (NDD) [2, 3] techniques have captured growing interests for the 5G communication system, since they can significantly increase the capacity of the cellular system as well as simplify the infrastructure of the cellular network. The basic philosophy of the No Division Duplex [2, 3] is that a radio device can transmit and receive on the same frequency at the same time; more explicitly, it can operate in a full-duplex fashion. However, as the radio signals attenuate quickly in an exponential way over the distance, the signal from a local transmitting antenna is hundreds of thousands of times stronger than transmissions from other nodes. Therefore, the challenge in implementing a full-duplex system is to recover the desired signal from the excessively strong local interferers. Hence, a complicated interference canceller [3] must be deployed in the NDD device to combat the local interference, in both the analog RF component as well as the digital baseband component.

In Code Division Multiple Access (CDMA) systems, the spreading sequences characterize the properties of the associated Intersymbol Interference (ISI) as well as the Multiple Access Interference (MAI) [4]. Traditional spreading sequences, such as m-sequences [4], Gold codes [4], and Kasami codes [4], exhibit non-zero off-peak auto-correlations and cross-correlations, which result in a high MAI in the system. To circumvent this issue, the ideal solution is to design the optimal sequences whose auto- and cross-correlations are zero for the infinite code offsets, as shown in Fig. 1a. However, this kind of perfect sequence does not exist in theory. Therefore, considerable research has been invested on the interference rejection code which can exhibit zero correlation values on the relative delay-induced code offsets. The area which has the value of zero correlation is the so-called Zero Correlation Zone (ZCZ) or Interference-Free Window (IFW) of the spreading code [5], as shown in Fig. 1d. The auto- and cross-correlation properties of four spreading codes are exhibited in Fig. 1. In this figure, we can observe that the correlation value of the interference rejection code is zero in IFW. Loosely Synchronous (LS) code [6, 7] is one of these specific interference rejection codes which exhibit an IFW, where the off-peak values of aperiodic auto- and cross-correlations are zero, resulting in zero ISI and zero MAI in the IFW. Given these conditions, a major benefit of the interference rejection codes is that they are capable of combating the mutual interference between the uplink and downlink without a complicated interference canceller. Hanzo, Wei, and Ni [8–10] investigated the performance of the CDMA system in the context of the deployment the LS codes. However, in their work, the power control [11] scheme is employed to avoid the near-far effect, and only the cellar communication scenarios are evaluated in previous work. In [9, 12], the interference rejection codes are deployed in the multi-carrier CDMA system and the performance is analyzed. The authors in [13] investigated the performance of the CDMA system in conjunction with the multiple antenna technique and the interference rejection code. Various chip waveforms [14] are investigated for the LS codes, and the network capacity in the cellular scenario is analyzed in [15].

Fig. 1
figure 1

Cross-correlations of four different codes. a Perfect sequence. b Walsh code. c Gold code. d Interference rejection code (LS code)

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In this paper, to circumvent the implementation of complicated interference canceller, Code Division Duplex (CDD) is proposed for a synchronous communication system. As shown in Fig. 2, the basic concept of the CDD is that the downlink and uplink signals are distinguished by their unique spread sequence C i , i.e., the CDD radio can transmit and receive simultaneously at the same frequency band with different spreading signatures. The state of the art CDD device can operate in the same fashion as the NDD device. However, the advantage of the CDD receiver is to employ a simple matched filter or RAKE receiver rather than a complicated interference canceller in the NDD device owing to the cross-correlation properties of the interference rejection code, which is beneficial in the engineering implementation and power saving.

Fig. 2
figure 2

Code Division Duplex system

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In this paper, we will investigate the performance of CDD communication systems in the context of two practical scenarios. The first scenario is a cellular-like communication system, where a base station node operates as the centric node and all the other CDD nodes must communicate with the center node. In this specific scenario, the center node has a much stronger transmission power than other nodes. The second scenario is an ad hoc communication system where each CDD device transmits on the same power in the network and can communicate with each other. In this contribution, we will investigate the Bit Error Rate (BER) performance and capacity of CDD systems in both scenarios when communicating over a Nakagami-m channel.

This paper is organized as follows. In Section 2, we will introduce the specific interference rejection code, namely LS code, while in Section 3, we will describe the model of the CDD system and fading channel. In Section 4, we will characterize the BER performance of the CDD communication, and in Section 5, we will discuss our findings. Finally, in Section 6, we will offer our conclusion.

2 Interference rejection codes

There exists a specific family of spreading codes, which exhibit an IFW; more specifically, Loosely Synchronized (LS) codes [6] exploit the properties of the so-called orthogonal complementary sets [6, 7]. Since the detailed construction method of binary LS codes was described in [6, 9, 10], here, we only focus our attention on the properties and the applications of the LS codes [16, 17].

Generally, an LS code can be denoted as LS (N,P,W 0), which denotes the family of LS codes generated by applying a (P×P)-dimensional Walsh-Hadamard (WH) matrix to an orthogonal complementary code set of length N and inserting W 0 number of zeros between two orthogonal complementary sets [6, 9, 10]. The constructed LS (N,P,W 0) codes can be constructed for any arbitrary length of the codes by selecting the proper parameter N, and then, the generated LS codes can exhibit an IFW of interference rejection code as shown in Fig. 1; more explicitly, the spreading signature function c i (t),c j (t) of the constructed LS codes satisfy:

$$\begin{array}{@{}rcl@{}} \int_{-\infty}^{-\infty} c_{i}(t)c_{j}(t-\tau)d t = 0, ~~~\tau<W_{I}, ~~~i\neq j \end{array} $$
(1)

where c i (t) and c j (t) are the signature functions in the generated LS (N,P,W 0) code set and W I is the width of the IFW. More explicitly, the constructed LS (N,P,W 0) codes are still orthogonal to each other even in the scenarios of the multipath environment provided that the delay spread of the multipath is less than the width of IFW. Hence, the LS code with inherent IFW can effectively suppress the Multipath Interference (MPI) and multiuser access interference (MAI), which also significantly simplify the design of receiver as well as reduce the complexity of the receiver in CDMA system.

For example, the LS (N,P,W 0) codes can be generated based on the complementary pair of [6, 16], and the total number of available codes in the family of LS (N,P,W 0) is given by NP. The total number of NP codes can be classified by N sets of P number of LS codes through their Walsh-Hadamard Matrix, and each set has P number of LS codes. The LS codes in the same set exhibit an IFW length of [ −W I ,+W I ], where we have W I = min{W 0,N−1}. The aperiodic auto-correlation and cross-correlation function ρ kk (τ), ρ jk (τ) of the codes belonging to the same set will be zero, provided that we have τ≤W I T c . Furthermore, the LS codes belonging to the N different sets are still orthogonal to each other at zero offset, namely in a perfectly synchronous environment. However, the LS codes belonging to the N different sets will lose their orthogonality, when they have a non-zero code offset.

To combat the interference incurred by its own transmission, the spreading codes in the CDD device have the following assignment policy: the spreading codes of the uplink and downlink are wisely chosen to belong to different sets of Walsh-Hadamard. The benefit of this assignment is that the spreading codes between the uplink and downlink have the maximum width of the IFW, which can guarantee the orthogonality between the uplink and downlink in one CDD node so that their mutual interference can be negligible.

3 System model

3.1 System model

Assume that the system supports K synchronous users and each user is assigned two unique spreading signature waveforms for its uplink and downlink, respectively. The spreading signature waveform can be noted as \(\mathbf {c}_{k}(t)=\sum \limits _{i=0}^{G-1}c_{ki} \psi _{T_{c}}(t-iT_{c})\), where G is the spreading gain and \(\psi _{T_{c}}(t)\) is the rectangular chip waveform, which is defined over the interval [0,T c ). Consequently, when the K users’ signals are transmitted over the frequency-selective fading channel, the complex low-pass equivalent signal received at a given RX antenna can be expressed as:

$$\begin{array}{@{}rcl@{}} R(t)&=&\sum_{k=1}^{K}\sum_{l=0}^{L_{p}-1}\sqrt{2P_{k}}{\mathbf{c}_{k}}(t-lT_{c}-\tau_{k}){b}_{k}(t-lT_{c}-\tau_{k})\\ &&\times\, h_{kl}\exp(j\theta_{kl}) +N(t), \end{array} $$
(2)

where N(t) is the complex-valued low-pass-equivalent AWGN (additive white Gaussian noise) having a double-sided spectral density of N 0 and τ k is the propagation delay between the transmitter and receiver for user k. In the same CDD device, τ k is assumed to be 0, while in different CDD devices, τ k is determined by the propagation distance: without loss of generality, \(\tau _{k}=\frac {d_{k}}{c} \), where c is the speed of light. L p is the total number of resolvable paths. P k is the received power, which is determined by the large-scale fading of the wireless channel.

3.2 Wireless large-scale fading model

The complexity of wireless signal propagation makes it difficult to obtain a single model that characterizes path loss accurately across a range of different environments. In the textbook [1], various propagation models of the wireless channel are proposed and investigated in wireless communications, such as the well-known free-space channel model, Okumura Model [1], Hata Model [18], and COST207 [1, 18] which are traditionally used in system analysis. However, for the convenience of analysis of various system performance, we simplify the channel model as well as capture the essence of signal propagation without resorting to complicated path loss models, which is sufficiently accurate to approximate the real channel model. In this contribution, the following simplified models [1] for path loss as a function of distance are commonly used for system design:

$$ P_{r} = P_{t} G_{a} \left [ \frac{d}{d_{0}} \right ]^{-\gamma}, $$
(3)

where d 0 is the wavelength of the radio signal, d is the distance between the transmitter and receiver, and G a is the constant which is related to the characteristic of the antenna, such as the height and direction. Parameter γ is the exponential attenuation factor which is related to the practical environment. Generally, the attenuation factor γ is about 2 to 4 according to the free-space, indoor, urban, or hilly environments.

3.3 Channel model

The DS-CDMA signal experiences independent frequency-selective Nakagami-m fading. The complex low-pass equivalent representation of the Channel Impulse Response (CIR) encountered by the kth user is given by [19]:

$$ h_{k}(t)=\sum_{l=0}^{L_{p}-1}h_{kl}\delta(t-lT_{c})\exp\left(j\theta_{kl}\right), $$
(4)

where h kl represents the Nakagami-distributed fading envelope, l T c is the relative delay of the lth path of user k with respect to the main path, while L p is the total number of resolvable multipath components. Furthermore, θ kl is the uniformly distributed phase-shift of the lth multipath component of the channel and δ(t) is the Kronecker delta function. More explicitly, the L P multipath attenuations {h kl } are independent Nakagami-distributed random variables with a Probability Density Function (PDF) of [20–22]:

$$\begin{array}{@{}rcl@{}} p(h_{kl})&=&M(h_{kl},m_{kl},\Omega_{kl}),\\ M(R,m,\Omega)&=&\frac{2m^{m}R^{2m-1}}{\Gamma(m)\Omega^{m}}e^{\left(-m/\Omega\right)R^{2}}, \end{array} $$
(5)

where Γ(·) is the gamma function [19] and m kl is the Nakagami-m fading parameter, which characterizes the severity of the fading for the lth resolvable path of user k [23]. Specifically, m kl =1 represents Rayleigh fading, m kl →∞ corresponds to the conventional Gaussian scenario, and m kl =1/2 describes the so-called one-sided Gaussian fading, i.e., the worst-case fading condition. The Rician and log-normal distributions can also be closely approximated by the Nakagami distribution in conjunction with values of m kl >1. The parameter Ω kl in Eq. 5 is the second moment of h kl , i.e., we have Ω kl =E[ (h kl )2]. We assume a negative exponentially decaying Multipath Intensity Profile (MIP) given by Ω kl =Ω k0 e −ηl, η≥0,l=0,…,L p −1, where Ω k0 is the average signal strength corresponding to the first resolvable path and η is the rate of average power decay.

4 BER performance analysis

As shown in Fig. 3, let the first user be the user of interest and consider a receiver using de-spreading as well as multipath diversity combining. The conventional matched filter-based RAKE receiver using maximum ration combining (MRC) can be invoked for detection, where we assume that the RAKE receiver combines a total of L r number of diversity paths, which may be more or possibly less than the actual number of resolvable components at the current chip rate. It is advocated that the system has achieved perfect time synchronization and perfect estimates of the channel tap weights. Then, after appropriately delaying the outputs of the individual matched filter, in order to coherently combine the L r number of path signals with the aid of the RAKE receiver, the output Z kl of the RAKE receiver’s lth finger sampled at t=T+l T c +τ k can be expressed as:

$$\begin{array}{@{}rcl@{}} Z_{kl}=D_{kl}+I_{kl}, \end{array} $$
(6)
Fig. 3
figure 3

CDD receiver structure

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where D kl represents the desired direct component, which can be expressed as:

$$\begin{array}{@{}rcl@{}} D_{kl}=\sqrt{2P}T_{s}b_{k}[\!0]h_{kl}^{2}, \end{array} $$
(7)

where T S is the symbol duration.

The MRC’s decision variable Z k , which is given by the sum of all the RAKE fingers’ outputs, can be expressed as:

$$\begin{array}{@{}rcl@{}} Z_{k}= \sum_{l=0}^{L_{r}-1} Z_{kl}. \end{array} $$
(8)

The RAKE fingers’ output signal Z kl is a Gaussian distributed random variable with a mean of D kl .

The term I kl in Eq. 6 represents the total interference incurred by the Multipath Interference (MPI) as well as the Multiple Access Interference (MAI), which can be expressed as:

$$\begin{array}{@{}rcl@{}} I_{kl}=I_{kl}[\!S]+I_{kl}[\!M]+N_{k}, \end{array} $$
(9)

where I kl [ S] represents the multipath interference imposed by the user of interest, which can be expressed as:

$${} \begin{aligned} I_{kl}[\!S]&=\sqrt{2P}T_{s}h_{kl}\sum\limits_{\stackrel{ l_{p}={0}}{l_{p}\neq l}}^{L_{p}-1}\frac{h_{kl_{p}}\cos\theta_{kl_{p}}}{T_{s}}\\ &\quad\times\! \int_{0}^{T_{s}}b_{k}\left[\!t\,-\,\left(l_{p}\,-\,l\right)T_{c}\right]\cdot c\left[\!t-\left(l_{p}-\!l\right)T_{c}\right]\!c[\!t]\!dt. \end{aligned} $$
(10)

Furthermore, I kl [ M] represents the multiuser interference inflicted by the K−1 number of interfering signals, which can be expressed as:

$$\begin{array}{@{}rcl@{}} I_{kl}[\!M]&=&\sqrt{2P}T_{s}h_{kl}\sum\limits_{\stackrel{ k'=1}{k'\neq k}}^{K}\sum\limits_{l_{p}=0}^{L_{p}-1}\frac{h_{k'l_{p}}\cos\theta_{k'l_{p}}}{T_{s}}\\ &{}&\cdot\int_{0}^{T_{s}}b_{k'}[\!t-(l_{p}-l)T_{c}-(\tau_{k'}-\tau_{k})] \times \\ &{}& c_{k'}[\!t-(l_{p}-l)T_{c}-(\tau_{k'}-\tau_{k})]c[\!t]dt. \end{array} $$
(11)

In Eqs. 10 and 11, the cos(·) terms are contributed by the phase differences between the incoming carrier and the locally generated carrier used in the demodulation. Finally, the noise term in Eq. 9 can be expressed as:

$$\begin{array}{@{}rcl@{}} N_{kl}=h_{kl}\int_{0}^{T_{s}}n(t)c[\!t]\cos(2\pi f_{c}t+\theta_{kl})dt, \end{array} $$
(12)

which is a Gaussian random variable with a zero mean and a variance of \(N_{0}T_{s}h_{kl}^{2}\), where {h kl } represents the path attenuations.

Furthermore, I kl [ S] represents the MPI imposed by the user of interest. Without loss of generality, we can assume that the width of the IFW is longer than the channel spread time L p T c , and in this case, the MPI becomes zero since the auto-correlation of the spreading code is zero.

Hence, the performance of the CDD system is mainly determined by the MAI term I kl [ M], which is affected by channel delay profile, distance amongst the nodes, and the number of users supported in a cell.

Again, the uplink TX and downlink RX have different spreading sequences for the kth user in one CDD device, and recall that the system can assign different sets of the spreading codes for the uplink and downlink; more specifically, the spreading codes of the uplink and downlink have the maximum width of IFW in the generated LS codes. And the distance between the TX antenna and RX antenna is very close, i.e., τ k =0. Hence, after the matched filter and RAKE receiver, the interference introduced by its own transmission can be ignored due to the inherent orthogonality of the spreading codes. Thus, the MAI are mainly introduced by other CDD device nodes.

Since the power control is difficult to be implemented in the CDD system, the near-far effect must be taken into account for the BER performance analysis. Assume that the power of the interested signals is P k while the power of the interference is P j , and P k and P j can be obtained from Eq. 3.

As shown in Fig. 4, if one path of the channel impulse response of the interference user is located within the IFW, this specific path will not introduce any interference to the interested signals. More explicitly, only these paths which are located outside the interference-free window will introduce the MAI to the desired signal. Without loss of generality, we can assume that the multipath intensity profile of all users obeys the same distribution, i.e., Ω k0=Ω 0,k=1,2,...K, and E b =P T s is the energy per bit. Thus, the variance of the lth RAKE finger’s output samples Z kl for a given set of channel amplitudes {h kl } may be approximated as [23, 24]:

$$\begin{array}{@{}rcl@{}} \sigma_{kl}^{2}=2PT_{s}^{2}\left[\frac{\chi_{kl}}{3G}+\left(\frac{2\Omega_{0} E_{b}}{N_{0}}\right)^{-1}\right]\cdot \Omega_{0} h_{kl}^{2}, \end{array} $$
(13)
Fig. 4
figure 4

MAI effect with IFW

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$$\begin{array}{@{}rcl@{}} \chi_{kl} = \sum\limits_{j=1}^{K}\sum\limits_{l=0}^{L_{p}-1} \frac{P_{j}}{P_{k}}e^{-\eta l} u\left(W_{I}-|\tau_{j}-\tau_{k}| -l\right) \end{array} $$
(14)

where W I is the width of the IFW; τ j and τ k are the propagation delay of the interference user j and the interested user k, respectively; L p is the total number of resolvable paths; and u(x) is the step function, i.e., u(x)=1,x≥0, otherwise u(x)=0,x<0. According to the large-scale fading model, P j and P k are the strength of the received signals, which can be derived in Eq. 3 as \(P_{j} = P_{tj}G_{a}\left [ \frac {d_{j}}{d_{0}} \right ]^{-\gamma }, P_{k} = P_{tk} G_{a}\left [ \frac {d_{k}}{d_{0}} \right ]^{-\gamma } \). Moreover, the propagation delay obeys τ j =d j /c,τ k =d k /c.

Hence, the MRC’s output sample Z k can be approximated by a Gaussian random variable with a mean of \(\mathrm {E}\left [Z_{k}\right ]=\sum \limits _{l=0}^{L_{r}-1} D_{kl}\) and a variance of \( \text {Var}\left [Z_{k}\right ] = \sum \limits _{l=0}^{L_{r}-1} \sigma _{kl}^{2}\) [21, 24], where we have:

$$\begin{array}{@{}rcl@{}} \mathrm{E}\left[Z_{k}\right]&=& \sum_{l=0}^{L_{r}-1} \sqrt{2P}T_{s}b_{k}[0]h_{kl}^{2}, \end{array} $$
(15)
$$\begin{array}{@{}rcl@{}} \text{Var}\left[Z_{k}\right]=\sum_{l=0}^{L_{r}-1} \sigma_{kl}^{2}. \end{array} $$
(16)

Hence, the BER using BPSK modulation conditioned on a set of fading attenuations {h kl ,l=0,1,…,L r −1} can be expressed as [1, 18]:

$$\begin{array}{@{}rcl@{}} P_{b}(\gamma) = Q\left(\sqrt{\frac{\left(\mathrm{E}\left[Z_{k}\right]\right)^{2}}{\text{Var}\left[Z_{k}\right]}}\right)= Q\left(\sqrt{\sum_{l=0}^{L_{r}-1}2\gamma_{l}}\right), \end{array} $$
(17)

where Q(x) represents the Gaussian Q-function which can also be simplified as [1, 23]:

$$\begin{array}{@{}rcl@{}} Q(x)=\frac{1}{\pi}\int_{0}^{\pi/2}\exp\left(-\frac{x^{2}}{2\sin^{2}\theta}\right)d\theta,x\geq 0. \end{array} $$
(18)

Furthermore, 2γ l in Eq. 17 represents the output Signal to Interference plus Noise Ratio (SINR) at the lth finger of the RAKE receiver.

Let us now substitute Eqs. 13, 15, and 16 into Eq. 17, and the expressions under the square-root functions must be equal, which allows us to express γ l as follows:

$$\begin{array}{@{}rcl@{}} \gamma_{l}=\left[ \frac{\chi_{kl}}{3G} +\left(\frac{\Omega_{0} E_{b}}{N_{0}}\right)^{-1}\right]^{-1} \cdot\frac{h_{kl}^{2}}{\Omega_{0}}. \end{array} $$
(19)

For the convenience of the BER derivation, we define a term γ c as:

$$\begin{array}{@{}rcl@{}} \gamma_{c}=\left[ \frac{\chi_{kl}}{3G} +\left(\frac{\Omega_{0} E_{b}}{N_{0}}\right)^{-1}\right]^{-1}. \end{array} $$
(20)

With the aid of Eq. 20, we have \(\gamma _{l}={\gamma }_{c}\cdot \frac {(h_{l})^{2}}{\Omega _{0}}\) and h l obeys the Nakagami-m distribution characterized by Eq. 5, and the PDF of γ l can be formulated as:

$$\begin{array}{@{}rcl@{}} p_{\gamma_{l}}(\gamma_{l})=\left(\frac{m_{l}}{\overline{\gamma}_{l}}\right)^{m_{l}}\frac{\gamma^{m_{l}-1}}{\Gamma(m_{l})}\exp\left(-\frac{m_{l}\gamma_{l}}{\overline{\gamma}_{l}}\right),~~\gamma_{l}\geq 0, \end{array} $$
(21)

where \(\overline {\gamma }_{l}={\gamma }_{c}e^{-\eta l}\) for l=0,1,…,L r −1. The average BER, P b (E), can be obtained by the weighted averaging of the conditional BER expression of Eqs. 17 and 21 over the joint PDF of the instantaneous SNR values.

5 Simulation performance of CDD

As we mentioned before, two scenarios are investigated in this paper. The first scenario is a cellular system which has a center node, and all other nodes besides the center node only communicate with the center one. The second scenario is a relay and ad hoc communication environment, where any two of the nodes can communicate with each other. For the CDD performance evaluation, we assume that the system has a chip rate of 1.2288 M chips which is the same as the CDMA2000 and EVDO systems [25]. The bandwidth of our system is 1.25 MHz. The carrier frequency of the system is f c =2 GHz, and the large-scale fading factor γ=2.5. Moreover, the radius of a cell is configured to 2000 m. All the CDD nodes are operated in synchronous transmission mode, i.e., all nodes transmit simultaneously. We also assume that the maximum delay spread of the channel is τ=3 μs [11]. Thus, the number of resolvable paths is considered as \(L_{p}=\lfloor \frac {\tau }{T_{c}} \rfloor +1 = 4\), where τ=3 μs, and the channel’s delay spread profile is negatively exponentially distributed with the factor η in the range of [ 0.3,3] μs. The RAKE receiver combines L r =3 number of fingers for BER performance evaluation, and the channel’s exponential decay factor is η=0.2.

In our simulation, the LS(4,32,4) codes are deployed as the spreading codes. More explicitly, the total length of the LS(4,32,4) code is L=N P+2W 0=136 chips, and its spreading factor may be calculated by simply noting that each bit to be transmitted is spread by one of the constituent LS(4,32,4) codes. Although the length of this LS code is L=N P+2W 0=136, the effective spreading gain of the LS code is identical to G eff=128 since the inserting zeros must not be taken into account. The LS(4,32,4) codes are constituted by a total of 128 spreading codes. The width of the IFW W I and the number of supported user K satisfies that:

$$\begin{array}{@{}rcl@{}} W_{I} &=& 4, K \leq16; \\ W_{I} &=& 2, 16 < K \leq 32;\\ W_{I} &=& 1, 32 < K \leq 64;\\ W_{I} &=& 0, 64< K \leq 128. \end{array} $$

First, we investigated the BER performance of CDD when there existed a strong interference CDD node. In this scenario, a cell has three CDD nodes, the source node, the destination node, and the interference node. In our simulation, the interference node has a transmission power 20 dB higher than the source node. It is obvious that the distance between the interference node and destination node dramatically affects the performance of the RAKE receiver since the strength of interference is inversely proportional to the distance. Figure 5 exhibits the BER performance of the destination node in conjunction with the distance between the interference node and destination node. More explicitly, we can observe that when the width of IFW W I is below 2 and the distance of the interference node is less than 300 m, the effect of the interference node cannot be completely rejected by the interference spreading code so that the MAI leak inevitably in the RAKE receiver due to the insufficient width of the IFW. However, when the W I increases to 3 or 4, the interference is suppressed significantly and the effect of the interference node is negligible.

Fig. 5
figure 5

BER vs distance of interference node in the cellular CDD communication system

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The performance of CDD in two common scenarios is shown in Fig. 6. In the cellular scenario, a center node or Base Station (BS) exists in the network, which has a high transmission power than other nodes. In our simulation, we assume that the base station node has a 20- dB higher transmission power. The second is an ad hoc scenario where each node has the same transmission power. Figure 6 plots the BER performance of the CDD communication system in the context of a various number of users K. In Fig. 6, we can observe that the performance in the ad hoc scenario is superior than that in the cellular scenario. This phenomenon is reasonable since the cellular environment exists as a strong base station which also acts as a strong interference node to other destination nodes and then degrades the BER performance in the cellular condition.

Fig. 6
figure 6

BER vs E b /N 0 performance in the context of the ad hoc and cellular scenarios

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Figures 7 and 8 exhibit the performance of the CDD system in conjunction with various supported users K. From Fig. 7, we can observe that the near interference-free performance can be achieved when the number of users K is less than 16 since the spreading codes have the maximum width of the IFW in this condition. As the number of users is increased to 32, the performance of the CDD system is degraded moderately, since the width of the IFW is still maintained in W I ≤2. However, when the number of users is increased to more than 64, the performance is degraded significantly since the width of IFW is decreased to 0∼1, which reduces the capability of combating the MAI. Figure 8 portrays the BER performance of the CDD in conjunction with various users K. Two different SNR parameters are configured in our simulation, i.e., SNR=10 dB and SNR=20 dB, respectively. From Fig. 8, we can conclude that the CDD system exhibits a near interference-free performance in the case K≤32. In Figs. 7 and 8, it is advocated that the CDD system should be deployed in a relatively-low load of users. For a spreading gain of 128 systems, the CDD system can support K=40 user communications simultaneously without the mutual interference; this load is promising since the linear Minimum Mean Square Estimator (MMSE) multiuser detector with a much higher complexity can only support to K=40 to achieve an ideal performance. Similarly, the performance of the linear MMSE multiuser detector [26, 27], the Parallel Interference Canceller (PIC) [26, 27], and Serial Interference Canceller (SIC) [26, 27] deteriorates significantly for the high load scenario in a CDMA system; furthermore, the traditional MMSE detector, PIC, and SIC suffer the serious near-far effect and degrade the performance in our CDD scenarios. The ideal Maximum Likelihood detector [26, 27] can solve this issue; however, the complexity of the ML detector is excessively high so that it is impossible to implement in practical communication systems.

Fig. 7
figure 7

BER performance of the CDD communication system with various number of users K

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Fig. 8
figure 8

BER performance vs. the number of supported users K in a cell

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Finally, Fig. 9 plots the BER performance of different channel fading models. The Nakagami-m channel is a generic model for the wireless channel, where the choice of the parameter m determines the fading type of the channel model. In Fig. 9, three different fading models are simulated and analyzed in our investigation. More explicitly, the Rayleigh fading [1, 18], Rician fading [1, 18], and Gaussian Fading [1, 18] are evaluated in the simulation. In the practical applications in a cellular communication, the Rayleigh fading and Rician fading model are commonly adopted for the performance evaluation. Hence, in this detection, we evaluate the CDD performance in conjunction with the real scenario of the cellar communications. The simulation results in Fig. 9 conform to the mathematic analysis, and we can conclude that the performance of Rayleigh fading is the worst and the performance of Gaussian fading is the best amongst these three fading models.

Fig. 9
figure 9

BER performance of the CDD communication system in adjunction with different fading channel models

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6 Conclusions

In this paper, we investigated the BER performance of the CDD communication system, which can exhibit the same operation fashion as the NDD device. Moreover, the advantage of the CDD is that the receiver can be simplified as the matched filter or RAKE receiver by carefully selecting our proposed interference rejection codes—LS codes. Hence, our proposed method can significantly reduce the complexity and difficulty in the implementation of the NDD fashion. Our simulation results show that the CDD system can achieve a near interference-free performance in a low-user load scenario due to the limited number of available LS codes having a certain IFW width; more explicitly, our simulation and analysis show that the CDD system can achieve an ideal performance when the number of user K≤32, and in this condition, a simple matched filter or RAKE receiver is sufficient to combat all the interference. For future research, the method of the smart assignment of the spreading codes will be investigated to reduce interference between neighboring nodes and increase the capacity and spectrum efficiency of the CDD network.