A multiuser Ultra Wide Band (UWB) channel suffers seriously from realistic impairments. Among this, multipath fading and interferences, such as Multiple Access Interference (MAI) and Inter Symbol Interference (ISI), that significantly degrade the system performance. In this paper, a polar coding technique, originally developed by Arikan, is suggested to enhance the BER performance of indoor UWB based Orthogonal Frequency Division Multiplexing (OFDM) communications. Moreover, Interleave Division Multiple Access (IDMA) scheme has been considered for multiuser detection depending on the turbo type Chip-By-Chip (CBC) iterative detection strategy. Three different models as Symmetric Alpha Stable (SαS), Laplace model and Gaussian Mixture Model (GMM), have been introduced for approximating the interferences which are more realistic for UWB system. The performance of the proposed Polar-coded IDMA OFDM-based UWB system is investigated under UWB channel models proposed by IEEE 802.15.3a working group and compared with Low Density Parity Check (LDPC)-coded IDMA OFDM-based UWB system in terms of BER performance and complexity under the studied noise models. Simulation results show that the complexity of the proposed polar-coded system is much lower than LDPC-coded system with minor performance degradation. Furthermore, the proposed polar-coded system is robust against noise and interferences in UWB indoor environment and gains a significant performance improvement by about 5 dB compared with un-coded IDMA-OFDM-UWB system under the studied noise models.
In the world of wireless communication, Ultra Wide Band (UWB) is considered as an attractive technology due to its important features as robustness towards multipath fading, high data rates, improved channel capacity, low cost, low power consumption and low-complexity devices [
The traditional design approach is known as Impulse-Radio UWB (IR-UWB) which based on a very short duration pulse in transmission. Although the impulse architectures offer relatively simple radio designs, their signals are usually impaired by fading and Inter Symbol Interference (ISI) due to multipath delay spread phenomenon which in turns leads to degradation of the overall system performance. Moreover, they provide little flexibility in spectrum management [
Regarding to a multiuser UWB indoor environment, the performance is mainly limited by inherent noise and interferences as Multiple Access Interference (MAI) and ISI which are poorly approximated by a Gaussian distribution because of their impulsive nature. Therefore, Gaussian approximation is inefficient for UWB systems. Several alternative distributions for approximating impulsive noise in UWB systems are introduced and compared [
For multiuser detection in UWB indoor environment, a recently proposed spread spectrum multiple access scheme, Interleave Division Multiple Access (IDMA) was introduced in [
Recent papers investigate the combination of OFDM and IDMA to avoid their individual disadvantages [
Low Density Parity Check (LDPC) codes are considered as one of the most powerful FEC techniques due to their superior error correction capability with simple iterative decoding. Hence, these codes have been utilized in most of current UWB systems [
Recently, Polar codes are presented as an active research which are firstly proposed by Arikan [
Therefore, Polar codes will be employed in this paper, as a channel coding, with IDMA scheme over OFDM based UWB channel corrupted by non Gaussian (impulsive) noise. Moreover, a comparative study is held between the proposed polar-coded IDMA OFDM-based UWB system and LDPC-coded system in [
The remainder of the paper is organized as follows: Section 2 presents the UWB channel and noise models. Section 3 describes the Polar-coded OFDM-IDMA-UWB system model. The simulation results of the proposed system are introduced in Section 4 and finally Section 5 concludes the paper.
The design of any communication system depends on studying the characteristics of the channel and noise nature. Indoor UWB environments are subjected to multipath fading and impulsive noise which have a strong effect on the system performance assessment. Hence, the accurate modeling of channel and noise is a very important issue. Such models create the facility for calculation of large and small-scale characteristics which are necessary for efficient system design [
The standardized channel model for indoor UWB environments has been proposed by the channel modeling subcommittee of the IEEE 802.15.3a Task Group. Such model is a modified version of the Saleh-Valenzuela (S-V) model, where the Rayleigh distribution of the channel coefficient amplitude is replaced by the log-normal distribution [
The wide spread of electronic devices in indoor environments causes high level noise impulses as a form of the non-Gaussian noise [
n = n a + n I (1)
where n is the total noise and can be divided into n a which is background (AWGN) noise with zero mean and variance σ 2 , and n I which is the Impulsive noise with zero mean and variance σ I 2 . Impulsive noise n I can be accurately modeled by three different models, symmetric alpha stable, Laplace model, Gaussian mixture model, which will be introduced below.
According to [
f Cauchy ( r ) = γ π 1 γ 2 + ( r − μ ) 2 (2)
where μ is the location parameter of the distribution.
The pdf of any random variable r of the Laplace noise model is given by [
f Laplace ( r ) = 1 2 C exp ( − | r − μ | C ) , − ∞ < x < ∞ (3)
where μ is the location parameter (mean), and C is the scale parameter ( 2 C 2 = variance ) and its value is always positive. By varying the scale parameter C, different tail behaviors can be obtained [
A simple two term mixture pdf of any random variable r is given by [
f G M M ( r ) = ( 1 − ϵ ) g ( r ) + ϵ h ( r ) = ( 1 − ϵ ) 2 π σ g 2 e − ( r − μ ) 2 2 σ g 2 + ϵ 2 π σ h 2 e − ( r − μ ) 2 2 σ h 2 (4)
where g ( . ) is the nominal Gaussian pdf with variance σ g 2 and h ( . ) is the heavy tailed Gaussian with variance σ h 2 = η σ g 2 , where η ≥ 1 is the impulsive part’s relative variance with respect to (w.r.to) nominal Gaussian noise variance. The parameter ϵ ∈ [ 0 , 1 ] controls the contribution of impulsive component to the whole pdf [
The proposed Polar-coded IDMA OFDM-based UWB system model for multi-user communication scenario is depicted in
For simplicity, synchronous BPSK signaling is considered over time in-variant multipath channel. For the user-k, the information data sequence d k ∈ { 0 , 1 } of length D is encoded using polar encoder into a coded sequence c k of length N. A polar code can be completely described by three-tuples (N, D, Ƒ), where N is the code length in bits, D is the number of information bits encoded per codeword, and Ƒ is a subset of N – D integer indices called frozen bit locations from { 0 , 1 , ⋯ , D − 1 } [
The construction of polar code depends on the choice of the subset Ƒ. This is corresponding to the selection of best D bit-channels among N, in terms of the
bit error rate (BER) at a given value of ( R E b / N 0 ), where R is the code rate, defined as the design-SNR [
The n^{th} coded bit of user-k, c k ( n ) , n = 1 , 2 , ⋯ , N is spread using a balanced spread sequence of length S. The same spreading code is applied to all users S k ∈ { + 1 , − 1 } . The obtained chip sequence is written as { c k ( j ) , j = 1 , 2 , ⋯ , J } , where J = N × S is the chip length. A specific distinct chip level random interleaver { π k , k = 1 , 2 , ⋯ , K } is employed for user separation [
The interleaved chip sequences are mapped using BPSK onto the modulated symbols { x k ( j ) , j = 1 , 2 , ⋯ , J } which are the elements of BPSK constellation. For simplicity, the symbol mapping process is not shown in
As a result, the time-domain data sequence, including CP, has the form:
v k = 1 N c ∑ n = 0 N c − 1 x k , n e i 2 π n t / N c , t = − L g , ⋯ , 0 , ⋯ , N c − 1 (5)
where x k , n is the modulated data of the OFDM symbol of the user-k which is mapped to the n^{th} sub-carrier. The OFDM signal v k is transmitted through the multipath channel. It is assumed that the multipath channel impulses of each user, h k = { h k ( 0 ) , h k ( 1 ) , ⋯ , h k ( L − 1 ) } , are stationary in a frame period and mutually independent, where L denotes the number of resolvable paths. The output signal r ( j ) at any time instant j is a linear superposition of independently symbols for each user. The output of multipath channel is written as [
r ( j ) = ∑ k = 1 K h k ( j ) ∗ v k ( j ) + n ( j ) (6)
where * denotes the convolution, h k the UWB channel coefficient for user-k and n ( j ) = n a ( j ) + n I ( j ) the total noise.
The turbo-type iterative receiver structure of the proposed system is depicted in
R ( j ) = ∑ k = 1 K H k ( j ) x k ( j ) + N ( j ) (7)
where x k ( j ) and H k ( j ) are respectively the data chip over the n^{th} subcarrier and the corresponding channel tap for the user-k estimated from H k ( j ) = ∑ l = 0 L − 1 h k ( l ) ⋅ e − i 2 π l n / N c ⋅ N ( j ) , FFT of n ( j ) , are samples of noise [
The ESE performs coarse chip-by chip estimation. Re-write Equation (7), the received signal at time instant j is:
R ( j ) = ∑ k = 1 K H k ( j ) x k ( j ) + ξ k ( j ) (8)
where ξ k ( j ) = ∑ m ≠ k H m ( j ) x m ( j ) + N ( j ) represents a distortion (including interference plus noise) with respect to user-k. It is assumed that the channel coefficients H k are known a priori at the receiver. Moreover, it is supposed that the received signal is perfectly synchronized. A CBC iterative detection based on Log Likelihood Ratio (LLR) of the received chip sequence approximated as either Gaussian or non-Gaussian noise. In case of Gaussian approximation (i.e. n ( j ) = n a ( j ) only) and x k ( j ) is treated as a random variable with mean E ( x k ( j ) ) and variance V a r ( x k ( j ) ) (initialized to 0 and 1 respectively) [
E ( R ( j ) ) = ∑ k = 1 K H k ( j ) E ( x k ( j ) ) (9.a)
V a r ( R ( j ) ) = ∑ k = 1 K | H k ( j ) | 2 V a r ( x k ( j ) ) + σ 2 (9.b)
where σ 2 is the power of the background (AWGN) occurring during the j^{th} chip. Using the central limit theorem [
E ( ξ k ( j ) ) = E ( R ( j ) ) − E ( H k ( j ) x k ( j ) ) (10.a)
V a r ( ξ k ( j ) ) = V a r ( R ( j ) ) − | H k ( j ) | 2 V a r ( x k ( j ) ) (10.b)
The ESE outputs are the (LLRs) about { x k ( j ) } computed based on Equation (9) (using Equation (10)) as [
L ( x k ( j ) ) ≡ log ( Pr ( x k ( j ) = + 1 | R ( j ) ) Pr ( x k ( j ) = − 1 | R ( j ) ) ) = log ( exp ( − ( R ( j ) − E ( ξ k ( j ) ) − H k ( j ) ) 2 2 V a r ( ξ k ( j ) ) ) exp ( − ( R ( j ) − E ( ξ k ( j ) ) + H k ( j ) ) 2 2 V a r ( ξ k ( j ) ) ) ) = ( 2 H k ( j ) ( R ( j ) − E ( ξ k ( j ) ) ) V a r ( ξ k ( j ) ) ) ∀ k , j (11)
In case of non-Gaussian approximation, ξ k ( j ) can be approximated using three different non-Gaussian models as, Cauchy, Laplace, and Gaussian mixture model. As Cauchy and Laplace models considered a good approximation for impulsive noise only (i.e. n ( j ) = n I ( j ) ) and ξ k ( j ) in Equation (8) has either Cauchy or Laplace distribution, so we have:
E ( R ( j ) ) = ∑ k = 1 K H k ( j ) E ( x k ( j ) ) (12.a)
V a r ( R ( j ) ) = ∑ k = 1 K | H k ( j ) | 2 V a r ( x k ( j ) ) + σ I 2 (12.b)
where σ I 2 is the power of the impulsive noise occurring during the j^{th} chip. While, GMM is considered a good fit for both Gaussian and non-Gaussian noise (i.e. n ( j ) = n a ( j ) + n I ( j ) ), ξ k ( j ) has GMM distribution, hence from Equation (8) we have:
E ( R ( j ) ) = ∑ k = 1 K H k ( j ) E ( x k ( j ) ) (13.a)
V a r ( R ( j ) ) = ∑ k = 1 K | H k ( j ) | 2 V a r ( x k ( j ) ) + σ 2 + σ I 2 (13.b)
For Cauchy distribution (α = 1), the ESE outputs are the (LLRs) about { x k ( j ) } as follow [
L { x k ( j ) } = log [ γ 2 + ( r ( j ) − E ( ξ k ( j ) ) + h k ( j ) ) 2 γ 2 + ( r ( j ) − E ( ξ k ( j ) ) − h k ( j ) ) 2 ] ∀ k , j (14)
The Cauchy detector has been used as a suboptimal detector to model the impulsive noise only with a robust performance, but it is still complex because of the need to calculate the log operation [
For Laplace model, the case in which the noise samples ξ k ( j ) have a Laplace distribution [
L { x k ( j ) } = 2 C ( | r ( j ) 2 − E ( ξ k ( j ) ) − h k ( j ) 2 | − | r ( j ) 2 − E ( ξ k ( j ) ) + h k ( j ) 2 | ) = { E ( ξ k ( j ) ) − h k ( j ) r ( j ) ≥ E ( ξ k ( j ) ) − h k ( j ) r ( j ) E ( ξ k ( j ) ) + h k ( j ) < r ( j ) < E ( ξ k ( j ) ) − h k ( j ) E ( ξ k ( j ) + h k ( j ) ) r ( j ) ≤ E ( ξ k ( j ) ) + h k ( j ) (15)
Laplace distribution has been used as a good approximation for impulsive noise also with complexity lower than Cauchy model since it needs to calculate the sum operation only. For GMM distribution, the case in which the noise samples ξ k ( j ) have a GMM distribution, the ESE outputs are the (LLRs) about { x k ( j ) } as follow [
L { x k ( j ) } = log ( 1 − ϵ σ e − ( r ( j ) − E ( ξ k ( j ) ) − h k ( j ) ) 2 2 V a r ( ξ k ( j ) ) + ϵ σ I e − ( r ( j ) − E ( ξ k ( j ) ) − h k ( j ) ) 2 2 η V a r ( ξ k ( j ) ) ) ( 1 − ϵ σ e − ( r ( j ) − E ( ξ k ( j ) ) + h k ( j ) ) 2 2 V a r ( ξ k ( j ) ) + ϵ σ I e − ( r ( j ) − E ( ξ k ( j ) ) + h k ( j ) ) 2 2 η V a r ( ξ k ( j ) ) ) (16)
GMM is considered an excellent fit to simulation because it can adapt between AWGN and impulsive noise but with high complexity since many parameters and operations need to be estimated. After the end of ESE process, The LLR-valued chip sequence L ( x k ( j ) ) is then de-interleaved to produce LLRs of c k ( j ) , { L ( c k ( j ) ) } which is delivered to DECs part discussed as follows.
The DECs in
E ( x k ( j ) ) = tanh ( E x t ( x k ( j ) ) 2 ) (17.a)
V a r ( x k ( j ) ) = 1 − E ( x k ( j ) ) 2 (17.b)
As discussed in Equation (10), E ( x k ( j ) ) and V a r ( x k ( j ) ) will be used in the ESE to update the interference mean and variance [
In this section, the simulation results demonstrate the performance of Polar-coded IDMA OFDM-based UWB system. The discrete time channel model proposed by the IEEE 802.15.3a working group [
The simulation parameters are summarized as follows: Each user data length is 128 bits, Polar encoding is applied with rate = 1/2. FFT size = 256 and cyclic prefix = 32 was added to each OFDM symbol block to avoid the effect of inter-chip interference. A common length-32 spreading sequence is assigned to all users, and a randomly generated chip interleaver is allocated to each user. The maximum iteration number of IDMA receiver is 3. The simulation is performed for (1 - 32) users. It is assumed that all users initially are synchronous with equal power allocation. The receiver is assumed to have perfect knowledge of the channel state information.
The simulation has three cases: The first case studies the BER performance of the proposed polar-coded IDMA OFDM-based UWB system for multi-users (k = 1, 8, 16, 32) under Gaussian and non-Gaussian noise models. The second case presents the performance and complexity comparison between the proposed system and LDPC-coded IDMA OFDM-based UWB system introduced in [
The BER performance of the proposed system for multi-users under GMM effect is introduced in
Furthermore, we found that by increasing the number of users to 32, the performance of the proposed system degraded by significant amount because of increasing the MAI and inter-chip interference as shown in
From the previous study, it can be noticed that Cauchy and Laplace models achieves a good performance compared with AWGN model than GMM. Furthermore, they are considered a good and simple approximation for impulsive noise while GMM can adapt between AWGN and impulsive noise but with high complexity since many parameters and operations needed to be estimated.
The second case of simulation shows a comparative analysis between the proposed system and LDPC-coded IDMA-OFDM-UWB system in [
In case of polar code, the recursive structure of channel polarization construction imposes a low complexity encoding and decoding algorithms. As in [
Hence the overall complexity of the system (both encoder and decoder including the iteration of IDMA) for polar codes is 3 ∗ [ 2 ∗ O ( N log N ) ] = 6 ∗ O ( N log N ) while for LDPC is
System Performance | E_{b}/N_{0} [dB] at BER = 10^{−4} | ||
---|---|---|---|
Noise Models | LDPC-Coded IDMA OFDM-UWB System | Polar-Coded IDMA OFDM-UWB System | Un-Coded IDMA OFDM-UWB System |
AWGN Cauchy Laplacian GMM | 2.5 3.65 4.3 5.3 | 3.35 4.2 4.65 6 | 7.95 9.11 10 11 |
3 ∗ [ O ( N 2 ) + 7 ∗ O ( N log N ) ] = 3 ∗ O ( N 2 ) + 21 ∗ O ( N log N ) So, despite of the performance of LDPC-coded system is better than polar-coded system, the complexity of LDPC-coded system exceeds the complexity of the polar-coded system by 3 ∗ O ( N 2 ) + 15 ∗ O ( N log N ) which make the polar codes do better from the complexity point of view.
Finally, as shown in
Furthermore, our work outperforms the work introduced in [
In this paper Polar code is introduced instead of LDPC code in IDMA-OFDM based UWB system to mitigate the interferences and reduce the complexity. Moreover, a comparative analysis is held between the two systems from the aspects of performance and complexity using non-Gaussian approximation for the interferences which are more realistic for UWB system.
The comparison shows that the proposed system is robust against noise and interferences with complexity much lower than LDPC-coded system. Therefore, polar coding technique can be considered a potential candidate for the emerging 5G communications due to its reliability with low decoding complexity.
El. Matary, D.E., Hagras, E.A. and Abdel-Kader, H.M. (2018) Performance of Polar Codes for OFDM-Based UWB Channel. Journal of Computer and Communications, 6, 102-117. https://doi.org/10.4236/jcc.2018.63008
APP: A Posteriori Probability
AWGN: Additive White Gaussian Noise
BER: Bit Error Rate
BWA: Broad-band Wireless Access
CBC: Chip-By-Chip
CM: Channel Model
CP: Cyclic Prefix
DECs: Decoders
ESE: Elementary Signal Estimator
FEC: Forward Error Correcting
FFT: Fast Fourier Transform
GMM: Gaussian Mixture Model
ICI: Inter Carrier Interference
IDMA: Interleave Division Multiple Access
IFFT: Inverse Fast Fourier Transform
IR: Impulse Radio
ISI: Inter Symbol Interference
LDPC: Low Density Parity Check
LLR: Log Likelihood Ratio
LOS: Line of Sight
MAI: Multiple Access Interference
MC: Multicarrier
MUD: Multi User Detection
OFDM: Orthogonal Frequency Division Multiplexing
pdf: probability density function
SCD: Successive Cancellation Decoder
SNR: Signal to Noise Ratio
SPA: Sum Product Alghorithm
S-V: Saleh-Valenzuela
SαS: Symmetric Alpha Stable
UWB: Ultra Wide Band
5G: fifth Generation.