BER estimation for STBC-MC-DS-CDMA-4 antennas system by varied wavelet-carriers features via AWGN-flat channels

ABSTRACT


INTRODUCTION
The recent advancements in data network gave rise to many applications that are involved with almost every aspect of life ranging from top secret military operation to merely sending greetings for a birthday [1], [2].These networks are becoming the backbone of societies because people are relying on them to spread news all over the world [3]- [5].Even the elections consider social networks as a reliable source of information to study the citizen's point of views regarding a certain party or a candidate.Presently, these networks are growing to be self-sustained economic systems and a pillar of modern societies [6], [7].
To satisfy the higher bandwidths, the engineers could follow the traditional way by providing the extra bandwidths for the networks.Although this might be satisfactory it suffers from by unavoidable limitations.First limitation is the scares affordable free bandwidth because the radio frequency (RF) spectrum is highly congested.The second limitation is the bandwidth over a wide range will increase the influence of nonlinearity and deformations effect on the transmitted signals [8]- [11].Differnt approaches will invented to modulate the data that used narrower bandwidth with higher data rates, like the multi-carrier transmission (MC) or its modern version the orthogonal frequency division multiplexing (OFDM) [12], [13].[14] suggested a system of using discrete wavelet packet transform (DWPT) in OFDM instead of discret fourier transform (DFT), then compared the performance between them based on bit error rate (BER) values and their corresponding signal to noise ratio (SNR), while Soni et al. [15], used a discret wavelet transform (DWT) instead of fast fourier transform (FFT) in the OFDM system and test the proposed system by sending various images through different channes.They found that, the DWT is more reliable and has less SNR as compared to FFT.The use of these techniques is very important in many application as Roy et al. [16] depictes in his paper, so that a comparison is made between the FFT, discrete cosine transform (DCT), DWT and wals-hadarmard transform (WHT) in many applications like electrocardiogram and photo-plethysmography signals.This paper aims to improve the BER of the code division multiple access (CDMA) multicarrier system by adding a DWPT instead of FFT as a suggestion to the system.Here the effected thing is reducing the SNR of the proposed system so that it reaches to the same BER but with low power, which consequently reduce the complexity and cost.

MC AND OFDM SYSTEMS
The idea of OFDM is based on dividing high bit rate stream of data into low bit stream segments.These segments are then modulated using orthogonal subcarriers to ensure neither overlapping between adjacent.These sub-streams are then combined together and transmitted using a main carrier before transmission.The block diagram of typical OFDM system is shown in Figure 1.Due to the fact that the resultant subs-terms have low rate with a low bandwidth requirement, then the OFDM requires less channel compared with the classical system.The fact that the subcarriers are orthogonal also allows the channels to overlap without interference because the orthogonality ensures complete separation among channels [12].The carriers and pilots illustration are shown in Figure 2, so that, Figure 2(a) depicts the user and pilot subcarriers and how the packets are performed, while Figure 2(b) states the effect of orthogonality on the subcarrier [13].

Figure 1. OFDM block diagram
Consider a multi-carrier (MC) OFDM system with  subcarriers and each subcarrier   is used to modulate   sub-frame, then () transmitted signal is (1): Here   is equally spaced subcarriers and   = /,  is the bandwidth.Let  to be 2/, then according to Nyquist theorem each sample occurs at /2 or 1/.Therefore, any sub-stream subcarrier will exist at time  then (1) becomes: In ( 2) is exactly the inverse discrete fourier transform (IDFT) or inverse fast fourier transform (IFFT).At the receiver, an FFT circuit can be used to retrieve the transmitted signals from the incoming samples.This is quite clear from Figure 1, where the symbols are modulated, possibly using quadrature amplitude moduation (QAM) [13]. ) or in other words a low pass filter bank.The frequency location for each subcarrier   occurs at every2  ×  0 , where f0 is the fundamental frequency and   are its harmonics.This means that the power spectral density (PSD) will be widely spread over the channel as the sinusoidal functions extends from -∞ to +∞.This consequently reduces the system spectral efficiency.In order to solve this problem, the analyzing function  (2   ) can be replaced by a more compact or spectral efficient function to ensure higher PSD concentration [14].The DWT can ensure the achievement of this goal.Hence; by replacing DFT or FFT with DWT, the spectral efficiency can be increased due to time limited period of DWT functions resulting in a more reliable and higher signal quality [15], [16].
The wavelets are orthonormal functions such that [17], [18]: Where Ψ() is the mother wavelet, and ⟨⋅⟩ is inner product.The  vanishing moments can be found as (4): The vanishing moments describe the speed of decay of the wavelet function.So, the DWT output is: Where ,  is the scale and translation.This means that the wavelet can capture the details of () in both time and frequency unlike FFT, which focuses only on the frequency domain.The translation means the wavelet is shifted in time while the scale is related to the width of the finite wavelet function.The scale and translation are shown in Figure 3 [19].Assume an  samples discrete signal []( = 0, 1, ⋯ ,  − 1) want to apply first level DWT to it, then: Figure 3.The scale and translation of wavelet In ( 6), the   is the signal coming from the low pass filter (LPF) bank with its impulse response  0 [2-], while,  ℎℎ is the signal coming from the high pass filter (HPF) bank with its impulse response  1 [2-].It is noted that the out coming signal has double the sampling rate  as the input signal.Thus, a decimation circuit by 2 (↓2) is required to switch back to the original sampling rate.Figure 4 shows the filter bank of the multi-level DWT [20].The DWT functions  0 and  1 suggest that the incoming signal will be transformed to orthogonal frequencies in the rate of  2  where  = 1, 2, ⋯ ,  which is the number of analysis levels.The reconstruction functions  0 and  1 are the inverse for the analyzing functions  0 and  1 .In reconstruction filter, the sampling rate is up by a factor of 2 (↑2) to reverse the decimation process.The DWT can be modified to be identical to FFT and further increases the final signal PSD concentration.The modification comes from decomposing  ℎℎ 1 in the same manner   1 at every stage.This analysis is called DWPT.The filters used for DWPT is shown in Figure 5 [21], [22].To generate the orthonormal subcarriers by using inverse discrete wavelet packet transform (IDWPT), consider (6) and by using mallat's pyramid algorithm (MPA) for multi-resolution analysis (MRA), in ( 6) can be written as (7), so that () is called the wavelet coefficient function and is related to ℎ(. ) (10).Then for a scaling function of order , in ( 11) can be depicted.In addition, the dilation and wavelet equations can be written as seen in ( 12) and ( 13): Therefore, by using MPA and extending it to the DWPT, series of [] vector can be analyzed using the following transformation matrix for  number of stages [23], [24].By substituting (11) in ( 14), the transformation matrix becomes: Using the orthonormal property, the reconstruction matrix is simply the transpose of the analysis function.
Hence, at the construction side the following matrix is used ( 16):

OFDM AND CDMA SYSTEMS
The OFDM can ensure a complete separation for overlapped channels due to the orthogonality property.This property is tightly adhered to the location of subcarriers.If the location of these subcarriers are dispositioned, then the orthogonality will fall apart and the system will experience the inter channel interference (ICI) [24].The ICI will reduce the system performance due to the decrease in the signal quality.Therefore, certain modifications should be taken, and one of them is using CDMA [25].The CDMA is built upon the spread spectrum system (SSS) technology by which it allows all the users to use the channel all the time.A typical CDMA is shown in Figure 6 [26].In the CDMA, each user is assigned by a pseudo noise-sequence (PN-sequence).The word pseudo comes from the fact that the sequence will repeat itself after a period of time depending on the linear feedback shift register (LFSR) constraint length.The generated PN pulses are almost random.The auto correlation function (ACF) of the PN sequence has the property given in (17) [27].

𝐴𝐶𝐹(𝑡) = ∫ 𝑃𝑁(𝑡)𝑃𝑁(𝑡 + 𝜏
Where  is the maximal length and equals to 2 n -1 and n is the LFSR constraint length. When the PN sequence is aligned with itself then the ACF will have 1 value otherwise the sequence will cancel the incoming data.Unfortunately, the PN-sequence has poor cross correlation function (CCF), so gold code is used.If the sequences are chosen carefully, then the CCF will have only three states, which are: for  odd for  even (18) At the receiver side, the incoming streams are multiplied by the same gold code, then passed through a coherent detector as seen in Figure 7 [26]- [28].Let the transmitted OFDM signal to be (), then the received signal () is seen in (19): where () is the spreading code at the transmitter side, () is the impairments from the channel like noise or jamming or interference or multipath.

FREQUENCY FLAT FADING CHANNEL AND SPACE TIME BLOCK CODING
The practical channels are usually imperfect and thus are described using its impulse response.assume a frequency selective fading channel is ℎ of length  then the received signal () is given in (22) [29]: Where () is the transmitted signal, in (22) shows that the channel with such channel impulse response (CIR) which will cause inter-symbols interference (ISI) or in the case of OFDM will be worse which is inter-block interference (IBI) or ICI.One of the methods that used to overcome the fading effect is the multiple-input and multiple-output (MIMO)-space time block coding (STBC) [30].MIMO is a technique by which the symbol is transmitted through different paths in order to null the CIR.Here Alamouti's STBC is used to avoid the continuous need to update the CIR [31].The simplest form of Alamouti scheme is the multiple-input single-output (MISO) of 1×2 which means that the number of receiving antennas is 1 and the transmitting antennas is 2 as in Figure 8.Let two symbols  1 and  2 to be transmitted by this MISO over two channels of CIR or sometimes called channel state information (CSI) of ℎ 1 and ℎ 2 , then the received symbols are: Figure 8.A 2×1 single input multiple output (SIMO) system At the second instance, the first antenna will transmit  2 * () and the second antenna will transmit− 1 * (), so the received symbols at the second instance are: Taking the conjugate of ( 24) and arranging the result to look like: Where  1 () and  2 () are the AWGN in the first and second transmission instances respectively.Combining (24) with (25), then it can be shown that: Here, the Alamouti orthognal space time block code (OSTBC) can be extended from MISO to MIMO easily [32], [33]. 1 and  3 is depicted in (27) and so on for  4 .

RESULTS AND DISCUSSION
The suggested system combines all the above concepts and integrate them into a single communication system as shown in Figure 9. First, the signals are mapped using quadrature phase shift key (QPSK) scheme.An OSTBC is added then the resultant stream is fed to IDWPT section to generate the orthogonal subcarriers.These pilots are used for synchronization and estimating the CSI of the channel in order to provide the necessary equalization if needed.This s erial stream is then fed to a MIMO of 2×2 antennas then send over the channel.
The transmitter is assumed to be a mobile travelling at speeds 2 km/hr, 45 km/hr, and 100 km/hr creating maximum doppler shift (MDS) of 10.7 Hz, 241.7 Hz, and 537 Hz respectively.The channel is flat fading rayleigh channel with AWGN.The number of transmitted bits per test is (102,400 bits) coming from (100 packet*64 characters *16 bit/char).The OFDM physical layer settings are shown in Table 1.Here, the mother wavelets utilized are as follows: Haar, Daubechies=db4, Symlets=sym4, Cohen-Daubechies-Feauveau=cdf, and B-spline 3=bs3.The results are as follows: a. MDS 10.7 Hz, the BER are shown in Figure 10  From Figure 10, it can be seen that to obtain a 10 -3 BER, the SNR must be at least 28 db, while by using the proposed system it decreaded to about 5 dB when using 512 subcarriers, and so on for the athors.The percentage improvent is about 80 percent, which is very good.The big difference between the results of the proposed system and the traditional method beyond to the excellent orthogonality of DWPT over FFT, which reduces the effect of ICI and ISI on the transmited data.
A comparison is made when motor driven systems (MDS) equal to 10.7 Hz [30], they reach to 10 -3 BER (128 subcarrier) at 8 dB SNR in AWGN, while the proposed system reaches the same value at 5 db.In addition, Ali et al. [34] for the same sub carrier reches to 7 dB SNR, while at 1,024 subcarriers the SNR is 8 dB and the proposed system of this work gives 7 dB.In the same context, Kusumawardhania et al. [35] obtain SNR of about 7.5 dB over AWGN channel when using a multi-level coding CDMA.This comparison gives the pereference of the proposed system over different algorithms used to enhance the BER evaluation.

CONCLUSION
Here, IDWPT/DWPT has an excellent performance as compared to the IFFT/FFT in all tests.This justifies the assumption that the DWPT has higher concentrated signal PSD.But in all cases, the BER increases with the increase of bit rate and this event is inevitable.The doppler shift has high impact on system performance as it increases due to the motion of the mobile station.The doppler shift changes the locations of the orthogonal subcarriers which derifts the orthogonality that depends on the difference between adjacent subcarriers, although, the pilots were transmitted to resynchronize the system but even at certain bit rate and motion speed they lose their effectiveness.Neither the STBC nor the CDMA are now able to track and fight of the doppler shift effect.It is highly noticed that the wavelets have no close relation to the bit rate and inspite of that, there are some kinds of wavelets that gives the worst performance upon FFT like bs3.Therefore, the wavelet selection should be carefully done.

Figure 2 .
Figure 2. OFDM idea; (a) the subcarriers assignment to users with pilot carriers and (b) OFDM signal spectrum

Figure 9 .
Figure 9.The complete suggested communication system (see in Appendix) b.MDS 241.7 Hz, the BER are shown in Figure 11(see in Appendix) c. MDS 537 Hz, the BER are shown in Figure 12 (see in Appendix)

Table 1 .
OFDM physical layer simulation settings