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Self-encoded spread spectrum eliminates the need for traditional pseudo noise (PN) code generators. In a self-encoded multiple access (SEMA) system, the number of users is not limited by the number of available sequences, unlike code division multiple access (CDMA) systems that employ PN codes such as m-, Gold or Kassami sequences. SEMA provides a convenient way of supporting multi-rate, multi-level grades of service in multimedia communications and prioritized heterogeneous networking systems. In this paper, we propose multiuser convolutional channel coding in SEMA that provides fewer cross-correlations among users and thereby reducing multiple access interference (MAI). We analyze SEMA multiuser convolutional coding in additive white Gaussian noise (AWGN) channels as well as fading channels. Our analysis includes downlink synchronous system as well as asynchronous system such as uplink mobile-to-base station communication.

In CDMA communications, each user is assigned a unique PN spreading sequence that has a low cross-correlation with other users' sequences. This prevents code collisions between the users and controls MAI. PN code generators are typically linear feedback shift register circuits that generate maximal-length or related sequences. These deterministic sequences provide low cross-correlations that are critical for achieving good system performance. Although random codes have often been employed for analysis purposes [

Our approach abandons the use of PN codes in SEMA that can reduce MAI, and provide a multi-rate and multi-level grade of service for multimedia communications and prioritized networks [3–6]. A realization of the self-encoding principle for a direct sequence spread spectrum systems is illustrated in

Transmitte’s spreading codes required for signal despreading. Data recovery is by means of a correlation detector. Notice that the contents of the delay registers in the transmitter and receiver should be identical at the start of the transmission. This is accomplished as part of the initial synchronization procedure. In the following, we develop SEMA multiuser convolutional coding, and investigate the performance with and without precoding or multiuser detection. Convolutional codes with Viterbi decoding have been studied for decades and applied in practical communication systems such as wide area networks (IS-95, CDMA2000) and local area networks (IEEE 802.11a and b). In order to improve the performance, we present the shift generator matrix concept that provides lower cross-correlations among users and reduces the MAI in the system. We present the performance analysis and simulation both in uplink asynchronous and downlink synchronous channels.

Acquisition and tracking of self-encoded sequences can be performed in a similar manner to PN sequences with the proviso that the chip updates are enabled once data transmission has commenced following code acquisition. At the chip rate, the self-encoded chips are latched at the output register by shifting the registers serially, with the output being fed back to the input register. The input feedback is switched to the data during the last chip period of the current symbol for a new chip input. This resembles a simple linear feedback register circuit of length, with zero valued taps except for the input and output taps, where the input register is updated periodically by the data and the output register provides the spreading sequence.

The conventional convolutional codes in

For a multiuser system with K +1 users (K interferers), the received signal at the matched filter is

where x(t) is the transmitted signal, and (t) is AWGN noise with a two-sided power spectral density of . The transmitted signal in Equation (1) is given by

where is the symbol duration and is a spreading sequence during for the user. is the amplitudes of the user, is the encoded symbol of the user during the symbol interval, and is the time delay of user signal, with 0≤≤ is zero for synchronous systems. For simplicity we do not consider carrier offset in uplink asynchronous systems. The output of the convolutional encoder for user and symbol is given by [

where and are the set of the user data sequences and the indices denoting the column and row in the generator matrix, respectively. The generator matrix is

where m is the memory size in the encoder, and r is the code rate.

Assuming that the signature waveforms have unit energy, the output of the matched filters of the user signature waveform during the symbol interval is

Equation (5) consists of the signal , Gaussian noise , and the multiple access interference. is the cross-correlation of the spreading sequences of the user and user during the symbol interval. In our analysis, the MAI is modeled as noise [

At the receiver, the despreading codes are updated by the detected data. If the data are incorrectly recovered, the incoming signals are correlated with an erroneous sequence set. This may lead to additional errors at the receiver and cause SI, which can be serious at a low signal-to-noise ratio (SNR). To combat self-interference, a longer spreading sequence is desired [

Precoding: Decorrelating and precoding techniques have been developed for multiuser detections [6,8,9]. Decorrelating detector is used for multiuser detection at the receiver, whereas precoding is employed at the transmitter to eliminate or reduce MAI. To reduce MAI, we examine the precoding system with interleaver. From Equations (1) and (2), we consider a synchronous system and rewrite (2) as

≤t≤T_{b} (6)

where is the signature waveforms vector, and is the transpose of . Then, the output of the matched filters can be expressed as

Equation (7) can be rewritten in a vector form, with as follows

A is the diagonal matrix of amplitudes, R is the crosscorrelation matrix, and h is the vector of the data symbols of K+1 users. The basic concept of precoding is to eliminate MAI at the receiver before transmitting signals. In other words, the transmit signals in Equation (2) become

where the precode matrix T is chosen as. Then, the output of the bank of matched filters at the receiver will be

In order to maintain the average power with precoding the same as without precoding, we modify the precode transformation matrix as [6,8,10]

such that

Multiuser Detection: Decorrelation detection is a suboptimal multiuser detection with comparatively low complexity. Receiver-based decorrelator can be found in [

Due to detection errors, the despreading sequence may not be identical to the spreading sequence at the transmitter. Since the recovered symbols are used to despread the signals, a chip error will remain in the shift registers and affect the following symbol decision until it is shifted out of the registers. This results in error propagation and causes SI: the bit error rate (BER) of SEMA is a dynamic quantity that depends on the signal-to-noise ratio (SNR), spreading factor, the number of users and transmitted symbols. The effect of SI is reduced as the spreading factor or the SNR increases.

The average bit error probability, , can be described by a Bernoulli distribution in terms of v and l, where l is the number of chip errors in the despreading registers and v is the spreading length. When v is large, the BER of SEMA can be well approximated by [12,13]

where the conditional bit error probability is

To ameliorate the effect of error propagation, differential encoding as shown in

quence, as v we have. Therefore, with differential encoding, the BER for large spreading length approaches the following

The downlink cellular system can be described as a synchronous system. The delay in synchronous transmission is zero for all users (). With the assumptions that the information sequences are independent and identically distributed, the probability density function (pdf) of MAI and noise is [

with variance [

The probability of a bit error in synchronous channels is [

where is. is the bit energy-to-noise ratio.

Signals in asynchronous systems arrive with different delays for all users as in uplink cellular systems. Thus, when the delay factor for user j is 0≤≤, the pdf of the MAI and noise is shown to be [

In asynchronous systems, the carriers of users are not synchronized. Therefore an additional term, , 0≤≤, should be included in MAI. However, we do not include the term in our discussion for simple presentation. In fact, incorporating the carrier mismatch will reduce MAI and improve the system performance a little. As a result, our analysis is somewhat conservative.

The variance of MAI and noise in the asynchronous channels is

and the probability of a bit error in asynchronous channels is given as

Notice that from Equations (18) and (21), the variance of asynchronous systems is less than that of synchronous systems, by a factor of 2/3.

power of N tells us the weight of the information bit weight. From this diagram, we derive the transfer function using Mason’ formula [

It can be shown from Equation (23) that d_{free} of this system is 8. For the hard-decision maximum likelihood decoder, Viterbi decoding algorithm for the binary symmetric channel (BSC) is used. We apply the transfer function upper bounds derived from the union bound computation for analytical comparison to the simulation results. From [15,16], we obtain the first-event error probability and the bit error probability:

＜＜

＜ (24)

where is the coefficient of the transfer function, and p is the probability of a bit error for BSC. The BER can be calculated from Equation (24) by replacing p with (19) and (22) (using symbol energy-to-noise ratio E_{s}/N_{o} instead of E_{b}/N_{o}) for synchronous and asynchronous systems, respectively. For moderate and high signal-to-noise ratios, it is well known that d_{free} in the union bound for the BER dominates the bound [

From Equations (11), (12) and (13), the bit error probability for precoding and decorrelating detector, respectively, is given by [

denotes the row and column of . The performance of SEMA with precoding/multiuser detection and convolutional coding in AWGN channels can be derived from equations (19) and (22) as [

which replaces p in Equation (24) to find the BER. The BER of SEMA with precoding/multiuser detection in Rayleigh fading channel can be obtained as

where for Rayleigh fading channels. Equation (28) (using instead of ) replaces p in Equation (25) to find the BER in fading channels.

In Subsection 3.1, we observed that SI is dominant at low SNR regions with small spreading length. The differential encoding was employed to mitigate the effect of error propagation. In fact, SI becomes negligible under high SNR and with a sufficiently large spreading length. The BER performance then approaches random spread spectrum (RASS).

approaches RASS not only asymptotically but also iteratively for≥2.5 dB. The results indicate that we can ignore SI in examining the asymptotic behavior of the system.

^{-4} BER, the performance with G1 = [5 7 7] applying to both users is approximately 2dB worse than that with shifted matrices G1 = [5 7 7] for user 1 and G2 = [7 5 7] for user 2.