The act of finding or constructing a model for a portion of a given polynomial or Bezier parametric surface from the whole original one is an encountered problem in surface modeling. A new method is described for constructing polynomial or Bezier piecewise model from an original one. It is based on the "Parametric Piecewise Model," abbreviated to PPM, of curve representation. The PPM representation is given by explicit expressions in terms of only control points or polynomial coefficients. The generated piecewise model behaves completely as a normal, polynomial or Bezier model in the same way as the original one for the piece of region considered. Also it has all characteristics, i. e, order and number of control points as the original one, and satisfies at the boundaries all order continuities. The PPM representation permits normalization, piecewise modeling, PPM reduction and systematic processes.

Time spread (TS) pulse position modulation (PPM) signals have been proposed for CDMA applications, where the envelope detection is employed instead of coherent detection for easier synchronization of PPM. In this paper, a new method of deriving symbol error probability (SEP) of TS PPM signals in the presence of interference is introduced. The analysis is based on the moment technique. The maximum entropy criterion for estimating an unknown probability density function (PDF) from its moments is applied to the evaluation of PDF of envelope detector output. Numerical results of SEP are shown for 4, 8 and 16PPM in the practical range of signal-to-noise power ratio (SNR) and signal-to-interference power ratio (SIR) of 5, 10 and 20 dB. SEP by the union bound is also given for comparison. From the results it is noted that when PPM multilevel number is small, the union bound goes near to SEP by the proposed method, but when it increases the difference of the SEP by the bound and proposed method becomes larger. The effect of central frequency offset of TS-filter is evaluated as an illustrative example.

Cutoff rate of overlapping multi-pulse pulse position modulation (OMPPM) is analyzed in the quantum-limited and the background noise cases. Our results suggest that the derived cutoff rate is higher than conventional one because of the infinite quantization at the demodulator and the definition of the erasure event in conventional analysis.

We propose an access timing controlled derectdetection optical code-division multiple-access (ATC-CDMA) system with pulse position modulation (PPM) signaling to improve the performance of direct-detection optical CDMA systems with PPM signaling without narrowing the chip width. In the ATC-CDMA system, each user is allowed to access the network Tu times out Tt times according to the scheduled access timing pattern, and the number of users accessing the network simultaneously is decreased; the effect of the channel crosstalk is thus reduced. We analyze the performance of the proposed system under the assumption of Poisson shot noise model for the receiver photodetector including the noise due to the detector dark currents. We compare the performance of the ATC-CDMA system using PPM signaling with that of a conventional CDMA system using PPM signaling under a fixed bit rate, almost the same chip width and a constraint on the transmitted energy per pulse. It is shown that the ATC-CDMA system has better performance than the conventional CDMA system. Moreover, it is shown that the ATC-CDMA system with fewer slots per symbol and lower access timing probability to the network has better performance at the fixed bit rate and the chip width. Therefore, controlling an access timing of each user to the netwark is shown to be more effective for improving the bit error probability performance than the pulse position multiplicity with the increase in the number of slots per symbol under the constraint on the bit rate and the chip width on direct-detection optical CDMA systems with PPM signaling.

The performance of direct-detection optical synchronous code-division multiple-access (CDMA) systems using pulse position modulation (PPM) signaling (PPM/CDMA) with interference canceller is analyzed. In optical CDMA systems, it is known that the maximum number of simultaneous users in CDMA systems is limited by the maximum tolerable interference among users. If the receiver is able to estimate this interference and cancel or reduce its effect, the capacity of CDMA systems can be increased and the system performance can be improved. There are some ways to increase the system performance, that is, using PPM and interference canceller. However, the system using both PPM and interference canceller has not been analyzed. In this paper, the upper bound on the bit error probability of optical synchronous PPM/CDMA systems with interference canceller is derived, and the bit error probability of optical synchronous PPM/CDMA systems is evaluated under the assumption of number-state light field where the background noise is negligible. We compare the bit error probability of the optical synchronous PPM/CDMA systems with interference canceller to that of the optical synchronous PPM/CDMA systems without interference canceller and to those of optical synchronous OOK/CDMA systems with and without interference canceller. We show that optical synchronous PPM/CDMA systems with interference canceller have better bit error probability performance.

The bit error rate (BER) performance of Multi-pulse pulse position modulation (MPPM) using an avalanche photodiode (APD) is evaluated in optical intersatellite links (ISL's) . We derive theoretical expression of BER of MPPM consisting of M slots and P pulses, referred to as (M, P) MPPM with output statistics of the APD which is approximated by Gaussian distribution. The BER performance of (7, 2) MPPM and (9, 2) MPPM is compared with that of 4-ary PPM under the conditions that bit rate is constant (200 Mbits/s) and that bandwidth is limited (slot time period is 2.5 ns) . It is shown that (9, 2) MPPM using the APD can achieve better BER performance than 4-ary PPM using the APD with broadening slot time period by about 10% under the condition that bit rate is constant or with improving bit rate by about 10% under the condition that bandwidth is constant when average photons/nat is somewhat large in achieving BER of 10-9.

The effect of an optical hard-limiter on the performance of direct-detection optical synchronous code-division multiple-access (CDMA) systems with M-ary pulse position modulation (PPM) signaling is analyzed. Moreover, the effect of the error correction coding on the performance of direct-detection optical synchronous CDMA systems with PPM signaling is analyzed: Reed-Solomon (RS) codes and convolutional codes (CC's) with soft-decision Viterbi decoding are employed. We analyze the performance under the assumption of Poisson shot noise model for the receiver photodetector and the noise due to the detector dark currents is considered. We analyze the performance under average power and bit rate constraints. Our results show that the optical hard-limiter is not effective for improving the performance of the optical CDMA systems with PPM signaling. Moreover, RS codes are shown to be more effective than CC's with soft-decision Viterbi decoding to reduce an asymptotic floor to the error probability of the system with large M, while CC's with soft-decision Viterbi decoding is more effective than RS codes for the system with small M. Furthermore, we show that as the code rate of the error correction code increases, the required average energy to achieve the bit error probability Pb105 for the RS coded PPM/CDMA system appreciably increases compared with that for the convolutional coded PPM/CDMA system when M16.

We analyze the effect of overlapping technique on differential pulse position modulation (DPPM) in optical direct-detection channel when the pulsewidth and the average power of the channel are constrained. We refer to the modulation scheme employing an overlapping technique in DPPM as differential overlapping PPM (DOPPM). To avoid frame synchronization problems, we analyze the performance of DOPPM under the window scheme that results in lower bounds on the capacity and the cutoff rate of DOPPM but is easy to analyze. Under this scheme, we analyze the lower bounds on the capacity and the cutoff rate of DOPPM. It is shown that DOPPM with the window scheme has higher capacity and cutoff rate than PPM and DPPM, and also than OPPM when the average received number of photons per slot is somewhat large. The overlapping technique is thus shown to be effective on DPPM under the pulsewidth constraint when the average received number of photons per slot is somewhat large.

We analyze the error performance of overlapping multipulse pulse position modulation (OMPPM) in optical direct-detection channel with existing noise. Moreover we analyze the error performance of trellis-coded OMPPM with the small overlap index N=2 in optical direct-detection channel to achieve significant coding gains over uncoded PPM, uncoded MPPM and the trellis coded overlapping PPM (OPPM) with the same pulsewidth. First we analyze the symbol error probability of OMPPM in both the quantum-limited case and the quantum and background noise case by using the distance defined as the number of nonoverlapped pulsed chips between symbols. Second by using this distance, we partition the OMPPM signals and apply the four-state and the eight-state codes described by Unger-boeck to OMPPM. It is shown that the trellis coding over OMPPM is effective in optical direct-detection channel: the eight-state trellis coded (4,2,2) OMPPM can achieve gains of 3.92dB and 3.23dB over uncoded binary PPM in the quantum-limited case and in the quantum and background noise case with noise photons per slot time is one, respectively.

Overlapping multi-pulse pulse position modulation (OMPPM) is a modulation scheme having higher capacity and cutoff rate than other conventional modulation schemes when both off-duration between pulses shorter than a laser pulsewidth and resolution better than a laser pulsewidth are realized [1],[2]. In Refs. [1],[2] erasure events of a few chips that can be decoded correctly is defined as an erasure event. This results in lower bounds on the performance of OMPPM in optical-direct-detection channel in quantum limited case. This paper analyzes more exact performance of OMPPM in optical direct-detection channel in quantum limited case when both off-duration between pulses shorter than a laser pulsewidth and resolution better than a laser pulsewidth are realized. First we derive the error probability of OMPPM with considering what chips are detected or erased. Then we derive the capacity and the cutoff rate of OMPPM using the error probability. It is shown that OMPPM outperforms on-off keying (OOK), pulse position modulation (PPM), multi-pulse PPM (MPPM), and overlapping PPM (OPPM) in terms of both capacity and cutoff rate for the same pulsewidth and the same duty cycle. Moreover, it is shown that OMPPM with fewer slots and more pulses per block has better cutoff rate performance when the average received power per slot is somewhat large.

We analyze the error probability performance of multi-pulse pulse position modulation (MPPM) in noisy photon counting channel. Moreover we investigate the error perofrmance of convolutional coded MPPM and RS coded MPPM in noisy photon counting channel. We define a distance between symbols as the number of nonoverlapping pulses in one symbol, and by using the distance we analyze the error performance of MPPM in noisy photon counting channel. It is shown that MPPM has better performance than PPM in the error probability performance in noisy photon counting channel. For PPM in noisy photon counting channel, convolutional codes are more effective than RS codes to reduce the average transmitting power. For MPPM in noisy photon counting channel, however, RS codes are shown to be more effective than convolutional codes.

In this paper, we propose a spread spectrum pulse position modulation (SS-PPM) system, and describe its basic performances. In direct sequence spread spectrum (DS/SS) systems, pseudo-noise (PN) matched filters are often used as information demodulation devices. In the PN matched filter demodulation systems, for simple structure and low cost of each receiver, it is desired that each demodulator uses only one PN matched filter, and that signals transmitted from each transmitter are binary. In such systems, on-off keying (SS-OOK), binary-phase-shift keying (SS-BPSK) and differential phase-shift keying (SS-DPSK) have been conventionally used. As one of such systems, we propose the SS-PPM system; the SS-PPM system is divided into the following two systems: 1) the SS-PPM system without sequence inversion keying (SIK) of the spreading code (Without SIK for short); 2) the SS-PPM system with SIK of the spreading code (With SIK for short). As a result, we show that under the same bandwidth and the same code length, the data transmission rate of the SS-PPM system is superior to that of the other conventional SS systems, and that under the same band-width, the same code length and the same data transmission rate, the SS-PPM system is superior to the other conventional SS systems on the following points: 1) Single channel bit error rate (BER) (BER characteristics of the SS-PPM system improve with increasing the number of chip slots of the SS-PPM system, and as the number of chip slots increases, it approaches Shannon's limit); 2) Asynchronous CDMA BER; 3) Frequency utilization efficiency. In addition, we also show that With SIK is superior to Without SIK on these points.

A model for time spread-pulse position modulation (TS-PPM)/code division multiple access (CDMA) systems is presented. A TS signal is produced by a TS-filter, whose characteristic is a pseudonoise sequence in frequency domain. The error probability performance is analyzed and compared with those of on-off keying (OOK) and binary phase shift keying (BPSK). It is shown that at the same transmission speed TS-PPM is superior to TS-OOK and TS-BPSK due to the dramatic decrease of multiple access interference. The throughput of the network is analyzed, and its relation to the length of pseudonoise sequence and the packet length is also discussed.

We propose parallel rate-variable punctured convolutional coded PPM in photon communication to achieve high energy information efficiency Ie for desired bit error rate (BER) and transmission bandwidth. We theoretically show the BER performance, bandwidth expansion factor β and necessary Ie to achieve BER=10-6 of the proposed systems for some combinations of code rates. It is found that the proposed system can achieve high Ie for desired BER and β by selecting a suitable combination of code rates depending on the channel conditions. Moreover, it is shown that the proposed system has better BER performance than RS-coded PPM in the range of small β.

We propose parallel rate-variable punctured convolutional coded PPM in photon communication to achieve high energy information efficiency Ie for desired bit error rate (BER) and transmission bandwidth. We theoretically show the BER performance, bandwidth expansion factor β and necessary Ie to achieve BER10-6 of the proposed systems for some combinations of code rates. It is found that the proposed system can achieve high Ie for desired BER and β by selecting a suitable combination of code rates depending on the channel conditions. Moreover, it is showm that the proposed system has better BER performance than RS-coded PPM in the range of small β.