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Interference is a serious cause of performance degradation for IEEE802.15.4 devices. The effect of concurrent transmissions in IEEE 802.15.4 has been generally investigated by means of simulation or experimental activities. In this paper, a mathematical framework for the derivation of chip, symbol and packet error probability of a typical IEEE 802.15.4 receiver in the presence of interference is proposed. Both non-coherent and coherent demodulation schemes are considered by our model under the assumption of the absence of thermal noise. Simulation results are also added to assess the validity of the mathematical framework when the effect of thermal noise cannot be neglected. Numerical results show that the proposed analysis is in agreement with the measurement results on the literature under realistic working conditions.

Among the possible medium access control (MAC) techniques for wireless communication systems, the simplicity of random access schemes (

The first papers in the literature about the capture effect mostly consider Frequency Modulation (FM) demodulators [

Although the experimental results in [

The impact of interference in wireless sensor networks plays a very important role and can severely degrade the overall performance of the network and the efficiency of the upper layers. In our opinion, this aspect has not been sufficiently addressed in past years. In a dense sensor network deployment, where many nodes are periodically sending data to the sink, concurrent transmissions are highly probable. However, the probability of having a collision because of more than two concurrent transmissions is relatively low [

On the other hand, we believe this study can also contribute to the identification of signal reception models for network simulators. In particular, as shown in

In SNRT-based models, the packet is correctly received if the signal-to-noise ratio (SNR) is larger than a given threshold, whereas, in BER-based models, the packet reception decision is based on the BER, which is determined probabilistically depending on the value of the SNR. These models are rather simple, but have some drawbacks. In particular, SNR-based models do not take into account the impact of interference. This latter effect can be considered, in principle, by BER-based models, but the impact of the waveform of the interferer signals should be carefully considered, as it plays a significant role. Typically, conventional interference models are based on the assumption that the disturbance can be modeled as a Gaussian random variable; unfortunately, this is not the case of IEEE 802.15.4 systems, where only a limited number of strong interferers is present. To counteract this problem, we mathematically analyze the impact of the waveform of the interferer on packet reception and obtain curves that are organized as specific look-up tables. Figures, such as those derived in

The computational complexity of the model for the coherent detection is O(1) (in big O notation). This makes it usable without intensive computational effort. For the non-coherent case, we show that the performance curve has a step-like behavior with the threshold at 0 dB. This simple model can capture the behavior of the non-coherent case without any computational effort.

The rest of the paper is organized as follows: Section 2 describes CSMA-CA and the 2.4 GHz PHY of the IEEE 802.15.4 Standard. Sections 3 and 4 evaluate CER for coherent and non-coherent offset quadrature phase shift keying (O-QPSK), respectively. Section 5, obtains PRR in concurrent transmission by finding an upper bound, which transforms CER to data symbol error rate (DSER).

Two different approaches are foreseen in IEEE 802.15.4 to coordinate the data traffic: beacon-enabled mode and non-beacon-enabled mode [

IEEE 802.15.4 PHY uses direct sequence spread spectrum (DSSS), and it operates over different ISM bands: the 868 MHz (EU) ISM band, the 915 MHz (US) ISM band and the 2.4 GHz global ISM band. Because of the global availability of the higher frequency band, currently transceiver manufacturers mostly have products working in this band (_{c}_{C}

For each chip sequence, even-indexed chips are modulated to the in-phase (I) carrier, and odd-indexed chips are modulated to the quadrature-phase (Q) carrier of O-QPSK. The offset between the I-phase and Q-phase is formed by delaying the Q-phase symbols one chip duration with respect to the I-phase.

In this section, we consider coherent communication and evaluate the chip error probability when there is one interferer with an unknown carrier phase and symbol timing. As the experiments in [_{c}_{c}_{C}_{I}_{C}_{I}_{C}_{i}

In the case of coherent demodulation, the I and Q components should be sampled at the instance corresponding to the maximum signal amplitude; hence for a sequence of chips, the I-phase should be sampled at the instants _{In}_{c}_{Qn}_{c} (see _{c}_{d}

When we take the case of no pulse shaping into account, the shaping function,

Accordingly, the complex envelope of the received signal in _{d}_{I}_{I}_{d}_{d}

With respect to the transmitted symbol, the first addend in _{d}_{I}_{d}_{d}_{c}_{s}

When

When
_{0} is the arc of the circle shown in _{0}/2, and

The sampling point of the received signal can only pass to the adjacent quadrants, and the receiver makes a one chip error per each transmitted symbol; as a result, _{c}_{s}

For increasing values of

When the received signal is in _{2}, the receiver makes two chip errors per each transmitted symbol. On the other hand, when it is in _{0} or in _{1}, the receiver makes one chip error per each transmitted symbol. Regarding the number of chip errors in the different quadrants, _{s}_{c}

The computational complexity of _{s}_{c}

Using the pulse shaping given by _{d}_{I}_{d}

When there is perfect knowledge of the useful carrier phase and symbol timing, there is no difference, from the demodulator point of view, between half-sine-shaped or not shaped useful symbols, since the sampling is done at the maximum amplitude instance. As a consequence, the first addends in

The random symbol sampling instance, _{I}_{I}_{X}_{Y}

The solution of the convolution integral in _{3} or _{0}, the slope of the diagonal line is +1; on the other hand, if the interferer is transmitting _{1} or _{2} then the slope is − 1. The probability distribution of the positions of the interferer sampling point on the dashed lines are symmetric on the opposite sides of the origin, because of the cosine function. The amplitude in a quadrant is:

Using _{i}

Without loss of generality, as we did in Section 3.1, it can be assumed that the transmitter is transmitting m_{3}. Therefore, to obtain _{c}_{s}

When
_{c}_{s}

When _{c}_{2} or _{1} will result in one chip error per symbol, while a transition towards _{0} will result in two chip errors per symbol. Therefore, the error rates when

In general, _{s}_{c}_{s}_{c}_{i}_{i}

The computational complexity of _{s}_{c}

The validity of our analytical framework has been tested using Monte-Carlo simulations. The simulation tool has been implemented in MATLAB (The MathWorks Inc., Natick, MA, USA) using the complex envelopes of the signals. Asynchronous symbols are obtained by shifting the complex envelope in the time axis, while the random carrier phase is obtained through rotating the complex envelope on the complex plane. SIR values with a step of 0.3 dB from −10 dB to 5 dB are simulated. At each specific SIR value, 10, 000 random O-QPSK symbols are generated for both the interferer and useful transmitter. The complex baseband representation of the O-QPSK symbol is generated by using 100 points; therefore, the resolution of the time for the asynchronous O-QPSK symbols was 10 ns. On the other hand, the resolution for the carrier phase was 2_{c}

Normally, O-QPSK requires coherent detection [_{c}

Chip intervals in _{C}_{1},_{C}_{2}, …. In each even chip interval, _{C}_{2},_{C}_{4},…, a symbol (_{0}, _{1}, _{2}, _{3}) is transmitted, and in the odd chip intervals, _{C}_{3},_{C5}_{4}, _{5}, _{6}, _{7}), which can be considered as the transitions between two sequential O-QPSK symbols. For instance, in _{C}_{2} chip interval (_{C}_{C}_{3}. In the next chip interval, _{C}_{3}, the phase reaches the _{C}_{5}. The sequential chip intervals send the phase of the complex envelope, _{c}_{C}_{R}_{R}

When useful and interfering signals are synchronous, the received signal can be written as:
_{C}_{I}_{C}_{I}_{C}, b_{I}_{eq}_{C}_{I}_{dif}_{C}_{I}_{c}

Note that if the phase of the interferer signal turns toward the direction of the useful signal, the received signal in the Q-I plane will turn toward the direction of the useful signal, as well. As a consequence, _{eq}_{dif}_{C}_{I}_{C}_{I}_{C}_{I}_{R}

With no loss of generality, we can assume _{c}_{R}_{R}_{I}

It can be showed that _{dif}

The condition in which interfering symbols can be asynchronous with respect to useful symbols increases the number of possible cases to be considered for the evaluation of Θ_{R}_{t}_{t}_{t}_{C}_{I1} and _{I2} are the turning directions of the sequential chips of the interferer. In _{t}_{mix}_{eq}, P_{dif}_{mix}_{C}_{I1} and _{I2} are shown in

By considering these occurrences, the chip error probability can be evaluated as:

To obtain _{mix}_{C}_{I}_{1} = +1, _{I}_{2} = −1, and _{I}_{t}_{mix}

_{mix}

Simulation results for the cases of synchronous and asynchronous useful and interfering symbols are shown in

The received path of the commercial 802.15.4 compatible transceivers, such as TICC2420 [_{c}_{d}_{r}

As discussed previously, in the 2.4 GHz PHY of IEEE 802.15.4, data symbols are mapped to the chips after the demodulation. Because of the symmetry in the data symbol set used by the 802.15.4 2.4 GHz PHY layer, the error probability of the data symbol, _{0}, is equal to the symbol error probability, _{d}_{d}_{0}) expressed as:
_{i}_{0}) represents the probability of deciding for the symbol, _{i}_{0} has been transmitted. Each probability in the summation of _{i}_{0} and _{i}_{c}

By substituting

In the case of coherent detection, we can calculate the Hamming distance of each chip sequence. In particular, the recurrences of the Hamming distances of the data symbols is given in

Using these recurrences, the symbol error probability can be bounded as follows:
_{i}, P_{c}_{i}_{i}

Note that, for a fixed value of _{c}

In the non-coherent case, we can only identify the phase transitions during the chip intervals. For a 32-chip long sequence, there will be 31 phase transitions, as shown in

Where + means a

The union upper bound is obtained as:
_{i}_{i}

Again, the computational complexity of the upper bound is

In 802.15.4 2.4 GHz PHY, four data bits are mapped to a data symbol [_{r}

It is worth noting that the upper bounds on _{d}_{r}

Experimental results shown in

The effect of concurrent transmission on the performance, in terms of chip, symbol and packet error probability, of IEEE 802.15.4 systems was investigated. To this aim, an analytical model, which starts from the PHY characterization of the IEEE 802.15.4 standard, was proposed, and the results were validated using simulations. Since the system performance does not depend on the number of interfering devices, but rather, on the overall amount of interfering power [

The authors declare no conflicts of interest.

The impact of the interferer. (_{d}_{d}_{d}

Coherent Cases. (

The impact of of the interferer: the asynchronous case. (_{d}_{d}_{d}

Chip error rate: the case of coherent O-QPSK demodulation. CER, chip error rate; SIR, signal-to-interference ratio.

Phase transitions of the complex envelope. (

Non-coherent chip error rate.

Packet reception rate (PRR).

Signal reception models in network simulators [

GloMoSim | ns-2 | OPNET | |
---|---|---|---|

SNRT-based, BER-based | SNRT-based | BER-based |

Useful and Interferer Chip Phase Combinations.

_{C} |
_{I}_{1} |
_{I}_{2} |
||
---|---|---|---|---|

+1 | +1 | +1 | _{I} |
_{eq} |

+1 | +1 | −1 | _{I}_{t} |
_{mix} |

+1 | −1 | +1 | _{I}_{t} |
_{mix} |

+1 | −1 | −1 | _{I} |
_{dif} |

−1 | +1 | +1 | _{I} |
_{dif} |

−1 | +1 | −1 | _{I}_{t} |
_{mix} |

−1 | −1 | +1 | _{I}_{t} |
_{mix} |

−1 | −1 | −1 | _{I} |
_{eq} |

Recurrences of unique Hamming distance values.

_{I} |
12 | 14 | 16 | 18 | 20 |

Reoccurrence (_{I} |
2 | 2 | 3 | 2 | 6 |

Phase transitions of chip sequences in 2450-MHz PHY of IEEE 802.15.4.

_{0} |
0000 | ++------+++-++++-+-+++--++-++-- |

_{1} |
1000 | +--+++------+++-++++-+-+++--++- |

_{2} |
0100 | ++-++--+++------+++-++++-+-+++- |

_{3} |
1100 | ++--++-++--+++------+++-++++-+- |

_{4} |
0010 | -+-+++--++-++--+++------+++-+++ |

_{5} |
1010 | ++++-+-+++--++-++--+++------+++ |

_{6} |
0110 | +++-++++-+-+++--++-++--+++----- |

_{7} |
1110 | ----+++-++++-+-+++--++-++--+++- |

_{8} |
0001 | --++++++---+----+-+---++--+--++ |

_{9} |
1001 | -++---++++++---+----+-+---++--+ |

_{10} |
0101 | --+--++---++++++---+----+-+---+ |

_{11} |
1101 | --++--+--++---++++++---+----+-+ |

_{12} |
0011 | +-+---++--+--++---++++++---+--- |

_{13} |
1011 | ----+-+---++--+--++---++++++--- |

_{14} |
0111 | ---+----+-+---++--+--++---+++++ |

_{15} |
1111 | ++++---+----+-+---++--+--++---+ |

Recurrences of unique Hamming distance values.

_{i} |
13 | 14 | 15 | 16 | 17 | 18 | 31 |

_{i} |
2 | 2 | 3 | 3 | 2 | 2 | 1 |