# Send-On-Delta Sensor Data Transmission With A Linear Predictor

## Abstract

**:**

## 1. Introduction

## 2. Send-on-delta transmission with a linear predictor

_{last}(t) is larger than specified δ. In this method, the error (x(t) − x

_{last}(t)) in the monitoring station (where sensor data are received) is always smaller than δ assuming there is no delay and packet loss during transmission.

#### 2.1. New send-on-delta algorithm with a linear predictor

_{k}is defined by x

_{k}= x(kT). Even if the sensor value is sampled with the period T, the actual transmission rate is not T since not all sampled sensor data are transmitted.

_{k}= f(x̂

_{k}

_{−1}, ⋯, x̂

_{k}

_{−}

_{M}

_{−1}) indicates a linear predictor, where M denotes the memory length. If M = 1, x̂

_{k}is computed based on x̂

_{k}

_{−1}and x̂

_{k}

_{−2}. How to choose a linear predictor f (·) is discussed later. Note that we use f (x̂

_{k}

_{−1}, ⋯, x̂

_{k}

_{−}

_{M}

_{−1}) instead of f (x

_{k}

_{−1}, ⋯, x

_{k}

_{−}

_{M}

_{−1}). If we used f (x

_{k}

_{−1}, ⋯, x

_{k}

_{−}

_{M}

_{−1}), we would have obtained more accurate x̂

_{k}; however in that case we have to transmit x

_{k}to the monitoring station at every T seconds. Thus we cannot reduce the number of transmission.

_{k}and the predicted value x̂

_{k}is larger than δ, we transmit x

_{k}, ⋯, x

_{k}

_{−}

_{M}instead of just transmitting x

_{k}. Why we have to transmit x

_{k}, ⋯, x

_{k}

_{−}

_{M}is discussed in Section 4. In the conventional send-on-delta method, we only need to transmit x

_{k}. In this sense, the transmission data size is larger than that of the conventional send-on-delta method. Despite this, we will see later in Section 4 that the overall network overhead of the proposed method is significantly smaller than the conventional send-on-delta method.

_{k}= f(·). From the transmitter algorithm, it is guaranteed that the error between x

_{k}and x̂

_{k}is smaller than δ.

#### 2.2. Linear predictor

_{k}= x(kT); this corresponds to M = 1 in the algorithm of Fig. 3 and Fig. 4. Similarly, we can obtain a second order predictor for x

_{k}= x(kT).

## 3. Mean rate of messages analysis

**Theorem 1**(Mean-Value Theorem) Assume that x has a derivative at each point of an open interval (a, b), and assume also that x is a continuous function at both endpoints a and b. Then there is a point c in (a, b) such that

_{k}(kT < α

_{k}< (k + 1)T) satisfying

_{k}is not unique; in that case, we can choose any α

_{k}.

_{k}and x

_{k}

_{−1}are transmitted for the first order predictor. Thus it is satisfied that x̂

_{k}= x

_{k}and x̂

_{k}

_{−1}= x

_{k}

_{−1}.

_{k}

_{+}

_{N}− x̂

_{k}

_{+}

_{N}|} so that we can see how often |x

_{k}

_{+}

_{N}− x̂

_{k}

_{+}

_{N}| exceeds δ. From the definition of α

_{k}in (4), we have

_{k}= x

_{k}and x̂

_{k}

_{−1}= x

_{k}

_{−1}. If we proceed to the next time step, we obtain

_{k}− α

_{k}

_{−1}is T, which can be verified from the following relationship:

_{max}be the maximum value satisfying (T is given)

## 4. Simulation

_{1}, x

_{2}, and x

_{3}are all 5 second. For the oscillating signal x

_{4}, 5 second corresponds to a quarter cycle.

_{1}, x

_{2}, and x

_{4}, the numbers of transmission are the same. There is just one transmission reduction for x

_{3}. Thus we believe that the first-order predictor is good enough for most signals.

_{1}plots are given in Fig. 6. The left plot is the result of the send-on-delta method and the right plot is the result of the proposed method.

_{k}and x

_{k}

_{−1}, while only x

_{k}is transmitted in the periodic sampling and the send-on-delta method. The impact of this increase on the overall network transmission is small since a packet-based transmission is used in most networks. For example, in CAN 2.0A network [14], the packet overhead is at least 65 bits and in ZigBee [15], it is at least 120 bits.

_{k}is encoded in 8 bits, total numbers of transmitted bits can be computed as in Table 5.

_{k}and x

_{k}

_{−1}should be transmitted.

_{k}instead of transmitting x

_{k}and x

_{k}

_{−1}in the proposed algorithm. In most cases, a predictor output would be oscillating and thus the number of transmission becomes very large. An example is given in Fig. 8. We can see the oscillation phenomenon and the number of transmission is 250 while only 11 for the proposed method.

## 5. Conclusion

## Acknowledgments

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algorithm | λ |
---|---|

periodic sampling [8] | $\lambda =\frac{max\dot{x}(t)}{\delta}$ |

conventional send-on-delta method [8] | $\frac{\stackrel{-}{\dot{x}}}{\delta}-2\nu \le \lambda \le \frac{\stackrel{-}{\dot{x}}}{\delta}$ |

proposed method | $\lambda \le \frac{1}{{N}_{max}T}$ |

type of signal | signal |
---|---|

step response of the first-order system | x_{1}(t) = 1 − e^{−}^{t} |

step response of the second-order underdamped system | x_{2}(t) = 1 − (1 + t) e^{t} |

step response of the second-order underdamped system | ${x}_{3}\left(t\right)=1+\frac{{T}_{1}}{{T}_{2}-{T}_{1}}{e}^{-t/{T}_{1}}+\frac{{T}_{2}}{{T}_{2}-{T}_{1}}{e}^{-t/{T}_{2}}$ |

step response of the second-order undamped system | x_{4}(t) = 1 − cos(ω_{n}t) |

_{1}= 1, T

_{2}= 5/7, ω

_{n}= π/10)

signal | periodic sampling λ | send-on-delta method upper bound of λ | proposed method (1st order) upper bound of λ |
---|---|---|---|

x_{1}(t) | 50.00 | 9.96 | 2.22 |

x_{2}(t) | 18.39 | 9.57 | 1.88 |

x_{3}(t) | 350.00 | 59.97 | 5.88 |

x_{4}(t) | 15.70 | 9.99 | 1.25 |

signal | periodic sampling | send-on-delta method | proposed method (1st order) | proposed method (2nd order) |
---|---|---|---|---|

x_{1}(t) | 250 | 44 | 11 | 11 |

x_{2}(t) | 919 | 46 | 9 | 9 |

x_{3}(t) | 1750 | 175 | 25 | 24 |

x_{4}(t) | 78 | 48 | 7 | 7 |

send-on-delta method | proposed method | |
---|---|---|

CAN 2.0A | 65 + 8 × # | 65 + 2 × 8 × # |

ZigBee | 120 + 8 × # | 120 + 2 × 8 × # |

CAN 2.0A | ZigBee | |||
---|---|---|---|---|

signal | send-on-delta | proposed | send-on-delta | proposed |

x_{1} | 417 | 241 | 472 | 296 |

x_{2} | 433 | 209 | 488 | 264 |

x_{3} | 1465 | 465 | 1520 | 520 |

x_{4} | 449 | 177 | 504 | 232 |

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**MDPI and ACS Style**

Suh, Y.S. Send-On-Delta Sensor Data Transmission With A Linear Predictor. *Sensors* **2007**, *7*, 537-547.
https://doi.org/10.3390/s7040437

**AMA Style**

Suh YS. Send-On-Delta Sensor Data Transmission With A Linear Predictor. *Sensors*. 2007; 7(4):537-547.
https://doi.org/10.3390/s7040437

**Chicago/Turabian Style**

Suh, Young Soo. 2007. "Send-On-Delta Sensor Data Transmission With A Linear Predictor" *Sensors* 7, no. 4: 537-547.
https://doi.org/10.3390/s7040437