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Hence, with a mechanism that allows each node to determine the angles sustained between reference nodes A and B, and B and C, resepctively, the network can selflocalize without the need for collaboration or centralized computations. A method for doing this was proposed in reference 20 and a prototype implementation of the scheme using the Crossbow mica2 wireless sensor nodes was presented in reference 18. The principle of operation of this AOA-based localization scheme is presented next. 12.5.1 AOA Based Localization Using Rotating Directional Beacons The principle of operation is somewhat similar to VOR stations that are used in aircraft navigation. The system uses three specially constructed beacon generators to facilitate angle determination at the sensor nodes. Each beacon generator transmits a directional beacon signal that rotates with a xed angular velocity. The sensor nodes calculate the angles and by determining the times of arrival of the directional beams from different beacon generators. It is assumed that the directional beacon signals have very narrow beamwidths on the horizontal plane so as to enable accurate estimation of times when the beams are pointed toward a sensor. They should have suf ciently large vertical beam angles, so that the beams cover the entire sensor network area; that is, they cover sensors located at different vertical angles with respect to the transmitter. The beacon signal can be RF or optical, as long as it is possible to transmit it at a narrow beam. A schematic illustration of the proposed implementation is depicted in Figure 12.7a, which assumes the special case where the beacon generators (reference points) are located at the vertices of a rectangle.

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Figure 12.4. Outage example for Toronto and Phoenix, 1-W SMAP loading. FIB: full initial battery charge.

Figure 12.7. Illustration of (a) the system model for AOA-based localization and (b) the times of arrival of beacon signals at a sensor node.

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Commonly referred to as the purchase and sale of the same or similar instruments in different months simultaneously, spreads have their own trading patterns and conventions and require experience to become familiar with them. Some New York market traders refer to spreads as a switch or straddle order. The concept of a spread is to buy and sell the same or related product, different products, different months, different exchanges, or some combination of all of these. The goal is to profit from the difference in the price movements of the two sides. The two sides of a spread order, the buy side and the sell side, are usually called legs. You can enter a spread one side at a time, called legging into a spread; you can enter both positions with market orders; or you can enter a limit order specifying the exact price difference between the two sides. Common examples of spreads include the S&P 500 and Nasdaq, gold and silver, Chicago wheat versus Kansas City wheat, July soybeans versus November soybeans (new crop/old crop spread) and cattle versus hogs. The list goes on and on. Exchanges recognize a number of spreads, generally offering reduced margin requirements. For example, the historic Chicago Board of TradeChicago Mercantile Exchange common clearing link that became fully operational on January 2, 2004, cleared the way for substantially lower margin requirements and more effective clearing procedures for spread trades involving e-mini S&P and mini-sized Dow futures, an outstanding

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The main considerations required for successful implementation of this scheme are as follows: r Identical Angular Velocities. All three beacon signals must be rotating with identical angular velocities. However, any arbitrary angular velocity may be used, which can be measured at the sensor nodes by noting the time period of rotations. r Beacon Identi cation. The sensor nodes should be capable of identifying the beacons from their signals. This can be implemented by various mechanisms, such as using coding or modulation of the beacon signals. r Phase Reference. The phase differences between the rotating beacon signals ( 1 between A and B and 2 between B and C in Figure 12.7) must be known at the sensors. To remove this problem, 1 , 2 , and 3 may all be made zero by appropriate initial con gurations. Each sensor will then receive the beacon signals periodically as shown in Figure 12.7b. The required angle estimates and can then be obtained by measuring the times t1 , t2 , and t3 as follows: = (t2 t1 ) 1 = (t3 t2 ) 2 (12.7)

Possible Sources of Error. The primary source of error in the above technique is due to inaccurate angle estimation that can be caused by nonzero beamwidth of the beacon signals. It is dif cult to get very narrow beams of RF signals unless large antenna arrays are used at the beacon generators. To avoid such expensive infrastructure in case RF signals are used, the sensor nodes may be allowed to detect the center of an incoming RF beam by detecting the time at which the received signal

TABLE 12.1. Cost (Normalized) for Different Outage Probabilities and Load Powers for City of Toronto POut 10 10 2 10 3 10 4

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