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If the a-frame is selected to be the inertial frame, eqn (1124) reduces to = f i + Gi (r) (1127)

to learn about amplifiers and oscillators that use these devices As a prelude to that, let s define the most common unit that engineers use to evaluate changes in signal strength

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because ii = 0 Eqn (1127) is the same as eqn (115) With knowledge of ri (0), vi (0), and Gi (r), the speci c force vector f i could be integrated to determine ri (t) and vi (t) For strapdown inertial systems, the speci c force vector would be measured in body frame and transformed into the i-frame This requires that the INS maintain the rotational transformation Ri based on integration of b the gyro measurements Then eqn (1127) can be computed as d2 ri dt2 = Ri f b + Gi (r) b

Choosing the direction cosine attitude representation, the inertial frame kinematic equations are ri vi Ri

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The human senses of hearing and vision don t perceive the intensity of sound and light in a linear manner Instead, our perception of these phenomena is logarithmic Electronic circuits behave in a similar way when subjected to signals that change in amplitude Because of this, scientists and engineers invented a unit called the decibel (symbolized dB), in which amplitude changes are expressed according to the base-10 logarithm of the actual change

= = =

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More or Less In the decibel scheme, increases in amplitude are assigned positive gain values, and decreases in amplitude are assigned negative gain values A negative gain figure is sometimes expressed as a loss by removing the minus sign If the output signal amplitude from a circuit is +6 dB relative to the input signal amplitude, then the output is stronger than the input If the output is at 14 dB relative to the input, then the output is weaker than the input In the first case, the circuit has a gain of +6 dB In the second case, we can say that the circuit has a gain of 14 dB or a loss of 14 dB How large a unit is the decibel, anyway The answer is that it has no absolute size; it is a relative unit The decibel has meaning only when two or more signals are compared An amplitude increase or decrease of plus or minus one decibel ( 1 dB) is roughly the smallest change a listener can detect if the change is expected If the change is not expected, the smallest difference a listener can notice is about 3 dB

(1128) (1129) (1130)

Let the a-frame be selected to represent the ECEF frame of reference and let the i-frame and e-frame origins be coincident In this case, eqn (1125) reduces to d2 re dt2 = Re f b + Ge (r) 2 e ve e e re ie ie ie b

For Voltage Consider a circuit with an rms ac input voltage of Ein and an rms ac output voltage of Eout, specified in the same units as Ein Then the voltage gain of the circuit, in decibels, is given by this formula:

where we have used the fact that ie can be considered to be zero for the e purpose of navigation applications Again, choosing the direction cosine attitude representation, the e-frame kinematic equations are re ve e R

= = =

Gain (dB) = 20 log(Eout /Ein ) The base-10 logarithm of a value x is written log x Log functions of base 10 and base e (another type of logarithm sometimes used in physics and engineering) can be determined easily using scientific calculators From now on, when we say logarithm, we mean the base-10 logarithm unless otherwise specified

(1131) (1132) (1133)

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