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The instantaneous noise power as a function of time Pn(t) is given by:
Pn(t) = [Vn(t)]^2/Ri
where Vn(t) is the instantaneous noise voltage and Ri is the circuit impedance.

The noise voltage Vn(t) can be broken into its orthogonal Fourier components V1(t)=V1sin(wt), V2(t)=V2sin(2wt), ....

When the average value of noise power Pn(t) is evaluated, all the cross terms cancel giving the average noise power as:
Pn = (1/2Ri)[(V1)^2 + (V2)^2 + (V3)^2 + ....]
Note that V1,V2,... are peak, not rms, voltages and the sampling frequency interval is uniform.

Hence, within a filter bandpass the noise power is additive. Thus if P1, P2, P3 are the powers of various mathematically orthogonal noise signals, then average noise power Pn is given by:
Pn = P1 + P2 + P3 + ....

Assume that on the source side of a filter the noise power is spectrally uniform.

If the voltages V1, V2, V3 ... are expressed in terms of the frequency dependent filter transfer function T(F):
V1 = Vo T(F1), V2 = Vo T(F2), ...
then the noise power output from the filter is given by:
Pn = (Vo^2/2Ri){[T(F1)]^2 + [T(F2)]^2 + [T(F3)]^2 + ....}

Let dF = F2-F1 = F3-F2 = F4-F3 = .....
as required for validity of the Fourier analysis.

dPn/dF = (Vo^2/2Ri)[T(F)]^2

Note that (Vo^2/2Ri)[T(F)]^2 is the noise power contained between frequencies F and F+dF.

For the purposes of this analysis assume that near 127 kHz the noise power on the electrical bus is uniform with frequency, so that T(F) is purely the result of an input filter to the noise voltage sensing device. This assumption may not be perfectly true, but this assumption is necessary to achieve a practical solution.

The total sensed noise power is given by:
Pn =Integral{(Vo^2/2Ri)[T(F)]^2}dF

Recall that:
Pn(t) = ([Vn(t)]^2)/Ri

Equating the two expressions for Pn gives the sensed RMS noise voltage Vn as:
[Vn]^2 = Integral{(Vo^2/2)[T(F)]^2}dF

Recall that Vo is a peak voltage. For ease of understanding it is convenient to let:
Vo^2/2 = Vb^2
where Vb is a pseudo RMS voltage that is indicative of the amplitude of the broad band noise. Note that (Vb)^2 has units of volts^2 / kHz.
[Vn]^2 =(Vb)^2Integral{[T(F)]^2}dF

This equation allows a filter network with a measured transfer ratio Tm(F) to be used to find (Vb)^2 from a measurement of RMS noise voltage Vnm at the filter output.

If T(F) is replaced by the filter function Tr(F) for a Systel PLC receiver, then the noise voltage seen by the Systel receiver Vnr can be obtained from the previously measured value of (Vb)^2 using the equation:
[Vnr]^2 = (Vb)^2Integral{[Tr(F)]^2}dF
where Tr(F) is the transfer function of the PLC receiver input circuit relative to a monofrequency interfering signal of known amplitude that causes a known Bit Error Rate (BER). Data for Tr(F) can be obtained by varing the frequency and amplitude of the interfering signal while maintaining the same BER.

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This web page last updated September 20, 2005

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