Rayleigh distribution

This is the 2.5 dB factor that we accounted for in the previous preamplifier
discussion, whenever the noise power out of the preamplifier was
approximately equal to or greater than the analyzer's own noise.

Figure 5-7. The envelope of band-limited Gaussian noise has a Rayleigh distribution

Another factor that affects noise measurements is the bandwidth in which
the measurement is made. We have seen how changing resolution bandwidth
affects the displayed level of the analyzer's internally generated noise.
Bandwidth affects external noise signals in the same way. To compare
measurements made on different analyzers, we must know the bandwidths
used in each case.

Not only does the 3 dB (or 6 dB) bandwidth of the analyzer affect the
measured noise level, the shape of the resolution filter also plays a role.
To make comparisons possible, we define a standard noise-power bandwidth:
the width of a rectangular filter that passes the same noise power as our
analyzer's filter. For the near-Gaussian filters in Agilent analyzers, the
equivalent noise-power bandwidth is about 1.05 to 1.13 times the 3 dB
bandwidth, depending on bandwidth selectivity. For example, a 10 kHz
resolution bandwidth filter has a noise-power bandwidth in the range of
10.5 to 11.3 kHz.

If we use 10 log (BW 2 / BW 1) to adjust the displayed noise level to what we
would have measured in a noise-power bandwidth of the same numeric value
as our 3 dB bandwidth, we find that the adjustment varies from:

10 log (10,000/ 10,500) = 0.21 dB
to
10 log (10,000/ 11,300) = 0.53 dB

In other words, if we subtract something between 0.21 and 0.53 dB from the
indicated noise level, we shall have the noise level in a noise-power bandwidth
that is convenient for computations. For the following examples below, we
will use 0.5 dB as a reasonable compromise for the bandwidth correction 9 .

9. ESA Series analyzers calibrate each RBW during
the IF alignment routine to determine the noise
power bandwidth. The PSA Series analyzers
specify noise power bandwidth accuracy to within
1% ( Â± 0.044 dB).