Averaged correlation spectra for sideband noise measurements with Linrad.
(Dec 17 2010)


Correlation is the standard method for extending the measurement range in signal analyzers.

The fourier transforms that Linrad produces when two RF signals are received are normally used to produce power spectra that can be averaged over a large number of transforms. With Linrad-03.19 there is an option to also show the correlation spectrum with the same averaging. (Part of this was already present in Linrad-03.18)

For each FFT bin we have two complex amplitudes that can be denoted Re_1 and Im_1 for channel 1 and Re_2 and Im_2 for channel 2.

The complex product is a complex number with a magnitude and a phase.

Re_C = Re_1*Re_2 + Im_1*Im_2
Im_C =-Re_1*Im_2 + Re_2*Im_1

When the two signals are completely independent the phase angle will vary at random and when accumulating Re_C and Im_C separately the magnitude of the sum will grow as the square root of the number of summed values.

When the two signals are equal it is different. Then the phase is always the same and the magnitude will grow linearly with the number of summed values.

The average of random signals will become reduced by 3 dB when the number of averaged spectra is increased by a factor of 4. 10000 averages would give a 20 dB improvement.

Data files.

A typical test setup is shown in figure 1.

Fig 1. Test arrangement used to generate the wideband recordings presented on this page.

Figure 1 is by SM0ERR, Mart, who used it to produce a couple of .raw files to be used for testing the correlation function in Linrad.

The test object is a HP8644 signal generator set to 9520 kHz. According to its specifications the sideband noise in the range 8 to 16 MHz should be like this:

Separation       Sideband noise
  (kHz)            (dBc/Hz)
    1                -130
   20                -145
  100                -145

Mart made several recordings with the setup of figure 1. Three of them are available here. They might be useful as input data for a program that could separate the noise modulation into phase noise and amplitude noise.

No modulation into the HP 8644: nomod-err.raw (103741641 bytes)
The HP8644 is AM modulated (6%) am-err.raw (103723209 bytes)
The HP8644 is FM modulated (300Hz swing) fm-err.raw (103704777 bytes)
Figure 2 shows the spectrum obtained from nomod-err without averaging. The correlation spectrum is enabled but the number of averages is set to 1.

Fig 2. The file nomod-err.raw played without averaging. The bin bandwidth is 1 kHz.

By use of the S-meter at 100 Hz bandwidth (and with appropriate window parameters for the first FFT) the following levels were measured with nomod-err as input and with polarisation set to 45 degrees giving the summed noise of the two generators. The S-meter reading is the averaged number in the coherent graph that appears when F2 is depressed. By setting the polarization angle to 0 degrees and reading the two curves from the two generators one gets readings with 3 dB more sideband noise. Those values represent the performance of the SMT03 generators.

Separation   Level in 100 Hz     Level
   (kHz)        (dB)            (dBc/Hz)
     0          143.1              0
     1           72.1            -91.0
     2           68.2            -94.9
     5           53.6           -109.5
    10           43.8           -119.3
    20           36.9           -126.2
    50           29.2           -133.9    (ev -3dB=>130.9) 

The uncorrelated spectrum shows essentially the average noise from the two R&S SMT03 generators with a -3 dB shift. Specifications say that the sideband noise at 20 kHz should be -120 dBc/Hz (or better.) The measurement shows they are about -123 dBc/Hz. The two generators differ by less than 0.5 dB at all frequency separations.

Measurements with correlation.

Figure 3 shows the nomod-err file with averaging 4 times. The yellow curve shows the correlated spectrum. As expected it is about 3 dB lower than the white curve which is the average power spectrum, the same as in figure 1.


Fig 3. The file nomod-err.raw played with averaging 4 times. The bin bandwidth is 1 kHz.

Figures 4 to 7 show averaging 16, 64, 256 and 1024 times with an expected improvement of 3 dB for each factor 4 in averaging.

Fig 4. The file nomod-err.raw played with averaging 16 times. The bin bandwidth is 1 kHz.

Fig 5. The file nomod-err.raw played with averaging 64 times. The bin bandwidth is 1 kHz.

Fig 6. The file nomod-err.raw played with averaging 256 times. The bin bandwidth is 1 kHz.

Fig 7. The file nomod-err.raw played with averaging 1024 times. The bin bandwidth is 1 kHz.

Figures 3 to 7 compute the true average of the specified number of transforms. This is rather cpu-intensive and the increasing work load is indicated by the cpu load on the screen thread which is 45% in figure 7. Linrad has an option to compute the average of N spectra first and then to compute the average of such averages. That allows a significant reduction of the cpu load. Figure 8 shows the averaging of 1024 spectra as the average of 128 averages of 8 spectra each. The result should be identical to that of figure 7, but the cpu load is much smaller. The CPU load is reduced from 45.23 % to 1.85 % for the screen thread. where 1024 averages is reduced to 128 averages. The CPU load in the wideband thread is increased from 6.50 % to 11.25 % because this thread has to form averages in groups of 8 spectra. Figures 9 to 11 with 4096 to 65536 averages would not have been possible without pre-averaging. Surely a mode with continuous averaging until a stop button would be pressed would be very CPU efficient and allow an unlimited number of averages. It would be easy to add, but it would increase the complexity of the user interface a little. A bad idea if nobody needs it...

Fig 8. The file nomod-err.raw played with averaging 1024 times in groups of 8 averages. The bin bandwidth is 1 kHz.

Figures 9 to 11 show 4096, 16384 and 65536 averages respectively. The noise from the SMT03 generators should be suppressed by 18, 21 and 24 dB in respective images. The yellow curve in figure 11 is however not more than about 18 dB below the white curve at the best point and therefore the yellow curve represents the noise of the device under test or possibly the noise of the LO used in the RX2500. There could also be a contribution from the Delta44 where some noise source could be present in both channels.

Fig 9. The file nomod-err.raw played with averaging 4096 times in groups of 8 averages. The bin bandwidth is 1 kHz.

Fig 10. The file nomod-err.raw played with averaging 16384 times in groups of 8 averages. The bin bandwidth is 1 kHz.

Fig 11. The file nomod-err.raw played with averaging 65536 times in groups of 8 averages. The bin bandwidth is 1 kHz.

The white curve was measured with the Linrad S-meter. See table above. The correlated measurement of the HP 8644 is obtained by subtraction of the difference between the white and the yellow curves.
Separation    Uncorrelated  Diff  Correlated 65536
   (kHz)       (dBc/Hz)     (dB)    (dBc/(Hz)
     0           0            0         0
     1          -91.0        18       -109
     2          -94.9        18       -113
     5         -109.5        18       -128
    10         -119.3        18       -137
    20         -126.2        19       -145
    50         -133.9        15       -148

The result is in good agreement with the specifications for the HP 8644 at 20 kHz, but at 1 kHz the discrepancy is 11 dB. The discrepancy can not be caused by the RX2500/Delta44 because the noise floor of the system at 1 kHz is below -140 dBz/Hz.

Fig 12. The file am-err.raw played with averaging 65536 times in groups of 8 averages. The bin bandwidth is 1 kHz.

Fig 13. The file fm-err.raw played with averaging 65536 times in groups of 8 averages. The bin bandwidth is 1 kHz.

AM, FM and composite noise.

The figures above show the composite noise, the total power emitted. In many applications one is interested in the phase noise only. Then the signal is typically a square wave with very fast rise and fall times and phase jitter on the transitions is of interest. Some noise on the amplitude would not affect the system so it is of less interest.

In amateur radio where spectral purity, the ability to produce a signal that will not cause interference to others, the composite noise is the adequate quantity to measure.

It would be possible to separate the sideband noise into an AM component and a FM component. One way would be to multiply the time function from each channel with a suitable time function that would bring the frequency of the dominating signal to zero with a phase that would cause all of its energy to appear in the I channel. That would be the equivalent of the standard method to use a mixer to convert to DC in a quadrature configuration where the Q channel would contain the phase modulation.

It should be possible to apply a transformation on the fourier transforms that are already available in Linrad to achieve the same result. Then the AM modulation would be present in the real part of the correlated spectrum while the FM part would be present in the imaginary part. The RMS sum would then give the composite noise. The transformation is probably trivial, but it is unknown to me so at the present time Linrad will only present the composite noise. The files available on this page should however be useful for testing algorithms that separate AM and FM noise.

High resoultion spectra with correlation.

Figure 14 shows the correlated spectrum of nomod-err.raw at a bin bandwidth of about 15 Hz. The number of averages is only 1024 because the file does not contain data for more transforms. The yellow curve, the correlated spectrum, is about 15 dB below the average spectrum so the only conclusion one can draw is that the HP8644A opt 004 is at least 15 dB better than the R&S SMT03 at close range.

It is obvious from the baseband spectrum that the two SMT03 units differ in their spurious emissions. The green curve is channel 1 (which is displayed in the baseband waterfall).

At close range the sideband noise of the SMT03 is lower. That is probably because of correlated noise between the two SMT03 units. They are locked to a common reference with PLL loop bandwidths in the order of a couple of hundred Hz as can be seen in figure 14.

Fig 14. The file nomod-err.raw played with averaging 1024 times in groups of 8 averages. The bin bandwidth is about 15 Hz.

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