Uncle Bob (the one who likes beer so much) mentioned noise bandwidth of the filters – and now I have to admit that I was a bit sloppy about that when evaluating the phase noise of my signal generator. Back then, I just took the -60dB bandwidth, which clearly is much wider than the actual noise bandwidth.
So how do we know what the noise bandwidth in our FFT analysis is?
FFT can be viewed as being equivalent to a number of bandpass filters, that is half the number of sample points. These filters obviously are pretty steep and from the textbook we know that in this case the noise bandwidth will roughly be the same as the -3dB bandwidth.
So what’s the actual -3dB bandwidth in our FFT then?
This depends on the window function we’ve chosen. Pico Technology was kind enough to show a table of all available window functions in their user manual, so I’ll repeat the relevant information here:
Window | Rel. BW | Side Lobe |
Blackman | 1.68 | -58dB |
Gaussian | 1.33 ~ 1.79 | -42 ~ -69dB |
Triangular (Barlett, Fejer) | 1.28 | -27dB |
Hamming | 1.30 | -41.9dB |
von Hann | 1.20 ~ 1.86 | -23 ~ -47dB |
Blackman-Harris | 1.90 | -92dB |
Flat-top | 2.94 | -44dB |
Rectangular | 0.89 | -13.2dB |
As can be seen from the table above, Blackman-Harris is the general purpose window, particularly if a high dynamic range is required, where it is the only window function with a reasonable side lobe suppression (apart from the Kaiser-Bessel window with high values for the Bessel parameter [beta], which is not available in the Pico software).
Most of these window functions have their specific uses because of specific merits, but due to their poor side lobe suppression cannot be recommended for any applications where dynamic range is of importance. This is particularly true for rectangular and triangular, as they create a lot of noise even in an 8 bit system. Rectangular has the narrowest bandwidth and maximal sharpness of all window functions though.
The performance of Gaussian and von Hann windows depends on window-specific parameters, which aren’t specified in the Pico documentation. This is most likely true with other scope manufacturers as well, where we can select these window functions from a menu but don’t actually know what we get. The table above just lists the performance for the usual range of these parameters. Consequently, these window functions should be avoided unless someone actually evaluates their specific properties on the scope in question beforehand.
It is clear, we should use Blackman-Harris for the phase noise measurement, but first let’s check whether the -3dB bandwidth of 1.9 times the bin width as specified in the table is plausible. The following screenshot shows both the -3dB and -60dB bandwidth by cursor measurement on the high frequency side of the filter curve. This is the zoomed picture from a 64k FFT at 5MHz bandwidth and 152.6Hz bin-width (FFT16_64k_2MHz_+3dBm_avg_zoom)
The automatic measurements confirm that the signal is indeed 2MHz at +3dBm. The -3dB point is at 2.000152MHz, hinting on a total -3dB bandwidth of 2 x 152Hz = 304Hz. Using the factor 1.9 from the table, we get 152.6 x 1.9 = 290Hz. Well, close enough. The difference is most likely due to a slight asymmetry in the top of the filter curve.
The -60dB point is at 2.0005631MHz, hence -60dB bandwidth is 2 x 563Hz = 1126Hz. That’s pretty steep indeed, but is still some 3.75 times the noise bandwidth, thus introducing almost 6dB error if this is used for noise calculation.
I will use 300Hz as the noise bandwidth for the following measurement.
Here’s the 64k point 16 bit FFT of a 2MHz signal from the analog synthesizer at +4dBm, slightly zoomed to show the range 1MHz to 3MHz (FFT16_64k_2MHz_+4dBm_avg)
I’m measuring the point of strongest phase noise here, which is at about 180kHz distance from the carrier. At only 20kHz distance, it would be some 4dB better.
Worst case phase noise at that frequency is -96.2dBc for 300Hz bandwidth. Normalized to 1Hz we can add 10 log(300) = 24.8dB, so we get a phase noise of -121dBc/Hz.
The specification of the synthesizer says -120 to -126dBc for the entire frequency range and the phase noise curve is only given for 480MHz, where it is indeed a flat line at -125dBc from 20kHz to some 500kHz distance from the carrier. So at least at the lower end of the frequency range, phase noise is 4dB higher than that, which is still low – certainly lower than what most ‘real’ spectrum analyzers can measure.