The corrections made to the measurements described in step 16 of the

Frequency Discriminator test setup procedure are somewhat inscrutable. For those who wish to understand the rationale behind them, I provide the mathematical justification in this message. Those who don't like or can't be bothered with math are encouraged to skip the remainder of this message.

Warning: Algebra aheadHP product note 11729C-2 provides the mathematical justification for the corrections specified in step 16. However, this justification is not isolated to a single section of the product note; although most of the material is found on pgs 16, 24-27 and 40. One complication that arises is the product note includes some corrections required when using a swept-tuned spectrum analyzer, corrections that are not applicable to an FFT spectrum analyzer. Specifically, the Noise Bandwidth of analog HP spectrum analyzers is used for one correction; whereas the Effective Noise Bandwidth correction of the FFT windowing function is already applied by the PicoScope 6 software and requires no further correction. In addition, a correction factor for the "log-shaping and detection circuitry of an analog spectrum analyzer" is applied. Again, this is not applicable to FFT spectrum analyzers. Consequently, the correction procedure given in the test setup description elides these steps and the mathematics justifying their use is modified to eliminate terms corresponding to them.

In

the derivation of the equation for the frequency discriminator constant the final result is: \$\nu(t)=K_{d}\varphi(t)\$, where \$\nu(t)\$ is the voltage output of the Phase Detector after low-pass filtering, \$K_{d}\$ is the (frequency) discriminator constant, and \$\varphi(t)\$ is the instantaneous frequency corresponding to the output voltage.

In the HP product note, this is presented in a slightly different form: \$\Delta V = K_{d}\Delta f\$, where \$\Delta V\$ is the change in output voltage and \$\Delta f\$ is the change in instantaneous frequency. These are mathematically equivalent formuations.

The equations above are time domain descriptions, whereas the spectrum measurements are in the frequency domain. Consequently, additional notation is necessary to identify these measurements. (Subsequent page references are citations to the

HP product note).

On page 6, two symbols are defined to identify the relevant spectral data. First, \$S_{v}(f_{m})\$ identifies the "power spectral density of the voltage fluctuations out of the detection system" at the offset frequency \$f_{m}\$. \$S_{\Delta f}(f_{m})\$ is the spectral density of the frequency fluctuations at the offset frequency \$f_{m}\$ . Thus, \$S_{v}(f_{m})\$ represents the power spectral density of the signal \$\nu(t)\$ (this is what is measured by the low frequency spectrum analyzer during an experiment) and \$S_{\Delta f}(f_{m})\$ represents the frequency spectral density of the signal \$\varphi(t)\$.

While discussing symbols representing spectral densities, it is convenient to mention another quantity that plays an important role in the characterization of phase noise, \$\mathscr{L(\mathcal{\mathrm{f_{m}}})=\frac{\mathcal{P_{SSB}}(\mathrm{f_{m}})}{\mathcal{P_{\mathrm{Carrier}}}}}\$. \$\mathcal{P_{\mathrm{Carrier}}}\$ is the power of the (oscillator) carrier signal. \$\mathcal{P_{SSB}}(\mathrm{f_{m}})\$ is the single side-band power of a phase modulation sideband at the offset frequency \$f_{m}\$. When discussing phase noise, most specifications provide values of \$\mathscr{L(\mathcal{\mathrm{f_{m}}})}\$, so there is a requirement to convert the spectra measured by the frequency discriminator into the form \$\mathscr{L(\mathcal{\mathrm{f_{m}}})}\$. On page 6 is derived the relationship between \$S_{\Delta f}(f_{m})\$ and \$\mathscr{L(\mathcal{\mathrm{f_{m}}})}\$:

\$\mathscr{L(\mathcal{\mathrm{f_{m}})}} = \mathcal{\frac{S_{\Delta f}(f_{m})}{{2f_{m}}^2}}\$.

Spectral densities are continuous functions of frequency. When the set of frequencies associated with a power measurement is countable, this is not true and the result is referred to as a spectrum. Since the measurements by a frequency discriminator test setup quantize frequency, phase noise characterizations based on measurement deal with spectra rather than spectral densities. We continue to use the notation \$S_{v}(f_{m})\$, \$S_{\Delta f}(f_{m})\$, and \$\mathscr{L(\mathcal{\mathrm{f_{m}}})}\$ to identify the spectra arising from quantization of the identically named spectral densities.

Phase noise is canonically described as arising from FM modulations of the carrier signal by stochastic (noise) processes within the oscillator. The products of these processes add linearly to create the total phase noise spectrum or spectral density.

Consider the value associated with \$S_{v}(f_{m})\$, for a particular offset frequency \$f_{m}\$. This is the power of the \$f_{m}\$ component of the spectrum and the total phase noise spectrum is mathematcially equivalent to a sum of spectra, where each comprises a single tone spectrum for the frequency \$f_{m}\$ (m ranging over all values for \$S_{v}(f_{m})\$). To be clear, the single tone spectra are

**not** those produced by the noise processes, which generally create multi-tone spectra. The single tone spectra are a mathematical decomposition useful when considering how to measure the discriminator constant. In particular, measurement of the system response to a single tone input contains all the information needed to compute the discriminator constant. This is the objective of the calibration steps described in the test setup procedure presented previously, specifically in steps 3-5.

The mathematical justification for the corrections specified in step 16 is found on page 40. It begins with the casual assertion that for m<0.2rad, where m is the modulation index of the modulation:

\$\frac{P_{ssb}}{P_{carrier}} = \frac{m^2}{4}\$

I searched for hours on the internet to find a justification for this result without success (it may be that it exists in some textbook, of which I do not have a copy). Finally, I found enough information to dervive it. On

this web page, it is noted that for sufficiently small FM modulation indices (which on other web pages is given as m<0.2), the Bessel J coeffients are: \$J_{0} = 1\$, \$J_{1} = \frac{m}{2}\$, and \$J_{n} = 0, n>1\$. As an aside, the correct constraint on the modulation index is m<.2, not m<0.2rad, since the modulation index is defined as: \$ m = \frac{\Delta f_{peak}}{f_{m}}\$, where \$\Delta f_{peak}\$ is the peak frequency deviation of the FM modulation and \$f_{m}\$ is the FM rate. This is a unitless ratio. The constraint m<0.2rad is appropriate for Phase Modulation and I found several references on the internet where it is erroneously cited for FM modulation.

In the calibration procedure (step 3), the FM rate is set to 1 KHz and the frequency deviation to 100 Hz. This yields a modulation index of .1, which satisfies the given constraint.

In a

slide presentation available on the internet (on slide "Angle and Pulse Modulation - 7"), the total power \$P_{T}\$ of an FM modulated signal with carrier \$P_{C}\$ is given as:

\$P_{T} = P_{C}({J_{0}}^2 + 2({J_{1}}^2+{J_{2}}^2+ ...))\$

Noting the values of \$J_{i}\$ when the constraint m<.2 holds and substituing into this equation:

\$P_{T} = P_{C}(1 + 2(\frac{m^2}{4})) = P_{C} + P_{C}\frac{m^2}{2}\$

In the last expression to the right of the equal sign, the first term represents the carrier power and the second term represents the power of the double sideband. The single sideband power is 1/2 of this, i.e., \$P_{ssb} = P_{C}\frac{m^2}{4}\$. Dividing this expression by \$P_{C}\$ (aka \$P_{Carrier}\$) yields the assertion made at the beginning of page 40.

Given \$\frac{P_{ssb}}{P_{carrier}} = \frac{m^2}{4}\$, we can substitute the values measured during calibration in steps 3-5 and develop an expression for the single-sideband to carrier power ratio in terms of these values. First, to ensure clarity, the derivation on page 40 identifies the calibration values as \$\Delta f_{peak_{cal}}\$ and \$f_{m_{cal}}\$ and uses them to express the modulation index, \$m = \frac{{\Delta f_{peak_{cal}}}}{f_{m_{cal}}}\$, so:

\$\frac{P_{ssb}}{P_{carrier}} = \frac{m^2}{4} = \frac{1}{4}\frac{{(\Delta f_{peak_{cal}})}^2}{(f_{m_{cal}})^2}\$

At this point it is useful to mention that, so far, we have not been dealing with values expressed in logrithmic units. Rather, the values used in the expressions are in linear units. This is important in the next step of the derivation. The measurement of

**P-cal** and

**Delta_SB-cal** on the spectrum analyzer generally will be made in logrithmic units (e.g. dBm). To use these values in the derivation, we must express them in linear units. To retain clarity, the distinction between these values in linear and logrithmic units is made by appending them with either [Lin] or [dBm]. Thus, from this point,

**P-cal** in linear units (e.g. milliwatts) is represented by the symbol \$P_{cal}[Lin]\$ and in logrithmic units by the symbol \$P_{cal}[dBm]\$. Simlarly

**Delta_SB-cal** in linear units is represented by \$\Delta SB_{cal}[Lin]\$ and in logrithmic units by \$\Delta SB_{cal}[dBm]\$. The [Lin] and [dBm] notation applies to the other symbols as well.

In step 4 of the test setup procedure, the difference between the carrier and sideband power is measured by subtracting the former from the latter (this assumes the SSA3032X is displaying results in dBm). In other words, \$P_{ssb}[dBm] - P_{carrier}[dBm] = \Delta SB_{cal}[dBm]\$. In linear units the subtraction becomes division and therefore:

\$\frac{P_{ssb}[Lin]}{P_{carrier}[Lin]} = \Delta SB_{cal}[Lin]\$.

Consequently, we can write:

\$\frac{P_{ssb}}{P_{carrier}} = \frac{m^2}{4} = \frac{1}{4}\frac{{(\Delta f_{peak_{cal}})}^2}{(f_{m_{cal}})^2} = \Delta SB_{cal}[Lin]\$

The last equality can be re-expressed as:

\$\Delta {f}^2_{peak_{cal}} = 4 {f}^2_{m_{cal}} 10^{{\frac{\Delta SB_{cal}[dBm]}{10}}}\$, since \$10^{{\frac{\Delta SB_{cal}[dBm]}{10}}} = \Delta SB_{cal}[Lin]\$

The

derivation of the equation for the frequency discriminator constant specifies voltage amplitudes for the oscillator (DUT),\$V_{DUT-AMP}\$ and the referenced signal, \$V_{R-AMP}\$. However, it fails to indicate whether these amplitudes are peak-to-peak values or RMS values. This follows the formuation in Appendix A (page 34), on which the derivation is based, which also does not indicate whether peak-to-peak or RMS voltages are meant. On page 40, however, the discriminator constant is defined implicitly as:

\$K_{d} = \frac{\Delta V_{rms}}{\Delta f_{rms}}\$

So far, we have derived the equivalent expression for \$\Delta {f}^2_{peak_{cal}}\$, not \$\Delta {f}^2_{rms_{cal}}\$. This is easily fixed, since \$\Delta {f}_{peak_{cal}} = \sqrt{2} \Delta {f}_{rms_{cal}}\$ and therefore:

\$\Delta {f}^2_{rms_{cal}} = 2 {f}^2_{m_{cal}} 10^{{\frac{\Delta SB_{cal}[dBm]}{10}}}\$

Substituting this expression into the definition of \$K_{d}\$ yields:

\$K^2_{d} = \frac{\Delta V^2_{rms}}{2 {f}^2_{m_{cal}} 10^{{\frac{\Delta SB_{cal}[dBm]}{10}}}} = \frac{P_{cal}[Lin]}{2 {f}^2_{m_{cal}} 10^{{\frac{\Delta SB_{cal}[dBm]}{10}}}}\$, since \$P_{cal}[Lin]\$ is the response expressed as power (\$\Delta V^2_{rms}\$) to the calibration input.

Applying \$10 log_{10}()\$ to both sides of the equation re-expresses it in terms of dB:

\$2K_{d}[dBm] = P_{cal}[dBm] - (\Delta SB_{cal}[dBm] + 20 log_{10}(f_{m_{cal}})+3dB)\$

Note: There is a mistake on page 40, which is corrected in the equation given above. On page 40, the left hand side of the equals sign is given as \$K_{d}[dBm]\$, rather than \$2K_{d}[dBm]\$. It turns out that this mistake is cancelled out by an error in the equation given for \$S_{\Delta f}(f_{m})\$ on page 16, which should be:

\$S_{\Delta f}(f_{m}) = S_{v}(f_{m}) - 2K_{d}\$

I am deliberately leaving off the units in this equation, since as stated on page 16, \$S_{\Delta f}(f_{m})\$ is in units of [dBHz/Hz], whereas both \$S_{v}(f_{m})\$ and \$K_{d}\$ are given in units of [dBm]. How one gets a quantity in [dBHz/Hz] by subtracting two quantities in [dBm] is beyond my comprehension. In fact the whole document is riddled with equations that combine units in such a way as to be completely baffling.

Anyway, the desired final result is \$\mathscr{L(\mathcal{f_{m}})}\$ and this is expressed in terms of \$S_{\Delta f}(f_{m})\$ on page 7:

\$\mathscr{L(\mathcal{f_{m}})} = S_{\Delta f}(f_{m}) - 20 log_{10}(\frac{f_{m}}{1 Hz}) - 3 dB\$

\$\;\;\;\;\;\;\;\;\;= S_{\Delta f}(f_{m}) - 20 log_{10}(f_{m}) - 3 dB\$,

where \$20 log_{10}(f_{m})\$ in the last expression to the right of the equal sign is written without the explicit reference to its units.

Substituing the equation for \$S_{\Delta f}(f_{m})\$ and in that the equation for \$2K_{d}\$ gives:

\$\mathscr{L(\mathcal{f_{m}})} = S_{v}(f_{m}) - (P_{cal}[dBm] - (\Delta SB_{cal}[dBm]\$

\$\;\;\;\;\;\;\;\;\;\;\;\;\; + 20 log_{10}(f_{m_{cal}})+3dB)) - 20 log_{10}(f_{m}) - 3 dB\$

\$\;\;\;\;\;\;\;\;\;= S_{v}(f_{m}) - P_{cal}[dBm] + \Delta SB_{cal}[dBm] - 20 log_{10}(\frac{f_{m}}{f_{m_{cal}}})\$

Recalling that \$S_{v}(f_{m})\$ is what is measured by the low frequency spectrum analyzer during an experiment, the last equation to the right of the equal sign justifies the corrections made in step 16 (except adding 10 dBm, which is already justified in the description of the step).