Demystify Integrated-Phase-Deviation Results In Phase-Noise Measurements

Figure 1 shows the curves of a Bessel function for J₀ to J₃. A vertical line is drawn at a modulation index of 3, which is equivalent to a Δφ_peak of 3 rads. With the intersection of this vertical line with the individual curves of the Bessel function, an indication is given of the relative amplitude of the modulated carrier and sidebands to the unmodulated carrier on a voltage scale.

This example includes the following relative amplitudes:

Modulated carrier      J₀ = –0.27
First sideband             J₁ = 0.33
Second sideband        J₂ = 0.48
Third sideband           J₃ = 0.33

Figure 2 is the spectrum of a phase-modulated signal carrier, F_c, at a frequency of 1 GHz. It has a modulating frequency (f_m) of 400 Hz and a peak phase deviation of 3 rads. The amplitude scale is in volts with the amplitude of the unmodulated carrier set to 100 mV. The spectrum of the unmodulated carrier is the yellow trace. The blue trace shows the modulated signal. The marker readouts for the modulated carrier and the first three sidebands are close to the predicted values of J₀ through J₃. Note that this is the magnitude of the spectrum. As a result, the negative values from the Bessel function are shown as positive.

2. This spectrum of an angle-modulated signal was taken from an Agilent X-Series signal analyzer.

Determining RMS phase deviation from the spectrum of a carrier with sinusoidal phase modulation

In spectrum analysis, there also is a need to determine the Δφ_{peak k} peak phase deviations from the spectrum of the signal. This would be somewhat difficult for large values of Δφ_peak. For small values of Δφ_peak (Δφ_{peak k} < 0.5 rad), however, examining the curves for the Bessel function will show that the amplitude of the modulated carrier and the unmodulated carrier are close to equal. In addition, the amplitudes of the second-order and higher sidebands are close to zero.

For small peak phase deviations, another advantage is that the first sideband’s relative amplitude is very close to one-half of the peak phase deviation. The Δφ_peak of phase noise is very small (<<0.5 rad). To determine Δφ_peak using the simple relationship detailed in Eq. 2, it permits these three factors to simply measure the relative amplitude of the first sideband to the amplitude of the modulated carrier:

Equation 2.

where V_ss is the amplitude of the single sideband (in voltage) and V_c is the amplitude of the carrier (in voltage).

The amplitude values of most modern spectrum analyzers are on a power not voltage scale. Equation 2 can therefore be used to convert voltage to relative power in Eq. 3. Also, the phase-noise result obtained from most spectrum analyzers is the RMS phase deviation (Δφ_rms) and not the peak phase deviation (Δφ_peak ). Equation 4 converts the peak deviation to RMS. Finally, Equation 5 shows how to determine the carrier’s RMS phase deviation when modulated with a sinusoidal signal and measured on a relative power scale.

Equation 5.

where P_ss is the SSB amplitude (in power) and P_c is the amplitude of the carrier (in power).

Calculating the integrated phase deviation from the phase noise present on a carrier

This calculation needs to be tailored so that it can be used to measure the integrated phase deviation from the phase noise. To do this, the term £ (f) is introduced. It is the noise density of the power present in the SSB phase noise of a signal relative to the amplitude of the carrier. The method used to determine the phase noise is the integrated bandwidth method. The user selects a range of offset: the interval f_start to f_stop. The power relative to the carrier is then integrated between these two frequency points and computed as shown in Eq. 6:

Equation 6.

where £ (f) is in power-density ratio units and Φ_SSB is the SSB integrated RMS phase noise (in rads). (Of course, the equation is implemented as a summation—not an integration—in the spectrum analyzer.)

In dBc units, the integrated RMS single-sideband phase noise is:

In Eq. 5, it can now be recognized that P_SS/P_c is the same single-sideband power ratio computed as Φ_SSB. Therefore:

Equation 9 is the calculation made in spectrum analyzers to report the integrated phase-noise measurement results in terms of the RMS phase deviation in units of radians. Now that this has been determined, some simple computations from the RMS phase deviation might be useful in other applications. The jitter (in s) can be calculated by relating the integrated phase deviation to one cycle in time of the unmodulated carrier at a frequency of F_c:

Figure 3 is a display of a log measurement of an unmodulated carrier’s phase noise at a frequency close to 1 GHz (taken on an X-Series signal analyzer). The measurement was set to record phase noise at offsets of 100 kHz to 1 MHz from the carrier. Markers 1, 2, and 3 display the phase noise in units of dBc_SSB—rads and s, respectively. Markers 4 and 5 display the slope of the phase noise in units of dB/octave. Similarly, markers 6 and 7 display the slope in units of dB/decade.

3. Shown is an X-Series signal analyzer log plot measurement.

Most modern spectrum analyzers will automate the calculation of the results found in the equations discussed so far. It is beneficial, however, to understand how these equations were derived. After all, it may be useful in better understanding or computing these results, which may not be available in other spectrum-analyzer phase-noise measurements. Some of these also can be automated using built-in band power markers to develop a specific measurement for a task that needs to be accomplished.

Additional Information

Agilent X-Series Signal Analyzers.

N9068A and W9068A Phase Noise Measurement Application, technical overview.

Agilent’s Phase Noise Measurement Solutions, selection guide.