Differential noise figure has been an important measurement topic in recent years with the increasing use of fully, or nearly full, differential chains in receivers and transmitters. While a number of measurement approaches are being used, many include an external balun at the device under test (DUT) output so noise can be measured in a single-ended sense, with some de-embedding of the balun’s effects, or by simply treating the DUT outputs as single-ended. While the balun-based method can be accurate, it can ignore more subtle influences of the balun on noise correlation and gives the user less visibility into the multi-mode behavior of the DUT, which can have noticeable system-level impact. Treating the DUT outputs as single-ended can miss correlated behavior entirely. A more careful treatment of the DUT’s correlation behavior without over-complicating the measurement can sometimes lead to more accurate results and a clearer picture of DUT behavior. Such measurements will be analyzed in this article, both with and without the use of an external balun, showing that improved treatments can lead to noise figure results changing by 1 dB or more.

With an increasing plethora of differential devices in development and on the market, including those operating in the mmWave range, there is more of a need for differential (and common-mode, in some cases) noise figure measurements. The concept itself has had some definitional challenges,^{1} but one can follow the lead of the literature and assume it is the output noise power in the given mode divided by the output noise power in that mode due to noise from uncorrelated terminations at the input at temperature T_{0} = 290K, borrowing from the IEEE definition. The uncorrelated termination input assumption is practical and will arise from any passive network at thermal equilibrium.^{2} The noise at the output due to input terminations requires some suitable gain definition for the mode, as many have discussed.^{3} The basic noise power measurement can proceed in many different ways, although a cold source approach will be used here,^{4} where an absolute power calibration is used with a mean square summation of noise wave measurements. We will also assume a vector network analyzer (VNA) is be- ing used as the receiver. Suppressing the receiver calibration coefficients, which establish the power accuracy, and leakage signal corrections, one can write

where b_{i} is the wave received at the i^{th} port of the VNA, ∗ denotes the complex conjugate and N equals the number of measurements. Typically, N is very large to reduce the amount of data variation, but not so large that DUT drift has an effect (usually many thousands). Similar processes are used in noise calculations on many other receivers and instruments.

For differential noise measurement, several approaches have been commonly used. The most obvious is to connect a balun to the device output so the noise power measurement can proceed as in the two-port case, with the loss of the balun de-embedded. Procedurally, sometimes there are a few challenges: finding a sufficiently broadband balun—although this has become much easier in recent years—and interfacing to the balun in an on-wafer or fixtured environment. As we will see, handling balun imperfections is another layer to the story.

Another approach has been to sim- ply measure the DUT outputs in a single- ended fashion, i.e., terminating the output not being measured and perform some average analysis on the noise powers using an appropriate gain definition. For certain devices, this works quite well and is fundamentally making the assumption that the output noise waves are uncorrelated. If the noise-dominant structure of the device is similar to that shown in **Figure 1a**, the noise mechanisms are not arising from a common node and there is weak coupling, this assumption is perfectly reasonable; there are many amplifier designs where this is the case. If, however, the noise-dominant stage is more like a differential pair (see **Figure 1b**), the uncorrelated assumption may be more problematic. The question is whether there are additional measurement options to get around these potential issues.

**METHODS: OLD AND NEW**

As discussed, the use of a DUT output balun is a simple, obvious and generally successful solution to performing the measurement—as would the use of an in-phase combiner, should common-mode noise figure be needed. Many authors have pointed out^{3,5-7} that there are a number of ways that even this measurement can be processed, and there are some error sources to consider. Even the most basic approach will de-embed the differential loss of the balun and perhaps consider its kTB thermal noise addition to the measurement—via the receiver noise calibration stage, for example. In terms of gain distortion, any imbalances in the balun will be captured by such an approach, but a simple method may not capture distortion in noise power correlation. Imbalances will lead to common-mode DUT noise power being coupled into the single-ended balun output and other mis-allocations. At even another level of analysis, the match interaction between the balun and the DUT could change the degree of noise power correlation.

An improved, corrected balun approach would compute the effective correlated power that is delivered to the common balun node, taking into account the balun imbalances. If the single-ended DUT noise powers are separately known—these add measurements of the DUT but require no added hardware—then the total dif- ferential power can be computed. A variety of permutations of techniques exist to do this correction.^{3,5} One such approach is to calculate the amount of common-mode DUT noise power that makes it to the balun output and the amount of differential mode DUT noise power that does not. This can be done with a combination of single- ended noise power measurements and either two measurements using the balun^{3} or by using a real correlation model of the DUT output and a single balun measurement.

It would be useful to have some understanding of the order of errors introduced by not doing an improved correction such as this. If the DUT outputs were uncorrelated, the error would be vanishingly small, from Bosma’s theorem.^{2} If the DUT outputs were highly correlated, then one can perform a simulation versus balun imbalance to see the size of the error. The results of such a simulation for a single-ended balun with nominal 4 dB insertion loss and a 5 dB noise figure, 20 dB differential gain DUT are shown in **Figure 2**. While levels of balun imbalance vary, 10 degrees of phase imbalance for a high frequency, broadband balun may not be un- usual, which could add 0.5 dB of error. Even in terms of cable length matching, 10 degrees at 50 GHz (using cables with expanded PTFE dielectric) arises from only a 125 μm length difference, so this is a practical concern. Note that the sensitivity to magnitude imbalance, at least on the levels commonly encountered, is somewhat less.

Another approach may be to directly measure the correlation signal. Expanding the mean-of-sum-of-squares of the differential signal yields the following equation, where ports 3 and 4 are defined to be the DUT output ports and receiver calibration and leakage signal calibration terms are omitted for simplicity:

The first two terms in the numerator will form the average of the single-ended powers, just what the uncorrelated method ended up with. The last term is related to the mean of the real part of the correlation of the two waveforms. This is a physically reasonable result: if the waveforms are indeed uncorrelated, the last term will sum to zero given enough measurements, and the whole equation reduces to the simple “assume disjoint amplifiers” case discussed earlier.

Instead of using a balun and corrections to get at this quantity, it could be useful to measure it directly and, if one has multiple synchronized receivers in the VNA, this would seem to be possible, i.e., b3 and b4 waveforms can be sampled at the same time. There are, how- ever, some caveats:

- Since these are complex quantities, a phase calibration is now required, as well as a power calibration. This can be performed using a common sinusoidal source but, of course, the reference plane must be consistent relative to the DUT for both receiver paths.
- It is generally assumed that the receiver chains are single-ended, but a coupled chain could be possible with additional characterization.
- There is a danger of losing cor- relation information if the runs from the DUT to the receivers are too long (more than ~10 m) or in certain receiver noise level situations. The receiver paths should be kept relatively short, with gain levels not appreciably higher than those of the DUT.
- Low levels of decorrelation can be corrected by comparing inter-channel responses over small frequency scales.

A simplified diagram of the measurement setup is shown in **Figure 3**. The key differences from a conventional setup are the coherently clocked receivers and the phase reference plane. Note that the two receiver chains need not be identical, but they typically will have similar net gain and noise levels. Since only one measurement is necessary and no balun characterization is needed, this approach does have a simplicity advantage. Also, differential and common-mode noise figures are available simultaneously, should that be important.

**MEASUREMENT COMPARISONS**

A logical next step is to compare measurements using the various methods discussed. As an initial comparison, consider a differential amplifier with a dominant noise stage at the output that is highly correlated (i.e., a differential pair). One might expect that an uncorrelated approach would not be appropriate since it will understate the actual differential noise power and overstate the actual common-mode noise power. The uncorrelated results are shown in **Figure 4** and compared to those using the direct correlated method and a balun based method with two different levels of correction: 1) a basic approach only taking into account differential insertion loss affecting the noise pow- er and 2) a more corrected approach taking into account the decorrelating influence of balun imperfections.

The uncorrelated approach does indeed produce a lower, erratic value and one that was quite unexpected based on the characteristics of the elements in the amplifier. The result is off by multiple dB. The other methods were more comparable in mean terms, with the basic balun approach showing an elevated level of scatter, on the order of 1 dB additional peak scatter. The corrected balun and direct correlated measurements agree more closely. Some disagreement is to be expected, since the balun is terminating the DUT slightly differently than the two noise receivers, for the correlated method, and the methods only partially correct for mismatch errors. The mean value for these latter methods are more consistent with the ~5 dB noise figure that was expected from the DUT.

Another comparative example is shown in **Figure 5**, this time just be tween the direct correlated method and the corrected balun method, so that differences can be examined more closely. For this example, the uncertainties of both methods are on the order of 0.5 dB, based on

- Noise power repeatability, limited by finite record length.
- S-parameter uncertainties of DUT gain.
- Balun characterization for the corrected balun method.
- Phase calibration process for the direct correlated method.
- Residual mismatch errors.
- Power calibration uncertainty.

Even though some of those aspects are identical for the two measurements (i.e., same DUT S-parameters and same receiver calibration), the level of agreement in the data still seems consistent. Again, the DUT is terminated slightly differently for the two measurements, so the variance may be higher in some cases.

**mmWAVE MEASUREMENTS**

As application frequencies in- crease, there is the need to measure differential noise figure at mmWave frequencies. A W-Band example is discussed next, showing the uncorrelated method with the differential mode and common-mode noise figure using the direct correlated approach (see **Figure 6**). As before, the uncorrelated method understates the noise power and noise figure. The common-mode noise power is actually 10 to 15 dB below the differential noise power, but since the commonmode gain of the device is very low, the common-mode noise figure does end up being quite large. Whether this common-mode noise figure will be a problem depends greatly on the system where the DUT is used. The common-mode noise power may be of interest in cases where it has a swamping effect on later gain stages. The noise power uncertainties on the non-dominant mode will be higher since the levels are typically closer to the receiver limit, and the calculation presents a subtraction-of-nearly-equal-numbers situation.

**OUTLOOK**

Noise figure by itself has been described as a simple figure of merit that does not take into account source impedance effects. Indeed, that is the measurement that has been discussed here. A more complete picture is given by the multiport form of noise parameters that has been an active research area in recent years.^{8} This article has focused on the underlying noise measurement and how

some improvements can be made to the differential/common-mode noise power measurement process to mitigate errors and simplify the measurement. For a highly correlated DUT, accuracies can be improved by more than a dB in some cases, with little additional hardware and with a few changes in procedure and calibration.

**References**

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