Heisenberg Uncertainty and RF Measurements
by Dr. Don Metzger, Chief Technologist, Constant Wave

The Heisenberg Uncertainty Principle
Most engineers are exposed to the Heisenberg Uncertainty Principle in a physics class. The basic notion of the Uncertainty Principle is that the uncertainty in a particle’s position multiplied by the uncertainty in the particle’s momentum is greater than a constant. Mathematically stated, the Uncertainty Principle is:

where:
Qx is the standard deviation of the particle’s position,
Qp is the standard deviation of the particle’s momentum and
h is Planck’s constant (6.63 x 10-34 J sec).

What the Uncertainty Principle means is that the more accurately you know a particle’s position, the less accurately you know its momentum, and vice versa. This leads to the conclusion that uncertainty with regard to one parameter is a trade-off with uncertainty in the other parameter.

Heisenberg for Time-Frequency Spaces
Many of us learn the Uncertainty Principle just well enough to get through the test in our physics class and then let it gently fade into the distant memory of “things I had to know to get the degree, but will never use again.” What we don’t realize is that measurements we make every day are affected by this principle. That’s because the Heisenberg Uncertainty Principle has an expression for measurements made in the time domain or frequency domain. The mathematical expression is:

where:
Qt is the standard deviation in the time domain in seconds, and Qf is the standard deviation in the frequency domain in Hz.

In this context, the Uncertainty Principle can be interpreted to mean that the more accurately you know when a signal occurred in time, the less accurately you know its frequency domain characteristics. Likewise, the more accurately you know its spectrum, the less accurately you know when that spectrum occurred in time. Before we explore this more, let’s first put this in terms more familiar to engineers.

RF measurement equipment has long used the notion of accuracy in the frequency domain. This is expressed through the quantity called resolution bandwidth (RBW). Figure 1 shows the spectrum of a single frequency sinusoid measured using an RBW of 100 kHz.

Figure 2 shows a zoomed in version of Figure 1. RBW is defined as the width of the spectrum at a point which is 3 dB down from the peak. The phrase that is used to describe this is “full width at half max”. That is, RBW is defined as the full width of the spectrum at a point which is 3 dB down from the peak (half max). (For mathematical purists, the width is actually calculated between the points which are 3.0103 dB down since 10 log(0.5) = -3.0103.)

While RF engineers have a quantity representing frequency domain resolution, we don’t have one for the time domain. To remedy this deficiency, let’s introduce an analogous quantity that we will call resolution time width (RTW). It will be defined equivalently to RBW, as the full width at half max in the time domain. With this definition, the Heisenberg Uncertainty Principle can be written in engineering terms as:

RTW × RBW ≥ 0.44

where:
RTW is the resolution time width in seconds and
RBW is the resolution bandwidth in Hz.

Spectrograms
A common means for displaying measurements in the time and frequency domains simultaneously is called a spectrogram. Spectrograms are particularly helpful for visualizing the constraint imposed by the Uncertainty Principle.

Figure 3 shows a spectrogram of a Bluetooth 1.0 signal. The horizontal axis of the spectrogram is frequency running from 1.5 MHz below the carrier, on the left, to 1.5 MHz above the carrier, on the right. The carrier frequency is at 0 Hz, in the center of the frequency axis. The vertical axis is time running from the bottom to the top. The bottom left corner of the plot is minimum time, minimum frequency. The colors represent the magnitude of the signal at a specific frequency and time. (It is important to note that although magnitude is being displayed, the data underlying the spectrogram is complex and could be displayed as real, imaginary or phase, as well as magnitude.)

Applications for Vector Signal Analyzers
Now that we have a means of viewing the time-frequency space, let’s look at how the Heisenberg Uncertainty Principle affects RF measurements. Let’s start with Vector Signal Analyzers (VSAs) and the closely related Spectrum Analyzers.

The commonly controlled resolution parameter is RBW. Figures 4, 5 and 6 show the measurement of a W-CDMA 3GPP signal at the output of an amplifier using various displays and various values of RBW. In each figure, Part (a) has an RBW = 100 kHz, Part (b) has an RBW = 400 kHz and Part (c) has an RBW = 1600 kHz. Figure 4 is a spectrogram representation, Figure 5 is an averaged spectrum and Figure 6 is power versus time. The horizontal and vertical scales for the spectrograms have been removed for clarity in visualizing the resolution trade-off. Because of the constraint imposed by the Uncertainty Principle, once the RBW has been chosen, the minimum value of RTW is determined. That is, RBW is the independent parameter and RTW is the dependent parameter. Thus, Part (a) has an RTW = 4.4 µsec, Part (b) has an RTW = 1.1 µsec and Part (c) has an RTW = 0.27 µsec.

Scenario (a) of the figures opted for better frequency resolution (smaller RBW) at the expense of time resolution. In the spectrogram, note the finer detail in frequency, but the smearing in time. Also note that while the spectrum has more detail, the power vs time plot has less.

Scenario (b) has a balanced trade-off between time and frequency domain resolution, as evidenced by the spectrogram. This balanced resolution allows us to plainly see the spectral re-growth from the amplifier. Notice that the spectrum has somewhat less resolution than Part (a), but the time domain is gaining detail.

Finally, scenario (c) opted for better time resolution (larger RBW). In this spectrogram we notice a smearing in frequency, but a finer resolution in time. The averaged spectrum is showing only the most basic character of the frequency domain. However, the time domain is extremely detailed.

The Uncertainty Principle imposes constraints on the resolution obtained from VSA measurements. Each signal type has a choice of RBW, which creates a balance between time and frequency domain resolutions. Generally, the best choice is one which creates spectral “pits” which are circular, neither smeared in time, nor in frequency. Figures 3 and 4(b) show this desired balanced choice of resolution.

Applications for Vector Network Analyzers
Spectrograms have not previously been applied to vector network analyzers (VNAs). Also, network analyzers measure system transfer functions rather than signals. Because of this, there is some flexibility in choosing the spectrogram layout for VNAs. Figure 7 shows a spectrogram with time on the horizontal scale and frequency on the vertical scale. (Note that this is a different choice of scales than for VSAs.) This figure depicts the S11 reflection coefficient as a function of both time and frequency. The colors represent the magnitude of S11. The data on which this spectrogram is based had 1601 points from 40 MHz to 20 GHz.

Figure 8 shows spectrograms for three values of RTW. As before, the horizontal and vertical scales have been removed for clarity in visualizing the resolution trade-off. Part (a) has an RTW = 100 psec, Part (b) has an RTW = 300 psec and Part (c) has an RTW = 900 psec. In accordance with the Uncertainty Principle, once an RTW has been chosen, the lower bound on the RBW is determined. That is, for VNAs, RTW is the independent parameter and RBW is the dependent parameter. Thus, Part (a) has an RBW = 4.4 GHz, Part (b) has an RBW = 1.47 GHz and Part (c) has an RBW = 0.49 GHz. Figure 9 shows the frequency domain characteristics of the discontinuity at 2 nsec. Figure 10 shows a time domain plot.

As will be explained in the section on windowing functions below, the trade-off between time domain resolution and frequency domain resolution for VNAs also involves the loss of data at the extremes of frequency. That is, the top and the bottom of the frequency range get cut off when more time domain resolution is desired. For VSAs, this situation is dealt with by simply acquiring more data. However, VNAs have a fixed frequency range over which they measure and it is not possible to acquire more data to compensate for this effect.

Scenario (a) opted for more time resolution (smaller RTW). Note the excellent time domain resolution in Figure 10. However, the low and high frequencies are not present in the spectrogram, nor in the frequency domain characteristics in Figure 9. This is part of the trade-off imposed by the Uncertainty Principle for VNAs.

Scenario (b) has a choice of RTW which yields a good balance of resolution for this DUT. Figure 9 has a larger range of frequency which can be displayed for the discontinuity at 2 nsec. Figure 10 has well defined discontinuities in time.

Scenario (c) opted for more frequency domain resolution (larger RTW). The discontinuities in the spectrogram are beginning to merge together, as can be seen in Figures 8 and 10. However, there is a large range of frequency for the frequency domain in Figure 9.

Each VNA measurement scenario involves a trade-off between time and frequency domain resolutions. It is important to set the RTW to a value which balances these resolutions optimally. In this case, the RTW is chosen to create adequate separation of discontinuities in the time domain, while preserving as much frequency domain information as possible.

Windowing Functions
You will notice that the Uncertainty Principle states that there is a lower bound on the product of RTW and RBW. Windowing functions have a significant influence on the lower bound which is achievable in practice. The properties of windowing functions are discussed in great detail elsewhere [1]. To cover them again here is beyond the scope of this article. However, because of their importance in time-frequency analyses, a few items should be noted. An example should help illustrate the important aspects of windowing.

Figure 11 shows a single timeslot of an EDGE waveform. Also shown is a windowing function which will be applied to the data. The windowing function is chosen to give an RBW of 30 kHz. This choice of RBW means a window which spans about 74.6 µsec. The shape of the window and the time span of the window yield an RTW (the “full width, half max” in the time domain) of 14.7 µsec. In the region where the window exists, it is multiplied by the input data. The windowed data is then run through a Fourier transform. The output of the transform is the spectrum of the signal. It is customary to associate the time at the peak of the window with the spectrum. Thus, the spectrum is said to be associated with the time 37.8 µsec. The log magnitude of the spectrum is plotted with colors as the lowest horizontal line in Figure 12. Note that the time is listed as 37.8 µsec.

The next horizontal line in the spectrogram results from sliding the window one data point to the right and repeating the windowing, transforming and displaying. This process is repeated until the last point in the window is aligned with the last data point. Notice that the final time in the spectrogram is 538.6 µsec. This time results from the final time of the input data, 576.5 µsec, minus the time at the window peak, 37.8 µsec (with a little tolerance for rounding of the time values).

From this example, we see that the width of the window in the time domain, which directly affects RTW, eliminates some of the first and last parts of the data. This is because the time is associated with the center of the window and not the beginning or the end. For VSAs, this truncation of the ends can be compensated for by either acquiring more data, or by zero padding before and after the data.

Compensating for the effects of data truncation for VNAs is not as simple. In this case, the input data is in the frequency domain and the transform is an inverse Fourier transform. It is often not possible to simply acquire more data by extending the frequency range over which the data is gathered. Zero padding is not an acceptable processing step, either. So, for VNAs, there is really no good way to compensate for the end effects. The result is that the choice of RTW affects both RBW and the frequencies over which the spectrogram can be computed.

Conclusion
The Heisenberg Uncertainty Principle states that there is a fundamental trade-off between time domain resolution and frequency domain resolution. The better the resolution is in one domain, the worse it is in the other. Of course, some measurements, such as a spectrum measurement, are only concerned with resolution in one domain. However, the informed engineer will always be aware that the Uncertainty Principle is lurking behind every measurement.

Additional examples and discussions concerning the Uncertainty Principle and time-frequency analyses can be found at Constant Wave’s website.

References
1. F. J. Harris, “On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform,” Proc. IEEE, Jan 1978, pp. 11-33.

Constant Wave
www.constantwave.com
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