# How to perform histogram analysis on your oscilloscope

Arthur Pini
-September 16, 2013

*Looking for histogram basics and how to apply histogram analysis to your project? This article from Art Pini has you covered.*

Oscilloscopes include analysis tools like Fast Fourier Transform (FFT) and histograms which provide alternative views of waveform data. FFTs add the frequency domain view while histograms provide an entry into statistical analysis. Viewing data in the statistical domain can provide great insight into the underlying processes behind a waveform or set of measurements.

**Histogram Basics**

There are two common ways to generate a histogram in an oscilloscope. The simpler is to base the histogram on a pixel map. A box, either horizontal or vertical, defines an area on a waveform. Pixel hits within the box are used to create the histogram. A better way is to create a histogram numerically from parameter measurements or waveform samples. The later method allows histogramming parameters defined by multiple edges, such as period or width.

A histogram is created numerically by breaking a range of potential values of samples or measurements into narrow bins. The number of values in each bin is counter and plotted again the mean value of the bin. Figure 1 shows a series of typical waveforms on the left, and the associated histograms of the sample values for those waveforms on the right.

**Figure 1: Three typical histograms: top sine wave and its histogram, middle triangle wave and a uniformly distributed histogram, bottom random noise and its Gaussian distributed histogram**

*(click on figure to download larger image in PDF format)*The histogram gives us an approximation of the probability density function (pdf) of a set of values. The shape of the pdf tells us about the underlying process. In the case of the sine wave, remember that the rate of change of the sine is maximum at the zero crossings and minimum at the positive and negative peaks. When we sample the waveform uniformly, we get more samples at the peaks and much fewer at the zero crossings. This is due to the differences in the rate of change, thus explaining the shape of the histogram. For the case of the triangle wave, the rate of change is constant, so samples are taken evenly. This results in what is called a uniform distribution. The final example is Gaussian distributed random noise. The pdf shows the classic bell-shaped Gaussian or normal distribution.

Let’s look at some common measurement examples that exhibit these common distributions.

**Jitter Measurements**

Figure 2 shows the measurement of a 100 MHz clock with a 250 kHz periodic jitter.

**Figure 2: a 100 MHz clock with a 250 kHz periodic jitter component.**

*(click on figure to download larger image in PDF format)*In Figure 2, the upper trace is the clock with an expanded (zoom) view beneath it. The measurement parameter P1 is the time interval error (TIE). The TIE is the time difference between the ideal clock edge location and the actual edge location. It is, in essence, the instantaneous phase of the clock signal on a cycle-by-cycle basis. Trace F2 (third from the top) is a plot of the track of the TIE parameter. The track is the measured TIE value versus time. The track shows the time variation of the clock’s instantaneous phase. The bottom trace is the histogram of the TIE. Since TIE is varying sinusoidally, the histogram has a major sine distribution component. It also has a random (Gaussian) component which is evident at the extrema points.

Histograms are characterized by a special set of measurement parameters. In Figure 2, parameter P2 shows the histogram mean or average value. P3 is the standard deviation of the histogram, and P4 is the range (maximum minus minimum value). The standard deviation corresponds to the rms value of the clock TIE jitter, while the range displays the peak-to-peak value over the 276,000 TIE measurements.

**Timing Analysis**

Consider the histogram of delta time at level (dt@lV) measurements shown in the bottom trace of Figure 3. This histogram shows the uniform distribution of the delay between the two source waveforms shown in the traces marked C1 and C2. These waveforms are from a trigger circuit which synchronizes an external event (trace 1) with an internal 400 MHz clock to produce a synchronous pulse output (trace C2).

**Figure 3: The histogram of a delay measurement shows the expected uniform distribution and an unexpected delayed output.**

*(click on figure to download larger image in PDF format)*The expected delay between the input (C1) and output (C2) is uniformly distributed over a range of 2.5 ns (one 400 MHz clock period) as shown in the histogram in math trace F1. Note that a small number of output pulses are delayed by an additional 2.5 ns clock period. This behavior was not expected.

By comparing the total population histogram parameters of the main histogram and that of the delayed outputs, it was determined that the delayed outputs occurred about 0.3% of the time (3 in 1000 events).

It would be useful to be able to capture these delayed input/output pairs. This can be easily accomplished using a scan and search tool which permits scanning for events that meet set measurement limits. In this case, we would like to examine waveforms with unusual values of delta time at level parameter. The histogram can be used to determine the search criteria.

In Figure 2, a cursor is used to determine the upper limit of the expected range of delay values. That cursor indicates the upper limit is 55.18 ns as indicated by the cursor horizontal readout (X1) under the timebase annotation box.

These are just a few example of how histograms can be applied to analyze waveforms and measurements. The next time you fire up your scope, you may want to spend a few minutes to investigate its histogram functionality.

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