Histograms are graphical representations of data which divides it into intervals or bins. These intervals/bins are plotted on a bar chart such that the bar height relates to the number of data points inside each interval/bin as a function of the data value. Figure 1 provides an example of a histogram. In this example it is the histogram of the period of a square wave. The range of period values, shown in the horizontal axis, is broken into 100 bins. The vertical axis represents the number of measured period values in each bin.
The histogram shows the distribution of period values and is the first order approximation to the probability density function (pdf) of the period measurement. Histograms are used to characterize random processes such as jitter or noise because these processes must be described statistically. In this tutorial we will investigate how to create and measure histograms.
WaveRunner 6Zi series oscilloscope
Displays shown in the tutorial are based on the following initial setup on a WaveRunner 6 Zi scope:
- Connect a coaxial cable from channel 1 to the Aux connector on the front panel.
- Recall the default setup: File pull down > Recall Setup> Recall Default.
- Turn off channel 2.
- Set the input coupling on Channel 1 to be 50 Ohms: Touch or click the channel 1 annotation box>touch or click on the coupling field >select DC 50 Ω.
- Set up the Aux output to be the Fast Edge signal. Utilities pull down > Utilities Setup >Aux Output Tab>touch or click on Fast Edge. The Fast Edge signal is a 5 MHz, 450 mVp-p, square wave.
- Auto Setup the scope: Press Scope Setup then select Auto Setup from the fly-out menu.
- Set the channel 1 signal amplitude using the C1 dialog box. Set the vertical scale to variable gain by checking the Var Gain checkbox. Adjust the vertical scale of channel 1 to maximize the C1 signal amplitude on the display. It should be 90% of full scale.
- This completes the initial setup. The scope display should be similar to Figure 2.
Setting up a Histogram
LeCroy Oscilloscopes can histogram data values from any trace or measured parameter values. In most applications engineers want to histogram the measured parameter values. In this tutorial we will histogram the period of the Fast Edge square wave available at the Aux connector on the front panel of the oscilloscope.
We start the process by setting up a measurement of the period at level (per@lv) of trace C1. Use the Measure pull down menu and select Measurement Setup (Measure > Measure Setup. Touch or click on the Clear all definition button. Touch or click the P1 tab on the Measure dialog box and set up parameter P1 to be the Period @ level of C1. Touch or click on the On checkbox. Return to the measure dialog box, click or touch the Show Table box. The screen should look like Figure 3.
Note that at the bottom of the P1 dialog box there is a set of Actions for P1 which allows the user to graph the parameters values using a histogram, a trend (plot of parameter values in the order that they are taken, or a track (plot of parameter values versus time). We could click the Histogram button and set up the histogram. We will defer this for a moment to shown some additional measurement features related to histograms.
Click or touch the Measure tab. On the right side of the measure tab, in the area titled Statistics, check the boxes labeled On. This will turn on the measurement statistics which include the mean, maximum, minimum, and standard deviation (sdev) values of the measured parameters as well as the total number of measurements included in the statistics. Statistics readouts summarize the key parameters of the distribution of measured period values. Using histograms we will enable 20 histogam based parameters to further characterize this distribution.
Check the box labeled Histicons. This will cause an iconic histogram (hence the name histicon) of the measured parameter to be displayed under the parameter readout on the screen. This is just a smaller image of the histogram of the Period at Level measurement. Histicons, shown in Figure 4, are a simple way of monitoring the distribution of all the measurements up on the display.
There are three ways to turn on a histogram display. The simplest method, using the interactive touch screen interface, is to click or touch the histicon on the display. Alternatively we could go back to the P1 measurement set up tab and click or touch the Histogram button under Actions for P1. Either of these actions results in a pop up box asking which math function trace you want to use for the histogram. Make that selection and the histogram will appear. The third, and least satisfactory due to the large number of steps required, method is to use the Math setup (Math > Math Setup) to define a functional trace to be the histogram of a selected parameter or trace.
Click or touch the histicon under the P1 parameter readout. Select to place the histogram in math function trace F1 as shown in Figure 5. After making the selection, press the Close button.
Touch or click the F1 trace annotation box to bring up the Math setup dialog box for trace F1. Touch or click the histogram tab on the right of the F1 dialog. The display will appear as shown in Figure 6.
The histogram tab shows the principal controls for setting up the histogram. The first, in the area labeled Buffer, sets the number of values to be included in the histogram. This defaults to 1000 values because for some scopes without the extended math option this is the maximum value. The WaveRunner 6 Zi series oscilloscopes allow this value to be set to two billion values. Double click or touch the # Value box. In the resulting popup press the Set to Max. The number of values will increase to 2,000,000,000.
The short table under the # Values box shows the number of values shown on the histogram display. If any values are outside the displayed range the number of values that are off the left hand side of the screen are listed as under. Likewise, those off the right hand edge will be counted and displayed as over. In the example in Figure 6 all 1000 values are in the display.
The Clear Sweep button will clear all stored values and reset the histogram buffer to zero values.
In the lower right hand side of Figure 6 you can see a heading labeled Scaling. Double touch or click on the # Bins box. This sets the number of bins. The default value is 100. This is a good number for histogramming amplitude parameters. Amplitude parameters in an 8 bit oscilloscope only have a resolution of 256 levels. If we use too many bins we will get unpopulated bins because the bin width is narrower than the amplitude resolution of the scope. Since we are histogramming a timing parameter we can use the maximum number of bins. Select Set to Max in the popup. This will give us 5000 bins.
Pressing Find Center and Width will reset the horizontal and vertical scales to include all values within the display. The check box marked Enable Auto Find enables this rescaling on each acquisition so all the data is always on the display. The Center and Width fields allow the user to set the histogram scaling manually (not recommended).
The final field in the histogram tab is marked Vertical Scale. There are two choices, Linear or lin(ear) Const(ant) Max(imum). The linear scale allows the histogram to build vertically as data accumulates. When the histogram reaches the top of the display it rescales the vertical axis to keep it on screen. LinConstMax keeps the histogram at near full scale and rescales the vertical axis as data is accumulated. Select LinConstMax.
The screen should now look like Figure 7.
Note that the F1 trace annotations box tells the user not only the vertical and horizontal scaling of the trace but also the number of values included in the histogram. This number will continue to increase as more and more measurements are being made.
Bring up the measure dialog box (Measure pull down > Measure Setup).
We will make some measurements on the histogram using histogram parameters.
Touch or click the P2 tab. Setup P2 to read the histogram Mean (Hist mean) of trace F1. In the Select Measurement scroll box set the category to Statistics to display all 20 histogram parameters.
Here is the list of histogram parameters:
|FWMH||full width (of largest peak) @ half of maximum bin|
|FWxx||full width (of largest peak) @ xx% of maximum bin|
|Hist ampl||histogram amplitude between two largest peaks|
|Hist base||histogram base or leftmost of two largest peaks|
|Hist max pop||population of most populated bin in histogram|
|Hist maximum||highest data value in histogram|
|Hist Mean||average of data values in histogram|
|Hist median||median data value of histogram|
|Hist mid||Mid of peak to peak range|
|Hist minimum||lowest data value in histogram|
|Hist mode||data value of most populated bin in histogram|
|Hist pop@x||Population at a bin for specified horizontal location|
|Hist range||difference between highest and lowest data values|
|Hist rms||rms value of data in histogram|
|Hist sdev||standard deviation of the data values in histogram|
|Hist top||histogram top or rightmost of two largest peaks|
|Hist X@peak||x-axis position of specified largest peak|
|Peaks||number of peaks in histogram|
|Percentile||Horizontal data value that divides a histogram so that the population to the left is xx% of the total|
|Total pop||total population in histogram|
In a similar manner set up P3 to measure the standard deviation of the histogram (Hist sdev) and P4 to read the Histogram Range (Hist range). These are the most commonly used statistical parameters and are shown in Figure 9.
In our example above we are characterizing the histogram of Period@ Level. The histogram standard deviation reads the rms value of the period jitter. Likewise the histogram range is measuring the peak to peak jitter. Note that the peak to peak jitter only has meaning when referenced to the total number of measurements included because the random jitter component is unbounded and increases with increasing the number of measurements. In our example the total population is shown in the Number of measurements in the P1 statistics, 1.342376E^6 in Figure 9.
LeCroy oscilloscopes can histogram any measurement parameter making statistical analysis a readily available tool for your signal analysis needs.
This completes the tutorial.