Histograms are used to display graphically continuous variables. They are relevant in many areas including digital photography, census data, actuarial statistics and, as we shall see, signal measurements in an oscilloscope.
In digital photography, an image histogram is a graphical representation of the tonal distribution. The graph depicts the number of pixels for each tonal value. In post-processing, the artist can, by using on-screen sliders or by typing in numeric values, alter various parameters such as hue and intensity, and observe the changing histogram as the photo’s appearance evolves.
In an oscilloscope, voltage (vertical) and time (horizontal) histograms may be invoked by simply pressing a button. This displays them as insets in oscilloscope automated measurements.
The word histogram is sometimes taken to refer to the historical nature of the diagram. But a more interesting if less obvious connection involves the ancient Greek words ἱστός (histos), which denotes rows of standing vertical items such as cornstalks, and γράμμα (gramma), drawing a record.
The histogram was conceived by Karl Pearson (1857-1936), an English mathematician and biostatician, who played a key role in the science of statistical analysis. The histogram resembles a bar graph, but it is significantly different. As a convention, and to distinguish between them, a bar graph is generally drawn with a space separating each bar, while the histogram consists of the rectangles in contact but not overlapping. The outside boundary, accordingly, has the appearance of an irregular line or generalized curve. The curve emphasizes the fact that the values are continuous. As they change to reflect changes in one or more electrical signals in an oscilloscope display, the histogram may become a moving, dynamic figure.
As a preliminary stage in constructing a histogram, the data is divided into appropriate intervals, which are then placed in bins. Each bin is translated into a rectangle, the height of which is proportional to the number of items in the bin.
Generally, the bins are all the same width. Their heights are then proportional to the population frequency. An alternate scheme is to have bars of differing widths. In this configuration, the area of each bin is proportional to the frequency or count of the elements contained therein, and the height of the bin represents the population density.
Because the data are continuous, there is no requirement as to the number of bins. A histogram can be created with varying numbers of bins, depending on its purpose and goals. Making the bins wide reduces noise due to sampling randomness. Greater precision and resolution, however, come from making the bins narrower. Besides having different bin widths, histograms can by symmetric, skewed right, skewed left, bimodal or unimodal.
To see histograms in a Tektronix MDO 3000 Series oscilloscope, first display the desired waveform. To do this, we have selected, from the internal arbitrary frequency generator, the sine wave. But this could just as well have been a utility power waveform taken from a premises branch circuit or one of the signals from the Tektronix Demo 1 board.
With the AFG sine wave fed into one of the analog input channels and displayed, we get a reasonable histogram reading by going to waveform settings and reducing the frequency to 60 Hz. At this point, it may be necessary to press Autoset, depending on what was last done on this oscilloscope. Then we press Measure and then the soft key associated with Waveform Histograms, the fifth menu item from the left below the display.
The waveform histograms menu appears down the side to the right of the display. At the top, we see that waveform histograms are off. Pressing the soft key allows display of either the vertical or the horizontal histogram. The vertical histogram refers to amplitude of the waveform that is the subject of this study. Notice that at the positive and negative peaks of the waveform, the histogram shows maximum and minimum values. If you press More in the Waveform Histograms menu, the histogram can be toggled between linear and log modes. For this particular display, linear is the more revealing. Going to the horizontal display, we see that the histogram, now situated along the time base, correlates not to amplitude, but to the rate of change.
We’ll take a look at another type of waveform. Press AFG to enter the waveform menu and use Multipurpose Knob a to scroll down to Square Wave. Once more we’ll have to adjust the frequency to 60 Hz and press Autoset if necessary. Press Measure as before and turn on vertical and horizontal displays. We see different histograms, again indicative of amplitude and rate of change.
Next on the agenda is Noise, an edifying topic. In the vertical view, the histogram conforms as you might expect to a Gaussian distribution, while in the horizontal view the histogram shows a uniform distribution along the time axis. If Menu Off is pressed so the entire display is visible, it will be seen that the superimposed rectangle represents the area that is depicted by the histogram, subject to constraints imposed by horizontal and vertical limits, which, when selected, may be adjusted by Multipurpose Knobs a and b.
One of the Measure menu items is DVM, for digital voltmeter. It is a useful adjunct to the conventional oscilloscope mode because it lets the user make voltage and frequency measurements of the signal being probed without having to fire up a separate instrument.
When the soft key associated with DVM is pressed, the Digital Voltmeter menu appears to the right of the display. The top item is Mode, which is controlled by Multipurpose Knob a. The default setting is off. Turning Multipurpose Knob a, the settings other than Off are AC + DC RMS (Root Mean Square), DC, AC RMS and Frequency. Each of these has a definite purpose, some going beyond the obvious.
AC + DC RMS is commonly used because it provides a lot of information about any given signal. When the positive and negative peaks are equidistant from a zero-point that is referenced to ground, we say that the dc component is zero. That is a bit of a misnomer because the signal is moving through that point and never stationary. That being said, AC + DC is a useful concept and we may say that there is a “dc component” provided the term is in quotes.
Everyone, at least those in the test and measurement business, knows that root-mean-square refers to how ac voltage is measured and that it is some fraction of peak-to-peak voltage. Actually the concept of root-mean-square arises in many disciplines, everything from actuarial statistics to the physics of gas molecules, even finding its way into the gambling casino.
In an electrical circuit, the RMS value of an alternating current is equal to the amount of direct current that would cause an equal amount of heat to be dissipated passing through a resistive load. As it happens, the RMS value of a continuous waveform is the square root of the arithmetic mean of the squares of the values or the square of the function that defines the continuous waveform.
Suffice to say that the RMS voltage is the metric of interest and is what is generally measured by a voltmeter. There are situations where our interests lie elsewhere, and the Measure modes then become relevant.
The dc mode is useful for checking the output of a battery or dc power supply. When the dc voltage of the sine wave output of our internal AFG is measured, the dc reading is found to be quite low, in the microvolt range.
AC RMS is useful in looking at the output and evaluating an ac power supply. Electrolytic filter capacitors are prone to loss of capacitance and this will manifest as an unacceptable amount of ripple, which can be readily seen in the ac RMS mode of the DVM. The question then becomes how much ripple is unacceptable. The answer varies depending on the size and type of filter network and the intended application. The way to go is to monitor the output ripple in a known good unit and keep records to see if there is an increase.
Finally, the Frequency mode is used to ascertain the frequency of an unknown signal and monitor its stability. And the DVM mode in the Measure function is frequently used and is user friendly.