Frequency counters at one time were considered fundamental measurement tools almost in the class of a voltmeter. They offered the capability of measuring a CW signal with fairly high resolution, depending upon modulation, and were largely used for calibrating oscillators. As frequency counters evolved, instruments were developed with pulse-measurement capability and wider measurement bandwidths capable of handling measurements on modulated waveforms. Modern frequency counters have achieved new levels of performance, with capabilities of characterizing dynamic signals that change frequency over time.

Frequency counters have changed dramatically in design and function over the past decade. Modern counters such as the model CNT-90 from Pendulum Instruments (www.pendulum-instruments.com) employ advanced measurement principles with high sampling speed to provide improved resolution compared to earlier instruments. This particular counter includes a graphical display that shows signal jitter and modulation as well as other key characteristics of a measured signal.

The evolution of frequency counters has been gradual but consistent since the first products were introduced over a quarter century ago. For example, the table shows the changes in frequency resolution over numerous generations of frequency counters. The model CNT-90 is a generation 4 product capable of making 250,000 measurements/s to internal memory and providing 12 digits/s frequency resolution. Since it is a combination timer/counter/analyzer, it also provides the capability of measuring period (time) and signal phase with resolutions of 100 ps and 0.001 deg., respectively.

The most common measurement approach for current-generation frequency counters is a technique known as reciprocal counting. This method is based on measuring the time (clock pulses) between two identical trigger points (trigger level and trigger slope) on the input signal. Between these start and stop trigger events, the instrument counts the number of signal cycles (N) and at the same time the number of clock pulses from the built-in reference oscillator, occurring during time TN. Then, the built-in microprocessor calculates the frequency according to the definition "N/TN" or the period time according to the definition "TN/N," where N is an integer number of cycles of the measured signal from the two trigger points. Fig. 1

In the reciprocal counting technique, the counter measures a multiple period time T_{N} (over exactly N cycles) that is converted to frequency by the microprocessor. The frequency is, of course, the *reciprocal *value of the period time (f = T^{-}1). It should be reinforced that N is an integer, since the measurement (start/stop of both counting chains) is synchronized with the input signal (Fig. 2).** ** But the measurement is instead totally unsynchronized with the clock pulses. As a result, it is possible for parts of clock pulse periods to be counted and parts that are not counted.

For a reciprocal frequency counter, the resolution in a time measurement T_{N} is therefore 1 clock pulse period. Most counters incorporate a 10-MHz reference oscillator, which translates to a time period of 100 ns. This reference provides relative resolution (which is equal to the absolute resolution divided by the measuring time) of seven digits in a measuring time of 1 s (100 ns/1 s = 10^{- }7), irrespective of the input signal frequency.

What if such resolution is not enough for a given application? The 7 digits/s resolution of classic reciprocal frequency counters can be improved in various ways. One approach is to increase the reference clock frequency. By multiplying the time base oscillator's clock frequency from 10 MHz to 100 MHz, for a 10-ns time period, the resolution is also improved. For a measurement time of 1 s, the resolution will be 8 digits (10 ns/1 s = 10^{-}8).

Interpolating frequency counters can bring further improvements to measurement resolution. An interpolating frequency counter starts with the basic capability of a reciprocal counter with its time measurement uncertainty of 1 clock cycle. This is due to the fact that it is not known where in the clock pulse cycle the measurement starts and stops. With a special interpolation circuit, it is possible to determine the phase angle of the clock pulse at the start and stop of the measurement. Two identical interpolators are always in operation at the same time: one for the start trigger event and one for the stop trigger event.

Such an interpolator can be built in various ways. A common implementation is the analog interpolator, where the time difference between trigger event and clock signal is converted to an analog-voltage that can be measured with an analog-to-digital converter (ADC). Through interpolation, the resolution for time measurements is theoretically improved from the 100 ns of the " digital clock pulse counting" approach to (1 clock pulse period)/(interpolation factor for interpolator counting).

In practice, it can be difficult to achieve this kind of precision, since several sources of errors exist that must be accurately characterized and controlled, including interpolator linearity. The interpolation factor usually stays between a factor of 100 and 500. With a clock frequency of 10 MHz, the typical time resolution will be 200 ps to 1 ns (compared with 100 ns for the conventional 10-MHz reference clock), which yields 9 to 10 digits of resolution for a 1-s measuring time. In the model CNT-81 timer/counter/ analyzer, the interpolation approach has been combined with a 100-MHz reference clock oscillator to achieve 50-ps time resolution and 11 digits of frequency resolution for a 1-s measurement time.

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Frequency Measurement

In a reciprocal frequency counter, with or without interpolation, the frequency measurement has a defined starting point (the start trigger event) and a defined stopping point (the stop trigger event). Between each starting point and stopping point of a measurement, there is an unavoidable dead time for transferring results to an output port, resetting the registers, and preparing the instrument's electronic circuitry for the next measurement. This scenario does not apply to continuously time stamping counters, however (Fig. 3).

Continuously time-stamping counters count the input signal trigger events and the time *continuously without interruptions*. In principle, counting continues indefinitely from a starting point until the counting chain overflows and starts from again from zero. With regular intervals (pacing intervals) it is possible to read the momentary content in the respective registers as well as the instantaneous interpolation values, without interrupting the operation of the counter. The values (trigger events or "Events" and time or "Time stamps") are stored in high-speed memory. The readout is always synchronized with the input trigger. That is, the number of counted input cycles is always an exact number. Just as in reciprocal counters, any uncertainty results from the imprecision of accurately monitoring time.

When a certain number of events have been storedor when a certain time has elapsedthe data will be analyzed, post-processed, and presented to an operator. While a common reciprocal counter simply delivers a final result after a frequency measurement performed over a certain time interval, such as 1 s, a continuously time-stamping counter saves and provides information on hundreds or thousands of timestamped events over that same interval (Fig 4).

A traditional reciprocal counter contains only one value at the end of the measurement period, after starting with an initial value of 0.0 at the start of the measurement. A continuously timestamping counter provides two axes of information: input cycles (events) on the x-axis and time (time stamps) on the y-axis. The inclination of the plotted line between the measurement start and stop points is the average period (total time divided by the number of input cycles). The resolution of the measurement, i.e., the uncertainty of the line's inclination, depends only on the uncertainty of the start and stop points used in the measurement. In time-stamping counters, a large number of events are represented along the straight line of a plot, each with a certain resolution.

By applying a well-known statistical methodlinear regressionit is possible to improve the resolution of the line's inclination (the time of the measurement period) by a factor of (N)^{0.5}/2.4, where N is the number of measurement points during a given measurement time. For example, with 1000 points, the resolution is improved by a factor of (1000)^{0.5}/2.4 = 31.622/2.4 = 13.176. (Fig. 5) and (Fig. 6) show how the resolution (or uncertainty of the line's inclination) can be improved through the application of linear regression compared to startstop measurements (see sidebar).

Continuously time-stamping frequency counters with linear regression can improve frequency resolution by using a large number of "intermediate" values between the measurement start and stop points. There are additional benefits to this measurement method, since it is two-dimensional in nature and provides an authentic time scale that can be referenced to individual trigger events in time relative to one another (classic reciprocal counters can only give individual values, without a mutual time relationship). Making measurements with an authentic time scale makes it possible to show frequency changes over time, compared to a traditional counter that makes measurements only in one dimension. This is analogous to the measurement results from a voltmeter when compared to measurements with an oscilloscope's time-varying voltage display. The authentic time scale of a continuously time-stamping frequency counter also makes it possible to perform improved postprocessing analysis of measurement data, such as a Fast Fourier Transform (FFT).

A continuously time-stamping frequency counter such as the CNT-90 (Fig. 7)** ** allows single period measurements to be made "back to back," without dead time between measurements. This capability is important when trying to detect "missing periods" and when trying to follow all cycles in a sequence. Applicationsfor this capability are diverse, from evaluation of serial data communications systems and components to the testing of mechanical rotational sensors. At best, a traditional reciprocal counter will only measure every other period, resulting in a practical measurement uncertainty (for these applications) on the order of 50 percent. By eliminating the dead time of reciprocal counters, a continuously timestamping frequency counter can calculate the Allan variance of a high-frequency oscillator, an essential measure of phase noise and frequency stability.