Instrumentation & Measurement Magazine 23-5 - 42

obtain the accurate frequency values of the low SNR FID signals is the key point to increase the precision of the PM. This
makes the research on high-precision frequency measurement
methods in limited time frames, for low SNR sine signals, very
important.
Counter method and its variants are the commonly used
method for frequency measurement [10]-[14], which include the MFM method, the discrete phase-delay detection
method, and the frequency measurement method based on
delay chains, and so on. The principle of these methods is
that the measured sine signal is converted to square wave
first at a comparator, and then the frequency of square wave
is measured by the method mentioned above. The precision of these methods is high enough if there is no noise in
the signal, and the method error is the main error source
for frequency measurement. However, for low SNR signal
such as FID signal, the performance will be decreased because the trigger error caused by noise is much larger than
the method error.
To increase measurement precision and speed of proton magnetometer, there are some reports for improving
the measurement precision of the FID signal. Dong [15] described a delay multi-channel parallel measurement (MPM)
method for increasing Overhauser magnetometer precision,
and the precision is improved because of adopting the average calculation. However, the rest of rising edge and all
falling edge of the square wave are missed because only partial rising edge of square wave is captured, and furthermore,
only ten times the average calculation is adopted. J. Ge [16]
developed a multichannel interpolation frequency measurement method based on FPGA for Larmor frequency of the
marine Overhauser sensor. The more channels present, the
higher the precision is. Nevertheless, the number of channels
is limited by the resource of FPGA. S. Zhang [17] introduced
a software zero crossing frequency measurement algorithm
to estimate the FID signal frequency. But only two points are
used in the mentioned method and the voltage zero crossing time cannot be obtained precisely. A Prony, SVD and
FDM method were proposed in [18], [19] and [20], respectively, to deal with FID signal frequency measurement. The
simulations showed that it took over ten seconds to obtain
a high-precision measurement. A high-order Prony Method
was presented in [21] to estimate the FID signal of NMR,
and the time consumption is longer than the common Prony
method.

Proposed Method
Multi-Average MFM Principle
Assume that the counting number of the frequency to be measured is n when one multi-cycle frequency measurement
(MFM) method is performed. For instance, the n cycles of the
measured frequency fx are selected from the first rising to the
n−th, which is shown as Fig. 2. Then the fx can be expressed as:


fr 1 (n k1 )  fs (2)

Fig. 2. The principle of the M-MFM method using Rising-Falling Edge. RE
stands for Rising Edge, FE stands for Falling Edge.

where fr1 is the first result when the MFM method is performed
for the rising edge of square wave, and k1 is the counting number of standard frequency fs at the first time the MFM method
is performed.
Similarly, the second result fr2 and the i−th result fri can be
written directly according to (2) and Fig. 2:


fr 2 ( n k2 )  fs (3)


fri (n ki )  fs (4)

Assume that the largest number of i is m, which indicate
that the MFM method is performed m times for the rising edge
of the square wave. Then, the result of average calculation for
rising edge can be expressed as follows:
m


fr

f

ri
n m 1

f
m
m s i 1 ki

i 1

(5)

Similarly, if the falling edge of the square wave is adopted
to calculate frequency in the same way as rising edge, and the
performed times is the same as the rising edge, too, then the
result of average calculation for the falling edge can be expressed as:
m


ff

f

fi
n m 1

f
m
m s i1 zi

i 1

(6)

where zi is the counting number of the standard frequency fs
while the i−th MFM method of falling edge is performed.
Then, the average frequency of both rising and falling edge
of the square wave is:

fx


m
fr  f f
n
1 1
f (  )

2
2 m s i1 ki zi

(7)

Ordering N = n + m, N is the total counting number of the
frequency to be measured. Generally, the rate of frequency
measurement based on the counter method is directly decided

IEEE Instrumentation & Measurement Magazine

August 2020

Instrumentation & Measurement Magazine 23-5

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