The CZT X(k) of an N-point sequence x(n) for n = 0, 1, 2, …, N−1 is given by Equation (1). In Equation (1), X(k) allows computing the frequency contents of x(n), sampled at a frequency rate of f_s, at a dense set of L frequencies in the range covered by the arc of the unit circle that begins at ω₀ = 2π f₀ and ends at ω₁ = 2π f₁ [17]: (1) X ( k ) = ∑ n = 0 N − 1 x ( n ) Z L kn

In Equation (1) the transformation kernel Z L kn is given by (2). (2) Z L kn = exp { − j 2 π   n f s [ f 0 + ( f 1 − f 0 ) k L ] } = cos ( ω n ) − jsen ( ω n ) where     ω = 2 π f s [ f 0 + ( f 1 − f 0 ) k L ]   and     k = 0 , 1 , … , L − 1

The transformation kernel Z L kn can be implemented as two discrete recursive functions described by Equations (3) and (4) for its real Z_R and imaginary Z_I components, respectively: (3) Z R   ( n ) = Z R   ( n − 1 )   cos ( ω n ) − Z I   ( n − 1 )   sin ( ω n ) (4) Z I   ( n ) = Z I   ( n − 1 )   cos ( ω n ) + Z R   ( n − 1 )   sin ( ω n )

The power spectrum of the CZT X(k) is given by (5): (5) [ X ( k ) ] 2 = [ X R ( k ) ] 2 + [ X I ( k ) ] 2

From Equation (5), the minimum value of [X(k)]² at which the CZT converges, on the high-resolution power spectrum, corresponds to the main frequency component of the discrete signal x(n). The frequency resolution Δ f of the CZT power spectrum depends on the length of the analyzed unit circle arc, and the number of frequency elements covered in this length L, as given in Equation (6): (6) Δ   f = ( f 1 − f 0 ) L

The operational architecture for the CZT computation unit of the proposed smart sensor is depicted in Figure 2. The transformation kernel block provides the real and imaginary components Z_R and Z_I of Z L kn implementing the recursive functions in Equations (3) and (4), respectively. The power spectrum magnitude [X(k)]² is obtained by adding the squared accumulation of Z_R and Z_I times the discrete input signal x(n). Finally, a magnitude comparison is carried out in order to identify the signal frequency with high resolution. The structure in Figure 2 for the CZT computation utilizes basic operations like addition, multiplications, and accumulations that provide an efficient architecture for hardware implementation.

In order to find an optimal algorithm that meets the international standard for frequency measurement [2], a computational complexity comparison between CZT, FFT, and Zoom-FFT, based on the number of operations, is developed in this section. The number of operations Op_FFT for computing the FFT is given by Equation (7) [17]; whereas for computing the Zoom-FFT the number of operations Op_ZFFT is given by Equation (8) [18]; finally the CZT requires Op_CZT operations, as given by Equation (9) [17]. N is the data length for all algorithms, and L is the length of the interest region for the Zoom-FFT and the CZT: (7) Op FFT = N log 2   N (8) Op ZFFT = N ( 1 + log 2   L ) (9) Op CZT = N ⋅ L

Table 1 summarizes the computational complexity and acquisition time T_acq required for complying international standard of frequency measurement with a resolution of Δ f = 0.01 Hz. The resolution Δ f for the FFT and the Zoom-FFT is given by Equation (10) with a sampling rate set to f_s = 655.36 Hz; whereas the resolution for the CZT is given by Equations (2) and (6) with a sampling rate set to f_s = 512 Hz in order to meet the standard. (10) Δ   f = f s N

From Table 1 it can be seen that the CZT requires less operations for computation than FFT and Zoom-FFT, but the most restricting parameter is the acquisition time that according to the standard is 10 s. which is only met by CZT.

The analyzed pure periodic signal is described in Equation (11), with normalized amplitude, and frequency f = 60 Hz, as shown in Figure 3a. Analyzing the artificially generated pure periodic signal and utilizing the proposed smart sensor for high-resolution frequency estimation, the CZT power spectrum in Figure 3b is obtained, giving the frequency estimation result of 60.0000 Hz for x(t): (11) x ( t ) = sin ( 2 π ft )

The periodic signal contaminated with white noise is described in Equation (12). The amplitude of the periodic signal is normalized, and it has a frequency f = 60 Hz, whereas three cases of severe white noise n(t) contamination are considered. The first case considers a normalized periodic signal with white noise contamination at 10% of its amplitude for a SNR = 17.0 dB, as shown in Figure 4a. The second case, shown in Figure 4b, analyzes a normalized periodic signal with white noise contamination at 20% of the signal amplitude for a SNR = 11.0 dB. Finally, the third case considers a white noise contamination at 30% of the periodic signal amplitude for a SNR = 7.4 dB, presented in Figure 4c. Table 2 shows the obtained result from the proposed smart-sensor frequency estimation for 40 runs of the periodic signal with added white noise. (12) x ( t ) = sin ( 2 π   f   t ) + n ( t )

The harmonic contamination of the periodic signal is described in Equation (13). Three different cases of harmonic contamination are treated. The first case considers a periodic signal contaminated with its 2^nd harmonic, as depicted in Figure 5a. The second case analyzes the periodic signal plus its 3^rd harmonic, as shown in Figure 5b. The last case considers the contamination of the periodic signal with its 3^rd + 5^th harmonics, as presented in Figure 5c. The amplitude of the periodic signal is normalized and has a frequency f = 60 Hz, whereas the harmonic signals h(t) added to the periodic signal have an amplitude of 10% of the main periodic signal amplitude for a signal-to-harmonic ratio SHR = 20.0 dB. Table 3 shows the obtained result from the proposed smart-sensor frequency estimation for 40 runs of a periodic signal with harmonic contamination. (13) x ( t ) = sin ( 2 π   f   t ) + h ( t )   where h ( t ) = 0.1 sin ( 2 π   f   l   t )   and     l = 2 ,   3 ,   5

The white noise n(t) and harmonic h(t) contamination of a periodic signal is described in (14), where the amplitude of the periodic signal is normalized, and has a frequency f = 60 Hz, whereas the white noise signal n(t) added to the periodic signal have a maximum amplitude of 10% of the main periodic signal amplitude and three different cases of harmonic contamination are considered. The first case considers a periodic signal contaminated with its 2^nd harmonic, as depicted in Figure 6a. The second case analyzes the periodic signal plus its 3^rd harmonic, as shown in Figure 6b. The last case considers the contamination of the periodic signal with its 3^rd + 5^th harmonics, as presented in Figure 6c. Where the signal-to-harmonic-plus-noise ratio SHNR = 15.2 dB. Table 4 shows the obtained result from the proposed smart-sensor frequency estimation for 40 runs of a periodic signal with white noise plus harmonic contamination. (14) x ( t ) = sin ( 2 π   f   t ) + h ( t ) + n ( t )   where h ( t ) = 0.1 sin ( 2 π   f   l   t )   and     l = 2 , 3 , 5

Figure 7 shows the effects of an instantaneous jump in the power line frequency over the proposed smart sensor readouts. The input signal was originally synthesized as a pure sinusoidal with a frequency of 60.0 Hz and at half of the measurement process the frequency was suddenly changed to 60.6 Hz, representing a 1% instantaneous jump. The international standard for frequency measurement [2], requires to give a readout every 10 s; the processing of the signal takes 1.0078 s and the instrument was set to provide a readout every 2 s. Figure 7a compares the input signal frequency against the smart-sensor readouts before and after the 1% frequency jump during the 10 s of the measurement required by the standard. Before the jump the frequency readout of the smart-sensor is 60.0 Hz as expected; during the jump, the readout is 60.18 Hz; once the signal frequency stabilizes, the readout is 60.6 Hz that corresponds to the magnitude of the jump. The final readout of the smart sensor follows the frequency change in the standard required time.

The proposed smart sensor for frequency measurement can be utilized with several primary sensors. For this study case, the standard off-the-shelf current clamp i200s from Fluke is utilized for current monitoring. A 12-bit 4-channel serial-output analog to digital converter ADS7841 [21] is used for signal acquisition. Figure 8 shows the experiment setup for high-resolution frequency measurement monitoring the power-line-supply current from a standard electrical load (1 hp induction motor) with a current clamp as primary sensor. The current signal is sensed during 1 s at a sampling frequency of 512 Hz. The signal conditioning system provides the FPGA implementation of the proposed smart sensor with 512 discrete samples for high-resolution frequency estimation, the result is shown on the 4-digit 7-segment LED display, where the user selects between the high- and low-part of the estimation result to be shown on the display. For the case treated here 60.002734 Hz. Figure 9 shows the sensed current signal from the power line. By taking 40 readouts of the power-line frequency, the calculated mean and standard deviation are 60.0008 Hz and 0.0013 Hz, respectively.

In this case the frequency is estimated from voltage signals generated by a Stanford Research DS345 synthesized function generator. Similar to the previous case of study, a 12-bit 4-channel serial-output analog to digital converter ADS7841 is used for signal acquisition. Figure 10 shows the experiment setup for high-resolution frequency measurement, monitoring voltage signals. The synthesized voltage signal shown in Figure 11 is sampled during 1 s at a sampling rate of 512 Hz. The signal conditioning system provides the FPGA implementation of the proposed smart sensor with 512 discrete samples for high-resolution frequency estimation. The estimation result is shown on the 4-digit 7-segment LED display, where the high- and low-part of the estimation result can be selected for being shown on the display. For the case treated here, the frequency estimation of a synthesized pure sinusoidal signal with a frequency of 59.973 Hz is 59.972656 Hz.

To test the performance of the proposed smart sensor, three experiments are set up considering a 60 Hz sinusoidal signal generated by the DS345 function generator with white noise contamination at 10%, 20%, and 30% of its amplitude for a SNR = 17.0, 11.0, 7.4 dB, and as shown in Figure 12a–c, respectively. Table 5 summarizes the obtained results for 40 readouts from the proposed smart sensor.

A test of linearity to the proposed smart sensor for high resolution frequency estimation is carried out in this section. Figure 13 shows the proposed smart sensor behavior under small variations of the input signal frequency. The input signal frequency increases from 59.3 Hz in steps of 0.05 Hz up to 60.7 Hz obtaining the correspondence line in Figure 13; which shows a linear behavior of the proposed smart sensor. The frequency variation range from 59.3 Hz to 60.7 Hz corresponds to the international standard CEI/IEC 61000-4-30 [2]; which states that the power frequency of the supply signal must vary around ±1% (from 59.4 Hz to 60.6 Hz). The results show that the proposed smart sensor provides a very accurate solution with an uncertainty of ±0.0024 Hz, with a maximum estimation error of 0.0055 Hz and an integral linearity of 0.189 %.