Research on Cosine-Sum Windows with Maximum Side-Lobe Decay for High Precision ADC Spectral Testing

Abstract

Achieving coherent sampling has always been a major challenge in analog-to-digital converter (ADC) spectral testing. If the coherent sampling condition cannot be met, leakage appears in the spectrum, which results in inaccurate parameters. The windowing method is widely used to process ADC data to eliminate leakage. However, this method requires prior knowledge about the type of window, and some commonly used windows cannot provide accurate results for a high precision ADC. In this paper, some general principles for optimizing the cosine-sum window and some windows with maximum side-lobe decay have been presented. A test method for eliminating the leakage caused by non-coherent sampling is also proposed. The proposed method can accurately evaluate the dynamic parameters of an ADC with arbitrary non-coherency. Various simulation results and measurement data demonstrate the functionality and robustness of the proposed method. The proposed method significantly relaxes the condition of coherent sampling and decreases the test cost.

1. Introduction

The analog-to-digital converter (ADC) is one of the most widely used mixed-signal circuits. With the rapid development in semiconductor processing technology, the performance of ADCs has increased dramatically. Spectral testing is a commonly used method to estimate the dynamic parameters of ADCs, including the signal-to-noise ratio (SNR), signal-to-noise-and-distortion ratio (SINAD), total harmonic distortion (THD) and spurious-free dynamic range (SFDR) [1,2,3]. Obtaining accurate test results of a high precision ADC is a significant challenge in modern industry. In order to perform spectral testing, the IEEE standard [4,5,6] provides several stringent requirements that need to be met. The input signal should be coherently sampled to obtain accurate test results, otherwise, the leakage will appear in the spectrum. It is challenging to achieve coherent sampling for a high precision ADC. Especially for a built-in self-test (BIST) [7,8,9,10,11], it is difficult to integrate a high-performance signal and clock generator due to the limitation of the chip area. Therefore, there is an urgent demand for a testing method that eliminates the requirement of coherent sampling.

In the past decade, various test methods have been proposed to solve the issue, such as the interpolated discrete Fourier transform (IpDFT) [12,13] techniques, the four parameter sine fitting techniques [14,15] and the FIRE method (fundamental identification and replacement) [16,17,18,19]. The IpDFT methods perform interpolation in the frequency domain to eliminate the leakage, but this method cannot obtain an accurate value of the SFDR when a non-harmonic spur dominates the distortion. The four parameter sine fitting methods cannot provide accurate parameters when the maximum spur happens to be a non-harmonic component. The FIRE method (fundamental identification and replacement) shows excellent performance for eliminating any level of non-coherency. Nevertheless, this method consumes more testing time, especially when the sampled data length increases [20].

The windowing method [21,22,23,24,25] is computationally efficient. However, this method needs the prior knowledge of window function selection, which involves a trade-off of several factors, such as the main lobe width, side-lobe decay rate and the largest side-lobe level. A general principle is that the largest side-lobe level of the selected window should be lower than the noise power (noise floor) of the ADC under testing. In addition, some popular windows cannot achieve accurate results for a high precision ADC, such as Hamming and Blackman-Harris.

In this paper, a general method for optimizing and selecting an appropriate window function is proposed to estimate the parameters without requiring coherent sampling. This method relaxes the test environment’s stringent requirement and significantly reduces the test cost. The functionality and robustness of the proposed method are verified using both simulation and measurement results. The rest of the paper is organized as follows. Section 2 describes the standard spectral testing and the non-coherency issue. In Section 3, some general principles to optimize cosine-sum windows and the method to test high precision ADCs are proposed. Section 4 and Section 5 present extensive simulation and experimental results to validate the proposed method. Finally, this paper is concluded in Section 6.

2. ADC Spectral Testing

Spectral testing has been widely used to evaluate the performance of ADCs. For ADC standard spectral testing, the IEEE standards recommended test setup to obtain accurate parameters is shown in Figure 1. The master clock controls the frequency of the input signal and the sample clock to achieve coherent sampling. A pure sinusoidal signal is applied to the ADC as the stimulus. The ADC output data are sampled and collected to analyze.

The standard spectral testing can be described by the following formula. The input sinusoidal signal x(t) can be written as:

(1)$\mathrm{x}(\mathrm{t})={\mathrm{A}}_1\cos (2\pi \cdot {\mathrm{f}}_{\mathrm{in}}\cdot \mathrm{t}+{\mathsf{\varphi }}_1)$

where ${\mathrm{A}}_1$, ${\mathrm{f}}_{\mathrm{in}}$ and ${\mathsf{\varphi }}_1$ represent the amplitude, frequency and initial phase of the input signal, respectively.

After the input signal x(t) is sampled and quantified by the ADC under test, the output data x[n] is given by

(2)$\mathrm{x}[\mathrm{n}]={\mathrm{A}}_0+{\mathrm{A}}_1\cos (2\pi \cdot \frac{{\mathrm{f}}_{\mathrm{in}}}{{\mathrm{F}}_{\mathrm{s}}}\cdot \mathrm{n}+{\mathsf{\varphi }}_1)+{\displaystyle \sum _{\mathrm{h}=2}^{\mathrm{H}}{\mathrm{A}}_{\mathrm{h}}\cos (2\pi \cdot \mathrm{h}\frac{{\mathrm{f}}_{\mathrm{in}}}{{\mathrm{F}}_{\mathrm{s}}}\cdot \mathrm{n}+{\mathsf{\varphi }}_{\mathrm{h}})}+\mathrm{Noise}[\mathrm{n}]$

where ${\mathrm{F}}_{\mathrm{s}}$ is the sampling frequency. ${\mathrm{A}}_0$ is the DC component. ${\mathrm{A}}_{\mathrm{h}}$ and ${\mathsf{\varphi }}_{\mathrm{h}}$ represent the amplitude and initial phase of the h-th harmonic, respectively. H is the total number of harmonics. n = 0, 1, 2, … M − 1 and M represents the total number of output data. Noise[n] is the white noise.

Then performing discrete Fourier transform (DFT) of M to obtain the spectrum of ADC under test. The DFT of x[n] is given by

(3)$\mathrm{X}[\mathrm{k}]=\frac{1}{\mathrm{M}}{\displaystyle \sum _{\mathrm{n}=0}^{\mathrm{M}-1}\mathrm{x}[\mathrm{n}]\cdot {\mathrm{e}}^{-\mathrm{j}\frac{2\pi }{\mathrm{M}}\mathrm{n}\mathrm{k}}}$

where k = 0, 1, 2, … M − 1 represents the frequency bin’s index. The dynamic parameters could be accurately calculated from Equation (3) if the data x[n] is coherently sampled.

Let J be the number of periods of the input signal. The coherent sampling condition is represented by

(4)$\frac{{\mathrm{f}}_{\mathrm{in}}}{{\mathrm{F}}_{\mathrm{s}}}=\frac{\mathrm{J}}{\mathrm{M}}$

When the coherent sampling condition is satisfied, the power of each component can be accurately calculated from the spectrum such as fundamental, noise and distortion. If the input signal is not coherently sampled, the number of periods J in Equation (4) is not an integer, and severe leakage will be presented in the spectrum as shown in Figure 2.

3. The Maximum Side-Lobe Decay Windows

If the coherent sampling condition cannot be met, the windowing method is widely used to reduce spectral leakage [21,22,23,24,25]. Engineers should know how to select an appropriate window function and multiply the ADC output data by the window in the time domain. Choosing an optimal window requires prior knowledge about the performance of the window. The ideal window should have a narrow main lobe, low side-lobe level and rapid decay rate. A narrower main lobe can increase spectral resolution, and a lower level and faster decay rate of the side-lobe can significantly suppress the leakage caused by non-coherent sampling.

For an ideal N-bit ADC, the noise floor level is the sum of SNR and process gain. It can be represented by the signal-to-quantization-noise ratio (SQNR), which is given as

(5)$\mathrm{SQNR}=6.02\cdot \mathrm{N}+1.76+10\mathrm{l}\mathrm{o}{\mathrm{g}}_{10}(\mathrm{M}/2)$

where N is the resolution of ADC, and M is the data length. SQNR implies that the input signal amplitude covers the full dynamic available for the conversion. In ADC spectral testing, the conventional windowing method requires that the largest side-lobe level of the window should be lower than the noise floor of the ADC by some margin. Then, the spectral leakage can be completely suppressed within the range of the main lobe, and the distortion will be accurately detected. A minimum side-lobe window is commonly used to eliminate the leakage, such as Hamming, Blackman, Exact Blackman, etc. These windows have a low side-lobe level with the same number of terms. However, for a high-resolution ADC, few windows can provide enough side-lobe attenuation. For example, the side-lobe level of the Blackman window (3-term) allows engineers to test a 12-bit ADC accurately. However, this window is not proper for a 16-bit or higher resolution ADC as the largest side-lobe level of the window cannot be higher than the ADC noise floor. The 7-term Blackman-Harris window has enough attenuation level for a 24-bit resolution ADC, the largest side-lobe level of which is −163 dB. Nevertheless, this complex window will take more test time.

If the window is optimized in the aspect of the side-lobe decay rate, it can achieve the maximum side-lobe decay rate. The cosine-sum windows with maximum side-lobe decay (MSLD) will be a better choice for ADC spectral testing, although these windows’ largest side-lobe level will be higher. If the window has the same number of terms, the MSLD window has better performance for suppressing leakage. The derivation of the minimum side-lobe window was introduced in [21,24], and is not repeated in this paper. The expression for the MSLD window is derived as follows.

The cosine-sum window is generally given by

(6)$$\begin{gathered} \mathrm{w}(\mathrm{t})={\displaystyle \sum _{\mathrm{k}=0}^{\mathrm{K}-1}{\mathrm{a}}_{\mathrm{k}}\mathrm{c}\mathrm{o}\mathrm{s}(\frac{2\pi }{\mathrm{N}}\mathrm{k}\mathrm{t}),|\mathrm{t}|\le \mathrm{N}/2} \\ =0,\left|\mathrm{t}\right|>\mathrm{N}/2 \end{gathered}$$

and

(7)$\mathrm{w}(0)={\displaystyle \sum _{\mathrm{k}=0}^{\mathrm{K}-1}{\mathrm{a}}_{\mathrm{k}}=}1$

where N, K and ${\mathrm{a}}_{\mathrm{k}}$ are the length, the total term number and the window coefficient, respectively. The w(t) and its derivative are symmetrical about w(0). For convenience, the range of t is omitted.

Let W(f) be the Fourier transform of w(t), which determines the frequency property of the window function, such as the main lobe width and side-lobe level.

(8)$\mathrm{W}(\mathrm{f})={\displaystyle \sum _{-\mathrm{N}/2}^{\mathrm{N}/2}\mathrm{w}(\mathrm{t})\cdot {\mathrm{e}}^{-\mathrm{j}2\pi \cdot \mathrm{f}\mathrm{t}}\mathrm{d}\mathrm{t}}$

The value of w(t) at the end is given by

(9)$\mathrm{w}(\pm \frac{\mathrm{N}}{2})={\displaystyle \sum _{\mathrm{k}=0}^{\mathrm{K}-1}{(-1)}^{\mathrm{k}}{\mathrm{a}}_{\mathrm{k}}}$

The value of w(t) and its derivative at t = $\pm$N/2 are crucial. ${\mathrm{w}}^{(\mathrm{n})}(\pm \mathrm{N}/2)$ determines the frequency property of the window function. If both ends of w(t) are not zero, w(t) is discontinuous for all t. In this case, the side-lobe decay rate is relatively slow, but the lowest first side-lobe will be obtained after adjusting the coefficient. If both ends of w(t) are zero, w(t) is continuous for all t, and w′(t) is obtained.

(10)${\mathrm{w}}^{\prime }(\mathrm{t})=-\frac{2\pi }{\mathrm{N}}{\displaystyle \sum _{\mathrm{k}=0}^{\mathrm{K}-1}\mathrm{k}{\mathrm{a}}_{\mathrm{k}}\mathrm{s}\mathrm{i}\mathrm{n}(\frac{2\pi }{\mathrm{N}}\mathrm{k}\mathrm{t})}$

and

(11)${\mathrm{w}}^{\prime }(\pm \frac{\mathrm{N}}{2})=-\frac{2\pi }{\mathrm{N}}{\displaystyle \sum _{\mathrm{k}=0}^{\mathrm{K}-1}\mathrm{k}{\mathrm{a}}_{\mathrm{k}}\mathrm{s}\mathrm{i}\mathrm{n}}\left(\mathrm{k}\pi \right)=0$

It can be seen that w′(t) is always continuous (if w′(t) exists). Thus w″(t) is obtained as follows

(12)${\mathrm{w}}^{″}(\mathrm{t})=-{(\frac{2\pi }{\mathrm{N}})}^2{\displaystyle \sum _{\mathrm{k}=0}^{\mathrm{K}-1}{\mathrm{k}}^2{\mathrm{a}}_{\mathrm{k}}\mathrm{c}\mathrm{o}\mathrm{s}(\frac{2\pi }{\mathrm{N}}\mathrm{k}\mathrm{t})}$

As mentioned before, the continuity of w″(t) determines the frequency property of the window function. If both ends of w″(t) are zero, w″(t) is continuous for all t, and W(f) has a faster side-lobe decay rate.

Assuming w(t) has an n-order derivative, the n-order derivative of w(t) can be represented by

(13)${\mathrm{w}}^{\left(\mathrm{n}\right)}(\mathrm{t})={(\frac{2\pi }{\mathrm{N}})}^{\mathrm{n}}\cdot {\displaystyle \sum _{\mathrm{k}=0}^{\mathrm{K}-1}{\mathrm{k}}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{k}}\mathrm{c}\mathrm{o}\mathrm{s}(\frac{2\pi }{\mathrm{N}}\mathrm{k}\mathrm{t}+\frac{\pi }{2}\mathrm{n})}$

where n is the derivative order. This equation could be divided into odd and even derivatives as follows

(14)${\mathrm{w}}^{\left(2\mathrm{m}\right)}(\mathrm{t})={(-1)}^{\mathrm{m}}{(\frac{2\pi }{\mathrm{N}})}^{2\mathrm{m}}\cdot {\displaystyle \sum _{\mathrm{k}=0}^{\mathrm{K}-1}{\mathrm{k}}^{2\mathrm{m}}{\mathrm{a}}_{\mathrm{k}}\mathrm{c}\mathrm{o}\mathrm{s}(\frac{2\pi }{\mathrm{N}}\mathrm{k}\mathrm{t}),\mathrm{n}=2\mathrm{m}}$

(15)${\mathrm{w}}^{(2\mathrm{m}+1)}(\mathrm{t})={(-1)}^{\mathrm{m}+1}{(\frac{2\pi }{\mathrm{N}})}^{2\mathrm{m}+1}\cdot {\displaystyle \sum _{\mathrm{k}=0}^{\mathrm{K}-1}{\mathrm{k}}^{2\mathrm{m}+1}{\mathrm{a}}_{\mathrm{k}}\mathrm{s}\mathrm{i}\mathrm{n}(\frac{2\pi }{\mathrm{N}}\mathrm{k}\mathrm{t}),\mathrm{n}=2\mathrm{m}+1}$

It can be observed that the odd derivative is always continuous. Thus the decay rate is determined by the even derivative. If the n-order derivative of w(t) is discontinuous and the (n − 1)-order derivative is continuous for all t, the side-lobe decay rate of W(f) is 6*(n + 1) dB/Octave. For example, w(N/2) of the Hamming window is 0.08 and it is discontinuous. Thus n is equal to zero, and the decay rate of Hamming is 6 dB/octave. For the Hanning window, w(N/2) and w′(N/2) are zero, but w″(t) is not continuous. Thus n is equal to 2, and the decay rate of Hanning is 18 dB/octave.

Therefore, in order to obtain a faster decay rate, it is necessary to satisfy both ends of the n-th derivative equal to zero. At the same time, Equation (7) also needs to be met. Thus the following equation can be obtained

(16)${\mathrm{w}}^{\left(2\mathrm{m}\right)}(\pm \frac{\mathrm{N}}{2})={(-1)}^{\mathrm{m}}{(\frac{2\pi }{\mathrm{N}})}^{2\mathrm{m}}\cdot {\displaystyle \sum _{\mathrm{k}=0}^{\mathrm{K}-1}{\mathrm{k}}^{2\mathrm{m}}{\mathrm{a}}_{\mathrm{k}}=0}$

and

(17)$\mathrm{w}(0)={\displaystyle \sum _{\mathrm{k}=0}^{\mathrm{K}-1}{\mathrm{a}}_{\mathrm{k}}=}1$

If the total term number K is determined, the coefficient of the MSLD window can be calculated from these equations.

For the same number of terms, the MSLD window has better performance for eliminating leakage at the cost of a higher first side-lobe. The two side-lobes closest to the main lobe can be regarded as the main lobe. The main lobe width of the window is determined by the term number K; it becomes wider as K increases, and then the frequency resolution decreases. The two ends of the window become much smoother as the number of orders n increases.

When the value of K is determined, the coefficient of the MSLD window can be calculated as shown in Table 1. The time domain and frequency domain of 2–5 term MSLD windows are also shown in Figure 3.

It can be seen that if the term number increases, the MSLD windows become smoother, the main lobe width increases and the side-lobe level decreases. In order to compare the performance of MSLD windows with other classic cosine-sum windows, the frequency domains of 2-term and 3-term MSLD windows and some classic cosine-sum windows are illustrated in Figure 4. The summary of classic window characteristics is presented in Table 2.

Although the largest side-lobe level is high, the side-lobe decay rate of the MSLD window is faster with the same number of terms. This property is important for eliminating leakage and detecting the component in the spectrum. Therefore, the test results can be accurately obtained with non-coherent sampled data.

4. Simulation Results

In this section, simulation results are presented to verify the functionality and robustness of the proposed windows.

4.1. Functionality

The simulation setup in MATLAB is given as follows. An 18-bit ADC is generated and the true parameters of the ADC are obtained by sending a pure sine wave, which is coherently sampled. DNL (Differential Non-linearity) and noise obey the normal distribution, the variances of which are 0.01 LSB (Least Significant Bit) and 0.5 LSB, respectively. The length of data M is 8192 and the number of periods J is 593. The input signal is set to be 98% of the ADC’s full range to avoid clipping. Then, the non-coherently sampled data with J = 593.12 is used to verify whether the proposed window accurately obtains the parameters. According to the proposed principle, the 3-term MSLD window is a better choice for 18-bit ADC. The spectra of the ADC with different windows are shown in Figure 5 and the parameters obtained are also illustrated in Table 3.

Figure 5 shows the spectra of the same ADC obtained using different methods, including directly performing DFT on the coherent and non-coherent data, Hamming, 2-term MSLD (Hanning), Exact Blackman and 3-term MSLD. The blue spectrum is obtained when the input signal is coherently sampled, and this spectrum is clean without any leakage. The red spectrum is obtained when DFT is directly performed on the non-coherent data, and severe leakage appears in the spectrum. In order to eliminate the leakage, different windows are used. It can be seen that the level of leakage suppression is strongly dependent on the selected window function. If the number of terms is identical, the MSLD window will achieve more accurate results.

The test results of the above different windows are listed in Table 3. It can be seen that the results obtained by 3-term MSLD windows are very close to the true parameters. It demonstrates that the proposed MLSD windows can accurately estimate the performance of an ADC even if the input signal is not coherently sampled.

4.2. Robustness

The robustness of the proposed method with different non-coherency is also presented. The fractional part of J is $\delta$, and 1000 values of $\delta$ are randomly generated. The values of $\delta$ range from −0.5 to 0.5. The selected window function is the 3-term MSLD window. The errors obtained in estimating the parameters of the ADC with different $\delta$ are shown in Figure 6. It can also be seen that the proposed method is robust over the whole range of $\delta$. Hence, the method can achieve accurate testing results in the arbitrary level of non-coherency. The robustness eliminates the requirement of coherent sampling for spectral testing.

In order to compare the estimation error of the different windows, a classic Blackman-Harris window (3-term) is used to suppress the leakage. The simulation is performed 100 times, and the value of $\delta$ is randomly generated to verify the functionality of the proposed method. For each simulation, the estimation error of SINAD and SNR (absolute value) are plotted in Figure 7 for both the Blackman-Harris window (red) and the 3-term MSLD window (blue). It can be seen that the estimation error of the proposed window is smaller.

4.3. Computation Time

To demonstrate the efficiency of the proposed window, the comparison of the computation time with different methods is also presented in Table 4. In the simulation, the proposed 3-term MSLD window is compared with the Hamming and Exact Blackman windows with the method in [14,17]. The computation time is the time of the CPU occupied by the MATLAB simulation. The computation time of each method is repeatedly performed 30 times to obtain an average value. It can be seen that the proposed window can obtain accurate results with the least computation time. The method in [14,17] uses the Newton iteration method to achieve accurate results at the price of a long test time. Furthermore, its computational complexity becomes larger if the number of sampled data increases. As a result, the proposed window is more computationally efficient and accurate than other state-of-the-art methods.

5. Experimental Results

In this section, measurement data are used to validate the effectiveness of the proposed method. The ADC under test is AD4003, an 18-bit successive approximation register (SAR) ADC. For reference, the true parameters of the ADC are obtained with the coherently sampled data. The input signal frequency is 2 kHz and the sampling clock frequency is 64.503937 kHz for coherent sampling. The length of data M is 4096, i.e., J = 127. In order to verify the robustness of the proposed method, two sets of non-coherent data are also collected. The sampling clock frequencies are 64.44 kHz and 64.28 kHz, respectively. Thus, J1 = 127.13 and J2 = 127.45. The testing circuit for AD4003 is shown in Figure 8.

Figure 9 and Table 5 show the spectrum and the test results, respectively. The blue spectrum is obtained with coherently sampled data. It can be seen that the blue spectrum is clean without any leakage. Since J1 and J2 are not integers, severe leakage appears in the red spectrum when DFT is directly performed. The green spectrum obtained by the 3-term MSLD window matches the blue spectrum and provides accurate test results. Table 5 lists the values of dynamic parameters of the ADC for different cases.

It can be seen that the dynamic parameters are accurately obtained with the proposed method for any non-coherency. Hence, the functionality and robustness of the proposed method are validated with experimental results.

6. Conclusions

When evaluating the dynamic performance of high precision ADCs using DFT analysis, coherent sampling provides the best results. If coherent sampling cannot be achieved, a window function with a low side-lobe is required to detect the noise and harmonic components in the spectrum. This paper describes how to design maximum side-lobe decay windows with better properties to eliminate the leakage caused by non-coherent sampling. Some MSLD windows were created by the proposed method, and the performance of selected windows is shown in this paper. The proposed windows completely eliminate the requirement of coherent sampling for high precision ADCs. Simulation and experimental results are presented to validate the functionality and robustness of the proposed method with any non-coherency. Because the requirement of coherent sampling is eliminated, the test setup is significantly simplified. The proposed method can save the test cost and effort to achieve coherent sampling.

Author Contributions

Conceptualization, J.F.; methodology, J.F.; software, J.F.; validation, J.F. and Z.Y.; investigation, J.S.; writing—original draft preparation, J.F.; writing—review and editing, J.F., Z.Y. and J.S.; supervision, Y.Z. and S.Q.; project administration, S.Q. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data can be provided upon request.

Acknowledgments

The authors would like to thank Minshun Wu, Xi’an Jiaotong University, for providing help in this work.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Tables

Figure 1. ADC standard spectral testing setup.

Figure 2. The ADC spectrum with coherent and non-coherent sampling.

Figure 3. MSLD windows with different terms: (a) comparison of time domain; (b) comparison of frequency domain.

Figure 4. Comparison of frequency domain: (a) 2-term windows; (b) 3-term windows.

Figure 5. The ADC spectra with different windows.

Figure 6. Error of estimation: (a) SINAD; (b) SNR; (c) THD; (d) SFDR.

Figure 7. Error of estimation (absolute value): (a) SINAD; (b) SNR.

Figure 8. Testing circuit of AD4003. (1) Differential input signal; (2) ADC evaluation board; (3) FPGA.

Figure 9. The ADC spectrum with different non-coherency: (a) J = 127.13; (b) J = 127.45.

Table 1

The coefficients of MSLD windows.

Window	Coefficient	Largest Side-Lobe Level (dB)	Decay Rate(dB/Octave)
2-term(Hanning)	a0 = 0.5a1 = 0.5	−31.47	18
3-term	a0 = 0.375a1 = 0.5a2 = 0.125	−46.74	30
4-term	a0 = 0.3125a1 = 0.46875a2 = 0.1875a3 = 0.03125	−62.54	42
5-term	a0 = 0.2734375a1 = 0.4375a2 = 0.21875a3 = 0.0625a4 = 0.0078125	−74.61	54

Table 2

The characteristics of some classical windows.

Windows	The Largest Side-Lobe Level (dB)	Decay Rate(dB/Octave)
Hamming (2-term)	−43.19	6
2-term MSLD	−31.47	18
Blackman (3-term)	−58.11	18
Exact Blackman (3-term)	−68.24	6
Blackman-Harris (3-term)	−62.05	6
Minimum Blackman (3-term)	−70.83	6
3-term MSLD	−46.74	30
Minimum Blackman (4-term)	−92.01	6
4-term MSLD	−62.54	42
Rife-Vincent (5-term)	−75.29	6
5-term MSLD	−74.61	54

Table 3

Estimation of dynamic parameters.

	SINAD (dB)	SNR (dB)	THD (dB)	SFDR (dB)
Coherent	102.733	102.888	−117.287	119.992
Non-coherent	23.007	23.007	−69.679	70.504
Hamming	45.789	45.789	−87.621	88.454
2-term MSLD (Hanning)	85.159	85.162	−117.253	119.418
Exact Blackman	66.226	66.227	−107.334	108.321
3-term MSLD	102.642	102.785	−117.519	119.307

Table 4

Comparison of computation time.

Method	Time (ms)	Functionality
Hamming	0.71	Inaccurate
Exact Blackman	1.16	Inaccurate
3-term MSLD (Proposed)	1.07	Accurate
Method in [17]	9.34	Accurate
Method in [14]	>15	Accurate

Table 5

Estimation of dynamic parameters.

	SINAD	SNR	THD	SFDR
Coherent	92.634	92.675	−112.851	117.659
Non-coherent (J = 127.13)	21.855	21.859	−53.186	54.496
3-term MSLD (J = 127.13)	92.836	92.885	−112.308	115.481
Non-coherent (J = 127.45)	14.292	14.296	−44.801	46.271
3-term MSLD (J = 127.45)	93.137	93.190	−112.325	115.283

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