An improved crest factor minimization algorithm to

Institute of Physics and Engineering in Medicine Physiol. Meas. 36 (2015) 895–910

Physiological Measurement doi:10.1088/0967-3334/36/5/895

An improved crest factor minimization algorithm to synthesize multisines with arbitrary spectrum Yuxiang Yang1, Fu Zhang1, Kun Tao1, Benjamin Sanchez2, He Wen3 and Zhaosheng Teng3 1

Department of Precision Instrumentation Engineering, Xi’an University of Technology, Xi’an 710048, People’s Republic of China 2 Department of Neurology, Division of Neuromuscular Diseases, Beth Israel Deaconess Medical Center, Harvard Medical School, Boston, MA 02215-5491, USA 3 Department of Instrumentation Science and Technology, Hunan University, Changsha 410082, People’s Republic of China E-mail: [email protected] Received 27 January 2015, revised 20 February 2015 Accepted for publication 4 March 2015 Published 2 April 2015 Abstract

Multisine signal with a low crest factor (CF) can bring a high signal-to-noise ratio for fast frequency response function (FRF) estimation. Synthesis of a low CF multisine with the given amplitude spectrum depends on optimum selection of the initial phases of its cosinusoidal components. The solutions investigated can be generally divided into two branches: (1) the analytical method based on direct formula calculation; and (2) the numerical method based on iterative computations. The analytical method works well only for an equidistant and flat amplitude spectrum, while the numerical method can generally output better results, even for a sparse or non-flat spectrum, but the number of iterations might be huge. This paper presents an improved CF minimization algorithm to synthesize multisine signals based on the combination of the previous Schroeder analytical method and the Van der Ouderaa (VDO) iteration procedure. The improved algorithm adopts the Schroder phases as the iterative initial phases, and employs a logarithmic clip function of the iterative index i in the VDO iteration procedure. Comprehensive experiments of multisine synthesis on three types of cosinusoidal amplitude spectra are performed, and the resulting CFs remain the lowest level in all cases compared with the earlier methods. The proposed algorithm provides a fast and efficient solution to synthesize multisine with the lowest CF for an arbitrary user-prescribed spectrum.

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Keywords: multisine, signal synthesis, crest factor (CF), phases selection, iteration procedure, logarithmic clip function (Some figures may appear in colour only in the online journal) 1. Introduction Estimating the frequency response function (FRF), which is defined as the ratio of the Fourier transform of the output response to the Fourier transform of the input excitation, is of continuous interest in numerous scientific branches such as system identification (Pintelon and Schoukens 2012, Pintelon et al 2014), acoustic properties analysis (Hye Jin et al 2013, Mignot et al 2014), fault detection (Zoubek et al 2008, Herrera Portilla et al 2014) and bioimpedance spectroscopy measurements (Sanchez et al 2013, 2014, Min and Paavle 2014). The FRF estimation generally involves an input signal which covers the frequency range of interest to excite the system. Traditionally, the mostly used excitation signal for the FRF estimation was the stepped-sine (sweeping frequency), which allowed high accuracy measurements to be made, but required a long measuring time due to the dead time waiting the transients to disappear after each frequency step (Schoukens et al 1988, Pintelon and Schoukens 2012). With the development of advanced digital signal processing algorithms based on the fast Fourier transform (FFT), more complex excitation signals with broadband spectrum have been introduced for a high speed FRF estimation, which results in a considerable reduction of the measurement time (Pintelon and Schoukens 2012). The multisine signal, which is defined as the sum of a finite number of harmonically related cosinusoids, has total flexibility to create an arbitrary spectrum, and has been regarded as the best suited broadband excitation signal for the FFT-based FRF estimation (Schoukens et al 1988, 1991). But the multisine with random cosinusoidal phases generally causes a high peak value of the waveform, which is usually measured by the crest factor (CF), the ratio of signal peak value to its root mean square (RMS) value. However, in practical applications, the peak value of the excitation signal is usually limited (e.g. ±1 V or ±1 mA) to keep the tested system close to linear. Under a certain peak value restriction, an excitation signal with a high level CF means a low level of RMS and thus less excitation energy injected, whereas an excitation signal with a minimal CF indicates the maximal signal-to-noise ratio (SNR) because of the maximal excitation energy injected (Schoukens et al 1988, 1991, Horner and Wun 2005, Pintelon and Schoukens 2012). Hence, in order to get the maximum FRF accuracy (i.e. maximal SNR to estimate the FRF), for the case of multisine signals, a minimal CF by a proper selection of the phases of their cosinusoidal components is most desirable, and various phase selection methods has been under investigation for many years (Guillaume et al 1991, Schoukens et al 1991). According to the literature, the solution to the multisine CF minimization is still an open mathematical problem (Van der Ouderaa et al 1988a), and can be generally divided into two branches: the analytical method which allows a direct calculation of the phases, and the numerical method which searches for an optimal set of phases using iterative algorithms (Guillaume et al 1991, Schoukens et al 1991). 1.1. Analytical method

Schroeder (Schroeder 1970) proposed an analytical solution for the multisine CF minimization through a simple formula to determine the initial phases (Schoukens et al 1988): 896

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k−1

ϕ (1) k = ϕ1 − 2π ∑ (k − i )pi (k = 2, 3, ⋯, K ), i= 1

where φk denotes the initial phase of the kth sinusoidal component in multisine, φ1 the initial phase of the fundamental component, K the number of cosinusoidal components, and pi the relative power of the ith component. Here φ1 could be a random value ranging in ±π. If the amplitude spectrum is flat, then (1) reduces to: k (k − 1) ϕ π (k = 2, 3, ⋯, K ). (2) k = ϕ1 − K

The Schroeder method is based on the assumption that the signal has a small spectral bandwidth relative to the center frequency and a smooth amplitude spectrum (Horner 2000), and the typical CF value could reach 1.6–1.7 when considering a flat and equally-spaced spectrum (Schoukens et al 1988). Other methods, such as those reported by Popovic (1991) and Boyd (1986) which could obtain CF of 1.78 ~ 2 (5 and 6 dB in the original text), don’t really give better results, while the Schroeder method seems to have better performance for all the flat and equally-spaced amplitude spectra. However, in the case of a non-flat or sparse spectrum, CF often reaches 3 or higher according to the Schroeder method, no better than a simple random generation (random phase multisine) (Schoukens et al 1991). Literature shows that there still has room to reduce the CF of the multisine, if using some numerical (iterative) methods at the cost of increased computations and lack of explicit mathematical formulas. 1.2. Numerical method

Van den Bos (1987) proposed an iterative procedure to select phases to synthesize a low CF multisine through maximally approximating to a two-level signal, which is called from now on the VDB method for simplicity in this paper. The VDB method starts with random initial phases and can obtain lower a CF (typically 1.55 (Van der Ouderaa et al 1988a)) than the Schroeder method at the modest cost of the iterative computation. However, it depends heavily on the initial conditions and often converges to a local optimum (Horner 2000). The author recommends repetition of the procedure for a number of times with different sets of initial phases and then to select the best solution. As a refinement of the VDB method, an iterative algorithm for the CF minimization was proposed by Van der Ouderaa et al (Van der Ouderaa et al 1988a, 1988b), which is called the VDO method in this paper. The algorithm starts from a given amplitude spectrum ak and random phases ϕk(0), and the multisine is computed using ak and ϕk(0) by the inverse discrete Fourier transform (IDFT), and the extremal value of the multisine signal in the time domain is then clipped by a fixed criterion (75 to 95%). The modified multisine is calculated again through the discrete Fourier transform (DFT) in the frequency domain, and the obtained phases ϕk(i ) is recognized as the improved phase solution if the multisine has a lower CF. The algorithm will repeat from IDFT to DFT using the new ϕk(i ) and the original ak, until the clipped multisine no longer has lower CF. Unlike the analytical approaches such as the Schroeder method, the VDO method has no explicit formula, but can generate lower CF multisines in practice (CF = 1.405 for two multisines with 15 and 31 equidistant and flat spectrum (Van der Ouderaa et al 1988a, 1988b)), even in the case of non-sequential and non-flat spectra (typically 2 (Schoukens et al 1988)). Generally, the algorithm needs a few hundred iterations to obtain a modest CF (for example, CF ≈ 1.5 for a multisine with equidistant and flat spectrum). However, in order to obtain a 897

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CF similar to a sine-wave (1.41), millions of iterations are more likely needed (Schoukens et al 1991). The reason for this situation is partly because there is no clear clip function in the algorithm which results in low convergence speed. Later, Guillaume et al (1991) proposed a CF minimization algorithm based on nonlinear Chebyshev approximation methods, which is called the Guillaume method in this paper. The Guillaume method starts iteration with the Schroeder phases, and the resulting CFs are significantly better than those provided by the earlier methods both in equidistant or logarithmic spectrum distribution (CF = 1.393 for a multisine with 31 equidistant and flat spectrum, and CF = 1.42 for a multisine with 13 logarithmic and flat spectrum (Guillaume et al 1991)). But the Guillaume method becomes time-consuming and even hardly comes to an optimum if the number of cosinusoidal components in the multisine is increased. Afterwards, Horner and Beauchamp (1996) presented a genetic algorithm which uses the VDB’s phases as the initial phases, and output moderate results for several types of amplitude spectra: CF = 1.70 (peak factor of 1.20 in (Horner and Beauchamp 1996), the same below) for the 31 equidistant and flat spectrum, CF = 1.47 (peak factor of 1.04) for non-uniform spectrum with 16 cosinusoidal components, and CF = 2.01 (peak factor of 1.42) for the 6 flat spectrum with 2nth cosinusoidal components (Here, the peak factor is defined as the ratio of signal peak value to 2 of its RMS value, so CF is equal to 2 of the peak factor). More recently, Ojarand et al (Ojarand et al 2014a, 2014b) proposed a CF optimization method based on an improved exhausting searching algorithm, and summarized some CF limit formulas for different spectra distributions. The Ojarand’s method can obtain the optimal CFs for the multisines containing several to a dozen of spectral components, but will become unaffordable for more spectral components (needing to run 58 h for a 20-component signal (Ojarand et al 2014b)). This paper proposes an improved CF minimization algorithm to synthesize low CF multisines. The improved algorithm is basically a modified VDO iteration procedure, which adopts the Schroder phases as the iterative initial phases, and designs a logarithmic function of the iterative index i as the clip criteria in the iteration procedure. Comprehensive multisine synthesis experiments are designed for three types of amplitude spectrum: equidistant and flat spectrum, equidistant and nonflat spectrum and quasi-logarithmic and flat spectrum. The resulting CFs are compared with those of the four earlier methods (Schroeder, VDB, VDO and Guillaume). The choices of the clip function, the clip range and the iteration times are explained detailedly in the discussion section. 2. The improved CF minimization algorithm 2.1. Preliminaries

Definition 1. A (real) continuous multisine signal x(t) is defined as K

x (t ) = ∑ ak cos(2π fk t + ϕk ), (3) k=1

where ak, ƒk and φk are the amplitude, frequency and initial phase of the kth cosinusoidal component, respectively, and K is the number of cosinusoidal components in the multisine. Accordingly, the discrete-time form of the multisine is given by: ⎛ ⎞ n xn = ∑ ak cos⎜2π fk + ϕk ⎟ (n = 0, 1, ⋯ , N − 1), (4) ⎝ ⎠ N K

k=1

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where N is the number of samples. Definition 2. The crest factor (CF) of a signal x(t) is defined by (Pintelon and Schoukens 2012) max x (t ) x peak t ∈ [0, T ] = , CF = T (5) xRMS 1 x (t ) 2 dt T

∫ 0

where xpeak and xRMS are the peak value and effective root-mean-square (RMS) value of the signal in the band of interest. Then, according to Parseval’s theorem (Oppenheim et al 1996) and (4), the discrete-time form of CF can be expressed as (Sanchez et al 2012) max ( xn ) max ( xn ) max ( xn ) n ∈ [0, N − 1] n ∈ [0, N − 1] n ∈ [0, N − 1] , CF = = = K K N−1 2 (6) a 1 ∑ k ∑ pk ∑ xn2 N 2 k=1 k=1 n=0

where pk denotes the power of the kth cosinusoid, and the amplitude of the kth cosinusoid can be expressed as a(7) k = 2pk .

Let the sum of pk equal to 1, namely K

pk = 1. ∑ (8) k=1

Then we call that the multisine has normalized power spectrum. According to (6) and (8), the CF of a multisine with normalized power spectrum can be simplified as CF = max ( xn ). (9) n ∈ [0, N − 1]

In each case a multisine’s spectrum is normalized, the computation of its CF is reduced to finding the peak value of the discrete signal xn. But it’s worth noting that xn must be sampled with an enough high sampling rate to ensure the peak of xn close to the true-value. Experimental tests show that if the sampling length (N in (4)) satisfies the condition as listed in (10), the resulting CF value may have a precision with four decimal places: N ≥ 64fK , (10)

where ƒK is the highest cosinusoid frequency in the multisine. 2.2. The improved CF minimization algorithm

The basic idea of the improved method is shown in figure 1, in which the iteration procedure is started from a given amplitude spectrum ak and the Schroder’s phases ϕk(0) calculated according to (1). The multisine is computed using ak and ϕk(0) by IDFT, and the extremal value of the resulting multisine in the time domain is then clipped by a well-planned logarithmic clip function as mentioned in (14). Then the modified multisine is calculated through DFT in the 899

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Figure 1. The flow chart of the improved multisine CF minimization algorithm. The

algorithm starts from a given amplitude spectrum ak and the Schroder’s phases ϕk(0), and the multisine is computed in the time domain using ak and ϕk(0) by IDFT. Then the extremal value of the signal is clipped according to the logarithmic clip function. Afterwards, the modified multisine is calculated through DFT in the frequency domain, and the obtained phases ϕk(i ) is recognized as the improved phase solution to the targeted multisine with lower CF. The algorithm will restart from IDFT to DFT using the new ϕk(i ) and the original ak, until the iteration index number i increasing from 0 to 1000.

frequency domain, and the obtained phase set ϕk(i ) is recognized as the improved phase solution to the targeted multisine which has lower CF. The algorithm will repeat from IDFT to DFT using the new ϕk(i ) and the original ak, until the iteration index number i increasing from 0 to 1000. The main improvement of our algorithm lies in two aspects: (1) it starts from the Schroder’s phases (according to (1)), (2) it adopts a logarithmic clip function in the iteration procedure, which is the function of the iteration index i and varies from 0.7 to 1 when i increasing from 0 to 1000. The logarithmic clip function is designed as the follows: yi = loga(i − b ) = log10(i − b ) /log10 a, (11)

where yi denotes the clip criteria in the ith iteration loop, i the iteration index, a and b are two constants. Here, we choose y0 = 0.7 (i = 0) and y1000 = 1.0 (i = 1000), so two equations can be deduced: 0.7 = loga( − b ) = log10( − b ) /log10 a, (12) 1(13) = loga(1000 − b ) = log10(1000 − b ) /log10 a. 900

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Figure 2. Example of the multisine with 31 equidistant and flat spectrum (Equi-

Flat-31). (a) Amplitude spectrum of multisine example Equi-Flat-31 (normalized power spectrum). (b) Waveform of Equi-Flat-31.

Combining (12) and (13), we can obtain a ≈ 1137.8, b ≈ − 137.8. So the final form of the logarithmic clip function is: yi = log10(i + 137.8) /log10(1137.8) . (14)

3. CF minimization experiments and results In order to verify the performance of the improved CF minimization algorithm, we prepare several multisine synthesis experiments on three types of amplitude spectra: (1) equidistant and flat (Equi-Flat) spectrum, (2) equidistant and nonflat (Equi-nonFlat) spectrum, and (3) quasi-logarithmic and flat (Quasi-Log-Flat) spectrum. The resulting CF values are compared with the results of the four earlier methods: the Schroeder method (Schroeder 1970), the VDB method (Bos 1987), the VDO method (Van der Ouderaa et al 1988a, 1988b) and the Guillaume method (Guillaume et al 1991). 3.1. Equidistant and flat (Equi-Flat) spectrum

The multisines with Equi-Flat spectrum have been the mostly investigated signal, whose normalized power spectrum can be expressed as 1 pk = (k = 1, 2, ⋯, K ), (15) K

where K denotes the number of cosinusoidal components. Multisine examples with 31 Equi-Flat spectrum (K = 31 in (15)), which are abbreviated as Equi-Flat-31 in this paper, are frequently reported in earlier literature (Schroeder 1970, Bos 1987, Van der Ouderaa et al 1988a, 1988b, Guillaume et al 1991, Horner and Beauchamp 1996). The amplitude spectrum and the synthesized waveform of Equi-Flat-31 are shown in figures 2(a) and (b), respectively, and the obtained phases are listed in the appendix ‘example 1’. In figure 2(b), the multisine signal has the normalized period 1, so the fundamental frequency is normalized to 1. The CF value based on the improved method is compared with those of four earlier methods as listed in table 1. Compared with the four earlier methods, a lowest CF of 1.375 is obtained based on our improved method. 901

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Table 1. CF comparison for multisines with Equi-Flat spectrum.

CFs of different methods Spectrum types

Schroeder

VDB

VDO

Guillaume

Our method

Equi-Flat-31

1.8385 (Schroeder 1970) 1.8892

1.7678 (Bos 1987) 1.7753

1.405 (Van der Ouderaa et al 1988a, 1988b) 1.4610

1.393 (Guillaume et al 1991) 1.5746

1.3750

Equi-Flat-1000

1.3997

Figure 3. Example of the multisine with 1000 equidistant and flat spectrum (Equi-

Flat-1000). (a) Amplitude spectra of the multisine example Equi-Flat-1000 (normalized power spectrum). (b) Waveform of Equi-Flat-1000.

In table 1, another relatively complex example Equi-Flat-1000, which contains 1000 equidistant and flat amplitude spectrum as shown in figure 3(a), has been tested and compared. The synthesized waveform is shown in figure 3(b), and a lowest CF of 1.3998 is obtained based on our improved method. It is worth noting that the CF result of the Guillaume method (1.5746 in table 1) may be not the global optimum value but a local minimum value, because the Guillaume algorithm becomes interminable to get an optimal CF when processing so huge number of cosinusoidal components. 3.2. Equidistant and nonFlat (Equi-nonFlat) spectrum

Multisines with Equi-nonFlat spectrum are also frequently reported in earlier literature, whose normalized power spectrum is defined as: ⎛ 2k − 1 ⎞ 2 ⎟ (k = 1, 2, ⋯, K ), pk = sin2⎜π (16) ⎝ 2K ⎠ K

where K denotes the highest order cosinusoidal component. For the convenience of contrast to the results of the earlier methods, a multisine example with 16 Equi-nonFlat spectrum (K = 16 in (16)) has been selected, which is abbreviated as Equi-nonFlat-16. The amplitude spectra and the synthesized waveform of Equi-nonFlat-16 are shown in figures 4(a) and (b), and the obtained phases are listed in the appendix ‘example 2’. Table 2 lists the CF values of the four earlier mentioned methods and our improved method, and a lowest CF of 1.4023 is obtained based on the improved method. 902

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Figure 4. Example of the multisine with 16 equidistant and nonflat spectrum (Equi-

nonFlat-16). (a) Amplitude spectra of the multisine example Equi-nonFlat-16 (normalized power spectrum). (b) Waveform of Equi-nonFlat-16.

Table 2. CF comparison for multisines with Equi-nonFlat spectrum.

Crest factors (CFs) of different methods Spectrum types

Schroeder

VDB

VDO

Guillaume

Our method

Equi-nonFlat-16

1.6546 (Schroeder 1970)

1. 5132 (Bos 1987)

1.4501

1.42 (Guillaume et al 1991)

1.4023

3.3. Quasi-logarithmic and flat (Quasi-Log-Flat) spectrum

It often occurs that we wish to have an excitation which covers several decades with limited spectral lines. For this situation, logarithmically spaced signal is commonly a good choice. Guillaume (Guillaume et al 1991) reported a multisine with quasi-logarithmic and flat spectra, which has 13 frequencies at 10, 12, 15, 18, 22, 27, 33, 40, 48, 58, 70, 84 and 100, abbreviated as Quasi-Log-Flat-13 here. A lowest CF of 1.96 is given by Guillaume, and a similarly low CF of 1.966 is obtained based on our method. The waveforms of the synthesized multisine of the above four examples are shown in figure 5, and the obtained phases are listed in the appendix ‘example 3’. The CFs found by the various methods are summarized in table 3. 3.4. Brief summary

For the above four examples: Equi-Flat-31, Equi-Flat-1000, Equi-nonFlat-16 and Quasi-LogFlat-13, we can conclude that our improved method can generally output the lowest level of CFs compared with the four earlier methods of the Schroeder, VDB, VDO and Guillaume. From the tables 1–3, we can also observe that the Schroeder method performs better in the example of Equi-Flat and Equi-nonFlat than in the example of Quasi-Log-Flat, because the spectra in the former examples are closer to the intensive and smooth assumption made by Schroeder (Schroeder 1970). Both the VDB and VDO methods have better performances than the Schroeder method over all examples, while the Guillaume method can output even better CFs that are nearest to the results of our improved method. In order to measure the execution efficiency of the Guillaume method and our improved method, we recorded the runtimes of the Guillaume method and our method for the four examples mentioned above. The runtime comparison is listed in table 4 (the data below is achieved by a computer with Intel (R) Core (TM) i3-2120 CPU @ 3.30 GHz 3.30 GHz), from 903

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Figure 5. Example of the multisine with 13 quasi-logarithmic and flat spectrum (Quasi-

Log-Flat-13). (a) Amplitude spectra of the multisine example Quasi-Log-Flat-13 (normalized power spectrum). (b) Waveform of Quasi-Log-Flat-13.

Table 3. CF comparison for multisine with Quasi-Log-Flat spectrum.

CFs of different methods Spectrum types

Schroeder

VDB

VDO

Guillaume

Our method

Quasi-Log-Flat-13

3.1700

2.2513

2.0517

1.9600 (Guillaume et al 1991)

1.9585

which we can see that our algorithm is much faster than the Guillaume method. In particular, for the Equi-Flat-1000 example which contains 1000 cosinusoidal components, the Guillaume method is hardly to achieve a global optimal value even after running dozens of hours, partly because of the excessive computations involved for this multivariable Chebyshev approximating based algorithm. Based on the analysis above, we conclude that our improved CF minimization method provides a fast and efficient solution to synthesize multisines with the lowest level of CFs in multiple cases of arbitrary user-prescribed spectra. 4. Discussion This section aims to discuss the design of the improved CF minimization algorithm: (1) the type of the clip function, (2) the choice of the clip change region, and (3) the number of iteration times. 4.1. The type of the clip function

By contrast, a linear function, a quadratic function, a root function, and the chosen logarithmic function, are adopted to act as the clip function in the iteration procedure of figure 1, respectively. Each of the four functions grows from 0.7 to 1 as the iteration number i increasing from 0 to 1000, but in different ways. Figure 6 illustrates the curves of the four clip functions, and table 5 lists the mathematical expressions of the four clip functions, their first-order derivatives and second-order derivatives. In order to test the performances of the four clip functions, the four examples of Equi-Flat-31, Equi-Flat-1000, Quasi-Log-Flat-13 and Equi-nonFlat-16 are chosen as the target multisines, and the resulting CF values are listed in the table 6. 904

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Table 4. Runtime comparison between the Guillaume methods and our improved

method (on a computer with Intel (R) Core (TM) i3-2120 CPU @ 3.30 GHz 3.30 GHz). Runtime (in second)

Spectrum types

Guillaume method

Our method

Equi-Flat-31 Equi-Flat-1000 Equi-nonFlat-16 Quasi-Log-Flat-13

70.71 Dozens of hours 13.41 13.41

0.94 3.73 0.28 1.66

Figure 6. Comparison of the logarithmic function, linear function, quadratic function

and root function. The four functions all grow from 0.7 to 1 as the iteration number i increasing from 0 to 1000, but in different ways.

From table 6, we can observe that the CF results based on the root and logarithmic clip functions are consistently lower than the results of the quadratic and linear functions. And in table 5, we also observe that both the root and the logarithmic functions have the characteristic of y ′ > 0 and y ′ ′ < 0, which demonstrates that the both functions grow in a diminishing speed as the iteration times i increasing from 0 to 1000. This phenomenon may contribute to the better performance of the two clip functions. Additionally, the performance of the logarithmic function is consistently better than the root function, which may partly because the logarithmic function grows in a more moderate way than the root function does as the iteration number i increasing. Summarily, a clip function with a diminishing uptrend has a better performance in our improved multisine CF minimization algorithm, and the logarithmic function is the best choice we have ever found. 4.2. The choice of the clip change region

In this section, different clip change regions are tested and the resulting CF minimization performances are compared. We choose five uptrend clip regions: 0.5 ~ 1, 0.6 ~ 1, 0.7 ~ 1, 0.8 ~ 1, 0.9 ~ 1, and a downtrend clip region: 1 ~ 0.7. These six clip regions are then experimented (using the logarithmic clip function and 1000 iterations) on the four types of 905

906

First-order derivatives (y′) Second-order derivatives (y″)

Clip function (y)

′ ″

y = 0.0000003 * i 2 + 0.7 y = 0.0000006 * i > 0 y = 0.0000006 > 0

Quadratic

′ ″

y = 0.009486 * i 0.5 + 0.7 y = 0.004743 * i−0.5 > 0 y = − 0.0023715 * i−1.5 < 0

y = (1 − 0.7) * i /1000 + 0.7 y = 0.7 > 0 y =0

′ ″

Root

Linear

′ ″

y = −1/(i − b )2 ln a < 0

y = loga(i − b )* y = 1/(i − b ) ln a > 0

Logarithmic

Mathematical expressions of the four clip functions (The clip coefficient y ranges from 0.7 to 1 as the iteration index i increasing from 0 to 1000)

Table 5. Mathematical expressions of the four clip functions.

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Table 6. CF comparison of the four clip functions (using the clip range from 0.7 to 1

in 1000 iterations).

CFs based on different clip functions Spectrum types

Quadratic

Linear

Root

Logarithmic


1.3825 1.4041 1.4056 1.9804

1.3788 1.4026 1.4037 1.9721

1.3756 1.4025 1.4025 1.9666

1.3750 1.3998 1.4023 1.9585

Table 7. CF comparison for the six different clip regions (using the logarithmic clip function and 1000 iterations).

Clip regions Spectrum types

0.5 ~ 1

0.6 ~ 1

0.7 ~ 1

0.8 ~ 1

0.9 ~ 1

1 ~ 0.7


1.3754 1.4040 1.4028 1.9675

1.3753 1.4008 1.4029 1.9669

1.3750 1.3998 1.4023 1.9585

1.3894 1.4094 1.4517 1.9988

1.4381 1.4234 1.4664 2.1678

1.4778 1.4797 1.5166 2.0704

Table 8. CF comparison of four different iteration times (using the logarithmic clip

function with the change region 0.7 ~ 1).

Number of iteration times Spectrum types

500

1000

2000

3000


1.3862 1.4140 1.4036 2.0094

1.3750 1.3998 1.4023 1.9585

1.3728 1.3920 1.4015 1.9576

1.3719 1.3882 1.4010 1.9565

spectrum examples, and the resulting CF values are listed in table 7. From table 7 we can see that the clip regions of 0.5 ~ 1, 0.6 ~ 1 and 0.7 ~ 1 output the similar results but the clip value from 0.7 ~ 1 has the best (smallest) CFs, while the downtrend clip region 1 ~ 0.7 outputs the worst (largest) CFs. So the change region 0.7 ~ 1 is chosen for the clip function in our improved CF minimization algorithm. 4.3. The number of iteration times

In this section, four typical numbers of the iteration times (i in (14)), 500, 1000, 2000 and 3000, are adopted to experiment (using the logarithmic clip function with the change region 0.7 ~ 1) on the four types of spectrum examples, and the CF results of the four different iteration times are listed in table 8. From table 8, we observe that the CFs of the four examples always have a tiny downtrend as the number of the iteration times increased, but the CF reduction trend becomes negligible when the iterative number is increased from 1000 to 2000 and 3000. Considering the efficiency of the algorithm and the typicality of the CF results obtained, we finally choose 1000 as the iteration times. 907

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5. Conclusion The multisine signal with a low CF can be very useful in numerous applications, and CF minimization with a given amplitude spectrum depends on an optimum selection of the initial phases of its sinusoidal components. This paper gives an overview of the existing CF minimization methods with a closer investigation of Schroeder’s analytical formula and Van der Ouderaa’s (VDO) numerical algorithm, and then proposes an improved CF minimization algorithm which combines the advantage of Schroeder’s formula with the VDO iterative algorithm. The employment of the Schroder phases as the iterative initial phases, the use of the logarithmic function as the clip function, the choice of the clip change region 0.7 ~ 1, and the choice of 1000 iteration times have significantly improved the performance of the algorithm. Multisine synthesis experiments on three types of spectrum distribution have shown that the proposed algorithm can synthesize multisines with the lowest level of CFs for arbitrary userprescribed spectra. Acknowledgment This study has been partly supported by two grants from the National Natural Science Foundation of China (No. 30900317, 61273271). Appendix A The reader can reconstruct the multisine examples mentioned in this paper according to (4) using the values of cosinusoidal frequencies (ƒk), amplitudes (ak) and phases (ϕk) listed as follows. The value of ak is chosen to make the synthesized multisine have normalized power spectrum pk (the sum of pk is equal to 1). Note that ϕk are in rad. Example 1. Equi-Flat-31 (CF = 1.3750) ƒk = [1, 2, 3, … , 31] ak = 2 pk = 2

1 31

ϕk = [3.1067, 0.0835, −2.2936, 2.0544, −1.5791, −3.0282, 1.6772, −2.2803, 2.4914, 1.7254, 2.7873, −2.0961, 1.3176, 2.0630, 2.6231, −0.9922, 0.7026, −0.6575, −2.0723, −2.4123, −2.0217, −1.2014, 0.8386, 0.3342, −0.3860, 2.0323, −2.7892, 0.9178, −2.0236, −0.9686, 2.6719]

Example 2. Equi-nonFlat-16 (CF = 1.4023) ƒk = [1, 2, 3, … , 16] ⎡ π (2fk − 1) ⎤ 1 ⎡ π (2fk − 1) ⎤ 2 sin ⎢ ak = 2 = sin ⎢ ⎥ ⎥⎦ ⎣ 2 × 16 ⎦ 2 ⎣ 32 16 ϕk = [0.8330, −0.9342, 0.7103, 1.4413, 2.7054, 2.0020, 2.9638, −2.2730, 1.3110, 1.6677, −0.2764, −2.2860, 0.5197, −2.9774, 2.0959, −1.2402] 908

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Example 3. Quasi-Log-Flat-13 (CF = 1.9585) ƒk = [10, 12, 15, 18, 22, 27, 33, 40, 48, 58, 70, 84, 100] ak = 2 pk = 2

1 13

ϕk = [1.4103, −0.3227, 1.1506, 0.7324, −1.5584, 0.6154, −0.9900, 0.2328, 2.2739, −1.4057, −0.1748, 0.5797, −2.9652]

References Bos A V D 1987 A new method for synthesis of low-peak-factor signals IEEE Trans. Acoust. Speech Signal Process. 35 C-2 Boyd S 1986 Multitone signals with low crest factor IEEE Trans. Circuits Syst. 33 1018–22 Guillaume P, Schoukens J, Pintelon R and Kollar I 1991 Crest-factor minimization using nonlinear Chebyshev approximation methods IEEE Trans. Instrum. Meas. 40 982–9 Herrera Portilla W, Aponte Mayor G, Pleite Guerra J and Gonzalez-Garcia C 2014 Detection of transformer faults using frequency-response traces in the low-frequency bandwidth IEEE Trans. Indust. Electron. 61 4971–8 Horner A 2000 Low peak amplitudes for wavetable synthesis IEEE Trans. Speech Audio Process. 8 467–70 Horner A and Beauchamp J 1996 A genetic algorithm-based method for synthesis of low peak amplitude signals J. Acoust. Soc. Am. 99 433–43 Horner A and Wun S 2005 Low peak amplitudes for group additive synthesis J. Audio. Eng. Soc. 53 475–84 Hye Jin K, Woo Seok Y and Kwangsoo N 2013 Effects of an elastic mass on frequency response characteristics of an ultra-thin piezoelectric micro-acoustic actuator IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60 1587–94 Mignot R, Chardon G and Daudet L 2014 Low frequency interpolation of room impulse responses using compressed sensing IEEE Trans. Audio Speech Lang. Process. 22 205–16 Min M and Paavle T 2014 Broadband discrete-level excitations for improved extraction of information in bioimpedance measurements Physiol. Meas. 35 997–1010 Ojarand J, Annus P, Min M, Gorev M and Ellervee P 2014a Optimization of multisine excitation for a bioimpedance measurement device Proc. IEEE Instrum. Meas. Technol. Conf. pp 829–32 Ojarand J, Min M and Annus P 2014b Crest factor optimization of the multisine waveform for bioimpedance spectroscopy Physiol. Meas. 35 1019–33 Oppenheim A V, Willsky A S and Nawab S H 1996 Signals and Systems 2nd edn (Englewood Cliffs, NJ: Prentice-Hall) Pintelon R, Louarroudi E and Lataire J 2014 Quantifying the time-variation in FRF measurements using random phase multisines with nonuniformly spaced harmonics IEEE Trans. Instrum. Meas. 63 1384–94 Pintelon R and Schoukens J 2012 System Identification: a Frequency Domain Approach 2nd edn (New York: Wiley) Popovic B M 1991 Synthesis of power efficient multitone signals with flat amplitude spectrum IEEE Trans. Commun. 39 1031–3 Sanchez B, Fernandez X, Reig S and Bragos R 2014 An FPGA-based frequency response analyzer for multisine and stepped sine measurements on stationary and time-varying impedance Meas. Sci. Technol. 25 015501 Sanchez B, Louarroudi E, Jorge E, Cinca J, Bragos R and Pintelon R 2013 A new measuring and identification approach for time-varying bioimpedance using multisine electrical impedance spectroscopy Physiol. Meas. 34 339–57 Sanchez B, Vandersteen G, Bragos R and Schoukens J 2012 Basics of broadband impedance spectroscopy measurements using periodic excitations Meas. Sci. Technol. 23 105501

909

Y Yang et al

Physiol. Meas. 36 (2015) 895

Schoukens J, Guillaume P and Pintelon R 1991 Design of multisine excitations Proc. Int. Conf. Control (Edinburgh) pp 638–43 Schoukens J, Pintelon R, van der Ouderaa E and Renneboog J 1988 Survey of excitation signals for FFT based signal analyzers IEEE Trans. Instrum. Meas. 37 342–52 Schroeder M R 1970 Synthesis of low-peak factor signals and binary sequences with low autocorrelation IEEE Trans. Inf. Theory 16 85–9 Van der Ouderaa E, Schoukens J and Renneboog J 1988a Peak factor minimization of input and output signals of linear systems IEEE Trans. Instrum. Meas. 37 207–12 Van der Ouderaa E, Schoukens J and Renneboog J 1988b Peak factor minimization using a timefrequency domain swapping algorithm IEEE Trans. Instrum. Meas. 37 145–7 Zoubek H, Villwock S and Pacas M 2008 Frequency response analysis for rolling-bearing damage diagnosis IEEE Trans. Indust. Electron. 55 4270–6

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