Patent Publication Number: US-8121815-B2

Title: Noise separating apparatus, noise separating method, probability density function separating apparatus, probability density function separating method, testing apparatus, electronic device, program, and recording medium

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This is a continuation in-part application of Ser. No. 11/836,784 filed on Aug. 10, 2007 the contents of which are incorporated herein by reference. 
    
    
     BACKGROUND 
     1. Technical Field 
     The present invention relates to a noise separating apparatus, a noise separating method, a probability density function separating apparatus, a probability density function separating method, a testing apparatus, an electronic device, a program, and a recording medium. More particularly, the present invention relates to an apparatus and a method for separating a deterministic component and a random component from a probability density function. 
     2. Related Art 
     A method for separating a probability density function with a deterministic component and a probability density function with a random jitter component can be used in an oscilloscope, a time interval analyzer, a universal time frequency counter, automated test equipment, a spectrum analyzer, a network analyzer, and so on. A signal under test may be an electrical signal or an optical signal. The signal under test may indicate information about the variances among products manufactured by the wafer fabrication process. 
     When amplitude of the signal under test is degraded, a probability by which a reception bit one is erroneously decided to a bit zero is increased. Similarly, when a timing of the signal under test is degraded, a probability of an erroneous decision is increased in proportion to the degradation. It takes longer observation time than T b /P e  to measure these bit error rates P e  (however, T b  shows a bit rate). As a result, it takes long measurement time to measure an extremely small bit error rate. 
     For this reason, as measures against amplitude degradation, there has been used a method for setting a bit decision threshold value to a comparatively large value to measure a bit error rate and extrapolate it into an area with an extremely small bit error rate. A deterministic component of a probability density function is bounded and causes a bounded bit error rate. On the other hand, a random component of a probability density function is unbounded. Therefore, a technique for accurately separating a deterministic component and a random component included in measured probability density function and causing bit error rate becomes important. 
     Conventionally, as a method for separating a deterministic component and a random component included in a probability density function or the like, for example, the invention disclosed in US 2002/0120420 has been known. According to this method, an estimate of variance of a probability density function over a predetermined time interval is computed and the computed estimate of variance is transformed into a frequency domain, in order to determine a random component and a period component constituting the variance. The method uses changing a measured time interval from one cycle to N cycles to measure an autocorrelation function of a period component and an autocorrelation function of a random component and making the Fourier transform respectively correspond to a line spectrum and a white noise spectrum. Here, the variance is a sum of a correlation coefficient of a period component and a correlation coefficient of a random component. 
     However, a probability density function is given by convolution integrating a deterministic component and a random component. Therefore, according to this method, it is not possible to separate a deterministic component and a random component from a probability density function. 
     Moreover, as another method for separating a deterministic component and a random component included in a probability density function or the like, for example, the invention disclosed in US2005/0027477 has been known. As shown in  FIG. 2  to be described below, according to this method, both tails of a probability density function are fitted to Gaussian distribution in order to separate two random components from the probability density function. In this method, random components and a deterministic component are performed fit of Gaussian curves under the assumption that both components do not interfere with each other, in order to separate a random component corresponding to Gaussian distribution. 
     However, it is generally difficult to uniquely determine a boundary between a random component and a deterministic component, and it is difficult to separate a random component with high precision in this method. Moreover, as shown in  FIG. 2  to be described below, according to this method, a deterministic component is computed based on a difference D(δδ) between two time instants corresponding to a mean value of each random component. 
     However, for example, when a deterministic component is a sine wave or the like, it is experimentally confirmed that this difference D(δδ) shows a smaller value than D(p−p) of a true value. In other words, according to this method, since only an ideal deterministic component by a square wave can be approximated, various deterministic components such as a deterministic component of a sine wave are not measured. Furthermore, a measurement error of a random component is also large. 
     SUMMARY 
     Therefore, it is an object of an aspect of the present invention to provide a noise separating apparatus, a noise separating method, a probability density function separating apparatus, a probability density function separating method, a testing apparatus, an electronic device, a program, and a recording medium which are capable of overcoming the above drawbacks accompanying the related art. The above and other objects can be achieved by combinations described in the independent claims. The dependent claims define further advantageous and exemplary combinations of the present invention. 
     A first embodiment of the present invention provides a noise separating apparatus that separates a probability density function of a predetermined noise component from a probability density function of a signal under test. The noise separating apparatus includes a domain transforming section that is supplied with the probability density function of the signal under test and transforms the probability density function into a spectrum in a frequency domain, and a standard deviation computing section that computes standard deviation of a random component of noise contained in the signal under test based on a level of a predetermined frequency component in a main lobe of the spectrum. 
     A second embodiment of the present invention provides a noise separating method for separating a probability density function of a predetermined noise component from a probability density function of a signal under test. The noise separating method includes being supplied with the probability density function of the signal under test and transforming the probability density function into a spectrum in a frequency domain, and computing standard deviation of a random component of noise contained in the signal under test based on a level of a predetermined frequency component in a main lobe of the spectrum. 
     A third embodiment of the present invention provides a noise separating apparatus that separates a probability density function of a predetermined noise component from a probability density function of a signal under test. The noise separating apparatus includes a domain transforming section that is supplied with the probability density function of the signal under test and transforms the probability density function into a spectrum in a frequency domain, and a standard deviation computing section that computes standard deviation of a random component of noise contained in the signal under test based on a level of a predetermined frequency component in a side lobe of the spectrum. 
     A fourth embodiment of the present invention provides a noise separating method for separating a probability density function of a predetermined noise component from a probability density function of a signal under test. The noise separating method includes being supplied with the probability density function of the signal under test and transforming the probability density function into a spectrum in a frequency domain, and computing standard deviation of a random component of noise contained in the signal under test based on a level of a predetermined frequency component in a side lobe of the spectrum. 
     A fifth embodiment of the present invention provides a probability density function separating apparatus that separates a predetermined component from a given probability density function. The probability density function separating apparatus includes a domain transforming section that is supplied with the probability density function and transforms the probability density function into a spectrum in a frequency domain, and a standard deviation computing section that computes standard deviation of a random component contained in the probability density function based on a level of a predetermined frequency component in a main lobe of the spectrum. 
     A sixth embodiment of the present invention provides a probability density function separating method for separating a predetermined component from a given probability density function. The probability density function separating method includes being supplied with the probability density function and transforming the probability density function into a spectrum in a frequency domain, and computing standard deviation of a random component contained in the probability density function based on a level of a predetermined frequency component in a main lobe of the spectrum. 
     A seventh embodiment of the present invention provides a probability density function separating apparatus that separates a predetermined component from a given probability density function. The probability density function separating apparatus includes a domain transforming section that is supplied with the probability density function and transforms the probability density function into a spectrum in a frequency domain, and a standard deviation computing section that computes standard deviation of a random component contained in the probability density function based on a level of a predetermined frequency component in a side lobe of the spectrum. 
     An eighth embodiment of the present invention provides a probability density function separating method for separating a predetermined component from a given probability density function. The probability density function separating method includes being supplied with the probability density function and transforming the probability density function into a spectrum in a frequency domain, and computing standard deviation of a random component contained in the probability density function based on a level of a predetermined frequency component in a side lobe of the spectrum. 
     A ninth embodiment of the present invention provides a testing apparatus for testing a device under test. The testing apparatus includes a noise separating apparatus that separates a probability density function of a predetermined noise component from a probability density function of a signal under test which is output from the device under test, and a deciding section that decides whether the device under test is acceptable based on standard deviation of the predetermined noise component separated by the noise separating apparatus. Here, the noise separating apparatus includes a domain transforming section that is supplied with the probability density function of the signal under test and transforms the probability density function into a spectrum in a frequency domain, and a standard deviation computing section that computes standard deviation of a random noise component contained in the probability density function based on a level of a predetermined frequency component in a main lobe of the spectrum. 
     A tenth embodiment of the present invention provides a program to cause a noise separating apparatus to function. Here, the noise separating apparatus separates a probability density function of a predetermined noise component from a probability density function of a signal under test. The program causes the noise separating apparatus to function as a domain transforming section that is supplied with the probability density function of the signal under test and transforms the probability density function into a spectrum in a frequency domain, and a standard deviation computing section that computes standard deviation of a random component of noise contained in the signal under test based on a level of a predetermined frequency component in a main lobe of the spectrum. 
     An eleventh embodiment of the present invention provides a recording medium storing thereon a program which causes a noise separating apparatus to function. Here, the noise separating apparatus separates a probability density function of a predetermined noise component from a probability density function of a signal under test. The program causes the noise separating apparatus to function as a domain transforming section that is supplied with the probability density function of the signal under test and transforms the probability density function into a spectrum in a frequency domain, and a standard deviation computing section that computes standard deviation of a random component of noise contained in the signal under test based on a level of a predetermined frequency component in a main lobe of the spectrum. 
     A twelfth embodiment of the present invention provides a testing apparatus for testing a device under test. The testing apparatus includes a noise separating apparatus that separates a probability density function of a predetermined noise component from a probability density function of a signal under test which is output from the device under test, and a deciding section that decides whether the device under test is acceptable based on standard deviation of the predetermined noise component separated by the noise separating apparatus. Here, the noise separating apparatus includes a domain transforming section that is supplied with the probability density function of the signal under test and transforms the probability density function into a spectrum in a frequency domain, and a standard deviation computing section that computes standard deviation of a random noise component contained in the probability density function based on a level of a predetermined frequency component in a side lobe of the spectrum. 
     A thirteenth embodiment of the present invention provides a program to cause a noise separating apparatus to function. Here, the noise separating apparatus separates a probability density function of a predetermined noise component from a probability density function of a signal under test. The program causes the noise separating apparatus to function as a domain transforming section that is supplied with the probability density function of the signal under test and transforms the probability density function into a spectrum in a frequency domain, and a standard deviation computing section that computes standard deviation of a random component of noise contained in the signal under test based on a level of a predetermined frequency component in a side lobe of the spectrum. 
     A fourteenth embodiment of the present invention provides a recording medium storing thereon a program which causes a noise separating apparatus to function. Here, the noise separating apparatus separates a probability density function of a predetermined noise component from a probability density function of a signal under test. The program causes the noise separating apparatus to function as a domain transforming section that is supplied with the probability density function of the signal under test and transforms the probability density function into a spectrum in a frequency domain, and a standard deviation computing section that computes standard deviation of a random component of noise contained in the signal under test based on a level of a predetermined frequency component in a side lobe of the spectrum. 
     A fifteenth embodiment of the present invention provides an electronic device for generating a predetermined signal. The electronic device includes an operational circuit that generates the predetermined signal and outputs the predetermined signal, a probability density function computing section that measures the predetermined signal and computes a probability density function of the predetermined signal, and a probability density function separating apparatus that separates a predetermined component of the probability density function. Here, the probability density function separating apparatus includes a domain transforming section that is supplied with the probability density function and transforms the probability density function into a spectrum in a frequency domain, and a standard deviation computing section that computes standard deviation of a random component contained in the predetermined signal based on a level of a predetermined frequency component in a side lobe of the spectrum. 
     The summary of the invention does not necessarily describe all necessary features of the present invention. The present invention may also be a sub-combination of the features described above. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a view exemplary showing configurations of a probability density function separating apparatus  100  according to an embodiment of the present invention. 
         FIG. 2  is a view exemplary showing a waveform of an input PDF. 
         FIG. 3  is a view exemplary showing a probability density function with a random component and a spectrum thereof. 
         FIG. 4A  is a view exemplary showing a probability density function with a deterministic component and a spectrum thereof. 
         FIG. 4B  is a view exemplary showing a probability density function with a deterministic component of uniform distribution. 
         FIG. 4C  is a view exemplary showing a probability density function with a deterministic component of sine wave distribution. 
         FIG. 4D  is a view exemplary showing a probability density function with a deterministic component of dual Dirac distribution. 
         FIG. 4E  is a view exemplary showing a probability density function with a deterministic component of triangular distribution. 
         FIG. 5  is a view exemplary showing a spectrum of a probability density function obtained by convolving a deterministic component and a random component. 
         FIG. 6A  is a view exemplary showing a probability density function with a random component, a spectrum of the probability density function, and the second-order derivative of the spectrum with respect to the frequency. 
         FIG. 6B  is a view exemplary showing a spectrum of a probability density function, which corresponds to the convolution of a deterministic component and a random component, and a result obtained by differentiating the spectrum with respect to frequency. 
         FIG. 7  is a view showing another example of a result obtained by differentiating a spectrum of a probability density function with respect to frequency. 
         FIG. 8  is a view exemplary showing a spectrum of a deterministic component of which a value of D(p−p) is different. 
         FIG. 9  is a view exemplary explaining a procedure for computing standard deviation of a random component. 
         FIG. 10  is a view exemplary showing a measurement result by a probability density function separating apparatus  100  described with reference to  FIG. 1  and a measurement result by a conventional curve fitting method described in  FIG. 2 . 
         FIG. 11  is used to explain an exemplary method to compute the standard deviation of a random component. 
         FIG. 12  illustrates, as an example, ideal spectra for a deterministic component of a sine wave distribution and a deterministic component of uniform distribution. 
         FIG. 13  illustrates an example of the measurement result obtained by the probability density function separating apparatus  100  described with reference to  FIGS. 11 and 12 . 
         FIG. 14  illustrates a different example of the measurement result obtained by the probability density function separating apparatus  100  described with reference to  FIGS. 11 and 12 . 
         FIG. 15  is a flowchart exemplary showing a method for directly computing a probability density function in a time domain of a random component from a Gaussian curve in a frequency domain. 
         FIG. 16  is a view exemplary showing an example of a configuration of a random component computing section  130 . 
         FIG. 17A  is a view illustrating a different exemplary configuration of the probability density function separating apparatus  100 . 
         FIG. 17B  is a flowchart showing an example of an operation of the probability density function separating apparatus  100  shown in  FIG. 17A . 
         FIG. 18A  is a view explaining the operation of the probability density function separating apparatus  100  described in  FIG. 17 . 
         FIG. 18B  is used to describe an exemplary case where a random component is computed based on the amount of attenuation of a predetermined frequency component in the main lobe of the spectrum. 
         FIG. 18C  is used to describe an exemplary case where a random component is computed based on the amount of attenuation of a predetermined frequency component in the side lobe of the spectrum. 
         FIG. 19A  illustrates, as an example, an input probability density function h(t), and the spectrum of the input probability density function |H(f)|. 
         FIG. 19B  illustrates, as an example, a different input probability density function h(t) and the spectrum |H(f)| of the input probability density function. 
         FIG. 19C  compares the value of the total jitter TJ which is computed in accordance with the probability density function separating method described with reference to  FIG. 17 , and the value of the total jitter which is measured by a bit error rate testing system. 
         FIG. 19D  shows a table indicating the relation between the coefficient which is to be multiplied by the value of the random jitter to compute the total jitter TJ and the corresponding threshold value of the bit error rate. 
         FIG. 19E  illustrates a different exemplary configuration of the probability density function separating apparatus  100 . 
         FIG. 20  is a view showing another example of a configuration of a probability density function separating apparatus  100 . 
         FIG. 21  is a view exemplary showing operations of a probability density function separating apparatus  100  shown in  FIG. 20 . 
         FIG. 22A  shows a probability density function with a deterministic component including only a sine wave as a deterministic jitter. 
         FIG. 22B  shows a spectrum obtained by transforming a probability density function shown in  FIG. 22A  into a frequency domain. 
         FIG. 23A  shows a probability density function with a deterministic component including a sine wave and a sine wave of which energy is relatively smaller than that of the sine wave as a deterministic jitter. 
         FIG. 23B  shows a spectrum obtained by transforming a probability density function shown in  FIG. 23A  into a frequency domain. 
         FIG. 23C  shows an asymmetric probability density function. 
         FIG. 23D  shows a spectrum obtained by transforming a asymmetric probability density function shown in  FIG. 23C  into a frequency domain. 
         FIG. 24A  shows a probability density function with a deterministic component consisting of two sine waves whose energies are equal to each other. 
         FIG. 24B  shows a spectrum obtained by transforming a probability density function shown in  FIG. 24A  into a frequency domain. 
         FIG. 25A  is a view showing uniform distribution obtained by performing a predetermined threshold process on a probability density function shown in  FIG. 24A . 
         FIG. 25B  is a view showing a spectrum obtained by transforming uniform distribution shown in  FIG. 25A  into a frequency domain. 
         FIG. 26  shows values of D(p−p) measured by a threshold process and D(δδ) measured by a conventional method for a probability density function including a plurality of deterministic jitters. 
         FIG. 27A  shows a spectrum of a probability density function with a deterministic component of a sine wave and a spectrum of a probability density function with a deterministic component in which two sine waves are convolution integrated. 
         FIG. 27B  is a view showing comparison for a main lobe. 
         FIG. 28  is a flowchart exemplary showing a method for obtaining the number of deterministic components included in a probability density function. 
         FIG. 29  is a view exemplary showing a configuration of a noise separating apparatus  200  according to an embodiment of the present invention. 
         FIG. 30  is a view exemplary showing a probability density function of a signal under test generated from a sampling section  210 . 
         FIG. 31  is a view explaining a deterministic component by a code error of ADC. 
         FIG. 32  is a view showing another example of a configuration of a noise separating apparatus  200 . 
         FIG. 33  is a view exemplary showing a configuration of a testing apparatus  300  according to an embodiment of the present invention. 
         FIG. 34  is a view exemplary showing a measurement result of jitter by a jitter separating apparatus  200  and a measurement result of jitter by a conventional method. 
         FIG. 35  is a view showing a conventional measurement result described in  FIG. 34 . 
         FIG. 36A  is a view showing an input PDF. 
         FIG. 36B  is a view showing a probability density function obtained by convolving a deterministic component and a random component which are separated by the probability density function separating apparatus  100 . 
         FIG. 37  is a view exemplary showing a configuration of a sampling section  210  described in  FIG. 33 . 
         FIG. 38  is a view exemplary showing a measurement result by a testing apparatus  300  described with reference to  FIG. 37  and a measurement result by a conventional curve fitting method described with reference to  FIG. 2 . 
         FIG. 39  is a view exemplary showing a configuration of a bit error rate measuring apparatus  500  according to an embodiment of the present invention. 
         FIG. 40  is a view showing another example of the configuration of the bit error rate measuring apparatus  500 . 
         FIG. 41  is a view showing another example of the configuration of the bit error rate measuring apparatus  500 . 
         FIG. 42  is a view exemplary showing a configuration of an electronic device  600  according to an embodiment of the present invention. 
         FIG. 43  is a view showing another example of the configuration of the electronic device  600 . 
         FIG. 44A  illustrates an exemplary configuration of a transfer function measuring apparatus  800  relating to an embodiment of the present invention. 
         FIG. 44B  illustrates another exemplary configuration of the transfer function measuring apparatus  800 . 
         FIG. 45  is a view exemplary showing a hardware configuration of a computer  1900  according to the present embodiment. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     An aspect of the invention will now be described based on the preferred embodiments, which do not intend to limit the scope of the present invention, but exemplify the invention. All of the features and the combinations thereof described in the embodiment are not necessarily essential to the invention. 
       FIG. 1  is a view exemplary showing configurations of a probability density function separating apparatus  100  according to an embodiment of the present invention. The probability density function separating apparatus  100  is an apparatus that separates a predetermined component from a given probability density function, and includes an domain transforming section  110 , a standard deviation computing section  120 , a random component computing section  130 , a peak to peak value detecting section  140 , and a deterministic component computing section  150 . 
     The probability density function separating apparatus  100  according to the present example separates a random component and a deterministic component from a given probability density function (hereinafter, referred to as an input PDF). Moreover, the probability density function separating apparatus  100  may separate either of a random component or a deterministic component from an input PDF. In this case, the probability density function separating apparatus  100  may have either combination of the standard deviation computing section  120  and the random component computing section  130  or the peak to peak value detecting section  140  and the deterministic component computing section  150 . 
     The domain transforming section  110  is supplied with an input PDF, and transforms the input PDF into a spectrum of an axis representing a predetermined variable (a predetermined variable axis). For example, the domain transforming section  110  may receive an input PDF in the time axis, and transform the input PDF into a spectrum in the frequency domain. The domain transforming section  110  may perform Fourier transform on the input PDF to generate the spectrum. 
     When the input PDF is a function of a real number, the operation of performing Fourier transform on the input PDF can be considered equivalent to the operation of performing inverse Fourier transform on the input PDF. When the input PDF is a function of a real number (or a histogram formed by a real number data sequence), the domain transforming section  110  may perform inverse Fourier transform on the input PDF to compute a spectrum in a predetermined variable axis. 
     Alternatively, the domain transforming section  110  may receive an input PDF in the frequency axis, and compute a spectrum in the time axis by performing inverse Fourier transform on the input PDF. The domain transforming section  110  may compute a spectrum in a predetermined variable axis by performing cosine transform on the input PDF. 
     For example, an input PDF may be a function showing probability by which a predetermined signal is likely to have an edge for each of timing. In this case, the probability density function separating apparatus  100  separates a random jitter component and a deterministic jitter component included in this signal. 
     As mentioned above, an input PDF is not limited to the function of a time variable. When the domain transforming section  110  receives an input PDF with a predetermined variable, the domain transforming section  110  may consider this variable as a time variable and generate a spectrum of a frequency domain of the input PDF. That is to say, the present invention relates to an apparatus, a method and so on for separating a predetermined component from an input PDF that is not a time-variable function. 
     Moreover, an input PDF may be digital data, and the domain transforming section  110  may have means for transforming an input PDF with an analog signal into a digital signal. The following explains how to transform the input PDF, which is a function of a real number in the time axis, into a spectrum in the frequency axis by Fourier transform or the like, for example. 
     The standard deviation computing section  120  computes standard deviation of a random component included in the input PDF based on a spectrum output from the domain transforming section  110 . Since the random component included in the input PDF follows Gaussian distribution, the standard deviation computing section  120  computes standard deviation of this Gaussian distribution. A concrete computation method will be below described in  FIGS. 2 to 7  and  FIGS. 17 to 19 . 
     The random component computing section  130  computes a probability density function of a random component based on the standard deviation computed from the standard deviation computing section  120 . For example, according to the probability density function separating apparatus  100  in the present example as described below in  FIGS. 2 to 7 , it is possible to uniquely determine a random component (Gaussian distribution) included in the input PDF based on the standard deviation. 
     The random component computing section  130  may output Gaussian distribution based on standard deviation, or may output this standard deviation. Moreover, the random component computing section  130  may output this Gaussian distribution or this standard deviation in a time domain. 
     The peak to peak value detecting section  140  detects a peak to peak value of the input PDF based on the spectrum output from the domain transforming section  110 . A concrete computation method will be below described in  FIGS. 2 to 7 . 
     The deterministic component computing section  150  computes a deterministic component of the input PDF based on the peak to peak value detected from the peak to peak value detecting section  140 . A concrete computation method will be below described in  FIGS. 2 to 7 . The deterministic component computing section  150  may output a probability density function with a deterministic component in a time domain, or may output this peak to peak value. 
       FIG. 2  is a view exemplary showing a waveform of an input PDF. In the present example, an input PDF includes a probability density function of a sine wave as a deterministic component. However, a deterministic component included in the input PDF is not limited to a sine wave. A deterministic component may be a probability density function with uniform distribution, triangular distribution, a probability density function with a dual Dirac model, a waveform prescribed by the other predetermined function. Moreover, a probability density function with a random component included in the input PDF follows Gaussian distribution. Furthermore, the deterministic component may be based on a combination of uniform distribution, sine wave distribution, triangular distribution and dual Dirac distribution. For example, the deterministic component may be represented by the following expression.
 
d1(t)−α−d2(β×t)
 
Here, α and β are any given coefficients, d 1 ( t ) and d 2 ( t ) are functions indicating any of the above-mentioned distributions.
 
     Moreover, a deterministic component is determined by a peak interval D(p−p) of the probability density function. For example, when a deterministic component is a sine wave, a peak appears at a position according to amplitude of a sine wave in the probability density function. Moreover, when a deterministic component is a square wave, a peak appears at a position according to amplitude of a square wave in the probability density function. Moreover, when a probability density function with a deterministic component is expressed by a dual Dirac model, a deterministic component is defined by an interval D(p−p) between two delta functions. When a deterministic component is triangular distribution, a peak appears at a position according to the spread of a triangle in the probability density function. 
     A composite component (an input PDF) obtained by convolving a deterministic component and a random component is given by a convolution integral of a probability density function with a deterministic component and a probability density function with a random component as shown in  FIG. 2 . For this reason, a peak interval D(δδ) of a composite component becomes smaller than the peak interval D(p−p) of a deterministic component. 
     According to a conventional curve fitting method, D(δδ) is detected as a interval between two peaks determining a deterministic component. However, as described above, since D(δδ) becomes a value smaller than D(p−p) of a true value, the separated deterministic component causes an error. 
     According to the conventional curve fitting method, each of the left-tail and right-tail peaks which are indicated by the solid line in the lowest graph in  FIG. 2  is approximated with a Gaussian distribution. After this, the square root of the sum of the squares of the standard deviations (σleft and σright) of the Gaussian distributions obtained from the left-tail and right-tail peaks is extracted. In this way, the standard deviation σ of the random component can be computed. As illustrated in  FIG. 2 , however, the standard deviations cleft and σright are larger than the true value σtrue. Therefore, the standard deviation σ obtained by the above computation becomes larger than the true value σtrue. Consequently, an error is inevitable. 
       FIG. 3  is a view exemplary showing a probability density function with a random component. A left waveform shown in  FIG. 3  shows a probability density function with a random component in a time domain and a right waveform shown in  FIG. 3  shows a probability density function with a random component in a frequency domain. A random component p(t) in a time domain is Gaussian distribution and is shown by the following Expression. 
                     p   ⁡     (   t   )       =       1     σ   ⁢       2   ⁢   π           ⁢     ⅇ       -       (     t   -   u     )     2       /     (     2   ⁢           ⁢     σ   2       )                   Expression   ⁢           ⁢     (   1   )                 
Here, σ shows standard deviation of Gaussian distribution, u shows time at which Gaussian distribution shows a peak.
 
     Then, a random component P(f) in a frequency domain obtained by performing Fourier transform on the random component p(t) in a time domain is shown by the following Expression.
 
 P ( f )= Ce   −f     2     /2σ     2     Expression (2)
 
     As shown in Expression (2), the result obtained by performing Fourier transform on Gaussian distribution also shows Gaussian distribution. At this time, Gaussian distribution in a frequency domain has a peak at zero frequency. 
       FIG. 4A  is a view exemplary showing a probability density function with a deterministic component. A left waveform shown in  FIG. 4A  shows a probability density function with a deterministic component in a time domain and a right waveform shown in  FIG. 4A  shows a probability density function with a deterministic component in a frequency domain. Moreover, it is assumed that a peak interval of a probability density function with a deterministic component in a time domain is 2 T 0 . 
     A spectrum obtained by performing Fourier transform on a waveform in this time domain has a first null at frequency obtained by multiplying a predetermined multiplier coefficient α by 1/(2 T 0 ). That is to say, it is possible to obtain a peak interval 2 T 0  defining a deterministic component by detecting a first null frequency of a spectrum in a frequency domain. In addition, a multiplier coefficient α can be determined according to the type of distribution of a deterministic component included in a probability density function. 
       FIG. 4B  is a view exemplary showing a probability density function with a deterministic component of uniform distribution. Moreover,  FIG. 4C  is a view exemplary showing a probability density function with a deterministic component of sine wave distribution. Moreover,  FIG. 4D  is a view exemplary showing a probability density function with a deterministic component of dual Dirac distribution. Moreover,  FIG. 4E  is a view exemplary showing a probability density function with a deterministic component of triangular distribution. 
     Left waveforms of  FIG. 4B ,  FIG. 4C ,  FIG. 4D  and  FIG. 4E  show a probability density function with a deterministic component in a time domain and right spectra of  FIG. 4B ,  FIG. 4C ,  FIG. 4D  and  FIG. 4E  shows a probability density function with a deterministic component in a frequency domain. Moreover, it is considered that a peak interval of a probability density function with a deterministic component in a time domain is 2 T 0 . 
     As shown in  FIG. 4B , a first null frequency of a spectrum obtained by performing Fourier transform on a probability density function with a deterministic component of uniform distribution is substantially ½T 0 . That is to say, it is possible to compute a peak interval 2 T 0  by multiplying a multiplier coefficient α=1 by an reciprocal number of the first null frequency. 
     Moreover, as shown in  FIG. 4C , a first null frequency of a spectrum obtained by performing Fourier transform on a probability density function with a deterministic component of sine wave distribution is substantially 0.765/2 T 0 . That is to say, it is possible to compute a peak interval 2 T 0  by multiplying a multiplier coefficient α=0.765 by an reciprocal number of the first null frequency. 
     Furthermore, as shown in  FIG. 4D , a first null frequency of a spectrum obtained by performing Fourier transform on a probability density function with a deterministic component of dual Dirac distribution is substantially 0.500/2 T 0 . That is to say, it is possible to compute a peak interval 2 T 0  by multiplying a multiplier coefficient α=0.500 by a reciprocal number of the first null frequency. 
     Furthermore, as shown in  FIG. 4E , a first null frequency of a spectrum obtained by performing Fourier transform on a probability density function of a deterministic component of triangular distribution is substantially 2.000/2 T 0 . That is to say, it is possible to compute a peak interval 2 T 0  by multiplying a multiplier coefficient α=2.000 by a reciprocal number of the first null frequency. 
       FIG. 5  is a view exemplary showing a spectrum of a probability density function obtained by convolving a deterministic component and a random component. A function obtained by convolving (a convolution integral) a probability density function with a deterministic component and a probability density function with a random component in a time domain becomes an input PDF. Moreover, a convolution integral in a time domain is multiplication of spectra in a frequency domain. That is to say, a spectrum of an input PDF is shown by a product of a spectrum of a probability density function with a deterministic component and a spectrum of a probability density function with a random component. 
     In  FIG. 5 , a deterministic component is shown with a dashed line and a random component is shown with a Gaussian curve of a solid line. When a random component is multiplied by a deterministic component, each peak spectrum of the deterministic component is attenuated in proportion to loss of a Gaussian curve. For this reason, it is possible to obtain a Gaussian curve that provides a random component in a frequency domain by detecting an input PDF, i.e., a level of predetermined frequency of a spectrum of a composite component. 
     The standard deviation computing section  120  may compute standard deviation for a Gaussian curve based on the variable component, in the predetermined variable axis, of the spectrum of the input PDF. The standard deviation computing section  120  relating to the present example may compute standard deviation for a Gaussian curve based on the level of the predetermined frequency component of the input PDF. The random component computing section  130  may compute a Gaussian curve in a frequency domain as shown in  FIG. 5 . At this time, as described in  FIG. 3 , a Gaussian curve in a frequency domain uses zero frequency as a reference. For this reason, the random component computing section  130  can easily compute this Gaussian curve based on the standard deviation computed from the standard deviation computing section  120 . 
     Moreover, as described in  FIG. 4 , D(p−p)=2 T 0  defining a deterministic component can be obtained from the first null frequency of the spectrum of the deterministic component. Since a peak to peak value of the spectrum of the deterministic component is preserved even when multiplying a Gaussian curve, a value of D(p−p) can be computed from the first null frequency of the spectrum of the input PDF. 
     The peak to peak value detecting section  140  detects a peak to peak value from the first null frequency of the spectrum of the input PDF. As described above, the peak to peak value detecting section  140  may multiply a multiplier coefficient ac according to the type of distribution of a deterministic component included in a given probability density function by the first null frequency of the spectrum of the probability density function and compute a peak to peak value of the probability density function with the deterministic component. 
     Moreover, the peak to peak value detecting section  140  may previously store a multiplier coefficient every type of distribution of a deterministic component and compute a peak to peak value by means of a multiplier coefficient corresponding to the reported type of distribution of the deterministic component. For example, the peak to peak value detecting section  140  may previously store a multiplier coefficient α for distribution of each deterministic component such as a sine wave, uniform distribution, triangular distribution, or a dual Dirac model. A multiplier coefficient α for each deterministic component can be previously obtained, for example, by performing Fourier transform on a probability density function with a deterministic component of which a peak to peak value has been known and detecting the first null frequency of the spectrum. 
     Moreover, the peak to peak value detecting section  140  may compute a peak to peak value when using each previously given multiplier coefficient α. The deterministic component computing section  150  may select a value that seems to be the most definite among the peak to peak values computed from the peak to peak value detecting section  140 . For example, the deterministic component computing section  150  may respectively compute a probability density function with a deterministic component based on each peak to peak value and compare the computed probability density function with the given probability density function, in order to select a peak to peak value. 
     Moreover, the deterministic component computing section  150  may compare a composite probability density function obtained by synthesizing a probability density function corresponding to each peak to peak value and a probability density function with a random component computed from the random component computing section  130  and the given probability density function, in order to select a peak to peak value. 
     Since a null value of a spectrum is sharply changed in comparison with a peak of a spectrum, it is possible to detect a peak to peak value with high precision in comparison with when a peak to peak value is computed based on peak frequency of a spectrum. Moreover, as an absolute value of frequency becomes large, null frequency has a large error for a peak to peak value. For this reason, it is possible to detect a peak to peak value with high precision by detecting a peak to peak value based on a first null frequency of which an absolute value is the smallest. 
     However, when detecting a peak to peak value, frequency is not limited to null frequency of which an absolute value is the smallest. For example, a peak to peak value may be detected based on at least one null frequency that is selected from the predetermined number of frequency having a small absolute value. 
     Moreover, a multiplier coefficient α is not limited to a value described in  FIG. 4B ,  FIG. 4C , and  FIG. 4D . The peak to peak value detecting section  140  can appropriately use a multiplier coefficient α substantially equal to this value. Moreover, the peak to peak value detecting section  140  may differentiate a spectrum of a probability density function by frequency and detect a first null frequency based on a differential result. That is to say, the null frequency is not limited to a null frequency which can be clearly detected in the spectrum. For example, when a null frequency is difficult to be clearly detected in the spectrum g(f) as illustrated in  FIGS. 6B and 7 , the frequency f 1  which is detected in the second-order derivative spectrum g″ (f) may be used as the null frequency. 
       FIG. 6A  illustrates, as an example, a second-order derivative (dB (2) (ω)) of a spectrum G(ω) of a probability density function g(t) with a random component with respect to the frequency. It should be noted that the probability density function g(t) illustrated in  FIG. 6A  does not contain a deterministic component. The second-order derivative spectrum dB (2) (ω) takes a constant value, and thus has no peaks. Therefore, a peak of a second-order derivative spectrum of a probability density function containing therein both random and deterministic components corresponds to a peak of a second-order derivative spectrum of the deterministic component, which is to say, a first null frequency in the spectrum of the deterministic component. 
       FIG. 6B  is a view exemplary showing a result obtained by differentiating a spectrum of a probability density function containing thereon random and deterministic components, with respect to frequency. In the present example, it is considered that a first null frequency of a spectrum is f 1 . As shown in  FIG. 4A , when a given probability density function has a few noises, the first null frequency of a spectrum can be precisely detected. Correspondingly, when a given probability density function has noises, as shown in a spectrum g(f) in  FIG. 6B , the first null may not be detected from frequency f 1  to be detected. 
     In this case, as shown in  FIG. 6B , it is possible to detect the first null frequency with high precision by differentiating this spectrum with respect to frequency. As shown in  FIG. 6B , a peak of a second-order derivative spectrum g″ (f) of this spectrum g(f) corresponds to a null of the spectrum g(f). For this reason, the peak to peak value detecting section  140  may second-order differentiate a spectrum of a probability density function and detect the first null frequency based on peak frequency of the derivative spectrum. 
       FIG. 7  is a view showing another example of a result obtained by differentiating a spectrum of a probability density function with respect to frequency. In this example, there is shown a result obtained by differentiating a spectrum of a probability density function without noises as shown in  FIG. 4A . 
     Since a null of a spectrum is a point at which an inclination of a spectrum of a spectrum changes from a negative to a positive, it is possible to detect a null of a spectrum by detecting a peak of a second-order derivative spectrum g″ (f). 
     By such a method, as shown in  FIG. 6B , it is possible to more precisely detect the first null frequency even if noises are large. The peak to peak value detecting section  140  may detect frequency of which an absolute value is the smallest as the first null frequency among peaks of the second-order derivative spectrum g″ (f). 
       FIG. 8  is a view exemplary showing a spectrum of a deterministic component of which a value of D(p−p) is different. A left waveform shown in  FIG. 8  shows a spectrum in case of D(p−p)=2 T 0  and a right waveform shown in  FIG. 8  shows a spectrum in case of D(p−p)=T 0 . Although the value of D(p−p) changes, a ratio between a level of a main lobe of zero frequency and a peak level of each side lobe does not change. That is to say, relative level of each spectrum of a probability density function with a deterministic component is uniquely determined according to whether the deterministic component is a sine wave, uniform distribution, triangular distribution, or a dual Dirac model. 
     For this reason, it is possible to obtain a spectrum of a random component by detecting a ratio between corresponding peak levels in a spectrum of a deterministic component and a spectrum of an input PDF. Here, it is noted that this level ratio depends on attenuation of a spectrum of a deterministic component caused by a random component. 
       FIG. 9  is a view exemplary explaining a method for computing standard deviation of a random component. A Gaussian curve in a frequency domain showing a random component is given by Expression (2). When a base takes a logarithm of e for Expression (2), a quadratic function of f is obtained like Expression (3). 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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     Here, as shown in  FIG. 9 , it is assumed that frequency of a first peak of a spectrum (a composite component) of an input PDF is f 1  and a level is A(f 1 ) and that frequency of a second peak is f 2  and a level is A(f 2 ). At this time, a level ratio between the first peak and the second peak is expressed by Expression (4). 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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     For this reason, it is possible to compute standard deviation based on a level ratio between two frequency components of a spectrum of an input PDF. The standard deviation computing section  120  may compute standard deviation based on a level ratio between a first frequency component and a second frequency component of the spectrum of the input PDF. According to Expression (4), precise measurement for dual Dirac is given. Moreover, an approximate solution for other deterministic components is given. 
     Moreover, it is preferable that these two frequency components are peaks of the spectrum of the input PDF. The standard deviation computing section  120  may compute standard deviation based on a level ratio between any two peaks of the input PDF. 
     A level of the peak of the spectrum of the input PDF shows resultant attenuation in a peak level of a spectrum of a deterministic component caused by a spectrum of a random component. For this reason, when a level of each peak of the spectrum of the deterministic component is constant, it is possible to compute standard deviation with high precision based on Expression (4). 
     Moreover, when a level of each peak of the spectrum of the deterministic component is not constant, the standard deviation computing section  120  may compute standard deviation further based on the level of the peak of the spectrum of the deterministic component. That is to say, the standard deviation computing section  120  may compute standard deviation based on a level ratio between a predetermined frequency component of the spectrum of the input PDF and a frequency component corresponding to a spectrum obtained by transforming a probability density function with a deterministic component into a frequency domain. In this case, the standard deviation computing section  120  may compute standard deviation based on Expression (5). Here, B(f 1 ) is a level of the first peak of the spectrum of the deterministic component, and B(f 2 ) is the second level of the spectrum of the deterministic component. Here, the frequency f 2  may be a frequency contained in the main or side lobe of the spectrum. 
     
       
         
           
             
               
                 
                   
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     In addition, standard deviation can be obtained according to a procedure equal to Expression (5). For example, in Expression (5), standard deviation is computed based on a value obtained by dividing a level ratio A(f 2 )/B(f 2 ) between the input PDF in the second frequency component and the spectrum of the deterministic component by a level ratio A(f 1 )/B(f 1 ) in the first frequency component. Similarly, standard deviation may be obtained based on a value obtained by dividing a level ratio A(f 2 )/A(f 1 ) between the second frequency component in the input PDF and the first frequency component by a level ratio B(f 2 )/B(f 1 ) between the second frequency component in the deterministic component and the first frequency component. 
     A spectrum of a probability density function of a random component and a spectrum of a probability density function of a deterministic component both take their maximum values at dc (f=0) as illustrated in  FIG. 3  and  FIGS. 4A to 4E . Therefore, when division is performed on the magnitudes of the respective frequency components by using the values of the spectra at f 1 =dc, A(f 1 )=B(f 1 )=1.0. Accordingly, A(f 2 )/A(f 1 )=A(f 2 ) and B(f 2 )/B(f 1 )=B(f 2 ). As a result, the standard deviation of the random component can be computed by using the magnitude of the component at the single frequency f 2 . 
     In this case, a ratio between a magnitude of the second frequency component and a magnitude of the first frequency component in the spectrum of the probability density function with the deterministic component may be given in advance. The standard deviation computing section  120  may store this magnitude ratio on a memory in advance. This magnitude ratio can be determined according to the type of distribution of the deterministic component included in the input PDF in advance. Particularly, when the deterministic component is given with a function of dual Dirac, this magnitude ratio is 1.0. 
     Moreover, a spectrum of a deterministic component can be obtained based on the above-described D(p−p). A deterministic component is determined by a value of D(p−p) as described above and whether the deterministic component is given by a sine wave, uniform distribution, triangular distribution, or dual Dirac. 
     The deterministic component computing section  150  may compute a deterministic component by being previously supplied with a function corresponding to a sine wave, uniform distribution, triangular distribution, dual Dirac, or the like defining the deterministic component and applying a peak to peak value detected from the peak to peak value detecting section  140  to this function. In this case, the random component computing section  130  computes a random component based on the spectrum of the deterministic component computed from the deterministic component computing section  150 . 
     Moreover, assuming that f 1 =0 in Expression (5), since the magnitude of the spectrum of the input PDF in f 1 =0 and the magnitude of the spectrum in the deterministic component are equal to each other, Expression (5) becomes like Expression (6). The frequency f 2  may be a frequency contained in the main or side lobe of the spectrum. 
     
       
         
           
             
               
                 
                   
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                   Expression 
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                     6 
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     The standard deviation computing section  120  may compute standard deviation based on Expression (6). That is to say, the standard deviation computing section  120  may compute standard deviation from a magnitude ratio between any corresponding peaks in the spectrum of the input PDF and the probability density function with the deterministic component. In this case, it is possible to compute standard deviation by simpler measurement and with high precision. 
     Moreover, the standard deviation computed based on Expression (5) and Expression (6) is standard deviation of Gaussian distribution in a frequency domain. The standard deviation computing section  120  may compute standard deviation σt in a time domain based on standard deviation of in a frequency domain. Relation between of and σt is expressed by Expression (7). 
                     σ   t     =       1     2   ⁢           ⁢   π       ⁢       2   ⁢           ⁢     σ   f   2                   Expression   ⁢           ⁢     (   7   )                 
In this way, it is possible to compute a probability density function in a time domain of a random component.
 
     It is possible to obtain a Gaussian curve in a frequency domain from Expression (2) by means σf. A Gaussian curve in a time domain of Expression (1) may be directly obtained by performing Fourier transform on the Gaussian curve in the frequency domain. In other words, a probability density function in a time domain of a random component can be directly obtained from the Gaussian curve in the frequency domain. 
       FIG. 10  is a view exemplary showing a measurement result by a probability density function separating apparatus  100  described with reference to  FIG. 9  and a measurement result by a conventional curve fitting method described in  FIG. 2 . In the present example, distribution in which a peak to peak value of a deterministic component is 50 ps and a random component is 4.02 ps is used as a probability density function to be measured. Moreover, for a sampling timing in sampling a measuring object, two cases was studied: one case of timing error in sampling and another case of zero timing error in sampling. As shown in  FIG. 10 , the probability density function separating apparatus  100  can obtain a measurement result with a smaller error than a conventional curve fitting method in any cases. 
       FIG. 11  is used to explain an exemplary method to compute the standard deviation of the random component. In  FIG. 11 , the frequency is plotted along the horizontal axis, and the magnitude spectrum of the probability density function is plotted along the vertical axis. In  FIG. 11 , a spectrum B(f) indicated by the dotted line is an ideal spectrum for the deterministic component contained in the probability density function, and a spectrum A(f) indicated by the solid line is the spectrum of a given probability density function. 
     The method described with reference to  FIG. 9  computes the standard deviation of the random component from the magnitudes of the side lobes. However, the given (or measured) probability density function contains errors therein due to measurement errors and the like. Such errors affect the side lobes more notably than the main lobe since the side lobes have smaller magnitudes than the main lobe. For this reason, when the standard deviation of the random component is computed from the magnitudes of the side lobes, the obtained standard deviation may have relatively large errors. Here, the main lobe of the spectrum may indicate the lobe which contains therein, for example, a frequency component at 0 Hz or the carrier frequency of the signal, and the side lobes may indicate lobes of the spectrum excluding the main lobe. 
     On the other hand, the probability density function separating apparatus  100  relating to the present embodiment computes the standard deviation of the random component based on the magnitude (A(fm)) of a component at a predetermined frequency (fm) in the main lobe of the spectrum of the probability density function. For example, the standard deviation computing section  120  may compute the standard deviation of the random component based on the magnitude (A(fm)) of the component at the predetermined frequency (fm) in the main lobe of the spectrum (A(f)) of a given probability density function and the magnitude (B(fm)) of the component at the same frequency (fm) in the main lobe of an ideal spectrum (B(f)) for the deterministic component of the probability density function. 
     Here, the ideal spectrum for the deterministic component can be obtained based on the type of the deterministic component contained in the probability density function and the first null frequency (fα). For example, the peak to peak value of the deterministic component can be computed based on the first null frequency (fα) and the type of the deterministic component as described with reference to  FIG. 4 . 
     It is then possible, as illustrated in  FIG. 4 , to uniquely determine a probability density distribution which has the computed peak to peak value and corresponds to the deterministic component type. By performing Fourier transform on the probability density distribution, the ideal spectrum for the deterministic component can be obtained. In the probability density function separating apparatus  100  relating to the present example, the deterministic component computing section  150  may compute the ideal spectrum for the deterministic component, and supplies the obtained ideal spectrum for the deterministic component to the standard deviation computing section  120 . 
     As mentioned above, the standard deviation computing section  120  computes the standard deviation of the random component based on the magnitudes A(fm) and B(fm) of the spectra. In more detail, the standard deviation computing section  120  may compute the standard deviation σ by using, for example, the following expression which is similar to Expression (6). 
     
       
         
           
             
               
                 
                   
                     - 
                     
                       1 
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
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                           2 
                         
                       
                     
                   
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                       1 
                       
                         f 
                         m 
                         2 
                       
                     
                     · 
                     
                       log 
                       ⁡ 
                       
                         ( 
                         
                           
                             A 
                             ⁡ 
                             
                               ( 
                               
                                 f 
                                 m 
                               
                               ) 
                             
                           
                           
                             B 
                             ⁡ 
                             
                               ( 
                               
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     Here, the magnitudes of the spectra are detected at the predetermined frequency fm which may be designated in advance by a user or the like. The standard deviation computing section  120  may use, as the predetermined frequency fm, a frequency in the main lobe of the ideal spectrum for the deterministic component which falls within such a range that the amount of attenuation of the component at the frequency fm is smaller than a predetermined value. This frequency range may be determined by the user or the like. 
       FIG. 12  illustrates, as an example, ideal spectra for a deterministic component of a sine wave distribution and a deterministic component of uniform distribution. In  FIG. 12 , the solid line indicates a spectrum of a deterministic component of a sine wave distribution, and the dotted line indicates a spectrum of a deterministic component of uniform distribution.  FIG. 12  shows the main lobes of the respective spectra. 
     As seen from  FIG. 12 , for the deterministic components of different types, the spectrum shapes of the main lobs are different from each other, even when the deterministic components are associated with the same first null frequency. For this reason, when the type of the deterministic component contained in the probability density function is unknown, the computed value of the standard deviation may have an error which is attributed to the different shapes. 
     As mentioned above, the standard deviation computing section  120  may use, as the predetermined frequency fm, a frequency in the main lobe of the ideal spectrum of deterministic component, which falls within such a range that the difference between the magnitudes of the components at the frequency fm (Δ(fm)) is smaller than a predetermined value. As seen from  FIG. 12 , the difference between the magnitudes (Δ(fm)) increases as the frequency increases. Therefore, when the standard deviation computing section  120  selects the predetermined frequency fm, a frequency fmax at which the difference between the magnitudes (Δ(fm)) becomes equal to a predetermined value may be set as the upper limit. 
     The ideal spectra of the respective deterministic components may be computed by the deterministic component computing section  150  based on the detected first null frequencies fα, and supplied to the standard deviation computing section  120 . The above-mentioned predetermined value may be set in accordance with the required measurement accuracy (e.g. tolerated measurement error or the like). 
     Referring back to  FIG. 11 , when the predetermined frequency fm is set in the vicinity of 0 Hz, the difference between the magnitude of the spectrum A(f) under test and the magnitude of the ideal spectrum B(f) is substantially equal to zero. This makes it difficult to compute the standard deviation. For this reason, when the standard deviation computing section  120  selects the predetermined frequency fm, a predetermined frequency fmin, which is not 0 Hz, may be set as the lower limit. The standard deviation computing section  120  may select, as the predetermined frequency fm, a frequency equal to substantially half the above-mentioned frequency fmax which may be set as the upper limit. 
     Even when having the same first null frequency, deterministic components of different types have spectra with different characteristics in their main lobes. To be specific, a variation in magnitude Δ(fm) in the main lobe for a deterministic component of a certain type may be larger than a variation in magnitude Δ(fm) in the main lobe for a deterministic component of a different type. The probability density function separating apparatus  100  relating to the present embodiment also addresses this issue, and thus can compute the standard deviation of the random component more accurately. 
       FIG. 13  illustrates, as an example, the measurement result obtained by the probability density function separating apparatus  100  described with reference to  FIGS. 11 and 12 .  FIG. 13  also shows the measurement results obtained by means of the conventional curve fitting methods (the tail-fit method and Q-scale method). 
     It should be noted that the probability density function separating apparatus  100  relating to the present example performed the measurement under the assumption that the probability density function contains a deterministic component of a sine wave distribution. As seen from  FIG. 13 , the values measured by the probability density function separating apparatus  100  relating to the present example provide the smaller standard deviations than the standard deviations of the measurement values estimated by using the two conventional curve fitting methods. As a consequence, the result of the measurement performed by the probability density function separating apparatus  100  relating to the present example is expected to be closer to the true values. 
       FIG. 14  illustrates a result of measuring the data dependent jitter measured by using the probability density function separating apparatus  100  described with reference to  FIGS. 11 and 12 . According to this example, a seven-stage pseudo random binary sequence (PRBS) generating circuit is used to generate a data pattern at the rate of 2.5 Gbps and 6400800 bits. In this example, the probability density function separating apparatus  100  separates jitter from the same probability density function under the assumption that the deterministic component has uniform distribution, and computes the measurement result. 
     The values measured by the probability density function separating apparatus  100  indicate the smaller standard deviations than the standard deviations of the measurement values estimated by using the two conventional curve fitting methods. As a consequence, the measurement result obtained by the probability density function separating apparatus  100  relating to the present example is expected to be closer to the true values. 
     The probability density function separating apparatus  100  also measures the standard deviation σ of the random component (RJ). In this case, the values measured by the probability density function separating apparatus  100  are smaller than the values measured by using the two conventional curve fitting methods. As explained with reference to  FIG. 2 , the values of the standard deviation of the random component which are measured by using the conventional curve fitting methods are larger than the true values. Considering this, the measurement result obtained by the probability density function separating apparatus  100  is expected to be closer to the true values and appropriate. 
     The values of the peak to peak value of the deterministic component (DDJ) which are measured by the probability density function separating apparatus  100  are equal to or larger than the values of the peak to peak value of the deterministic component (DDJ) which are measured by using the two conventional curve fitting methods. As explained with reference to  FIG. 2 , the values of the peak to peak value of the deterministic component which are measured by using the conventional curve fitting methods are smaller than the true values. Considering this, the measurement result obtained by the probability density function separating apparatus  100  is expected to be closer to the true values and appropriate. 
       FIG. 15  is a flowchart exemplary showing a method for directly computing a probability density function in a time domain of a random component from a Gaussian curve in a frequency domain. First, a Gaussian curve G(f) in a frequency domain is acquired by substituting standard deviation of in a frequency domain for Expression (2) (S 30 ). At this time, if required, in order to distribute a Gaussian curve in a time domain around a mean value p of the input PDF, a result obtained by multiplying exp(j2πμf) by G(f) is used considering time shifting property. 
     Next, a complex sequence (really, it is noted that it is a real sequence) of which a real part is G(f) and an imaginary part is zero is acquired (S 32 ). Then, a function g(t) in a time domain obtained by performing inverse Fourier transform on the acquired complex sequence is acquired (S 34 ). At this time, since an original signal is a real number, Fourier transform or cosine transform may be performed in place of inverse Fourier transform. 
     Next, a Gaussian curve in a time domain is acquired by extracting the square root of a sum of a square of a real part and a square of an imaginary part of the g(t) acquired in S 34  (S 36 ). In other words, a Gaussian curve in a time domain is acquired by computing a square root of sum of squares of real part and imaginary part of g(t). By such a process, a Gaussian curve in a time domain can be acquired. 
       FIG. 16  is a view exemplary showing a configuration of the random component computing section  130 . The random component computing section  130  according to the present example acquires a Gaussian curve in a time domain using a method described in  FIG. 15 . The random component computing section  130  has a frequency domain computing section  132 , a complex sequence computing section  134 , an inverse Fourier transform section  136 , and a time domain computing section  138 . 
     The frequency domain computing section  132  computes a Gaussian curve G(f) in a frequency domain based on standard deviation of a random component in a frequency domain computed from the standard deviation computing section  120 . At this time, the frequency domain computing section  132  may compute a Gaussian curve G(f) in a frequency domain in a manner similar to the step of S 30  described in  FIG. 15 . 
     The complex sequence computing section  134  computes a complex sequence of which a real part is G(f) and an imaginary part is zero. The inverse Fourier transform section  136  computes a function g (t) in a time domain obtained by performing inverse Fourier transform (or Fourier transform) on this complex sequence. The time domain computing section  138  extracts the square root of sum of squares of real part and imaginary part of the function g(t) in a time domain, and acquires a Gaussian curve in a time domain, that is, a probability density function in a time domain of a random component. 
     In addition, a process described in  FIGS. 15 and 16  is not limited to a process for a probability density function. That is to say, it is possible to suppose a waveform in a time domain from a spectrum in an arbitrary frequency domain by means of a process similar to that described in  FIGS. 15 and 16 . 
     In this case, the time domain computing section  138  described in  FIG. 16  is supplied with a magnitude spectrum of a signal under test. Then, the time domain computing section  138  computes a waveform in a time domain by transforming the magnitude spectrum into a function in a time domain. When transforming a magnitude spectrum into a function in a time domain, it is possible to obtain a function in this time domain by applying Fourier transform, inverse Fourier transform, cosine transform or the like to this magnitude spectrum. Then, the time domain computing section  138  can suppose a waveform in a time domain by extracting the square root of sum of squares of real part and imaginary part of this time domain. 
     In this manner, a computing apparatus for computing a waveform in a time domain from a spectrum in a frequency domain may further include a frequency domain measuring section for detecting a magnitude spectrum of a signal under test in addition to the time domain computing section  138 . The frequency domain measuring section supplies the detected magnitude spectrum to the time domain computing section  138 . By such a configuration, it is possible to suppose a waveform in a time domain of a signal under test based on only a magnitude spectrum of a signal under test. 
     As described above, according to the probability density function separating apparatus  100  in the present example, it is possible to separate a random component and a deterministic component from a given probability density function with high precision. For example, in case of a random component, it is possible to compute a random component with high precision based on standard deviation computed in a frequency domain without performing an approximation such as conventional curve fitting. Moreover, in case of a deterministic component, it is possible to detect a value D(p−p) closer to a true value for D( 55 ) having an error like a conventional method. 
       FIG. 17A  illustrates a different exemplary configuration of the probability density function separating apparatus  100 . The probability density function separating apparatus  100  relating to the present example includes therein the peak to peak value detecting section  140 , the standard deviation computing section  120 , the deterministic component computing section  150 , and the random component computing section  130 . The respective constituents may be the same as the corresponding constituents in  FIG. 1  which are assigned the same reference numerals. 
       FIG. 17B  is a flowchart showing an example of an operation of the probability density function separating apparatus  100  illustrated in  FIG. 17A . The probability density function separating apparatus  100  relating to the present example computes the probability density function corresponding to the deterministic component from the first null frequency of the spectrum of the probability density function, similarly to the description made with reference to the  FIGS. 4A to 4E . 
     An operation of the domain transforming section  110  is equal to that of the domain transforming section  110  described with reference to  FIG. 1 . In other words, the domain transforming section  110  transforms a given probability density function into a spectrum in a frequency domain (S 60 ). 
     After this, the peak to peak value detecting section  140  detects a first null frequency of the spectrum (S 62 ). For example, the peak to peak value detecting section  140  may detect the first null frequency of the spectrum from the second-order derivative waveform of the spectrum, as described with reference to  FIGS. 6B and 7 . 
     In addition, the peak to peak value detecting section  140  may compute the peak to peak value of the probability density function corresponding to the deterministic component, based on the first null frequency of the spectrum. For example, the peak to peak value detecting section  140  may compute the peak to peak value in the manner described with reference to  FIGS. 4A to 4D . 
     Subsequently, the deterministic component computing section  150  computes the probability density function corresponding to the deterministic component based on the first null frequency (or the peak to peak value) (S 64 ). The deterministic component computing section  150  may compute a spectrum in the frequency domain for the probability density function corresponding to the deterministic component. For example, the deterministic component computing section  150  may compute the spectrum which is indicated by the dotted line in  FIG. 5  or  11 . 
     Following this, the random component computing section  130  computes the spectrum of the probability density function corresponding to the random component, by dividing the spectrum of the input probability density function by the spectrum of the probability density function corresponding to the deterministic component (S 66 ). The random component computing section  130  may divide the absolute values of the spectrum of the input probability density function (the magnitude spectrum) by the absolute values of the spectrum of the probability density function corresponding to the deterministic component. For example, the random component computing section  130  may divide the absolute values of the spectrum of the input probability density function indicated by the solid line in  FIG. 5  or  11  by the absolute values of the spectrum indicated by the dotted line in  FIG. 5  or  11 . 
     In the above-described manner, the probability density function separating apparatus  100  can compute the probability density functions of the random and deterministic components. The standard deviation computing section  120  may compute the standard deviation of the random component from the computed spectrum of the probability density function corresponding to the random component. Here, the standard deviation computing section  120  may convert the spectrum of the probability density function corresponding to the random component into a spectrum plotted along a logarithmic axis. 
     Alternatively, in place of the operations performed in the steps S 64  and S 66 , the standard deviation computing section  120  may compute the standard deviation of the random component based on the magnitude of the predetermined frequency component in the main lobe of the spectrum of the input probability density function in the manner described with reference to  FIG. 11 . The random component computing section  130  may compute the probability density function corresponding to the random component based on the standard deviation of the random component. 
       FIG. 18A  is a view explaining an operation of the probability density function separating apparatus  100  described in  FIG. 17 . As described above, the domain transforming section  110  outputs a spectrum D(f)R(f) of a probability density function. A spectrum of a random component R(f) is given by dividing the spectrum D(f)R(f) by a magnitude spectrum |D(f)| with a deterministic component. 
     Here, it is not necessary to divide the entire range of the spectrum D(f)R(f) by the magnitude spectrum |D(f)|. Alternatively, the random component may be computed from the amount of attenuation of the predetermined frequency component as described with reference to Expressions (5) and (6). Which is to say, the random component can be computed from the ratio between the value of the spectrum D(f)R(f) of the input probability density function and the value of the spectrum D(f) of the deterministic component at the predetermined frequency f 2 . Here, the predetermined frequency f 2  may be a frequency in the main or side lobe of the spectrum of the input probability density function. 
       FIG. 18B  is used to explain an exemplary case where the random component is computed based on the amount of attenuation of the predetermined frequency component in the main lobe of the spectrum. The probability density function separating apparatus  100  may compute the spectrum of the probability density function corresponding to the random component, based on the magnitude of the predetermined frequency component f 2  in the main lobe of the spectrum of the input probability density function, in the manner described with reference to  FIG. 11 . 
     For example, when the input probability density function contains therein a sine wave distribution with a small amplitude as the deterministic component, the error component in the side lobe of the spectrum of the input probability density function is evident. When the deterministic component contained in the input probability density function is of a sine wave distribution and the energy of the sine wave is smaller than a predetermined value, the probability density function separating apparatus  100  may compute the standard deviation of the random component based on the ratio between the predetermined frequency components in the main lobes of the spectra of the input probability density function and deterministic component. For example, when an unexpected sine wave is generated as the deterministic component and when the energy of the sine wave is smaller than a predetermined value, the probability density function separating apparatus  100  may compute the standard deviation of the random component from the main lobes of the spectra. 
       FIG. 18C  is used to explain an exemplary case where the random component is computed from the amount of attenuation of the predetermined frequency component in the side lobe of the spectrum. The probability density function separating apparatus  100  may compute the spectrum of the probability density function corresponding to the random component, based on the magnitude of the predetermined frequency component f 2  in the side lobe of the spectrum of the input probability density function. When the deterministic component contained in the input probability density function is not of a sine wave, the probability density function separating apparatus  100  may compute the standard deviation of the random component based on the ratio between the predetermined frequency components in the side lobes of the spectra of the input probability density function and deterministic component. When the deterministic component contained in the input probability density function is of a sine wave distribution and the energy of the sine wave is larger than a predetermined value, the probability density function separating apparatus  100  may compute the standard deviation of the random component from the side lobe of the spectrum. 
     Furthermore, as shown in  FIG. 18A , an error component of the spectrum D(f)R(f) of a probability density function becomes large as frequency becomes high. For this reason, the deterministic component computing section  150  may compute a probability density function in a time domain with a deterministic component by transforming a spectrum in a predetermined frequency range including frequency of a main lobe into a function in a time domain among the spectra D(f) with the computed deterministic component. Moreover, the deterministic component computing section  150  may extract the predetermined number of side lobes in vicinity of the main lobe from the spectra D(f) with the computed deterministic component and transform the extracted main lobe and side lobe into a function in a time domain. By such a process, it is possible to reduce an influence of an error in a high-frequency area. 
       FIG. 19A  illustrates, as an example, an input probability density function h(t), and the spectrum |H(f)| of the input probability density function. In this example, a 15-stage pseudo random binary sequence (PRBS) is input into a coaxial cable, and a probability density function of jitter of a data sequence output from the coaxial cable is obtained as the input probability density function h(t). This data sequence has data dependent jitter (DDJ) caused therein in accordance with the length of the coaxial cable. In the present example, the coaxial cable has a length of 5 m. 
       FIG. 19B  illustrates, as an example, a different input probability density function h(t) and the spectrum |H(f)| of the input probability density function. The input probability density function h(t) and the spectrum |H(f)| of the input probability density function shown in  FIG. 19B  are obtained under the same conditions as the input probability density function h(t) and the spectrum |H(f)| shown in  FIG. 19A  except for that the coaxial cable has a length of 15 m. When compared with the example shown in  FIG. 19A , the data dependent jitter DDJ is more obvious in the example shown in  FIG. 19B . 
     The probability density function separating method described with reference to  FIG. 17  is used to separate the random jitter RJ and the deterministic jitter DJ in the input probability density function, so that the total jitter TJ is computed. The total jitter TJ can be computed based on the following expression, for example:
 
 TJ=DJ ( p−p )+12× RJ   Expression (8)
 
Here, the coefficient “12” is determined in accordance with the bit error rate threshold, and selected from the table shown in  FIG. 19D , for example. In this example, the coefficient associated with the bit error rate of 10 −9  is used.
 
       FIG. 19C  compares the values of the total jitter TJ which are computed in accordance with the probability density function separating method described with reference to  FIG. 17 , and the values of the total jitter which were measured by a bit error rate testing system. In  FIG. 19C , the values of the total jitter are plotted with respect to the value of 1/T b /f −3 dB , where T b  denotes the bit time interval of the pseudo random binary sequence, and f −3 dB  denotes the 3 dB bandwidth of the coaxial cable. 
     According to this example, the number of samples of data measured for the probability density function separating method is different from that captured by the bit error rate testing system. (For the probability density function separating method, the number of samples of data measured was 3×10 4 , while for the bit error rate test system and the number was 10 9 .) 
     Therefore, in the region where the value of 1/T b /f −3 dB  is small and the random jitter is a dominant factor, the error of the values obtained by the measurement performed in accordance with the probability density function separating method with respect to the values obtained by the measurement performed by using the bit error rate test system is approximately 50%. In the region where the value of 1/T b /f −3 dB  is large and the deterministic jitter is a dominant factor, the error is 10% or less. 
     The error of the measured values of the random jitter can be reduced by obtaining the histogram or the probability density function from the same number of samples of data as captured samples of data from which the bit error rate of the device under test was measured. Therefore, it has been confirmed that the measurement of the total jitter which is performed by using the probability density function separating method described with reference to  FIG. 17  is related to the measurement performed by using the conventional bit error rate test system. 
       FIG. 19E  illustrates a different exemplary configuration of the probability density function separating apparatus  100 . The probability density function separating apparatus  100  relating to the present example has a total jitter computing section  152  and a deciding section  154  in addition to the constituents of the probability density function separating apparatus  100  illustrated in one of  FIGS. 1 and 17A . The configuration shown in  FIG. 19E  is obtained by adding the total jitter computing section  152  and deciding section  154  to the probability density function separating apparatus  100  shown in  FIG. 17A . The probability density function separating apparatus  100  relating to the present example is supplied a probability density function indicating a noise component contained in a signal under test. 
     The total jitter computing section  152  computes the value of the total jitter contained in the signal under test based on the peak to peak value which is computed by the deterministic component computing section  150  (or the peak to peak value detecting section  140 ). The total jitter computing section  152  may compute the value of the total jitter by using the method described with reference to Expression (8). 
     For example, the total jitter computing section  152  may receive the random component computed by the random component computing section  130 , and compute the value of the total jitter based on the received random component and the above-mentioned peak to peak value. Alternatively, the total jitter computing section  152  may receive the value of the random component contained in the probability density function from the user or the like. If such is the case, the probability density function separating apparatus  100  may not necessarily include therein the standard deviation computing section  120  and random component computing section  130 . 
     The deciding section  154  decides whether the signal under test is acceptable or not based on the value of the total jitter computed by the total jitter computing section  152 . For example, the deciding section  154  may make the decision whether the signal under test is acceptable or not by judging whether the value of the total jitter falls within a predetermined range. 
       FIG. 20  is a view showing another example of a configuration of the probability density function separating apparatus  100 . The probability density function separating apparatus  100  according to the present example further includes a synthesizing section  160  and a comparing section  170  in addition to a configuration of the probability density function separating apparatus  100  described with reference to  FIG. 1 . Other components have a function equal to that of components that have been described using the same symbols in  FIG. 1 . 
     The synthesizing section  160  generates a composite probability density function (hereinafter, referred to as a composite PDF) obtained by convolving (convolution integrating) a probability density function of a random component computed from the random component computing section  130  and a probability density function of a deterministic component computed from the deterministic component computing section  150 . 
     The comparing section  170  compares a composite PDF output from the synthesizing section  160  and an input PDF. As described in  FIG. 9 , the deterministic component computing section  150  is previously supplied with a function of which a peak to peak value is unknown and substitute the peak to peak value detected from the peak to peak value detecting section  140  for the function, in order to compute a probability density function with a deterministic component. 
     At this time, this function is different according to whether a deterministic component is a sine wave, uniform distribution, triangular distribution, or dual Dirac. For this reason, in order to compute a probability density function with a deterministic component based on a peak to peak value, it is preferable to be able to decide which function is a function with a deterministic component. 
     The deterministic component computing section  150  may be supplied with which function is a function with a deterministic component in advance. Moreover, the deterministic component computing section  150  may be previously supplied with a plurality of functions according to the type of distribution of a deterministic component, substitute the peak to peak value detected from the peak to peak value detecting section  140  for each function, and respectively compute a probability density function for each type of distribution of a deterministic component. 
     In this case, the synthesizing section  160  respectively synthesizes each probability density function output from the deterministic component computing section  150  and a probability density function output from the random component computing section  130 . The comparing section  160  respectively compares the composite PDF each synthesized by the synthesizing section  160  and the input PDF. The comparing section  170  selects a function appropriate as a function showing a deterministic component included in the input PDF based on a comparison result for each composite PDF. For example, the comparing section  170  may select a function in which a difference between the composite PDF and the input PDF becomes smallest. 
     Then, the deterministic component computing section  150  may output a probability density function with a deterministic component corresponding to the function selected by the comparing section  170  as an appropriate probability density function. By such a process, although a type of distribution of a deterministic component is indefinite, it is possible to select appropriate distribution from distribution of a predetermined type and compute a probability density function with a deterministic component included in an input PDF. 
     Moreover, the peak to peak value detecting section  140  detects a peak to peak value with predetermined measurement resolution. In this case, the detected peak to peak value includes an error according to measurement resolution. The probability density function separating apparatus  100  in the present example can perform a process reducing this measurement error. Moreover, the probability density function separating apparatus  100  may perform both of selection of a function prescribing the deterministic component and a process reducing a measurement error to be described below. 
     For example, the deterministic component computing section  150  computes a deterministic component corresponding to each peak to peak value when sequentially changing the peak to peak values using the peak to peak value detected from the peak to peak value detecting section  140  as standard. At this time, the deterministic component computing section  150  may sequentially change the peak to peak values in a range according to measurement resolution. 
     For example, when measurement resolution is 2 a and the peak to peak value detected from the peak to peak value detecting section  140  is 2 T 0 , the deterministic component computing section  150  may sequentially change the peak to peak values in a range of 2 T 0 −a to 2 T 0 +a. At this time, it is preferable that resolution changing a peak to peak value is sufficiently smaller than measurement resolution. 
     The synthesizing section  160  sequentially generates composite PDF obtained by sequentially synthesizing a probability density function with each deterministic component sequentially output from the deterministic component computing section  150  and a probability density function with a random component. The comparing section  170  compares each composite PDF and the input PDF, and selects either of the peak to peak values as an optimum value based on the comparison result. By such a process, it is possible to reduce a measurement error caused by measurement resolution. 
       FIG. 21  is a view exemplary showing operations of the probability density function separating apparatus  100  shown in  FIG. 20 . In this example, it will be explained about an operation when reducing the measurement error. First, the domain transforming section  110  transforms the input PDF into a spectrum in a frequency domain. 
     Then, the standard deviation computing section  120  computes standard deviation of a random component included in the input PDF based on this spectrum (S 10 ). Then, the random component computing section  130  computes a probability density function with this random component based on this standard deviation (S 12 ). 
     Next, the peak to peak value detecting section  140  computes a peak to peak value of a spectrum of the input PDF (S 14 ). Then, the deterministic component computing section  150  computes a probability density function with a deterministic component based on this peak to peak value (S 16 ). 
     Next, the synthesizing section  160  generates composite PDF made by convolving a probability density function of a random component and a probability density function of a deterministic component (S 18 ). This synthesizing may be performed by convolution integrating probability density functions in each time domain. 
     Next, the comparing section  170  compares the input PDF and the composite PDF (S 20 ). The comparing section  170  may compute an error between the input PDF and the composite PDF. This error may be root mean square of an error on a time section respectively set. Tail sections on both ends of a probability density function may be designated as the time sections. 
     Next, the peak to peak value is changed in predetermined entire range, and it is determined whether comparison between the input PDF and the composite PDF has been completed (S 22 ). When there is a range in which the peak to peak value is not changed, the peak to peak value is changed into a value to be compared (S 24 ), and processes of S 16  to S 20  are repeated. 
     When the peak to peak value is changed in entire range, a peak to peak value having a small error is determined based on the comparison result in S 20  for each peak to peak value (S 26 ). 
     By such a process, it is possible to reduce a measurement error and determine an optimal peak to peak value. The B(f) of Expression (5) may be recalculated to compute standard deviation of a random component with high precision by means of a probability density function with a deterministic component having this peak to peak value. 
     Tails on both ends of a probability density function are decided by a random component. On the contrary, it is possible to compare a value of a probability density function with a predetermined threshold value from both ends to a central portion and detect a time width having probability density larger than this threshold value, in order to compute D(p−p). 
       FIG. 22A  shows a probability density function with a deterministic component including only a sine wave as a deterministic jitter. An expected value of D(p−p) of a sine wave in the present example is 50 ps. 
       FIG. 22B  shows a spectrum obtained by transforming a probability density function shown in  FIG. 22A  into a frequency domain. Null frequency of this spectrum is 15.3 GHz (0.765/50 ps) of an expected value. 
       FIG. 23A  shows a probability density function with a deterministic component including a sine wave and a sine wave of which energy is relatively smaller than that of the sine wave as a deterministic jitter. In this case, this probability density function is obtained by convolution integrating two sine waves. It is understood that a small sine wave acts on a probability density function as noises. 
     An expected value of D(p−p) of a large sine wave in the present example is 50 ps.  FIG. 23B  shows a spectrum obtained by transforming a probability density function shown in  FIG. 23A  into a frequency domain. Null frequency of this spectrum is 15.3 GHz. In other words, it is understood that noises of a probability density function does not act on null frequency. That is to say, according to the present method for detecting D(p−p) based on null frequency, it is possible to reduce an influence of noise of a probability density function to detect D(p−p). 
       FIG. 23C  shows a asymmetric probability density function.  FIG. 23D  shows a spectrum obtained by transforming a asymmetric probability density function shown in  FIG. 23C  into a frequency domain. In the present example, an expected value of D(p−p) is 50 ps, and null frequency of this spectrum is 16.5 GHz. In other words, a conventional method cannot detect reproducible D(p−p). However, the present method for detecting D(p−p) based on null frequency can detect D(p−p) with an error of 8%. 
       FIG. 24A  shows a probability density function with a deterministic component including a sine wave and a sine wave of which energy is equal to that of the sine wave as a deterministic jitter. An expected value of D(p−p) in the present example is 100 ps. 
       FIG. 24B  shows a spectrum obtained by transforming a probability density function shown in  FIG. 24A  into a frequency domain. Null frequency of this spectrum has an error of about 5 GHz for 10 GHz of an expected value. 
       FIG. 25A  is a view showing uniform distribution obtained by performing a predetermined threshold process on a probability density function shown in  FIG. 24A . In other words, a value larger than a predetermined threshold value is replaced by this threshold value among values of this probability density function and a value smaller than the predetermined threshold value is replaced by zero, in order to show a probability density function transformed into uniform distribution. 
       FIG. 25B  is a view showing a spectrum obtained by transforming uniform distribution shown in  FIG. 25A  into a frequency domain. It is possible to obtain 10.1 GHz substantially equal to an expected value as D(p−p) by performing a threshold process. A threshold value providing D(p−p) substantially identical with an expected value can be determined by, for example, sequentially changing a threshold value to compute D(p−p) for each threshold value and detecting a threshold value of which D(p−p) is not substantially changed. 
       FIG. 26  shows values of D(p−p) measured by a threshold process and D(δδ) measured by a conventional method for a probability density function including a plurality of deterministic jitters. As described in  FIGS. 24A ,  24 B,  25 A and  25 B, in case of measuring a probability density function made by convolution integrating two sine waves, in a conventional curve fitting method, a result of D( 55 )=80.5 ps is obtained when an expected value of a peak to peak value of a deterministic component is 100 ps. 
     Correspondingly, in the measurement performing a threshold process, it is possible to obtain D(p−p)=99.0 ps substantially equal to an expected value. Similarly, when measuring a probability density function made by convolution integrating two sine waves of a sine wave and a relatively small sine wave as a deterministic jitter, in the measurement performing a threshold process, it is possible to obtain D(p−p)=49.0 ps substantially equal to an expected value. Moreover, conventionally, each deterministic component cannot be separated from a probability density function in which a plurality of deterministic components is convolution integrated. 
       FIG. 27A  shows a spectrum of a probability density function with a deterministic component of a sine wave and a spectrum of a probability density function with a deterministic component in which two sine waves are convolution integrated. Since a spectrum of a probability density function in which two sine waves are convolution integrated is a square of a spectrum of a probability density function of one sine wave, a magnitude of a main lobe adjacent to 0 Hz changes. 
     In other words, as shown in  FIG. 27B , when raising a spectrum of a probability density function in which two sine waves are convolution integrated to 0.5th power, a probability density function of one sine wave and a main lobe are to be identical with each other. Using the above-described principle, it is possible to obtain the number of deterministic components included in a probability density function. 
       FIG. 28  is a flowchart exemplary showing a method for obtaining the number of deterministic components included in a probability density function. First, an input PDF is transformed into a spectrum in a frequency domain (S 50 ). The step S 50  may be performed by the domain transforming section  110 . 
     Next, a main lobe of a spectrum is raised to βth power (S 52 ). Then, it is decided whether a main lobe of a spectrum of a probability density function with a predetermined deterministic component and β power of the main lobe obtained in S 52  are identical with each other (S 54 ). That main lobes are identical with each other may be determined when an error between the main lobes is within a predetermined range. A probability density function with a predetermined deterministic component may be designated by a user. Moreover, as described in reference to  FIG. 10 , the deterministic component computing section  150  may select a probability density function with a deterministic component out of a previously given plurality of functions. 
     In S 54 , when it is determined that the main lobes are not identical with each other, β is changed (S 58 ), and then the processes of S 52  and S 54  are repeated. Moreover, in S 54 , when it is determined that the main lobes are identical with each other, the number of deterministic components is computed in S 56 . 
     In S 56 , 1/β is computed as the number of deterministic components. At this time, β is not limited to an integer number. A value of β after the decimal point shows that a deterministic component with the different size is included. 
     For example, when D(p−p) values of two sine waves described in  FIGS. 24A ,  24 B,  25 A and  25 B are 50 ps, the whole value of D(p−p) becomes 100 ps. Then, for example, when performing a threshold process described in  FIGS. 25A and 25B , a value substantially equal to 100 ps is measured as D(p−p) of a deterministic jitter. 
     Furthermore, by a method described in reference to  FIG. 28 , the number of deterministic components is computed. Since values of D(p−p) of two sine waves are substantially equal, β=0.5 is computed and the number of deterministic components becomes two. From the above-described result, it is possible to compute D(p−p) of each sine wave as 50 ps. 
     As described above, according to this method, it is possible to estimate the number of deterministic components from a probability density function including a plurality of deterministic components. The number of deterministic components may be computed by the deterministic component computing section  150  according to the method. 
       FIG. 29  is a view exemplary showing a configuration of a noise separating apparatus  200  according to an embodiment of the present invention. The noise separating apparatus  200  separates a probability density function with a predetermined noise component from a probability density function of a signal under test. For example, the noise separating apparatus  200  separates a random noise component and a deterministic noise component from a probability density function with noises included in the signal under test. 
     The noise separating apparatus  200  includes a sampling section  210  and the probability density function separating apparatus  100 . The probability density function separating apparatus  100  may have the same function and configuration as those of the probability density function separating apparatus  100  described in  FIGS. 1 to 28 . 
     The sampling section  210  samples the signal under test according to a given sampling signal, and generates a probability density function of the signal under test. For example, the sampling section  210  may generate a probability density function with jitter included in the signal under test, or generate a probability density function with amplitude noises of the signal under test. 
       FIG. 30  is a view exemplary showing a probability density function of a signal under test generated from the sampling section  210 . The sampling section  210  according to the present example outputs a probability density function of a signal under test as described in  FIG. 29 .  FIG. 30  shows an eye diagram of the signal under test assuming that a horizontal axis is a time and a vertical axis is a level of the signal under test. The sampling section  210  may acquire this eye diagram. 
     When generating a probability density function with jitter included in the signal under test, the sampling section  210  computes a probability by which an edge of the signal under test exists for each time. For example, the sampling section  210  may sample the signal under test by multiple times for each of relative timings for the signal under test in a transition timing of the signal under test. Then, a probability by which an edge exists at each of the relative timings may be acquired based on a sampling result. 
     Moreover, when generating a probability density function of amplitude noises in a signal under test, the sampling section  210  acquires a probability by which the signal under test is likely to have a particular amplitude value. For example, the sampling section  210  acquires an amplitude value of the signal under test at the generally same relative timing to the signal under test in a stationary timing of the signal under test. 
     When the sampling section  210  is a comparator for comparing a reference voltage and a level of the signal under test, the sampling section  210  may change this reference voltage and sample the signal under test for each reference voltage by multiple times. For each amplitude value, the sampling section  210  acquires a probability based on a sampling result. 
     The probability density function separating apparatus  100  separates a random component and a deterministic component from a probability density function provided from the sampling section  210 . For example, when this probability density function is a probability density function of jitter in a signal under test, the probability density function separating apparatus  100  can separate a random jitter from a deterministic jitter in the signal under test with high precision. 
     Moreover, when this probability density function is a probability density function of amplitude noises in a signal under test, the probability density function separating apparatus  100  can separate a random component from a deterministic component in amplitude noises of the signal under test with high precision. For this reason, according to the noise separating apparatus  200  in the present example, it is possible to separate a noises component of a signal under test with high precision and thus analyze the signal under test with high precision. 
     Moreover, the noise separating apparatus  200  can also separate a random component from a deterministic component in noises of a sampling signal given to the sampling section  210 . For example, the sampling section  210  has a comparator or an ADC for converting a level of a signal under test into a digital value according to the sampling signal. 
     When an analog sinusoidal jitter or amplitude noise is given as a signal under test, a probability density function of digital data output from the comparator or the ADC in the sampling section  210  shows a characteristic that both ends sharply attenuated as shown in  FIG. 2 . However, when internal noises occur in a sampling signal and measurement errors occur in digital data, this probability density function becomes a composite component of a random component and a deterministic component. 
     The sampling section  210  generates a probability density function of the signal under test based on a result obtained by sampling the signal under test with small noises. Then, the probability density function separating apparatus  100  separates a random component and a deterministic component included in this probability density function. In this way, it is possible to measure noises of a sampling signal with high precision. Moreover, the noise separating apparatus  200  can be also used for a test of the ADC. That is to say, it is possible to separate a deterministic component caused by a code error of the ADC. 
       FIG. 31  is a view showing probability density of each code of an ADC when the ADC samples a sine wave without noises. Here, a code of the ADC is a code corresponding to each digital value output from the ADC. The ADC determines which code corresponds to a level of a signal to be input, and outputs a digital value according to this code. 
     In the present example, the ADC has codes of 0 to 255. Here, it will be described, for example, about when an error occurs in the 213th code and a level corresponding to this code cannot be detected. In this case, as shown in  FIG. 31 , probability density of the code  213  deteriorates and probability density of a code (a code  214  in the present example) adjacent to the code  213  rises. The reason is that the code  214  detects a level of a sine wave to be originally detected by the code  213 . 
     A probability density function shown in  FIG. 31  includes a deterministic component by a sine wave to be input and a deterministic component caused by a code error of the ADC. As described in reference to  FIG. 28 , the probability density function separating apparatus  100  can separate these deterministic components. 
       FIG. 32  is a view showing another example of a configuration of the noise separating apparatus  200 . The noise separating apparatus  200  in the present example further includes a correction section  220  in addition to a configuration of the noise separating apparatus  200  described with reference to  FIG. 29 . The noise separating apparatus  200  in the present example reduces an influence by internal noises of the above-described sampling signal to separate a deterministic component and a random component from a probability density function of a signal under test. 
     For example, when reducing an influence by noises of a sampling signal, the sampling section  210  first functions as a sampling signal measuring section that computes a probability density function of a sampling signal itself as described above. At this time, it is preferable that the sampling section  210  is supplied with a reference signal with small noises. 
     Moreover, the sampling section  210  functions as a signal under test measuring section that computes a probability density function of a measurement signal to be measured. At this time, the sampling section  210  may perform an operation similar to that of the sampling section  210  described in  FIG. 24 . 
     The probability density function separating apparatus  100  separates a random component and a deterministic component from each of a probability density function of a signal under test and a probability density function of a timing signal. 
     Then, the correction section  220  separates a random component from a deterministic component in the signal under test with higher precision by correcting a parameter of the probability density function of the signal under test based on the probability density function of the timing signal. 
     For example, the correction section  220  may correct a random component according to the signal under test by subtracting energy of a random component according to the timing signal from energy of a random component according to the signal under test. Moreover, the correction section  220  may correct a deterministic component according to the signal under test by subtracting a deterministic component according to the timing signal from a deterministic component according to the signal under test. By such a process, it is possible to separate a random component from a deterministic component according to a signal under test with high precision. 
       FIG. 33  is a view exemplary showing a configuration of a testing apparatus  300  according to an embodiment of the present invention. The testing apparatus  300  is an apparatus for testing a device under test  400  and includes a noise separating apparatus  200  and a deciding section  310 . 
     The noise separating apparatus  200  has a configuration substantially equal to that of the noise separating apparatus  200  described in  FIGS. 29 to 32  and measures a signal under test output from the device under test  400 . In the present example, the noise separating apparatus  200  has a configuration substantially equal to that of the noise separating apparatus  200  shown in  FIG. 32 . The noise separating apparatus  200  may have a timing generator  230  for generating a timing signal as shown in  FIG. 32 . The other components are equal to components with the same symbol described in reference to  FIGS. 29 to 28 . 
     The deciding section  310  decides the good or bad of the device under test  400  based on a random noise component and a deterministic noise component separated from the noise separating apparatus  200 . For example, the deciding section  310  may decide the good or bad of the device under test  400  based on whether standard deviation of the random noise component is within a predetermined range. 
     Moreover, the deciding section  310  may decide the good or bad of the device under test  400  based on whether a peak to peak value of the deterministic noise component is within a predetermined range. The deciding section  310  may compute total jitter from the standard deviation of the random noise component and the peak to peak value of the deterministic noise component, and decide the good or bad of the device under test  400 . The deciding section  310  may compute total jitter, for example, given by 14*σ+D(p−p). Here, the coefficient “14” is associated with the bit error rate of 10 −12  in the table shown in  FIG. 19D . The coefficient may be selected in accordance with the bit error rate threshold for the device under test. 
     According to the testing apparatus  300  in the present example, since a probability density function of a signal under test can be separated with high precision, it is possible to decide the good or bad of the device under test  400  with high precision. Moreover, the testing apparatus  300  may further include a pattern generating section that inputs a test signal into the device under test  400  and outputs a predetermined output signal. 
       FIG. 34  is a view exemplary showing a measurement result of jitter by the jitter separating apparatus  200  and a measurement result of jitter by a conventional method. As shown in  FIG. 34 , the jitter separating apparatus  200  can obtain a measurement result with precision more preferable than a conventional method in any measurement result of a random jitter and a deterministic jitter, about when only a random jitter is included in a signal under test, when a random jitter and sine wave jitter (a deterministic jitter) are included in a signal under test, and when noises are included in a sampling signal. 
       FIG. 35  is a view showing a conventional measurement result described in  FIG. 34 . As described above, according to a conventional measuring method, tail portions of an input PDF shown with a wavy line in  FIG. 35  is curve fitted. As a result, random components as shown with a solid line in  FIG. 35  is detected. Moreover, an interval between two peaks of this random components is detected as a deterministic component. When using such a measuring method, since curve-fitting approximation is used, each component cannot be measured with high precision. For this reason, a measurement result has gross errors with respect to an expected value as shown in  FIG. 34 . 
     Moreover, this method cannot separate a deterministic component caused by the above-described error in sampling signal and a deterministic component caused by a code error of an ADC. For this reason, for example, when a sampling error occurs as shown in  FIG. 34 , it is not possible to perform measurement with high precision. 
       FIGS. 36A and 36B  are views showing a measurement result by the present invention described in  FIG. 34 .  FIG. 36A  shows an input PDF and  FIG. 36B  shows a probability density function obtained by convolving a deterministic component and a random component separated using the probability density function separating apparatus  100 . 
     The probability density function separating apparatus  100  can separate a random component from a deterministic component in the input PDF with high precision as described above. For this reason, as shown in  FIG. 34 , it is possible to obtain a measurement result with a small error for an expected value. Furthermore, since the present invention can separate a plurality of deterministic components, it is possible to separate, for example, a deterministic component of a sinusoid and a deterministic component caused by timing errors in a sampling signal. As a result, it is possible to perform measurement with higher precision. 
       FIG. 37  is a view exemplary showing a configuration of the sampling section  210  described in  FIG. 33 . The sampling section  210  has an amplifier  202 , a level comparing section  204 , a variable delay circuit  212 , a variable delay circuit  214 , a timing comparing section  216 , an encoder  226 , a memory  228 , and a probability density function computing section  232 . 
     The amplifier  202  receives an output signal from the device under test  400 , amplifies the signal at a predetermined amplification gain, and outputs the amplified signal. The level comparing section  204  compares a level of the output signal and a given reference value, and outputs a comparison result. In the present example, the level comparing section  204  has a comparator  206  and a comparator  208 . The comparator  206  is supplied with a reference value of a High level. Moreover, the comparator  208  is supplied with a reference value of a Low level. 
     The timing comparing section  216  samples the comparison result output from the level comparing section  204  according to a given timing signal, and converts it into digital data. In the present example, the timing comparing section  216  has a flip-flop  218  and a flip-flop  222 . 
     The flip-flop  218  receives the timing signal output from the timing generating section  224  via the variable delay circuit  212 . Moreover, the flip-flop  218  samples the comparison result output from the comparator  206  according to this timing signal. 
     The flip-flop  222  receives the timing signal output from the timing generating section  224  via the variable delay circuit  214 . Moreover, the flip-flop  222  samples the comparison result output from the comparator  208  according to this timing signal. 
     In the present example, the level comparing section  204  has two comparators  206  and  208 . However, the level comparing section  204  may output a comparison result by one comparator, or may output a comparison result by three or more comparators. In other words, the level comparing section  204  may output a comparison result with multiple values. The timing comparing section  216  may have flip-flops according to the number of comparators belonging to the level comparing section  204 . 
     The variable delay circuits  212  and  214  delay and output a timing signal. The variable delay circuits  212  and  214  adjust a phase of the timing signal to a predetermined phase to supply it to the timing comparing section  216 . 
     The encoder  226  encodes the digital data output from the timing comparing section  216 . For example, the encoder  226  may generate digital data with multiple values based on each of the digital data output from the flip-flop  218  and the flip-flop  222 . The memory  228  stores the digital data generated from the encoder  226 . 
     The probability density function computing section  232  computes a probability density function of the output signal based on the digital data stored on the memory  228 . For example, the probability density function computing section  232  may generate a probability density function with a jitter described in  FIG. 30 , or may generate a probability density function with an amplitude degradation component described in  FIG. 30 . 
     When generating a probability density function with a jitter, the timing generating section  224  generates a timing signal of which a phase for the output signal is sequentially changed. The phase of the timing signal may be adjusted by changing a delay amount in the variable delay circuits  212  and  214 . Moreover, the level comparing section  204  is supplied with a reference value. 
     The timing comparing section  216  samples a logical value of the output signal according to a timing signal of which a phase for the output signal is sequentially changed. The probability density function computing section  232  compares a sampled value sequence stored on the memory  228  and a given expected value sequence. 
     Moreover, the probability density function computing section  232  detects a phase of the output signal based on this comparison result. For example, the probability density function computing section  232  may detect a phase of an edge of the output signal based on this comparison result. Moreover, the probability density function computing section  232  may detect a timing at which a logical value of the output signal is changed. At this time, although consecutive data in the output sequence show the identical logical value, the probability density function computing section  232  can detect a boundary timing of each data section in the output signal. 
     Moreover, the timing comparing section  216  and the probability density function computing section  232  perform comparison between the logical value of the output sequence and the expected value at each phase of the timing signal for multiple times, and obtains an error count value. A probability by which the logical value of the output signal is generated at each phase can be computed from this error count value. In other words, it is possible to generate a probability density function with a jitter. For example, the timing comparing section  216  and the probability density function computing section  232  perform comparison between the logical value of the output sequence and the expected value at each phase of the timing signal for multiple times. Then, a probability density function may be obtained by computing a difference between error count values of adjacent phases of the corresponding timing signals. 
     Next, it will be described about when a probability density function with an amplitude degradation component in an output signal is generated. In this case, the timing generating section  224  generates a timing signal substantially synchronized with the output signal. In other words, an edge of the timing signal has a constant phase for the output signal. Moreover, the level comparing section  204  is sequentially supplied with different reference values. 
     The timing comparing section  216  samples the comparison result according to the timing signal synchronized with the output signal. In other words, the timing comparing section  216  detects a comparison result between a level of the output signal at an edge timing of the timing signal and a reference value. It is possible to generate a probability density function with an amplitude degradation component in the output signal by detecting this comparison result for each reference value for multiple times. 
     The probability density function computing section  232  supplies the generated probability density function to the probability density function separating apparatus  100 . By such a configuration, it is possible to separate a noise component from an output signal with high precision and thus test the device under test  400  with high precision. For example, in case of testing a random jitter included in an output signal from the device under test  400 , when a deterministic jitter occurs in a timing signal, the good or bad of the device under test  400  cannot be decided with high precision. However, according to the testing apparatus  300  in the present example, it is possible to simultaneously separate a component of a deterministic jitter caused by a timing signal and detect a component of a random jitter in an output signal. 
       FIG. 38  is a view exemplary showing a measurement result by the testing apparatus  300  described with reference to  FIG. 37  and a measurement result by a conventional curve fitting method described in  FIG. 2 . In  FIG. 2  shows an error between each measurement result and a measurement result to be expected. 
     In addition, the measurement result by a conventional method in the present example has been quoted from the following document: G. Hansel, K. Stieglbauer, “Implementation of an Economic Jitter Compliance Test for a Multi-Gigabit Device on ATE”, in Proc. IEEE int. Test Conf., Charlotte, N.C., Oct. 26-28, 2004, pp. 1303-1311. 
     Moreover, in measurement of the present example, a random component and a deterministic component in a probability density function of jitter in an output signal from the device under test  400  have been separated from each other. Moreover, the measurement result by the conventional method corresponds to a case of including a large sinusoidal component with amplitude of about 40 ps and a case including a small sinusoidal component with amplitude of about 5 ps as a deterministic component. As shown in  FIG. 38 , the testing apparatus  300  can obtain a measurement result with a smaller error than that of a conventional curve fitting method in any case. 
       FIG. 39  is a view exemplary showing a configuration of a bit error rate measuring apparatus  500  according to an embodiment of the present invention. The bit error rate measuring apparatus  500  is an apparatus for measuring a bit error rate of output data provided from the device under test  400  or the like, and includes a variable voltage source  502 , a level comparator  504 , an expected value generating section  510 , a sampling section  512 , an expected value comparing section  514 , a timing generating section  506 , a variable delay circuit  508 , a counter  516 , a trigger counter  518 , a probability density function computing section  520 , and a probability density function separating apparatus  100 . 
     The level comparator  504  compares a level of output data and a given reference value, and outputs comparison data. For example, the level comparator  504  outputs comparison data showing a magnitude relation between a level of output data and the given reference value with a binary logical value. The variable voltage source  502  generates this reference value. The sampling section  512  samples a data value output from the level comparator  504  according to a given timing signal. 
     The timing generating section  506  generates a timing signal, and supplies the generated signal to the sampling section  512  via the variable delay circuit  508 . The timing generating section  506  may generate a timing signal with a period substantially equal to that of the output data. The variable delay circuit  508  adjusts the timing signal to a predetermined phase. 
     The expected value generating section  510  generates an expected value that the data value output from the sampling section  512  should have. The expected value comparing section  514  compares the data value output from the sampling section  512  and the expected value output from the expected value generating section  510 . The expected value comparing section  514  may output, for example, an exclusive OR of this data value and this expected value. 
     The counter  516  counts the number of times by which a comparison result in the expected value comparing section  514  shows a predetermined logical value. For example, the counter counts the number of times by which the exclusive OR output from the expected value comparing section  514  is one. Moreover, the trigger counter  518  counts pulses of the timing signal. 
     By such a configuration, it is possible to count the number of erroneous timings by which a data value of output data at a particular phase of timing signal is different form expected value. Moreover, similarly to the testing apparatus  300  described in  FIG. 37 , an error count value is obtained for each phase of a timing signal by sequentially changing a phase of the timing signal. The probability density function computing section  520  may compute a probability density function of a jitter in output data by computing a difference between adjacent error count values. 
     In addition, similarly to the testing apparatus  300  described in  FIG. 37 , although consecutive data in an output sequence show the identical logical value, the probability density function computing section  520  can detect a boundary timing of each data section in output data. 
     Moreover, similarly to the testing apparatus  300  described in  FIG. 37 , the probability density function computing section  520  can compute a probability density function of an amplitude degradation component of output data by sequentially changing the reference value generated from the variable voltage source  502 . In this case, a phase of a timing signal for capturing output data is substantially constantly controlled. 
     The probability density function separating apparatus  100  is equal to the probability density function separating apparatus  100  described with reference to  FIG. 33 . That is to say, a deterministic component and a random component in a given probability density function are separated from each other. 
     By such a configuration, it is possible to generate a probability density function of given output data and simultaneously separate a deterministic component and a random component. In other words, it is possible to simultaneously separate and analyze a bit error caused by a deterministic component from a bit error caused by a random component. 
       FIG. 40  is a view showing another example of a configuration of the bit error rate measuring apparatus  500 . The bit error rate measuring apparatus  500  in the present example includes an offset section  522 , an amplifier  524 , a sampling section  526 , a comparison counting section  528 , a variable delay circuit  530 , and a processor  532 . 
     The offset section  522  adds a predetermined offset voltage to a waveform of output data. The amplifier  524  outputs a signal output from the offset section  522  at a predetermined amplification factor. 
     The sampling section  526  samples a data value of the signal output from the amplifier  524  according to a given timing clock. A timing clock may be, e.g., a recovered clock generated from output data. The variable delay circuit  530  adjusts a timing clock to a predetermined phase. 
     The comparison counting section  528  compares a data value output from the sampling section  526  and a given expected value, and counts a comparison result. The comparison counting section  528  may have a function equal to that of the expected value comparing section  514  and the counter  516  described in  FIG. 39 . 
     The processor  532  controls the offset section  522  and the variable delay circuit  530 . For example, the processor adjusts an offset voltage to a predetermined level and controls a delay amount in the variable delay circuit  530 . By such a configuration, it is possible to compute a probability by which a data value of output data corresponding to a phase of a timing clock is different from the expected value. 
     Moreover, the processor  532  functions as the probability density function computing section  520  and the probability density function separating apparatus  100  described in  FIG. 39 . Similarly to the testing apparatus  300  described in  FIG. 37 , the processor  532  can compute a probability density function of a jitter in output data by sequentially changing a phase of a timing clock. For example, it is possible to can change a phase of a timing clock by changing a delay amount in the variable delay circuit  530 . 
     Here, a jitter in output data may be a timing jitter at a boundary of each data section in output data. Although consecutive data in an output signal show the identical logical value, the probability density function computing section  520  can detect a boundary timing of each data section in output signal. 
     Moreover, it is possible to perform measurement equal to a case when changing a reference value described in  FIG. 39  by sequentially changing an offset voltage added by the offset section  522 . In this case, the processor  532  can compute a probability density function with an amplitude degradation component of output data. In this case, a phase of a timing clock for output data is substantially constantly controlled. 
     The probability density function separating apparatus  100  is equal to the probability density function separating apparatus  100  described with reference to  FIG. 33 . That is to say, a deterministic component and a random component in a given probability density function are separated from each other. 
     By such a configuration, it is also possible to generate a probability density function of given output data and separate a deterministic component from a random component in this probability density function. In other words, it is possible to simultaneously separate and analyze a bit error caused by a deterministic component and a bit error caused by a random component. 
       FIG. 41  is a view showing another example of a configuration of the bit error rate measuring apparatus  500 . The bit error rate measuring apparatus  500  in the present example includes a flip-flop  534 , a switch section  536 , a flip-flop  538 , a frequency measuring section  548 , a control section  546 , a probability density function computing section  540 , and a probability density function separating apparatus  542 . 
     The flip-flop  534  samples a data value of output data according to a given timing clock. The switch section  536  selects one path from a plurality of paths having path length different from one another, and delays and outputs the data value output from the flip-flop  534  in a fixed delay amount according to the selected path. The latch section  538  latches the data value of which a phase is adjusted by the switch section  536  according to a given timing clock. 
     In other words, the bit error rate measuring apparatus  500  shown in  FIG. 40  adjusts a relative phase of a sampling clock for output data by adjusting a phase of a timing clock. However, the bit error rate measuring apparatus  500  in the present example adjusts a relative phase of a sampling clock for output data by adjusting a phase of the output data. 
     As shown in  FIG. 40 , if a timing of a clock is controlled in a large range by means of a variable delay circuit, the variable delay element will output incomplete or partial clocks when delay setting changes are made. The bit error rate measuring apparatus  500  in the present example can reduce a delay range of the variable delay circuit  544  and thus reduce the generation of incomplete clocks. 
     The frequency measuring section  548  measures frequency of a timing clock. The control section  546  generates a first control signal controlling a delay amount in the variable delay circuit  544  and a second control signal controlling a delay amount in the switch section  536  based on frequency of a timing clock to be expected and a relative phase of a sampling clock to be set. 
     The probability density function computing section  540  computes a probability density function of output data based on a data value sequentially latched in the latch section  538 . For example, similarly to the bit error rate measuring apparatus  500  described in  FIG. 40 , it is possible to compute a probability density function with a jitter of output data by sequentially changing a relative phase of a timing clock for the output data. Moreover, similarly to the bit error rate measuring apparatus  500  described in  FIG. 40 , in the present example, the bit error rate measuring apparatus  500  may further include means for computing a probability density function with an amplitude degradation component. 
     The probability density function separating apparatus  542  is equal to the probability density function separating apparatus  100  described with reference to  FIG. 33 . That is to say, a deterministic component and a random component are separated from a given probability density function. 
     By such a configuration, it is also possible to generate a probability density function of given output data and separate a deterministic component from a random component in this probability density function. In other words, it is possible to simultaneously separate and analyze a bit error caused by a deterministic component and a bit error caused by a random component. 
     In addition, a configuration of the bit error rate measuring apparatus  500  is not limited to the configurations described in  FIGS. 39 to 41 . It is possible to simultaneously separate and measure a random component and a deterministic component of a probability density function corresponding to a bit error rate by adding a probability density function separating apparatus and a probability density function computing section to a configuration of a conventional bit error rate measuring apparatus. 
       FIG. 42  is a view exemplary showing a configuration of an electronic device  600  according to an embodiment of the present invention. The electronic device  600  may be a semiconductor chip for generating a predetermined signal. The electronic device  600  includes an operation circuit  610 , a measurement circuit  700 , a probability density function computing section  562 , and a probability density function separating apparatus  100 . 
     The operation circuit  610  outputs a predetermined signal according to a given input signal. In the present example, the operation circuit  610  is a PLL circuit having a phase comparator  612 , a charge pump  614 , a voltage controlled oscillator  616 , and a divider  618 . In addition, the operation circuit  610  is not limited to the PLL circuit. 
     The measurement circuit  700  has a selector  550 , a base delay  552 , a variable delay circuit  554 , a flip-flop  556 , a counter  558 , and a frequency counter  560 . The selector  550  selects and outputs either of an output signal from the operation circuit  610  or a round loop signal output from the variable delay circuit  554 . 
     The base delay  552  delays the signal output from the selector  550  in a predetermined delay amount. Moreover, the variable delay circuit  554  delays the signal output from the base delay  552  in a set delay amount. 
     The flip-flop  556  samples a signal output from the selector  550  according to a signal output from the variable delay circuit  554 . The flip-flop  556  can sample the signal output from the selector  550  at a desired phase by controlling a delay amount in the variable delay circuit  554 . 
     The counter  558  counts the number of times by which data output from the flip-flop  556  show a predetermined logical value. When the selector  550  selects an output signal from the operation circuit  610 , it is possible to obtain an existing probability of an edge at each phase of the output signal from the operation circuit  610  by changing a delay amount in the variable delay circuit  554 . 
     The probability density function computing section  562  computes a probability density function of an output signal based on the counted result output from the counter  558 . The probability density function computing section  562  may compute a probability density function in an operation similar to that of the probability density function computing section  232  described in  FIG. 37 . 
     The probability density function separating apparatus  100  separates a predetermined component from a probability density function computed from the probability density function computing section  562 . The probability density function separating apparatus  100  may have the same or similar function and configuration as or to those of the probability density function separating apparatus  100  described with reference to  FIGS. 1 to 31 . 
     Moreover, the probability density function separating apparatus  100  in the present example may include a part of a configuration of the probability density function separating apparatus  100  described with reference to  FIGS. 1 to 31 . For example, the probability density function separating apparatus  100  may output standard deviation of a random component or a peak to peak value of a deterministic component detected from the standard deviation computing section  120  or the peak to peak value detecting section  140  to an external apparatus, without including the random component computing section  130  or the deterministic component computing section  150  described in  FIG. 1 . 
     By such a configuration, it is possible to separate a predetermined component from a probability density function of a signal output from the operation circuit  610  by a circuit provided in a chip including the operation circuit  610 . It is possible to obtain standard deviation of a random component of a signal output from the operation circuit  610  with high precision without receiving an influence of a deterministic component depending on the base delay  552  and the variable delay circuit  554 . In this way, it is possible to easily perform analysis of the operation circuit  610 . 
     Moreover, when the selector  550  selects an output signal from the variable delay circuit  554 , the output signal from the variable delay circuit  554  is input into the base delay  552  by round looping. The frequency counter  560  measures frequency of a pulse signal by counting pulse signals transmitting this loop within a predetermined period. Since this frequency varies with a delay amount set in the variable delay circuit  554 , a delay amount in the variable delay circuit  554  can be measured by measuring this frequency. 
       FIG. 43  is a view showing another example of a configuration of the electronic device  600 . The electronic device  600  in the present example includes the same components as those of the electronic device  600  described in  FIG. 42 . However, connection relation between components is different. 
     In the present example, the selector  550  receives an input signal being split from the input into the operation circuit  610 . The selector  550  selects and outputs either of this input signal or an output signal from the variable delay circuit  554 . 
     Moreover, the base delay  552  is provided between the operation circuit  610  and the flip-flop  556 . In the present example, the base delay  552  delays a signal output from the divider  618 , and inputs it into the flip-flop  556 . 
     By such a configuration, similarly to the electronic device  600  described in  FIG. 42 , it is possible to compute a probability density function of a signal generated from the operation circuit  610 . Moreover, it is possible to separate a predetermined component from this probability density function. It is possible to obtain standard deviation of a random component of a signal output from the operation circuit  610  with high precision, without receiving an influence of a deterministic component depending on the base delay  552  and the variable delay circuit  554 . 
     In addition, a configuration of the measurement circuit  700  is not limited to a configuration described in  FIG. 42  or  43 . The measurement circuit  700  can adopt various configurations. For example, the measurement circuit  700  may have a configuration similar to that of the testing apparatus  300  described in  FIG. 37 , or may have a configuration similar to that of the bit error rate measuring apparatus  500  described in  FIGS. 39 to 41 . 
     Moreover, the probability density function separating apparatus  100  described above may input a high-purity signal into a circuit to be measured and compute a probability density function of a signal output from the circuit to be measured. A high-purity signal is, e.g., a signal of which a noise component is sufficiently small for a signal component. 
     Moreover, the probability density function separating apparatus  100  may input a signal of which a component such as jitter or amplitude degradation is known into a circuit to be measured. That is to say, a signal of which a random component of a probability density function is known may be input into a circuit to be measured. In this case, the probability density function separating apparatus  100  may separate the random component of the probability density function of the signal output from the circuit to be measured. Then, the random component generated in the circuit to be measured may be computed by comparing a random component of an input signal and a random component of an output signal. Any probability density function separating apparatus  100  included in the testing apparatus  200 , the bit error rate measuring apparatus  500 , or the electronic device  600  may have this function. 
       FIG. 44A  illustrates an exemplary configuration of a transfer function measuring apparatus  800  relating to an embodiment of the present invention. The transfer function measuring apparatus  800  includes therein the probability density function separating apparatus  100 , a transfer function computing section  820 , and a signal generating section  810 . The signal generating section  810  generates a test signal, and supplies the generated test signal to the device under test  400 . The signal generating section  810  has a function of injecting deterministic jitter to the test signal. The deterministic jitter is, for example, sinusoidal jitter. The signal generating section  810  has a function of adjusting the amplitude of the deterministic jitter. 
     The transfer function computing section  820  causes the signal generating section  810  to generate jitter having a predetermined amplitude. For example, the transfer function computing section  820  may cause the signal generating section  810  to generate deterministic jitter which has a constant peak to peak value, for example, sinusoidal jitter. 
     The probability density function separating apparatus  100  separates a deterministic component and a random component from a probability density function of jitter contained in a signal under test which is output from the device under test  400  in response to the test signal. The probability density function separating apparatus  100  may be the same as the probability density function separating apparatus  100  described with reference to  FIGS. 1 to 43 . 
     The probability density function separating apparatus  100  may receive the probability density function generated by the probability density function computing section  830 . The probability density function computing section  830  may be the same as the any of the probability density function computing sections ( 232 ,  520 ,  540 , and  562 ) which are described with reference to  FIGS. 37 to 43 . The probability density function computing section  830  may be provided between the device under test  400  and the probability density function separating apparatus  100 , and generate a probability density function indicating the jitter contained in the signal under test output from the device under test  400 . The probability density function computing section  830  may be alternatively provided within the transfer function measuring apparatus  800 . 
     The transfer function computing section  820  computes a jitter transfer function for the device under test  400 , based on the jitter generated by the signal generating section  810  and the jitter component separated by the probability density function separating apparatus  100 . For example, the transfer function computing section  820  may compute the jitter transfer function for the device under test  400 , based on the peak to peak value of the deterministic component generated by the signal generating section  810  and the peak to peak value of the deterministic component separated by the probability density function separating apparatus  100 . 
       FIG. 44B  illustrates another exemplary configuration of the transfer function measuring apparatus  800 . The transfer function measuring apparatus  800  relating to the present example may have the same constituents as the transfer function measuring apparatus  800  illustrated in  FIG. 44A . However, the probability density function separating apparatus  100  relating to the present embodiment has a channel for measuring the test signal output from the signal generating section  810  and a channel for measuring the signal under test output from the device under test  400 . The probability density function separating apparatus  100  relating to the present example may have the same configuration and functions as the probability density function separating apparatus  100  described with reference to  FIGS. 1 to 43 , for the respective channels. 
     The probability density function separating apparatus  100  may separate the deterministic component from the probability density function input from the probability density function computing section  830 , and separate the deterministic component from the probability density function indicating the jitter contained in the signal under test. The probability density function separating apparatus  100  may simultaneously perform appropriate operations respectively on the test signal and signal under test. 
     The transfer function computing section  820  computes the jitter transfer function for the device under test  400 , based on the jitter components separated from the test signal and signal under test by the probability density function separating apparatus  100 . For example, the transfer function computing section  820  may compute the jitter transfer function of the device under test  400  based on the peak to peak value of the deterministic component contained in the test signal and the peak to peak value of the deterministic component contained in the signal under test. 
       FIG. 45  is a view exemplary showing a hardware configuration of a computer  1900  according to the present embodiment. The computer  1900  functions as the probability density function separating apparatus  100 , the noise separating apparatus  200 , the computing apparatus, the testing apparatus  300 , the bit error rate measuring apparatus  500  and the transfer function measuring apparatus  800  described in  FIGS. 1 to 44 , based on a given program. 
     For example, when the computer  1900  functions as the probability density function separating apparatus  100 , the program may make the computer  1900  function as each component of the probability density function separating apparatus  100  described with reference to  FIGS. 1 to 28 . Moreover, when the computer  1900  functions as the noise separating apparatus  200 , the program may make the computer  1900  function as each component of the noise separating apparatus  200  described with reference to  FIGS. 29 to 36 . 
     Moreover, when the computer  1900  functions as the computing apparatus, the program may make the computer  1900  function as a computing apparatus including the time domain computing section  138  described in  FIGS. 11 and 12 . For example, when the computer  1900  functions as the computing apparatus that directly computes a probability density function in a time domain of a random component from a Gaussian curve in a frequency domain, the program may make the computer  1900  function as each component of the random component computing section  130  described in  FIG. 9 . 
     Moreover, when the computer  1900  functions as the computing apparatus that computes a waveform in a time domain from a spectrum in an arbitrary frequency domain, the program may make the computer  1900  function as the time domain computing section  138  and the frequency domain measuring section described with reference to  FIG. 12 . Moreover, this program may make the computer  1900  function as the probability density function computing section and the probability density function separating apparatus  100  described in  FIGS. 37 to 43 . 
     When the computer  1900  functions as the transfer function measuring apparatus  800 , the program may cause the computer  1900  to function as the constituents of the transfer function measuring apparatus  800  described with reference to  FIGS. 44A and 44B . For example, the program may cause the computer  1900  to function as the probability density function separating apparatus  100  and transfer function computing section  820 . 
     The computer  1900  according to the present embodiment includes a CPU peripheral section, an input-output section, and a legacy input-output section. The CPU peripheral section has a CPU  2000 , a RAM  2020 , a graphic controller  2075 , and a display apparatus  2080  that are interconnected by a host controller  2082 . The input-output section has a communication interface  2030 , a hard disk drive  2040 , and a CD-ROM drive  2060  that are connected to the host controller  2082  by an input-output controller  2084 . The legacy input-output section has a ROM  2010 , a flexible disk drive  2050 , and an input-output chip  2070  that are connected to the input-output controller  2084 . 
     The host controller  2082  connects the RAM  2020  to the CPU  2000  and the graphic controller  2075  that access the RAM  2020  at high transfer rate. The CPU  2000  operates based on a program stored on the ROM  2010  and the RAM  2020 , and controls each section. The graphic controller  2075  acquires image data to be generated by the CPU  2000  on a frame buffer provided in the RAM  2020 , and displays the data on the display apparatus  2080 . Alternatively, the graphic controller  2075  may include therein a frame buffer for storing image data generated from the CPU  2000 . 
     The input-output controller  2084  connects the host controller  2082  to the communication interface  2030 , the hard disk drive  2040 , and the CD-ROM drive  2060  that are a comparatively fast input-output apparatus. The communication interface  2030  communicates with other apparatuses via network. The hard disk drive  2040  stores a program and data to be used by the CPU  2000  within the computer  1900 . The CD-ROM drive  2060  reads a program or data from a CD-ROM  2095 , and provides it to the hard disk drive  2040  via the RAM  2020 . 
     Moreover, the ROM  2010  and the flexible disk drive  2050  and the input-output chip  2070  that are a comparatively low-speed input-output apparatus are connected to the input-output controller  2084 . The ROM  2010  stores a boot program to be executed by the computer  1900  on starting and a program or the like dependent on hardware of the computer  1900 . The flexible disk drive  2050  reads a program or data from a flexible disk  2090 , and provides it to the hard disk drive  2040  via the RAM  2020 . The input-output chip  2070  connects a various types of input-output apparatuses via the flexible disk drive  2050  and a parallel port, a serial port, a keyboard port, a mouse port, or the like. 
     A program provided to the hard disk drive  2040  via the RAM  2020  is stored on the flexible disk  2090 , the CD-ROM  2095 , or a recording medium such as an IC card, to be provided by a user. A program is read from a recording medium, is installed in the hard disk drive  2040  within the computer  1900  via the RAM  2020 , and is executed in the CPU  2000 . 
     This program is installed in the computer  1900 . This program works on the CPU  2000  or the like, and makes the computer  1900  functions as the probability density function separating apparatus  100 , the noise separating apparatus  200 , the computing apparatus, the testing apparatus  300 , or the bit error rate measuring apparatus  500 , that are previously described. 
     A program described above may be stored on an outside recording medium. A recording medium can include an optical recording medium such as DVD and CD, a magneto-optical recording medium such as MO, a tape medium, a semiconductor memory such as an IC card in addition to the flexible disk  2090  and the CD-ROM  2095 . Moreover, a storage device such as a hard disk or a RAM provided in a server system connected to a private communication network and Internet may be used as a recording medium, and a program may be provided to the computer  1900  via a network. 
     Although one aspect of the present invention has been described by way of an exemplary embodiment, it should be understood that those skilled in the art might make many changes and substitutions without departing from the spirit and the scope of the present invention. It is obvious from the definition of the appended claims that embodiments with such modifications also belong to the scope of the present invention. 
     As apparent from the above descriptions, according to the present invention, it is possible to separate a random component and a deterministic component from a given probability density function with high precision.