SPECTRAL LEAKAGE-BASED LOOPBACK METHOD FOR PREDICTING PERFORMANCE OF MIXED-SIGNAL CIRCUIT, AND SYSTEM THEREFOR

Abstract

The present invention relates to: a spectral leakage-based loopback method for a built-in self-test (BIST), achieving cost efficiency by accurately predicting the nonlinearity of a mixed-signal circuit in a loopback mode; and a system therefor, the method comprising the steps of: modeling a correlation by deriving the transfer function of a loopback path; generating a digitally synthesized single-tone sine wave input signal by means of an on-chip DSP core so as to sample same in a nonlinear DAC channel, and supplying a DAC output signal to a nonlinear ADC channel through an analog loopback path so as to measure each of the DAC channel and the ADC channel for a process test; and performing post-processing by means of the on-chip DSP core and predicting the harmonics of the two separate DAC and ADC channels.

Claims

1. A spectral leakage-driven loopback method for predicting performance of a mixed-signal circuit, the method comprising: generating, by an on-chip DSP(Digital Signal Processor) core, a digitally-synthesized single-tone sinusoidal stimulus, sampling in a nonlinear digital-to-analog-converter (DAC) channel, supplying a DAC output signal to a nonlinear analog-to-digital converter (ADC) channel through an analog loopback path, and measuring each of the DAC channel and the ADC channel for a production testing; and performing, by the on-chip DSP core, postprocessing and predicting harmonics of the two individual DAC channel and ADC channel.

2. The method of claim 1, wherein the measuring each of the DAC channel and the ADC channel comprises generating, by the on-chip DSP core, the digitally-synthesized single-tone sinusoidal stimulus, and the digitally-synthesized single-tone sinusoidal stimulus is sampled by the nonlinear DAC channel.

3. The method of claim 2, wherein the DAC output signal represents a correlation between DAC harmonics and spectral leakage introduced by incoherent sampling by a DAC harmonic coefficient.

4. The method of claim 3, wherein the measuring each of the DAC channel and the ADC channel comprises delivering a spectrally weighted DAC output signal to the nonlinear ADC channel through the analog loopback path without modification of the sampled DAC output signal.

5. The method of claim 4, wherein a loopback response through the analog loopback path represents a correlation between ADC harmonics and the DAC output signal according to a spectral leakage by the DAC harmonics and the DAC and ADC harmonic coefficients.

6. The method of claim 1, wherein the predicting the harmonics of the DAC channel and the ADC channel comprises predicting, by the on-chip DSP core, postprocessing and performing harmonics of the two individual DAC channel and ADC channel to calculate device under test (DUT) harmonics.

7. The method of claim 6, wherein the performing the harmonics of the DAC channel and the ADC channel comprises calculating a harmonic coefficient of each of the DAC channel and the ADC channel from a loopback response through the analog loopback path and the DAC and ADC harmonic coefficients.

8. The method of claim 1, further comprising: modeling a correlation by deriving a transfer function of a loopback path prior to measuring each of the DAC channel and the ADC channel.

9. The method of claim 8, wherein the modeling the correlation comprises acquiring a specific model by deriving the transfer function that is a path from the DAC channel to the ADC channel through the analog loopback path and thereby modeling the correlation.

10. The method of claim 9, wherein the specific model represents loopback responses that are determined by DAC and ADC harmonic coefficients and harmonics of the DAC channel and the ADC channel.

11. A spectral leakage-driven loopback system for predicting performance of a mixed-signal circuit, the system comprising: a measurer configured to generate, using an on-chip DSP(Digital Signal Processor) core, a digitally-synthesized single-tone sinusoidal stimulus, to sample in a nonlinear digital-to-analog converter (DAC) channel, to supply a DAC output signal to a nonlinear analog-to-digital converter (ADC) channel through an analog loopback path, and to measure each of the DAC channel and the ADC channel for a production testing; and a harmonic predictor configured to perform, using the on-chip DSP core, postprocessing and to predict harmonics of the two individual DAC channel and ADC channel.

12. The system of claim 11, wherein the measurer is configured to generate, using the on-chip DSP core, the digitally-synthesized single-tone sinusoidal stimulus, and the digitally-synthesized single-tone sinusoidal stimulus is sampled by the nonlinear DAC channel.

13. The system of claim 12, wherein the measurer is configured to deliver a spectrally weighted DAC output signal to the nonlinear ADC channel through the analog loopback path without modification of the sampled DAC output signal.

14. The system of claim 11, wherein the harmonic predictor is configured to predict, using the on-chip DSP core, postprocessing and to perform harmonics of the two individual DAC channel and ADC channel to calculate device under test (DUT) harmonics.

15. The system of claim 14, wherein the harmonic predictor is configured to calculate a harmonic coefficient of each of the DAC channel and the ADC channel from a loopback response through the analog loopback path and the DAC and ADC harmonic coefficients.

16. The system of claim 11, further comprising: a modeling unit configured to model a correlation by deriving a transfer function of a loopback path prior to executing the measurer.

17. The system of claim 16, wherein the modeling unit is configured to acquire a specific model by deriving a transfer function of the loopback path that is a path from the DAC channel to the ADC channel through the analog loopback path and thereby modeling the correlation.

Description

DESCRIPTION OF THE DRAWINGS

[0009] FIG. 1 is a flowchart illustrating a loopback method for predicting performance of a mixed-signal circuit according to an example embodiment.

[0010] FIG. 2 illustrates a loopback configuration proposed for a system on chip (SoC) according to an example embodiment.

[0011] FIGS. 3A and 3B illustrate sampled signals according to an example embodiment.

[0012] FIGS. 4A and 4B illustrate a spectrum of sampled signals according to an example embodiment.

[0013] FIG. 5 illustrates an example of a loopback configuration according to an example embodiment.

DETAILED DESCRIPTION

[0014] Hereinafter, example embodiments disclosed herein will be described with reference to the accompanying drawings. However, the present invention is not limited thereto or restricted thereby. Also, reference numerals used in the respective drawings refer to like elements throughout.

[0015] FIG. 1 is a flowchart illustrating a loopback method for predicting performance of a mixed-signal circuit according to an example embodiment.

[0016] Referring to FIG. 1, in operation 120, an on-chip DSP core generates a digitally-synthesized single-tone sinusoidal stimulus, samples in a nonlinear digital-to-analog converter (DAC) channel, supplies a DAC output signal to a nonlinear analog-to-digital converter (ADC) channel through an analog loopback path, and measures each of the DAC channel and the ADC channel for a production testing.

[0017] In operation 130, the on-chip DSP core performs postprocessing and predicts harmonics of two individual DAC channel and ADC channel. Next, end operation 102 is performed.

[0018] The spectral leakage-driven loopback method for predicting the performance of the mixed-signal circuit according to an example embodiment performs operation 110 prior to performing operation 120.

[0019] Hereinafter, operations 110 to 130 of the loopback method for predicting the performance of the mixed-signal circuit of FIG. 1 are described with reference to FIGS. 2 to 4B.

[0020] FIG. 2 illustrates a loopback configuration proposed for a system on chip (SoC) according to an example embodiment, FIGS. 3A and 3B illustrate sampled signals according to an example embodiment, and FIGS. 4A and 4B illustrate a spectrum of sampled signals according to an example embodiment.

[0021] Referring to FIG. 2, the SoC includes an antialiasing filter (AAF) to prevent aliasing and a smoothing filter (SF) to alleviate imaging. Here, an ADC channel and a DAC channel configured as a buffer are connected to an input terminal and an output terminal, respectively, in an operation mode. An analog loopback path is formed between the DAC channel and the ADC channel in a test mode, which is the same as in the existing loopback est.

[0022] Referring to FIG. 1, in operation 110 after start operation 101, a correlation may be modeled by deriving a transfer function of the loopback path.

[0023] In operation 110, a specific model may be acquired by deriving the transfer function of the loopback path that is a path from the DAC channel to the ADC channel through the analog loopback path and thereby modeling the correlation. Here, the specific model may represent the following Equation 13, and may describe loopback responses that are determined by individual DAC and ADC harmonic coefficients (Δw.sub.s) and harmonics of the DAC channel and the ADC channel.

[0024] For example, to mathematically identify the individual ADC channels and harmonic coefficients of the ADC channels, the specific model uses two sets of spectral loopback responses. Each model is derived in a form of a simultaneous equation model using an incoherent sampling factor (Δw.sub.s). Therefore, solutions for DAC and ADC harmonic coefficients may be formulated by solving two sets of models and the solutions are identified only once through operation 110 prior to production testing or operation 120.

[0025] Referring to FIG. 2, in operation 120, the on-chip DSP core may generate the digitally-synthesized single-tone sinusoidal stimulus x.sub.d(t)=cos(w.sub.0t). Here, the digitally-synthesized single-tone sinusoidal stimulus may be sampled for test purpose by a nonlinear DAC channel (a device under Test (DUT)) that is clocked at w.sub.s+Δw.sub.s.

[0026] An incoherently sampled DAC output signal y.sub.d(t) may represent a correlation between DAC harmonics and the spectral leakage introduced by an incoherent sampling factor Δw.sub.s. Here, an increase of the incoherent sampling factor Δw.sub.s introduces an increase of spectral leakage power that surrounds DUT harmonics present in the DAC output signal y.sub.d(t). Therefore, the harmonics needs to be decreased by the increased power of the spectral leakage.

[0027] In operation 120, the spectrally weighted DAC output signal y.sub.d(t) may be delivered to the nonlinear ADC channel (another DUT) through the analog loopback path. Here, the analog loopback path delivers the spectrally weighted DAC output signal y.sub.d(t) to another DUT (e.g., the nonlinear ADC channel) without any modification of the sampled DAC output signal y.sub.d(t).

[0028] Also, to synchronize the DAC output signal y.sub.d(t) sampled in the DAC channel, the ADC channel is also clocked at w.sub.s+Δw.sub.s. The loopback response through the analog loopback path may represent a correlation between the ADC harmonics and the DAC output signal y.sub.d(t) according to the spectral leakage through the DAC harmonics and the incoherent sampling factor Δw.sub.s.

[0029] Here, the loopback response α(t) is experimentally measured a second time using a value of the incoherent sampling factor Δw.sub.s is one at a time for postprocessing in operation 130. Synchronization between DAC sampling and ADC sampling is performed based on an edge of a reference sampling clock signal as in conventional testing. Subsequently, the decreased harmonics and the increased spectral leakage needs to be observed from spectral expression of the measured loopback response a(t) as in the DAC output signal .sub.d(t).

[0030] In operation 130, postprocessing may be performed by the on-chip DSP core to calculate the DUT harmonics and harmonics of the two individual DAC channel and ADC channel may be predicted.

[0031] In operation 130, the harmonic coefficients of the individual DAC and ADC channels may be calculated from the loopback response and the incoherent sampling factor Δw.sub.s through the analog loopback path.

[0032] Hereinafter, describing the spectral leakage according to an example embodiment, a spectral leakage theory that is processed from the digitally-synthesized single-tone sinusoidal stimulus x.sub.d(t) of FIG. 2 to the DAC output signal .sub.d(t) is described through comparison of sampling output results of FIGS. 3A and 3B.

[0033] The DAC output signal .sub.d(t) refers to a signal shown in an upper DAC of FIG. 3A and a signal that is dynamically sampled by the DAC for

[00001]$0<t<N_0{T}_0=\frac{2\pi N_0}{{\omega }_0}.$

Here, To denotes a single period time of a sinusoid.

[0034] Also, with the assumption that, in a lower DAC of FIG. 3A, .sub.d(t) is another DAC output sampled for an infinite time, acquiring the DAC output signal) (.sub.d(t)) is equivalent to multiplexing .sub.d(t) by a rectangular function {circumflex over (r)}(t). That is,

[00002]${\hat{y}}_d\left(t\right)=y_d\left(t\right).Math.\hat{r}\left(t\right)\mathrm{or}{\hat{Y}}_d\left(\omega \right)=\frac{1}{2\pi }{Y}_d\left(\omega \right)\hat{R}\left(\omega \right).$

Here, Ŷ.sub.d (w), Y.sub.d (w), {circumflex over (R)}(w) refer to spectral expressions of .sub.d(t), .sub.d(t), and {circumflex over (r)}(t) of FIG. 3A, respectively. denotes a spectral convolution operator. Therefore, .sub.d(t) is also represented using the rectangular function {circumflex over (r)}(t) with a time duration of N.sub.0T.sub.0.

[0035] Referring to FIG. 3B, the DAC output signal y.sub.d(t) refers to a signal that is sampled contradictory to

[00003]$0<t<\left(N_0-\Delta N_0\right)T_0=\frac{2\pi \left(N_0-\Delta N_0\right)}{{\omega }_0}.$

Also,

[0036] [00004]${\stackrel{\_}{y}}_d\left(t\right)=y_d\left(t\right).Math.\stackrel{\_}{r}\left(t\right)\mathrm{or}{\stackrel{\_}{Y}}_d\left(\omega \right)=\frac{1}{2\pi }{Y}_d\left(\omega \right)\stackrel{\_}{R}\left(\omega \right).$

Here, the DAC output signal (.sub.d(t)) refers to spectral expressions of .sub.d(t) and a rectangular function {circumflex over (r)}(t). Basically, the DAC output signal may be expressed with a period change ΔN.sub.0 that may replace variation of sampling frequency DAC and ADC harmonic coefficient Δw.sub.s. Here, {circumflex over (r)}(t) and r(t) are irrelevant to a window function used to alleviate the spectral leakage of a sampled signal.

[0037] The spectral leakage theory processed from the DAC output signal to the loopback response .sub.a(t) in FIG. 2 is discussed as follows.

[0038] The ADC of FIG. 2 is assumed to be an ideal digitizer clocked at w.sub.s+Δw.sub.s for better understanding and may sample .sub.d(t) or |Ŷ.sub.d(w)| and |Y.sub.d(w)|, respectively. Ŷ.sub.d(w)| and |Y.sub.d(w)| are compared in two boxes of FIG. 4A, focusing on their fundamental tones. Horizontal Bin axes in the two boxes are aligned together with w-axis, which is shown between the two boxes. For an upper box of FIG. 4A, a fast Fourier transform (FFT) frequency resolution

[00005]$\psi =\frac{{\omega }_0}{N_0}$

may be calculated for coherent sampling.

[00006]$\hat{Y}d\left(\omega \right)=\frac{1}{2\pi }{Y}_d\left(\omega \right)\hat{R}\left(\omega \right)$

is represented as a sinc function that is shifted to w.sub.0 in w-axis using an FFT multiplication property. Also, w.sub.0 in w-axis corresponds to N.sub.0. Similarly, for a lower box of FIG. 4A, |Y.sub.d(w)| is represented as a sinc function that is shifted to w.sub.0 in w-axis.

[0039] The FFT frequency resolution is expressed as

[00007]$\phi =\frac{{\omega }_s+{\mathrm{\Delta \omega }}_s}{N_s}=\frac{{\omega }_0}{N_0-\Delta N_0}$

for incoherent sampling. This results in φ>ψ. Therefore, a frequency corresponding to an N.sub.0-th bin is

[00008]$\frac{N_0}{N_0-\Delta N_0}{\omega }_0,$

which is higher than w.sub.0 for the N.sub.0-th bin for shown in the-axis and also which is greater than a center frequency of a main lobe of |Y.sub.d(w)|. If 0<ε<1 and Δw.sub.0 is a positive value, the frequency may be further simplified as follows:

[00009]$\phi N_0=\frac{N_0}{N_0-\Delta N_0}{\omega }_0=\left(1+\epsilon \right){\omega }_0\stackrel{△}{=}{\omega }_0+\Delta {\omega }_0.$

If w.sub.s−Δw.sub.s is set instead of, Δw.sub.0 may also be a negative value.

[0040] In addition, since harmonics may be buried in the spectral leakage power increased with Δw.sub.0 in loopback responses, Δw.sub.0 needs to be small enough to avoid the above situation. As a result, since the bin is invisible on the spectrum of |Y.sub.d(w)|, it can be seen that the fundamental tone is shifted from w.sub.0 to w.sub.0+Δw.sub.0 due to ΔN.sub.0 as a weighting factor.

[0041] Referring to FIG. 4B, it can be seen that, according to an increase in No, the main lobe of |Y.sub.d(w)| becomes wider in width and shorter in height and w.sub.0+Δw.sub.0 also increases. Therefore, |Y.sub.d(w.sub.0+Δw.sub.0)| additionally decreases and the same spectral process is performed for each harmonic.

[0042] FIG. 5 illustrates an example of a loopback configuration according to an example embodiment.

[0043] FIG. 5 shows a loopback configuration proposed herein for a quantitative model using a period variation, that is, ΔN.sub.0.

[0044] Harmonic distortion is considered up to third order for simplicity.

[0045] In a first stage, a digitally-synthesized single-tone sinusoidal stimulus x.sub.d(t)=cos(w.sub.0t) is applied to a DAC channel, hd(⋅), which is modeled with a symmetrical Taylor expansion. Here, .sub.d(t) and Y.sub.d(w) may be represented as the following Equation 1.

[00010]$\begin{array}{cc}\begin{array}{c}{y}_d\left(t\right)=h_d\left(x_d\left(t\right)\right)={.Math.}_i{\zeta }_i{\cos }^i\left({\omega }_{0}t\right)\\ ={\stackrel{\_}{\zeta }}_1\cos \left({\omega }_{0}t\right)+{\stackrel{\_}{\zeta }}_2\cos \left(2{\omega }_{0}t\right)+{\stackrel{\_}{\zeta }}_3\cos \left(3{\omega }_{0}t\right)\\ Y_d\left(\omega \right)=\underset{Y_{d1}\left(\omega \right)}{\underbrace{{\stackrel{\_}{\zeta }}_1\pi \delta \left(\omega -{\omega }_0\right)}}+\underset{Y_{d2}\left(\omega \right)}{\underbrace{{\stackrel{\_}{\zeta }}_2\pi \delta \left(\omega -2{\omega }_0\right)}}+\underset{Y_{d3}\left(\omega \right)}{\underbrace{{\stackrel{\_}{\zeta }}_3\pi \delta \left(\omega -3{\omega }_0\right)}}\end{array}& .Math.\mathrm{Equation}1.Math.\end{array}$

[0046] In Equation 1, ζ.sub.i denotes an i.sup.th harmonic coefficient of a DAC channel and

[00011]${\stackrel{\_}{\zeta }}_1={\zeta }_1+\frac{3{\zeta }_3}{4},{\stackrel{\_}{\zeta }}_2=\frac{{\zeta }_2}{2},\mathrm{and}{\stackrel{\_}{\zeta }}_3=\frac{{\zeta }_3}{4}.$

Also, Y.sub.d1(w), Y.sub.d2(w), and Y.sub.d3(w) represent a fundamental signal, a second-order harmonic, and a third-order harmonic of the DAC output in a spectral domain, respectively.

[0047] As discussed in FIG. 3B, .sub.d(t) and Y.sub.d(w) are modeled as the following Equation 2 and Equation 3.

<Equation 2>

[00012]$\begin{array}{cc}\begin{array}{c}{\stackrel{\_}{y}}_d\left(t\right)=y_d\left(t\right)\stackrel{\_}{r}\left(t\right)\\ {\stackrel{\_}{Y}}_d\left(\omega \right)=\frac{1}{2\pi }\left\{Y_{d1}\left(\omega \right)\stackrel{\_}{R}\left(\omega \right)+Y_{d2}\left(\omega \right)\stackrel{\_}{R}\left(\omega \right)+Y_{d3}\left(\omega \right)\stackrel{\_}{R}\left(\omega \right)\right\}\end{array}r\left(t\right)=\{\begin{array}{cc}1& \mathrm{if}0<t<\frac{2\pi \left(N_0-\Delta N_0\right)}{{\omega }_0}\\ 0& \mathrm{elsewhere}\end{array}.& .Math.\mathrm{Equation}3.Math.\end{array}$

[0048] Here, the spectral expression of the rectangular function for may be given as the following Equation 4.

[00013]$\begin{array}{cc}R\left(\omega \right)=\frac{2\pi \left(N_0-\Delta N_0\right)}{{\omega }_0}\sin c\left(\frac{N_0-\Delta N_0}{{\omega }_0}\omega \right)& .Math.\mathrm{Equation}4.Math.\end{array}$

[0049] Here, Y.sub.d1(w)R(w) of Equation 2 may be calculated using Y.sub.d1(w) in Equation 1 and of Equation 4 based on an FFT multiplication property, as in the following Equation 5.

[00014]$\begin{array}{cc}{Y}_{d1}\left(\omega \right)\stackrel{\_}{R}\left(\omega \right)=\frac{2{\pi }^2}{\phi }\left({\zeta }_1+\frac{3{\zeta }_3}{4}\right)\sin c\left(\frac{\omega -{\omega }_0}{\phi }\right)& .Math.\mathrm{Equation}5.Math.\end{array}$

[0050] In Equation 5,

[00015]$\phi =\frac{{\omega }_0}{N_0-\Delta N_0}.$

Similarly, other two convolution terms of Equation 2 are calculated through the following Equation 6.

<Equation 6>

[00016]$Y_{d2}\left(\omega \right)\stackrel{\_}{R}\left(\omega \right)=\frac{{\pi }^2{\zeta }_2}{\phi }\sin c\left(\frac{\omega -2{\omega }_0}{\phi }\right)Y_{d3}\left(\omega \right)\stackrel{\_}{R}\left(\omega \right)=\frac{{\pi }^2{\zeta }_3}{2\phi }\sin c\left(\frac{\omega -3{\omega }_0}{\phi }\right)$

[0051] The aforementioned Equation 2 may be re-formulated as in the following Equation 7.

[00017]$\begin{array}{cc}{\stackrel{\_}{Y}}_d\left(\omega \right)=\frac{\pi }{2\phi }\left\{\left(2{\zeta }_1+\frac{3{\zeta }_3}{2}\right)\sin c\left(\frac{\omega -{\omega }_0}{\phi }\right)+{\zeta }_2\sin c\left(\frac{\omega -2{\omega }_0}{\phi }\right)+\frac{{\zeta }_3}{2}\sin c\left(\frac{\omega -3{\omega }_0}{\phi }\right)\right\}& .Math.\mathrm{Equation}7.Math.\end{array}$

[0052] Therefore, ζ.sub.i and ΔN.sub.0 are correlated with Y.sub.d(w) as shown in Equation 7. Then, using Y.sub.d(w) discussed as follows in FIG. 4A, that is, Equation 5, the decreased fundamental tone of |Y.sub.d(w.sub.0+Δw.sub.0)| may be identified as in the following Equation 8.

[00018]$\begin{array}{cc}{Y}_{d1}\left(\omega \right)\stackrel{\_}{R}\left(\omega \right){.Math.}_{\omega =\phi N_0}=\frac{2{\pi }^2}{\phi }\left({\zeta }_1+\frac{3{\zeta }_3}{4}\right)\sin c\left(\Delta N_0\right)& .Math.\mathrm{Equation}8.Math.\end{array}$

[0053] Similarly, the decreased magnitudes of harmonics may be identified from Equation 6 through the following Equation 9.

[00019]$\begin{array}{cc}{Y}_{d2}\left(\omega \right)\stackrel{\_}{R}\left(\omega \right){.Math.}_{\omega =2N_0\phi }=\frac{{\pi }^2{\zeta }_2}{\phi }\sin c\left(2\Delta N_0\right)Y_{d3}\left(\omega \right)\stackrel{\_}{R}\left(\omega \right){.Math.}_{\omega =3N_0\phi }=\frac{{\pi }^2{\zeta }_3}{2\phi }\sin c\left(3\Delta N_0\right)& .Math.\mathrm{Equation}9.Math.\end{array}$

[0054] Next, .sub.d(t) (or Y.sub.d(w) in Equation 7) is applied to the ADC channel and h.sub.a(⋅) is represented as in the following Equation 10.

[00020]$\begin{array}{cc}{y}_a\left(t\right)=h_a\left({\stackrel{\_}{y}}_d\left(t\right)\right)={.Math.}_i{\rho }_i{\stackrel{\_}{y}}_d^i\left(t\right)Y_a\left(\omega \right)={.Math.}_i{\left(2\pi \right)}^{1-i}{\rho }_i{\stackrel{\_}{Y}}_d^{\left(i-1\right)}\left(\omega \right)& .Math.\mathrm{Equation}10.Math.\end{array}$

[0055] In Equation 10, ρ.sub.i denotes an i-th harmonic coefficient of the ADC channel and Y.sub.a(w) denotes a spectral representation of .sub.a(t). As in Equation 10, ζ.sub.i, ρ.sub.i, and ΔN.sub.0 are correlated with Y.sub.a(w).

[0056] Here, (w) in Equation 10 may be identified as in the following Equation 11.

[00021]$\begin{array}{cc}\begin{array}{c}{\stackrel{\_}{Y}}_d^1\left(\omega \right)={\left({\stackrel{\_}{Y}}_{d_F}\left(\omega \right)+{\stackrel{\_}{Y}}_{d_H}\left(\omega \right)\right)}^1\\ ={\stackrel{\_}{Y}}_{d_F}^1\left(\omega \right)+2{\stackrel{\_}{Y}}_{d_F}\left(\omega \right){\stackrel{\_}{Y}}_{d_H}\left(\omega \right)+{\stackrel{\_}{Y}}_{d_H}^1\left(\omega \right)\end{array}& .Math.\mathrm{Equation}11.Math.\end{array}$

[0057] In Equation 11, Y.sub.d.sub.f(w) and Y.sub.d.sub.H(w) represent and spectral responses of the fundamental term and a spectral sum of second and third harmonics, respectively. Frequency convolution power i itself indicates an i-fold iteration of the convolution. Similarly, (w) in Equation 10 may be identified as in the following Equation 12.

<Equation 12>

(w)=(Y.sub.d.sub.f(w)+Y.sub.d.sub.H

[0058] Finally, using and Equation 11, Equation 10 may be expressed as each harmonic magnitude as in the following Equation 13.

[00022]$\begin{array}{cc}\{\begin{array}{c}.Math.Y_a\left({\stackrel{\_}{\omega }}_0\right).Math.=.Math.\left({\stackrel{\_}{\omega }}_0\right)+\left({\stackrel{\_}{\omega }}_0\right)+\left({\stackrel{\_}{\omega }}_0\right).Math.\\ .Math.Y_a\left(2{\stackrel{\_}{\omega }}_0\right)=.Math.\left(2{\stackrel{\_}{\omega }}_0\right)+\left(2{\stackrel{\_}{\omega }}_0\right)+\left(2{\stackrel{\_}{\omega }}_0\right).Math.\\ .Math.Y_a\left(3{\stackrel{\_}{\omega }}_0\right).Math.=.Math.\left(3{\stackrel{\_}{\omega }}_0\right)+\left(3{\stackrel{\_}{\omega }}_0\right)+\left(3{\stackrel{\_}{\omega }}_0\right).Math.\end{array}& .Math.\mathrm{Equation}13.Math.\end{array}$

[0059] Here, =ρ.sub.1,=(2π).sup.−1p.sub.2, =(2π).sup.−2p.sub.3, and w.sub.0=φN.sub.0.

[0060] Here, Equation 13 represents a subset of Equation 10 for overall spectral response and thus, represents a characteristic model according to an example embodiment to characterize harmonics of DUTs.

[0061] Y.sub.a(iw.sub.0) in the characteristic model, that is, Equation 13 inherently has the identical correlation with ζ.sub.i, ρ.sub.i and ΔN.sub.0 as in Y.sub.a(w) in Equation 10.

[0062] Here, if DAC and ADC harmonic coefficients Δw.sub.s is are set on a specific value IΔw.sub.s, all equations from Equation 1 to Equation 10 are derived using IΔw.sub.s based on the correlation shown in FIG. 5 and R(w), Y.sub.d(w), and Y.sub.a(w) are accordingly specified as .sup.IR(w), .sup.IY.sub.d(w), and .sup.IY.sub.a(w), respectively.

[0063] Then, a first set of Equation 13 may be generated by replacing corresponding variables in three equations of Equation 13 with .sup.IY.sub.d(iw.sub.0) and .sup.IY.sub.a(iw.sub.0).

[0064] Similarly, if DAC and ADC harmonic coefficient Δw.sub.s) is set on another value .sup.IIΔw.sub.s, the corresponding variables in Equation 13 are replaced with .sup.IIY.sub.d(w) and .sup.II Y.sub.a(w) as a second set.

[0065] Finally, a total of six equations from the two sets are simultaneously solved to acquire solutions for a total of six unknown variables of ζ.sub.i and ρ.sub.i where i=1, 2, 3. To summarize, the number of measurement values, the sets of Equation 13, and DUTs are the same as two. The number of equations in Equation 13 and the highest-order of harmonics are the same as three. Such solutions are identified only once prior to production testing. In the production testing, two loopback measurements are performed using .sup.IΔw.sub.s and .sup.IIΔw.sub.s as the first stage. The two sets of harmonics from those measured loopback responses are plugged into .sup.I|Y.sub.a(iw.sub.0)| and .sup.II|Y.sub.a(iw.sub.0)| in the solutions as the second stage. Then, the solutions readily provide ζ.sub.i and ρ.sub.i, that is, accurate harmonic coefficients of individual DUTs, which indicates that a fault-masking problem may be resolved.

[0066] FIG. 6 is a block diagram illustrating a detailed configuration of a loopback system for predicting performance of a mixed-signal circuit according to an example embodiment.

[0067] Referring to FIG. 6, a spectral leakage-driven loopback system for predicting performance of a mixed-signal circuit according to an example embodiment precisely predicts nonlinearity of a mixed-signal circuit in a loopback mode.

[0068] To this end, a loopback system 600 for predicting performance of a mixed-signal circuit may include a measurer 620 and a harmonic predictor 630, and may further include a modelling unit 610.

[0069] The modelling unit 610 may model a correlation by deriving a transfer function of a loopback path prior to executing the measurer 620.

[0070] The modelling unit 610 may acquire a specific model by deriving the transfer function of the loopback path that is a path from a DAC channel to an ADC channel through an analog loopback path and thereby modeling the correlation. Here, the specific model represents the aforementioned Equation 13 and may describe loopback responses that are determined by individual DAC and ADC harmonic coefficients and harmonics of DAC and ADC channels.

[0071] The measurer 620 generates a digitally-synthesized single-tone sinusoidal stimulus using an on-chip DSP core, samples in a nonlinear DAC channel, supplies a DAC output signal to a nonlinear ADC channel through the analog loopback path, and measures each of the DAC channel and the ADC channel for production testing.

[0072] The measurer 620 may generate the digitally-synthesized single-tone sinusoidal stimulus using the on-chip DSP core, and the digitally-synthesized single-tone sinusoidal stimulus may be sampled for test purposes by the nonlinear DAC channel.

[0073] Then, the measurer 620 may deliver the spectrally weighted DAC output signal to the nonlinear ADC channel through the analog loopback path. Here, the analog loopback path delivers the spectrally weighted DAC output signal to another DUT (e.g., the nonlinear ADC channel) without any modification of the sampled DAC output signal.

[0074] The harmonic predictor 630 performs postprocessing using the on-chip DSP core and predicts harmonics of the two individual DAC channel and ADC channel.

[0075] The harmonic predictor 630 may perform postprocessing using the on-chip DSP core and thereby predict the harmonics of the two individual DAC channel and ADC channel to calculate DUT harmonics.

[0076] The harmonic predictor 630 may calculate a harmonic coefficient of each of the DAC channel and the ADC channel from the loopback response through the analog loopback path and DAC and ADC harmonic coefficients.

[0077] Although the example embodiments are described with a limited example embodiment and drawings, it will be apparent to one of ordinary skill in the art that various modification and alternations may be made from the description.