CN111983308B - Signal recovery method for incoherent sampling and harmonic distortion in ADC (analog to digital converter) spectrum test - Google Patents

Signal recovery method for incoherent sampling and harmonic distortion in ADC (analog to digital converter) spectrum test Download PDF

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CN111983308B
CN111983308B CN202010834610.7A CN202010834610A CN111983308B CN 111983308 B CN111983308 B CN 111983308B CN 202010834610 A CN202010834610 A CN 202010834610A CN 111983308 B CN111983308 B CN 111983308B
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harmonic
adc
fundamental wave
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CN111983308A (en
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马敏
曾钰琴
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University of Electronic Science and Technology of China
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Abstract

The invention discloses a signal recovery method for incoherent sampling and harmonic distortion in ADC frequency spectrum test, which comprises the steps of firstly, estimating a fundamental wave to obtain an amplitude estimation value and an initial phase estimation value, reconstructing an incoherent sampling fundamental wave signal, estimating the amplitude and the initial phase of each order of harmonic component after removing the incoherent sampling fundamental wave signal from an output signal of a tested ADC, reconstructing the harmonic signal and removing the harmonic signal from the output signal of the tested ADC, secondly estimating and reconstructing a fundamental wave part based on residual signals, secondly estimating and reconstructing the harmonic wave part, reconstructing the fundamental wave signal and the harmonic signal under the coherent sampling condition according to secondly estimated parameters, replacing the incoherent signal in the output signal of the tested ADC with a coherent signal, and recovering the incoherent sampling signal to obtain the coherent sampling signal. The invention can be used for carrying out accurate spectrum test on the ADC under the condition that the sampling signal of the ADC to be tested has incoherent sampling and harmonic distortion.

Description

Signal recovery method for incoherent sampling and harmonic distortion in ADC (analog to digital converter) spectrum test
Technical Field
The invention belongs to the technical field of ADC (analog to digital converter) spectrum testing, and particularly relates to a signal recovery method for incoherent sampling and harmonic distortion in ADC spectrum testing.
Background
An analog-To-Digital Converter (ADC) is a conversion tool from analog signals To Digital signals, is widely applied To signal acquisition links of various systems, is an important component of a signal acquisition and Digital signal processing system, and is also an important component of an integrated circuit test system. It is important for integrated circuit test systems to ensure accurate measurement of ADC parameters. The parameters of the ADC include static parameters and dynamic parameters, and the dynamic parameter measurement is also called a spectrum parameter test. As the resolution of the ADC is higher and higher, the sampling speed is higher and higher, and the difficulty and cost of performing spectrum parameter testing on the ADC are also increased. At present, ADC test is required to meet various indexes required by the tested ADC, and the test time and the test cost are reduced as far as possible on the premise of completely testing various functions of the ADC. The spectrum parameter test of the ADC is realized in a frequency domain, which is called a spectrum test or an AC test and mainly comprises the following parameters: Signal-to-Noise-Ratio (SNR), Signal-to-Noise-Ratio (SINAD), Total Harmonic Distortion (THD), Spurious-Free Dynamic Range (SFDR), and Effective Number (ENOB). Full spectrum testing is important for testing the system's SFDR subject to non-harmonic distortion, since full spectrum testing tests not only ADC dynamics but also the full spectrum range frequency characteristics including harmonics and noise.
At present, aiming at the verification of a plurality of dynamic performance indexes of an ADC chip, a common test method is that a perfect sinusoidal signal is input at the input end of the ADC, the ADC to be tested carries out quantitative conversion and output on the signal, and the dynamic parameter indexes of the ADC are analyzed by converting the signal into a frequency spectrum by using discrete Fourier transform. FIG. 1 is a block diagram of a standard ADC frequency domain parameter test flow. As shown in FIG. 1, for ADC accurate spectral parameter testing, the International standards the IEEE Standard for Digital Waveform Recorders (IEEE Std.1057) and the IEEE Standard for telematics and Test Methods for Analog-to-Digital Converters (IEEE Std.1241) require that the Test equipment satisfy the following five conditions:
first, the length of the data obtained from the test must be large enough;
second, the clock jitter must be controlled within a very small certain range;
thirdly, the input signal is subjected to coherent sampling, which is a strict condition for realizing accurate test of frequency spectrum parameters, if the input signal is not subjected to coherent sampling, a skirt effect appears on the frequency spectrum of the output signal, a quantization index corresponding to the fundamental frequency is not a unique index any more, a frequency spectrum leakage phenomenon appears, and a frequency spectrum parameter calculation formula under a standard condition is not applicable any more;
fourthly, the tested ADC has very high requirements on the quality of the input analog signal, and usually requires that the purity of the input analog signal is 3 to 4 bits higher than the significant digit of the tested ADC, and for a high-precision ADC test, it is expensive to introduce an excitation signal externally, and it is almost impossible to realize such a high-quality analog signal internally, and the requirements on hardware are very high. Harmonic distortion occurs to the frequency spectrum due to insufficient purity of the output signal, and the harmonic distortion covers the true harmonic distortion of the ADC to be tested;
fifthly, the amplitude of the input signal must be slightly lower than half of the full-scale range of the tested ADC, so that the output signal of the tested ADC is ensured not to generate a topping phenomenon, if the amplitude range of the input signal exceeds the range of the ADC, the signal is topped, the tested ADC samples the topping signal, and a large amount of higher harmonic distortion occurs in the frequency spectrum of the output signal, so that an error frequency spectrum parameter measurement result is caused.
Obviously, besides the first test condition being easy to implement, the implementation of the other test conditions is very difficult, especially when the ADC to be tested is a high-precision ADC, the coherent sampling condition and the harmonic distortion-free condition are almost impossible to implement.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a signal recovery method for noncoherent sampling and harmonic distortion in ADC frequency spectrum test, which breaks through the limitation of coherent sampling conditions and harmonic distortion conditions and recovers the sampling signals obtained by the tested ADC under the conditions of noncoherent sampling and harmonic distortion to obtain coherent sampling signals, so that the ADC can be accurately tested on frequency spectrum under the conditions of noncoherent sampling and harmonic distortion of the sampling signals of the tested ADC.
In order to achieve the above object, the signal recovery method for incoherent sampling and harmonic distortion in ADC spectrum test of the present invention comprises the following steps:
s1: when ADC spectrum test is carried out under the incoherent sampling condition, an input signal x (t) of the tested ADC is recorded as follows:
Figure BDA0002639216090000031
wherein A represents the amplitude of the fundamental wave, fSigDenotes the fundamental frequency of the input signal,. phi denotes the fundamental initial phase, AhRepresents the amplitude of the h-th harmonic component, phihDenotes the initial phase of the H-th harmonic component, H2, 3, …, H denotes the harmonic order; a, b are coefficients, a ═ Acos (Φ), b ═ Asin (Φ);
for the output signal x [ n ] of the ADC to be tested]Performing discrete Fourier transform to obtain frequency domain expression X of output signalk，XkThe expression of (a) is:
Figure BDA0002639216090000032
wherein n represents the sampling point serial number, M represents the number of sampling points, and j represents an imaginary unit;
sampling period of ADC to be tested by using frequency domain information of output signal
Figure BDA0002639216090000033
Integer part of
Figure BDA0002639216090000034
And fractional part
Figure BDA0002639216090000035
Estimating to obtain the estimated value of the total number of sampling periods
Figure BDA0002639216090000036
Wherein the integer part
Figure BDA0002639216090000037
And fractional part
Figure BDA0002639216090000038
Are respectively:
Figure BDA0002639216090000039
Figure BDA00026392160900000310
wherein argmax represents the variable value for making the function reach the maximum value, imag represents the complex imaginary part, and e represents the natural constant;
s2: calculating the following formula to obtain the estimated values of the coefficients a and b
Figure BDA00026392160900000311
Figure BDA00026392160900000312
Wherein the content of the first and second substances,
Figure BDA00026392160900000313
Figure BDA00026392160900000314
Figure BDA0002639216090000041
wherein the superscript H denotes the determination of the conjugate transpose, Y1、Y2、Y3The expression of (a) is as follows:
Y1＝[x[0],x[1],…,x[M-1]]T,
Figure BDA0002639216090000042
Figure BDA0002639216090000043
E＝αI
the matrix I is a unit matrix, and alpha is a preset minimum constant greater than 0, so that A + E is a symmetrical positive definite matrix;
s3: the amplitude estimation value is calculated by the following formula
Figure BDA0002639216090000044
Figure BDA0002639216090000045
The initial phase estimate is calculated using the following formula
Figure BDA0002639216090000046
Figure BDA0002639216090000047
Wherein the content of the first and second substances,
Figure BDA0002639216090000048
s4: using amplitude estimates
Figure BDA0002639216090000049
Initial phase estimation
Figure BDA00026392160900000410
And sampling period
Figure BDA00026392160900000411
Reconstructing incoherent sampling fundamental wave signals to obtain reconstructed incoherent sampling fundamental wave signals x _ nc1[ n [)]：
Figure BDA00026392160900000412
S5: at the output signal x [ n ] of the ADC to be tested]Removing incoherent sampling fundamental wave signal x _ nc1[ n [ ]]Obtaining a residual signal R1[n]：
R1[n]＝x[n]-x_nc1[n]
S6: for residual signal R1[n]Performing discrete Fourier transform to obtain residual signal R1[n]Frequency domain expression of (1)k，NkThe expression of (a) is:
Figure BDA00026392160900000413
order to
Figure BDA0002639216090000051
Figure BDA0002639216090000052
When rounding is shown, the parameters are set:
Figure BDA0002639216090000053
Figure BDA0002639216090000054
if it is not
Figure BDA0002639216090000055
Calculating an estimated amplitude of the h-th order harmonic component using the following formula
Figure BDA0002639216090000056
And estimating the initial phase
Figure BDA0002639216090000057
Figure BDA0002639216090000058
Figure BDA0002639216090000059
Wherein phase { } denotes an acquisition angle, λ ═ pi (M-1)/M;
if it is not
Figure BDA00026392160900000510
Calculating an estimated amplitude of the h-th order harmonic component using the following formula
Figure BDA00026392160900000511
And estimating the initial phase
Figure BDA00026392160900000512
Figure BDA00026392160900000513
Figure BDA00026392160900000514
S7: amplitude estimation using harmonic components
Figure BDA00026392160900000515
Initial phase estimation
Figure BDA00026392160900000516
And sampling period
Figure BDA00026392160900000517
Carrying out harmonic signal reconstruction to obtain a reconstructed harmonic signal x _ Harm _ nc1[ n [ ]]：
Figure BDA00026392160900000518
S8: at the output signal x [ n ] of the ADC to be tested]Middle removed harmonic signal x _ Harm _ nc1[ n]Obtaining a residual signal R2[n]：
R2[n]＝x[n]-x_Harm_nc1[n]
S9: for residual signal R2[n]Performing discrete Fourier transform to obtain residual signal R2[n]Frequency domain expression Zk，ZkThe expression of (a) is:
Figure BDA00026392160900000519
according to ZkSampling period number of ADC to be tested
Figure BDA00026392160900000520
Fractional part of
Figure BDA00026392160900000521
Performing a second estimation of the fractional part of the sampling period
Figure BDA00026392160900000522
The expression of (a) is:
Figure BDA0002639216090000061
the following formula is then used to calculate a quadratic amplitude estimate for the fundamental component
Figure BDA0002639216090000062
Figure BDA0002639216090000063
Calculating a secondary initial phase estimate of the fundamental component using the following equation
Figure BDA0002639216090000064
Figure BDA0002639216090000065
S10: using quadratic amplitude estimates of the fundamental part
Figure BDA0002639216090000066
Secondary initial phase estimation
Figure BDA0002639216090000067
And number of sampling cycles
Figure BDA0002639216090000068
Integer part of
Figure BDA0002639216090000069
Performing incoherent sampling fundamental wave signal reconstruction to obtain secondary reconstructionThe constructed incoherent sampling fundamental wave signal x _ Fund _2[ n [ ]]：
Figure BDA00026392160900000610
S11: at the output signal x [ n ] of the ADC to be tested]Removing twice reconstructed incoherent sampling fundamental wave signal x _ Fund _2[ n [ ]]Obtaining a residual signal R3[n]：
R3[n]＝x[n]-x_Fund_2[n]
S12: for residual signal R3[n]Performing discrete Fourier transform to obtain residual signal R3[n]Is expressed as Sk，SkThe expression of (a) is:
Figure BDA00026392160900000611
according to SkThe number of sampling cycles of the ADC to be tested corresponding to the h-th harmonic component
Figure BDA00026392160900000612
Fractional part of
Figure BDA00026392160900000613
And performing secondary estimation, wherein the expression is as follows:
Figure BDA00026392160900000614
order to
Figure BDA0002639216090000071
Calculating a second order amplitude estimate of the h-th order harmonic component using the following equation
Figure BDA0002639216090000072
And a secondary initial phase estimate
Figure BDA0002639216090000073
Figure BDA0002639216090000074
Figure BDA0002639216090000075
S13: second order amplitude estimation using harmonic components
Figure BDA0002639216090000076
Secondary initial phase estimation
Figure BDA0002639216090000077
And number of sampling cycles
Figure BDA0002639216090000078
Integer part of
Figure BDA0002639216090000079
Fraction of sampling period number obtained by secondary estimation
Figure BDA00026392160900000710
Carrying out harmonic signal reconstruction to obtain a reconstructed harmonic signal x _ Harm _2[ n]：
Figure BDA00026392160900000711
S14: the fundamental wave signal and harmonic wave signal reconstruction under the coherent sampling condition is carried out by utilizing the parameters obtained by the secondary estimation of the fundamental wave part and the harmonic wave part, and the fundamental wave signal x _ Fund _ C [ n ] and the harmonic wave signal x _ Harm _ C [ n ] under the coherent sampling condition are obtained:
Figure BDA00026392160900000712
Figure BDA00026392160900000713
s15: the incoherent fundamental wave signal x _ Fund _2[ n ] in the output signal x [ n ] of the ADC to be tested is replaced by a coherent fundamental wave signal x _ Fund _ C [ n ], the harmonic signal x _ Harm _2[ n ] under the incoherent sampling is replaced by the harmonic signal x _ Harm _ C [ n ] under the coherent sampling, and then the sampling signal can be restored to obtain a coherent sampling signal x _ new [ n ]:
x_new[n]＝x[n]-x_Fund_2[n]-x_Harm_2[n]+x_Fund_C[n]+x_Harm_C[n]
the invention relates to a signal recovery method for incoherent sampling and harmonic distortion in ADC frequency spectrum test, which firstly utilizes the frequency domain information of the output signal of the ADC to be tested to sample the period of the ADC to be tested
Figure BDA00026392160900000714
Integer part of
Figure BDA00026392160900000715
And fractional part
Figure BDA00026392160900000716
Estimating to obtain estimated values of coefficients a and b
Figure BDA00026392160900000717
And then obtaining an amplitude estimation value and an initial phase estimation value, reconstructing the incoherent sampling fundamental wave signal, estimating the amplitude and the initial phase of each order of harmonic component after removing the incoherent sampling fundamental wave signal from the output signal of the ADC to be tested, reconstructing the harmonic signal and removing the harmonic signal from the output signal of the ADC to be tested, secondly estimating and reconstructing the fundamental wave part based on the residual signal, secondly estimating and reconstructing the harmonic wave part, reconstructing the fundamental wave signal and the harmonic signal under the coherent sampling condition according to the secondly estimated parameters, replacing the incoherent signal in the output signal of the ADC to be tested with a coherent signal, and recovering the incoherent sampling signal to obtain the coherent sampling signal.
The invention has the following beneficial effects:
1) when the coefficients a and b are estimated, all fundamental wave information is utilized, but a part of fundamental wave information is not intercepted for estimation, so that the obtained coefficients a and b are more accurate;
2) the diagonal loading principle is creatively utilized when the coefficients a and b are estimated, a very small positive number alpha is selected, the problem of singular irreversible of the matrix A is solved, and the introduced error can be ignored due to the proper and very small alpha;
3) when the initial phase of the fundamental wave is evaluated for the first time, two times of estimation averaging is adopted, so that the estimation error is smaller;
4) the fundamental wave signal and the harmonic wave signal are respectively estimated twice to break through the limit of coherent sampling condition and harmonic wave signal, and the sampling signal obtained by the tested ADC under the condition of incoherent sampling and harmonic wave distortion is recovered to obtain coherent sampling signal, so that the ADC can be accurately tested in frequency spectrum under the condition of incoherent sampling and harmonic wave distortion of the sampling signal of the tested ADC.
Drawings
FIG. 1 is a graph of the output signal spectrum under incoherent sampling conditions;
FIG. 2 is a graph of the output signal spectrum in the presence of incoherent sampling and harmonic distortion of the input signal;
FIG. 3 is a flowchart of an embodiment of a signal recovery method for the presence of incoherent sampling and harmonic distortion in ADC spectral testing according to the present invention;
FIG. 4 is a graph of a spectrum of a non-coherently sampled fundamental signal reconstructed from the output signal of FIG. 2;
FIG. 5 is a graph of the spectrum of the residual signal of the output signal of FIG. 2 minus the non-coherently sampled fundamental signal of FIG. 4;
FIG. 6 is a graph of a harmonic signal spectrum reconstructed from the output signal of FIG. 2;
FIG. 7 is a spectral plot of a coherently sampled signal resulting from the recovery of the output signal shown in FIG. 2 using the present invention;
fig. 8 is a graph of an ideal signal spectrum obtained by sampling the input signal by the ADC under test in this embodiment.
Detailed Description
The following description of the embodiments of the present invention is provided in order to better understand the present invention for those skilled in the art with reference to the accompanying drawings. It is to be expressly noted that in the following description, a detailed description of known functions and designs will be omitted when it may obscure the subject matter of the present invention.
To better explain the technical scheme of the invention, firstly, the spectrum leakage problem caused by incoherent sampling and the harmonic distortion problem caused by impure signal sources are briefly explained.
According to the existing research results, the conditions for realizing coherent sampling are as follows:
Figure BDA0002639216090000091
wherein M is the total number of sampling points, generally taking the power of 2, J is the sampling period number of the input signal and is an integer, when J takes the odd number, M and J are both prime, fSigRepresenting the frequency, f, of the input signalSampRepresenting the sampling frequency.
When J is not an integer, non-coherent sampling occurs. Dividing the number of sampling periods J under non-coherent sampling conditions into an integer part JintAnd a decimal part delta, let J equal to Jint+δ,δ∈[-0.5,0.5]The remaining four test conditions were met. Discrete Fourier transform is to extend a periodic signal infinitely, and because the sampling period number J is not an integer under the incoherent sampling condition, that is, the sampled and intercepted signal is not an integral multiple of the period of the input signal, non-continuous points occur when the DFT transform is directly performed on the output signal, so that the frequency spectrum leaks. Fig. 1 is a graph of the output signal spectrum under non-coherent sampling conditions. As shown in fig. 1, a spectrum leakage situation may occur when sampling non-coherently. The fundamental frequency has a large skirt effect, and the spectrogram is directly used for calculating the frequency domain parameters of the ADC to be detected, so that the correct values of the frequency domain parameters of the ADC to be detected cannot be obtained.
The input signal x (t) of the ADC under test is:
Figure BDA0002639216090000092
wherein A represents the amplitude of the fundamental wave, fSigDenotes the fundamental frequency of the input signal,. phi denotes the fundamental initial phase, AhRepresents the amplitude of the h-th harmonic component, phihDenotes the initial phase of the H-th harmonic component, H2, 3, …, H denotes the harmonic order.
The trigonometric function transformation of the input signal can obtain:
Figure BDA0002639216090000093
where a is Acos (phi) and b is-Asin (phi).
The ADC to be tested collects input signals, and the analog expression of theoretical output digital signals x' n is as follows:
Figure BDA0002639216090000101
decomposing the first term on the right side of the equation can obtain:
Figure BDA0002639216090000102
wherein the content of the first and second substances,
Figure BDA0002639216090000103
representing harmonic components, w [ n ]]Representing noise, n-0, 1, …, M-1.
Fig. 2 is a graph of the output signal spectrum in the presence of incoherent sampling and harmonic distortion of the input signal. As shown in fig. 2, when an input signal source is impure, a harmonic part in the signal is not negligible, the power of the harmonic is above the noise power, and meanwhile, an incoherent sampling problem exists, which causes that an ADC output signal not only has a large amount of harmonic power distortion, but also both the harmonic and the fundamental have a spectrum leakage phenomenon, and it is difficult to obtain accurate spectrum parameters by directly calculating ADC frequency domain parameters based on such spectrum information.
FIG. 3 is a flow chart of an embodiment of a signal recovery method for the presence of incoherent sampling and harmonic distortion in ADC spectrum testing according to the present invention. As shown in fig. 3, the specific steps of the signal recovery method for incoherent sampling and harmonic distortion in the ADC spectrum test of the present invention include:
s301: estimating the number of sampling cycles:
when ADC spectrum test is carried out under the incoherent sampling condition, an input signal x (t) of the tested ADC is recorded as follows:
Figure BDA0002639216090000104
wherein A represents the amplitude of the fundamental wave, fSigDenotes the fundamental frequency of the input signal,. phi denotes the fundamental initial phase, AhRepresents the amplitude of the h-th harmonic component, phihDenotes the initial phase of the H-th harmonic component, H2, 3, …, H denotes the harmonic order; a and b are coefficients, a is Acos (phi), and b is-Asin (phi).
In actual test, the output signal x [ n ] of the ADC to be tested is tested]Performing discrete Fourier transform to obtain frequency domain expression X of output signalk，XkThe expression of (a) is:
Figure BDA0002639216090000105
wherein n represents the sampling point number, M represents the number of sampling points, and j represents the imaginary unit.
Because the sampling period under the incoherent condition is non-integer, the frequency domain information of the output signal is utilized to carry out sampling period measurement on the ADC to be tested
Figure BDA0002639216090000111
Integer part of
Figure BDA0002639216090000112
And fractional part
Figure BDA0002639216090000113
Estimating to obtain the total sampling periodEstimation of numbers
Figure BDA0002639216090000114
Wherein the integer part
Figure BDA0002639216090000115
And fractional part
Figure BDA0002639216090000116
Are respectively:
Figure BDA0002639216090000117
Figure BDA0002639216090000118
wherein arg max represents the variable value for making the function reach the maximum value, imag represents the complex imaginary part, and e represents the natural constant.
S302: estimating coefficients:
when the coefficient a needs to be estimated, the output signal x [ n ] of the ADC to be tested]And
Figure BDA0002639216090000119
multiplication, n is 0,1, …, M-1, and adding M points, ignoring the harmonic term and noise term yields:
Figure BDA00026392160900001110
where symbol denotes taking the conjugate.
When it is desired to estimate the coefficient b, x n is estimated, as such]And
Figure BDA00026392160900001111
multiplication, n-0, 1, …, M-1, adding M points, ignoring the harmonic and noise terms yields:
Figure BDA00026392160900001112
order to
Y1＝[x[0],x[1],…,x[M-1]]T,
Figure BDA0002639216090000121
Figure BDA0002639216090000122
Where the superscript T denotes transpose.
The expression of the above estimation coefficients a, b can be rewritten as follows:
Y1 HY2＝a*Y2 HY2+b*Y3 HY2
Y1 HY3＝b*Y3 HY3+a*Y2 HY3
due to Y1，Y2，Y3Are all M-dimensional column vectors, so Y3 HY2＝Y2 HY3And its value is constant. For the same reason Y1 HY2，Y1 HY3，Y2 HY2，Y3 HY3Is also constant, so the above equation is a system of linear equations in two variables, a is constant since a and b are constant when the initial phase is constant*＝a,b*B, the system of equations in one binary equation may be changed to:
Y1 HY2＝aY2 HY2+bY3 HY2
Y1 HY3＝bY3 HY3+aY3 HY2
therefore, it is not only easy to use
Figure BDA0002639216090000123
Order to
Figure BDA0002639216090000124
The above equation can be converted to solve the problem Ax ═ B, since the determinant of matrix a is 0, i.e.:
Figure BDA0002639216090000125
and the matrix A is a singular matrix, so that Ax is not solved to B, the matrix A is diagonally loaded, namely E is taken to be alpha I, wherein the matrix I is an identity matrix, alpha is a minimum constant (which can be set according to actual conditions) larger than 0, A + E is a symmetrical positive definite matrix, and x is (A + E) x is taken to B, so that the matrix A is a singular matrix, and the matrix A is a matrix with the condition that Ax is not solved to B, so that the matrix A is a matrix with the condition that E is not solved to alpha I, and the matrix A is a matrix with the condition that A + E is symmetrical positive definite, and the matrix A is a matrix with the condition that A + E is not solved to B, so that the matrix A is a matrix with the condition that E is not solved to alpha I, and the matrix A is a matrix with the condition that X is not solved to B, and the matrix A is a matrix with the condition that X is not solved to B, and the matrix A is a matrix A, and the matrix is a matrix with the minimum value of the maximum value of the minimum value of the maximum
Figure BDA0002639216090000126
Wherein
Figure BDA0002639216090000127
The estimation values of the coefficients a and b in the invention can be obtained according to the derivation process
Figure BDA0002639216090000128
The calculation formula of (a) is as follows:
Figure BDA0002639216090000131
wherein the content of the first and second substances,
Figure BDA0002639216090000132
Figure BDA0002639216090000133
Figure BDA0002639216090000134
wherein the superscript H denotes the determination of the conjugate transpose, Y1、Y2、Y3The expression of (a) is as follows:
Y1＝[x[0],x[1],…,x[M-1]]T,
Figure BDA0002639216090000135
Figure BDA0002639216090000136
s303: estimating the amplitude and initial phase of the fundamental component:
using the estimated coefficients of step S302
Figure BDA0002639216090000137
Namely, the amplitude and the phase of the fundamental wave part of the signal can be estimated, and the specific method is as follows:
the amplitude estimation value of the fundamental wave portion is calculated by the following formula
Figure BDA0002639216090000138
Figure BDA0002639216090000139
Due to the fact that
Figure BDA00026392160900001310
Therefore, the initial phase estimation value of the fundamental wave portion is calculated by the following formula
Figure BDA00026392160900001311
Figure BDA00026392160900001312
Wherein the content of the first and second substances,
Figure BDA00026392160900001313
s304: reconstructing incoherent sampling fundamental wave signals:
using amplitude estimates of the fundamental part
Figure BDA00026392160900001314
Initial phase estimation
Figure BDA00026392160900001315
And sampling period
Figure BDA00026392160900001316
Reconstructing incoherent sampling fundamental wave signals to obtain reconstructed incoherent sampling fundamental wave signals x _ nc1[ n [)]：
Figure BDA0002639216090000141
Fig. 4 is a spectrum diagram of an incoherent sampled fundamental signal reconstructed from the output signal shown in fig. 2.
S305: removing non-coherent sampled fundamental signals:
at the output signal x [ n ] of the ADC to be tested]Removing incoherent sampling fundamental wave signal x _ nc1[ n [ ]]Obtaining a residual signal R1[n]：
R1[n]＝x[n]-x_nc1[n]
Fig. 5 is a graph of the spectrum of the residual signal of the output signal of fig. 2 minus the non-coherently sampled fundamental signal of fig. 4. As can be seen from fig. 5, the fundamental wave is completely eliminated, the remaining signal mainly includes harmonics and noise signals, and it is obvious that the harmonics have leakage. For the input signal with an impure signal source, the harmonic part in the output signal x [ n ] of the ADC to be tested is not negligible, so that it is far from enough to reconstruct the fundamental wave.
S306: estimating the amplitude and initial phase of the harmonic part:
for residual signal R1[n]Performing discrete Fourier transform to obtain residual signal R1[n]Frequency domain expression of (1)k，NkThe expression of (a) is:
Figure BDA0002639216090000142
order to
Figure BDA0002639216090000143
Figure BDA0002639216090000144
When rounding is shown, the parameters are set:
Figure BDA0002639216090000145
Figure BDA0002639216090000146
if it is not
Figure BDA0002639216090000147
Calculating an estimated amplitude of the h-th order harmonic component using the following formula
Figure BDA0002639216090000148
And estimating the initial phase
Figure BDA0002639216090000149
Figure BDA00026392160900001410
Figure BDA00026392160900001411
Here, phase { } denotes an acquisition angle, and λ ═ pi (M-1)/M.
If it is not
Figure BDA00026392160900001412
Calculating an estimated amplitude of the h-th order harmonic component using the following formula
Figure BDA00026392160900001413
And estimating the initial phase
Figure BDA00026392160900001414
Figure BDA0002639216090000151
Figure BDA0002639216090000152
S307: and (3) harmonic signal reconstruction:
amplitude estimation using harmonic components
Figure BDA0002639216090000153
Initial phase estimation
Figure BDA0002639216090000154
And sampling period
Figure BDA0002639216090000155
Carrying out harmonic signal reconstruction to obtain a reconstructed harmonic signal x _ Harm _ nc1[ n [ ]]：
Figure BDA0002639216090000156
Fig. 6 is a graph of the spectrum of the harmonic signal reconstructed from the output signal of fig. 2.
S308: removing harmonic signals:
at the output signal x [ n ] of the ADC to be tested]Middle removed harmonic signal x _ Harm _ nc1[ n]Obtaining a residual signal R2[n]：
R2[n]＝x[n]-x_Harm_nc1[n]
S309: second order estimation of the amplitude and initial phase of the fundamental component:
the original signal is a signal with a large amount of harmonic distortion and incoherent sampling, then the estimation of the fundamental wave signal in step S303 is directly based on the original signal, which results in an error in the fundamental wave signal reconstructed in step S304 and further an error in the harmonic wave signal reconstructed in step S307, so that the fundamental wave and the harmonic wave information are secondarily estimated and reconstructed in the present invention to eliminate the spectrum leakage of the harmonic wave on the fundamental wave and the spectrum leakage of the fundamental wave on the harmonic wave.
For residual signal R2[n]Performing discrete Fourier transform to obtain residual signal R2[n]Frequency domain expression Zk，ZkThe expression of (a) is:
Figure BDA0002639216090000157
according to ZkSampling period number of ADC to be tested
Figure BDA0002639216090000158
Fractional part of
Figure BDA0002639216090000159
Performing a second estimation of the fractional part of the sampling period
Figure BDA00026392160900001510
The expression of (a) is:
Figure BDA00026392160900001511
the following formula is then used to calculate a quadratic amplitude estimate for the fundamental component
Figure BDA0002639216090000161
Figure BDA0002639216090000162
Calculating a secondary initial phase estimate of the fundamental component using the following equation
Figure BDA0002639216090000163
Figure BDA0002639216090000164
S310: performing incoherent sampling fundamental wave signal secondary reconstruction:
using quadratic amplitude estimates of the fundamental part
Figure BDA0002639216090000165
Secondary initial phase estimation
Figure BDA0002639216090000166
And number of sampling cycles
Figure BDA0002639216090000167
Integer part of
Figure BDA0002639216090000168
Performing incoherent sampling fundamental wave signal reconstruction to obtain an incoherent sampling fundamental wave signal x _ Fund _2[ n ] after secondary reconstruction]：
Figure BDA0002639216090000169
S311: removing the secondarily reconstructed incoherent sampling fundamental wave signals:
at the output signal x [ n ] of the ADC to be tested]Removing twice reconstructed incoherent sampling fundamental wave signal x _ Fund _2[ n [ ]]Obtaining a residual signal R3[n]：
R3[n]＝x[n]-x_Fund_2[n]
S312: second order estimation of the amplitude and initial phase of the harmonic part:
for residual signal R3[n]Performing discrete Fourier transform to obtain residual signal R3[n]Is expressed as Sk，SkThe expression of (a) is:
Figure BDA00026392160900001610
extracting the harmonic part of the ideal input signal x (t), and performing discrete Fourier transform on the harmonic part of the signal to obtain:
Figure BDA0002639216090000171
when M > 1024, the fundamental wave portion contains
Figure BDA0002639216090000172
The term part can be ignored, and similarly it can be assumed that when M > 1024, the fundamental part contains
Figure BDA0002639216090000173
The top is negligible, let λ ═ π (M-1)/M, since J ═ Jint+δ,δ∈[-0.5,0.5]Therefore, the harmonic part formula can be approximately transformed into the following formula:
Figure BDA0002639216090000174
according to SkThe number of sampling cycles of the ADC to be tested corresponding to the h-th harmonic component
Figure BDA0002639216090000175
Fractional part of
Figure BDA0002639216090000176
Performing a second estimation, expressed as：
Figure BDA0002639216090000177
Order to
Figure BDA0002639216090000178
round () represents rounding and rounding, and the following formula is used to calculate the second order amplitude estimate of the h-th harmonic component
Figure BDA0002639216090000179
And a secondary initial phase estimate
Figure BDA00026392160900001710
Figure BDA00026392160900001711
Figure BDA00026392160900001712
S313: and (3) secondary reconstruction of harmonic signals:
second order amplitude estimation using harmonic components
Figure BDA00026392160900001713
Secondary initial phase estimation
Figure BDA00026392160900001714
And number of sampling cycles
Figure BDA00026392160900001715
Integer part of
Figure BDA00026392160900001716
Fraction of sampling period number obtained by secondary estimation
Figure BDA00026392160900001717
Carrying out harmonic signal reconstruction to obtain a reconstructed harmonic signal x _ Harm _2[ n]：
Figure BDA0002639216090000181
S314: and (3) coherent sampling signal reconstruction:
the fundamental wave signal and harmonic wave signal reconstruction under the coherent sampling condition is carried out by utilizing the parameters obtained by the secondary estimation of the fundamental wave part and the harmonic wave part, and the fundamental wave signal x _ Fund _ C [ n ] and the harmonic wave signal x _ Harm _ C [ n ] under the coherent sampling condition are obtained:
Figure BDA0002639216090000182
Figure BDA0002639216090000183
s315: and (3) sampling signal recovery:
the incoherent fundamental wave signal x _ Fund _2[ n ] in the output signal x [ n ] of the ADC to be tested is replaced by a coherent fundamental wave signal x _ Fund _ C [ n ], the harmonic signal x _ Harm _2[ n ] under the incoherent sampling is replaced by the harmonic signal x _ Harm _ C [ n ] under the coherent sampling, and then the sampling signal can be restored to obtain a coherent sampling signal x _ new [ n ]:
x_new[n]＝x[n]-x_Fund_2[n]-x_Harm_2[n]+x_Fund_C[n]+x_Harm_C[n]
fig. 7 is a spectral diagram of a coherently sampled signal obtained by recovering the output signal of fig. 2 using the present invention. Fig. 8 is a graph of an ideal signal spectrum obtained by sampling the input signal by the ADC under test in this embodiment. Comparing fig. 2, fig. 7 and fig. 8, it can be known that the present invention can effectively recover the tested ADC sampling signal with non-coherent sampling and harmonic distortion, so that the recovered coherent sampling signal is substantially equal to the signal obtained under the coherent sampling condition, and when testing the ADC spectrum, the coherent sampling condition and the harmonic distortion condition do not need to be strictly executed, and the accurate measurement of the spectrum parameter can be realized, thereby reducing the test difficulty and the test cost of the ADC.
Although illustrative embodiments of the present invention have been described above to facilitate the understanding of the present invention by those skilled in the art, it should be understood that the present invention is not limited to the scope of the embodiments, and various changes may be made apparent to those skilled in the art as long as they are within the spirit and scope of the present invention as defined and defined by the appended claims, and all matters of the invention which utilize the inventive concepts are protected.

Claims (1)

1. A signal recovery method for incoherent sampling and harmonic distortion in ADC spectrum test is characterized by comprising the following steps:
s1: when ADC spectrum test is carried out under the incoherent sampling condition, an input signal x (t) of the tested ADC is recorded as follows:
Figure FDA0003058060790000011
wherein A represents the amplitude of the fundamental wave, fSigDenotes the fundamental frequency of the input signal,. phi denotes the fundamental initial phase, AhRepresents the amplitude of the h-th harmonic component, phihDenotes the initial phase of the H-th harmonic component, H2, 3, …, H denotes the harmonic order; a, b are coefficients, a ═ Acos (Φ), b ═ Asin (Φ);
for the output signal x [ n ] of the ADC to be tested]Performing discrete Fourier transform to obtain frequency domain expression X of output signalk，XkThe expression of (a) is:
Figure FDA0003058060790000012
wherein n represents the sampling point serial number, M represents the number of sampling points, and j represents an imaginary unit;
sampling period of ADC to be tested by using frequency domain information of output signal
Figure FDA0003058060790000013
Integer part of
Figure FDA0003058060790000014
And fractional part
Figure FDA0003058060790000015
Estimating to obtain the estimated value of the total number of sampling periods
Figure FDA0003058060790000016
Wherein the integer part
Figure FDA0003058060790000017
And fractional part
Figure FDA0003058060790000018
Are respectively:
Figure FDA0003058060790000019
Figure FDA00030580607900000110
s2: calculating the following formula to obtain the estimated values of the coefficients a and b
Figure FDA00030580607900000111
Figure FDA00030580607900000112
Wherein the content of the first and second substances,
Figure FDA0003058060790000021
Figure FDA0003058060790000022
Figure FDA0003058060790000023
wherein the superscript H denotes the determination of the conjugate transpose, Y1、Y2、Y3The expression of (a) is as follows:
Y1＝[x[0],x[1],…,x[M-1]]T,
Figure FDA0003058060790000024
Figure FDA0003058060790000025
E＝αI
the matrix I is a unit matrix, and alpha is a preset minimum constant greater than 0, so that A + E is a symmetrical positive definite matrix;
s3: the amplitude estimation value is calculated by the following formula
Figure FDA0003058060790000026
Figure FDA0003058060790000027
The initial phase estimate is calculated using the following formula
Figure FDA0003058060790000028
Figure FDA0003058060790000029
Wherein the content of the first and second substances,
Figure FDA00030580607900000210
s4: using amplitude estimates
Figure FDA00030580607900000211
Initial phase estimation
Figure FDA00030580607900000212
And sampling period
Figure FDA00030580607900000213
Reconstructing incoherent sampling fundamental wave signals to obtain reconstructed incoherent sampling fundamental wave signals x _ nc1[ n [)]：
Figure FDA00030580607900000214
S5: at the output signal x [ n ] of the ADC to be tested]Removing incoherent sampling fundamental wave signal x _ nc1[ n [ ]]Obtaining a residual signal R1[n]：
R1[n]＝x[n]-x_nc1[n]
S6: for residual signal R1[n]Performing discrete Fourier transform to obtain residual signal R1[n]Frequency domain expression of (1)k，NkThe expression of (a) is:
Figure FDA0003058060790000031
order to
Figure FDA0003058060790000032
Figure FDA0003058060790000033
When rounding is shown, the parameters are set:
Figure FDA0003058060790000034
Figure FDA0003058060790000035
if it is not
Figure FDA0003058060790000036
Calculating an estimated amplitude of the h-th order harmonic component using the following formula
Figure FDA0003058060790000037
And estimating the initial phase
Figure FDA0003058060790000038
Figure FDA0003058060790000039
Figure FDA00030580607900000310
Wherein phase { } denotes an acquisition angle, λ ═ pi (M-1)/M;
if it is not
Figure FDA00030580607900000311
Calculating an estimated amplitude of the h-th order harmonic component using the following formula
Figure FDA00030580607900000312
And estimating the initial phase
Figure FDA00030580607900000313
Figure FDA00030580607900000314
Figure FDA00030580607900000315
S7: amplitude estimation using harmonic components
Figure FDA00030580607900000316
Initial phase estimation
Figure FDA00030580607900000317
And sampling period
Figure FDA00030580607900000318
Carrying out harmonic signal reconstruction to obtain a reconstructed harmonic signal x _ Harm _ nc1[ n [ ]]：
Figure FDA00030580607900000319
S8: at the output signal x [ n ] of the ADC to be tested]Middle removed harmonic signal x _ Harm _ nc1[ n]Obtaining a residual signal R2[n]：
R2[n]＝x[n]-x_Harm_nc1[n]
S9: for residual signal R2[n]Performing discrete Fourier transform to obtain residual signal R2[n]Frequency domain expression Zk，ZkThe expression of (a) is:
Figure FDA0003058060790000041
according to ZkSampling period number of ADC to be tested
Figure FDA0003058060790000042
Fractional part of
Figure FDA0003058060790000043
Performing a second estimation of the fractional part of the sampling period
Figure FDA0003058060790000044
The expression of (a) is:
Figure FDA0003058060790000045
the following formula is then used to calculate a quadratic amplitude estimate for the fundamental component
Figure FDA0003058060790000046
Figure FDA0003058060790000047
Calculating a secondary initial phase estimate of the fundamental component using the following equation
Figure FDA0003058060790000048
Figure FDA0003058060790000049
S10: using quadratic amplitude estimates of the fundamental part
Figure FDA00030580607900000410
Secondary initial phase estimation
Figure FDA00030580607900000411
And number of sampling cycles
Figure FDA00030580607900000412
Integer part of
Figure FDA00030580607900000413
Performing incoherent sampling fundamental wave signal reconstruction to obtain an incoherent sampling fundamental wave signal x _ Fund _2[ n ] after secondary reconstruction]：
Figure FDA00030580607900000414
S11: at the output signal x [ n ] of the ADC to be tested]Removing twice reconstructed incoherent sampling fundamental wave signal x _ Fund _2[ n [ ]]Obtaining a residual signal R3[n]：
R3[n]＝x[n]-x_Fund_2[n]
S12: for residual signal R3[n]Performing discrete Fourier transform to obtain residual signal R3[n]Is expressed as Sk，SkThe expression of (a) is:
Figure FDA00030580607900000415
according to SkThe number of sampling cycles of the ADC to be tested corresponding to the h-th harmonic component
Figure FDA00030580607900000416
Fractional part of
Figure FDA00030580607900000417
And performing secondary estimation, wherein the expression is as follows:
Figure FDA0003058060790000051
order to
Figure FDA0003058060790000052
Calculating the second amplitude of the h-th harmonic component using the following formulaDegree of estimation
Figure FDA0003058060790000053
And a secondary initial phase estimate
Figure FDA0003058060790000054
Figure FDA0003058060790000055
Figure FDA0003058060790000056
S13: second order amplitude estimation using harmonic components
Figure FDA0003058060790000057
Secondary initial phase estimation
Figure FDA0003058060790000058
And number of sampling cycles
Figure FDA0003058060790000059
Integer part of
Figure FDA00030580607900000510
Fraction of sampling period number obtained by secondary estimation
Figure FDA00030580607900000511
Carrying out harmonic signal reconstruction to obtain a reconstructed harmonic signal x _ Harm _2[ n]：
Figure FDA00030580607900000512
S14: the fundamental wave signal and harmonic wave signal reconstruction under the coherent sampling condition is carried out by utilizing the parameters obtained by the secondary estimation of the fundamental wave part and the harmonic wave part, and the fundamental wave signal x _ Fund _ C [ n ] and the harmonic wave signal x _ Harm _ C [ n ] under the coherent sampling condition are obtained:
Figure FDA00030580607900000513
Figure FDA00030580607900000514
s15: the incoherent fundamental wave signal x _ Fund _2[ n ] in the output signal x [ n ] of the ADC to be tested is replaced by a coherent fundamental wave signal x _ Fund _ C [ n ], the harmonic signal x _ Harm _2[ n ] under the incoherent sampling is replaced by the harmonic signal x _ Harm _ C [ n ] under the coherent sampling, and then the sampling signal can be restored to obtain a coherent sampling signal x _ new [ n ]:
x_new[n]＝x[n]-x_Fund_2[n]-x_Harm_2[n]+x_Fund_C[n]+x_Harm_C[n] 。
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