CN107769781A - A kind of method for the analog signal sampling and reconstruct for ensureing the point-by-point maximum reconstructed error of time domain - Google Patents
A kind of method for the analog signal sampling and reconstruct for ensureing the point-by-point maximum reconstructed error of time domain Download PDFInfo
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- CN107769781A CN107769781A CN201711061794.2A CN201711061794A CN107769781A CN 107769781 A CN107769781 A CN 107769781A CN 201711061794 A CN201711061794 A CN 201711061794A CN 107769781 A CN107769781 A CN 107769781A
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- H—ELECTRICITY
- H03—ELECTRONIC CIRCUITRY
- H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
- H03M1/00—Analogue/digital conversion; Digital/analogue conversion
- H03M1/12—Analogue/digital converters
- H03M1/124—Sampling or signal conditioning arrangements specially adapted for A/D converters
- H03M1/1245—Details of sampling arrangements or methods
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- H—ELECTRICITY
- H03—ELECTRONIC CIRCUITRY
- H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
- H03M1/00—Analogue/digital conversion; Digital/analogue conversion
- H03M1/66—Digital/analogue converters
- H03M1/665—Digital/analogue converters with intermediate conversion to phase of sinusoidal or similar periodical signals
Abstract
A kind of analog signal sampling for ensureing the point-by-point maximum reconstructed error of time domain and the method for reconstruct, analog signal to be sampledAfter simulating first differential circuit several times, the maximum of its amplitude is tried to achieve by analog circuit to output signal, to calculate the sampling period, and with the sampling period to the analog signal to be sampled after delaySampled, in reconstruct, each sampled value is formed into a piecewise polynomial come approximate continuous analog signal to be sampled using Lagrange interpolation polynomial.The present invention is a method for being based purely on time domain, to carry out effective uniform sampling on the premise of point-by-point maximum reconstructed error is ensured.
Description
Technical field
The present invention relates to technical field of signal sampling, more particularly to a kind of simulation for ensureing the point-by-point maximum reconstructed error of time domain
Signal sampling and the method for reconstruct.
Background technology
In this digital Age, an analog signal is first converted into data signal, and the work such as to be handled or stored again several
Into a standard procedure.Signal sampling is a basic step in this transfer process.Up to the present it is used for true
The basic theories for determining sample rate is the classical Shannon Sampling Theory (such as document [1], [2]) for band-limited signal.The theory
Core content can be described as follows：To any one signal x (t) with a width of W, if sample rate fsAt least 2W, then x
(t) can is by its sampled point and sinc functionsIdeally reconstruct and.It is fragrant when carrying out sampler design
Agriculture theory can regard an instrument based on frequency domain as, because its analysis foundation is the frequency spectrum based on signal.In the past few decades
In have many (such as documents [1], [3]-[6]) along the follow-up work of this thinking.For the sampling of some class distinctive signals
Method is also suggested (such as document [7]-[10]).
Although Shannon framework is succinct beautiful in theory, may be made troubles during actual design sampler or
It is difficult.First, the realistic simulation signal of processing in need be all limited in time, then these signals are with regard to inevitable right and wrong
Band-limited signal [11].So the signal spectrum aliasing blocked after just necessarily bringing sampling of any frequency spectrum, is missed so as to produce
Difference.In fact, generally following four error is required for considering：Spectral aliasing error, range error, truncated error, and time
Jitter error, such as document [12], [13].These error analyses are also used on wavelet analysis, such as document [14].Fourier analysis
In the famous Gibbs phenomenons that are related to there may be larger instantaneous error, such as document [15].With the application of high-speed dsp
Increasingly wider, it is often desirable to sample as far as possible few number on the premise of point-by-point (pointwise) reconstructed error of time domain is ensured
Strong point.What Shannon theory do not provide and supports in this respect.Although there is method to be proposed to ensureing reconstructed error
Under the premise of reduce the number (such as document [16]) of sampled point as far as possible, be also a lack of at present easy and effective suitable for engineer applied
Method.In some application scenarios, we can not obtain the spectrum information of signal in advance, directly cannot also be managed using Shannon
By.Also, the circuit of reconstruct analog signal usually uses constant or linear interpolation, such as document [17], but in Shannon theory
In be used for interpolation sinc functions can not obtain completely in practice.Although the energy stability used in Shannon theory exists
Convenient use in derivation, but the stability of point-by-point (pointwise) is more often desired in engineering practice, is especially being studied
When signal transient feature, such as document [3].
The content of the invention
The present invention proposes a kind of analog signal sampling for ensureing the point-by-point maximum reconstructed error of time domain and the method for reconstruct, is one
The individual method for being based purely on time domain, to carry out effective uniform sampling on the premise of point-by-point maximum reconstructed error is ensured.
The technical solution adopted in the present invention is：
A kind of analog signal sampling for ensureing the point-by-point maximum reconstructed error of time domain and the method for reconstruct, analog signal to be sampled
(t tries to achieve the maximum of its amplitude to output signal to x after simulating first differential circuit several times by analog circuit, uses
To calculate sampling period T, and the analog signal x (t) to be sampled after delay is sampled with sampling period T, in reconstruct,
Each sampled value is formed into a piecewise polynomial come approximate continuous analog signal to be sampled using Lagrange interpolation polynomial
x(t)。
Further:
The computational methods of the sampling period T：
In formula：ε is maximum allowable reconstructed error, and unit is consistent with the unit of analog signal x (t) to be sampled；
K is to reconstruct polynomial top step number, k >=1, (k+1)！It is k+1 factorial；
ηk+1For the maximum of the k+1 order derivative amplitudes of continuous analog signal x (t) to be sampled.
Sampled using the sampling period T, signal reconfiguring method：
In formula：K is to reconstruct polynomial top step number, k >=1；
Integer m span is：n-k+1≤m≤n；
li(t) it is Lagrange's interpolation basic functionWherein ti=iT, tj=jT；
The all-order derivative of analog signal need not be sampled, only utilize the sampled point of analog signal to be sampled；In weight
During signal content on each sampling period [nT, (n+1) T] of structure, using including sampled point x (nT) and x ((n+1) T)
K+1 continuous sampling point carry out signal reconstruction.
Compared with traditional Shannon Sampling Theory, beneficial effects of the present invention：
1) it is easy to analysis and realizes.Set forth herein method allow sampler design completely time domain carry out.If letter
Number maximum rate of change (x'(t), x " (t) etc.), it is known that the effective sampling period can be easily calculated.If do not known,
As shown in figure 3, it can be automatically obtained by circuit.But in Shannon Sampling Theory, signal bandwidth can not easily pass through mould
Intend circuit to estimate to obtain.
2) the reconstructed error precision of each point is ensured.In the Shannon Sampling Theory of classics, this is generally in practice
What required function was almost missing from.But in process proposed herein, it can design and ensure that time domain is each put most
Big reconstructed error.As being embodied shown in the discussion of part, reconstructed error scope of the present invention is very rigorous.But in Shannon Sampling Theory
Middle time domain reconstruction error range is difficult to be estimated, and is changed greatly such as the change of signal.
3) analog signal can be led by being applied to non-band-limited.All physical signallings are all limited times, therefore they are inevitable
It is non-band-limit signal.So always it is related to approximation in the practical application of Shannon Sampling Theory, and it is every also to be difficult to analysis
The time domain reconstruction error of individual point.By contrast, method proposed by the invention can easily handle non-band-limited analog signal,
And reconstructed error is controllable.
4) it is applied to the sampled signal of unknown characteristics.In the Shannon Sampling Theory of classics, we need before sampling
Know the bandwidth of signal.But method proposed by the present invention realizes that structure does not need any information of sampled signal, because related
Parameter can obtain in sample circuit.
Brief description of the drawings
A, b, c, d are followed successively by four kinds of analog signal x (t) schematic diagrames to be sampled that the present invention uses in Fig. 1；
A, b, c, d are followed successively by reconstruct of four kinds of continuous analog signals to be sampled after Shannon theory is sampled in Fig. 2
Error simulation result schematic diagram；
Fig. 3 is the circuit theory diagrams of the present invention, wherein the T that is delayeddIt is suitable with the time for calculating T；
A, b, c, d are followed successively by reconstruct of the four kinds of continuous analog signal x (t) to be sampled of the present invention after over-sampling in Fig. 4
Error simulation result schematic diagram (k=1)；
A, b, c, d are followed successively by reconstruct of the four kinds of continuous analog signal x (t) to be sampled of the present invention after over-sampling in Fig. 5
Error simulation result schematic diagram (k=2)；
A, b, c, d are followed successively by reconstruct of the four kinds of continuous analog signal x (t) to be sampled of the present invention after over-sampling in Fig. 6
Error simulation result schematic diagram (k=3)；
A, b, c, d are followed successively by reconstruct of the four kinds of continuous analog signal x (t) to be sampled of the present invention after over-sampling in Fig. 7
Error simulation result schematic diagram (k=4).
Embodiment
The present invention and its effect are further illustrated below in conjunction with the accompanying drawings.
Our first brief overview Shannon Sampling Theories.An analog signal x (t) to be sampled with a width of W is given, is first defined
One sampling sequence of impacts signal
Wherein sampling period T is determined below, so as to ensure perfect reconstruction.
S (t) frequency spectrum can be write as
Wherein sample rate fs=1/T.
Then the sample sequence obtained is exactly
xs(t) frequency spectrum and then can is write as
Wherein * is convolution algorithm.
Formula (4) is taught that to obtain perfect reconstruction, it would be desirable to following two conditions：
1)
fs≥2W (5)
2) by xs(t) an ideal low-pass filter Th (t) is passed through
Wherein
And
W ＜ fc＜ fs-W (7)
Meet that two above condition will cause reconstruction signal xr(t) there is identical frequency spectrum with x (t).So we must
To in L2X under normr(t)=x (t).
The time domain impulse of wave filter responds h (t)
H (t)=2fcsinc(2fct) (8)
So utilize formula (3) and (8), xr(t) analytical expression is exactly
Formula (9) is the formula being reconstructed using x (t) sampled point.We sample Shannon and reconstruct side above
Method is summarized as algorithm AF.
We list four test signals in Fig. 1, and by algorithm AF quality reconstruction row in fig. 2.Four in Fig. 1
Individual signal is all defined on [0,1] section, and is followed successively by following form：
x1(t)=sin (2 π t) (10)
x2(t)=cos (2 π t) (11)
x4(t)=etsin(40πt2) (13)
In order to effectively sample (collection data point as few as possible), it is intended that on the premise of desired precision is ensured
Use sampling period T as big as possible.First difficulty is exactly to estimate the bandwidth W of signal.Even with the expression formula of signal, I
Following frequency spectrum is generally just obtained after arduous derivation：
WhereinIt is 1[0,1](t) frequency spectrum.
Substantially withIt is identical, and forWithWe hardly result in analytic expression
Notice that the bandwidth W of no one of this four signals is limited.So we just bandwidth W be defined as frequency spectrum from
The sufficiently small Frequency point of decay after this, i.e., for all f>W,We fix ε hereinf=10-3.Then
We can be obtained by signal xk(t) bandwidth WkIt is as follows：
W3=160
W4=138
For signal x3And x (t)4(t), we have done the over-sampling of 30000 points, then obtain its bandwidth W using FFT.
In order to meet formula (5) and (7), we select fs=2.1W and fc=fs/2。
We can carry out signal sampling and reconstruct according to formula (9) now.But before this, we can not to by
The reconstructed error of point has any guarantee, because Shannon theory is to pass through L2What norm was set up, or as document [3] describes
, Shannon theory only ensures energy stability.How by the upper error of frequency domain be converted into time domain error or one not yet
Solve the problems, such as.In fact, this in peak-to-average power ratio (PAPR) problem widely studied in 4G and 5G communications is a core
Difficult point (such as document [18]).The result shown in Fig. 2 also partly illustrates the complexity of this estimation error problem.We
See for signal x1(t), maximum reconstructed error is 10-4The order of magnitude, less than 10-3；It is but same for substantially, possessing
Bandwidth and the signal x in sampling period2(t), reconstructed error can be up to about 0.4.Because Gibbs phenomenons.Signal x3(t) and
x4(t) order of magnitude of the reconstructed error 0.05 and 0.2.
In the present invention it is proposed that one is answered following problem based on the method for time domain：A given finite length
Continuous analog signal x (t) to be sampled and a maximum point-by-point reconstructed error upper bound ε, how effectively x (t) to be carried out equal
Even sampling so that maximum point-by-point reconstructed error is no more than ε
Therefore, the method that the present invention uses is：By analog signal x (t) to be sampled by simulating first differential electricity several times
Lu Hou, the maximum (such as document [21]) of its amplitude is tried to achieve by analog circuit to output signal, to calculate sampling period T,
And the continuous analog signal x (t) to be sampled after delay is sampled with sampling period T, in reconstruct, utilize Lagrange
Each sampled value is formed a piecewise polynomial and carrys out approximate continuous analog signal x (t) to be sampled by interpolation polynomial.
Checking explanation is carried out to the above method below by specific embodiment.
A kind of analog signal sampling and the method for reconstruct, continuous analog to be sampled for ensureing the point-by-point maximum reconstructed error of time domain
Signal x (t) tries to achieve the maximum of its amplitude to output signal after first differential circuit several times by analog circuit, uses
To calculate the sampling periodAnd with sampling period T to the continuous analog signal x (t) to be sampled after delay
Sampled, in reconstruct, each sampled value is formed into a piecewise polynomial come approximate using Lagrange interpolation polynomial
Analog signal x (t) to be sampled.In formula：
ε is maximum allowable reconstructed error, and unit is consistent with the unit of continuous analog signal x (t) to be sampled；
K is to reconstruct polynomial top step number, k >=1, (k+1)！It is k+1 factorial；
ηk+1For the maximum of the k+1 order derivative amplitudes of continuous analog signal x (t) to be sampled.
Sampled using the sampling period T, signal reconfiguring method：
In formula：K is to reconstruct polynomial top step number, k >=1；
Integer m span is：n-k+1≤m≤n；
li(t) it is Lagrange's interpolation basic functionWherein ti=iT, tj=jT.
The all-order derivative of analog signal need not be sampled, only utilize the sampled point of analog signal to be sampled.In weight
During signal content on each sampling period [nT, (n+1) T] of structure, using including sampled point x (nT) and x ((n+1) T)
K+1 continuous sampling point carry out signal reconstruction.
Carry out what guarantee the sampling of signal and reconstruct can not have to point-by-point reconstructed error according to Shannon theory, it is such as attached
Shown in Fig. 2, the maximum reconstructed error difference of unlike signal is very big, and some reconstructed errors are 10-3The order of magnitude, some is in 0.01 number
Magnitude, also have in 0.1 order of magnitude, which show the complexity of estimation error problem under Shannon theory.The present invention can make
Obtain peak signal reconstructed error and fully meet design requirement, as shown in accompanying drawing 4 to accompanying drawing 7, reconstructing method of the invention is in difference
K values under reconstructed error be respectively less than the 10 of design requirement-3.Moreover, as shown in Table I, sampling period of the invention and Shannon
Sampling period is big in the same order of magnitude or more, and accompanying drawing 2 shows that the maximum reconstructed error after Shannon sampling reaches 0.4, much
Higher than the 0.001 of the present invention, that is to say, that, can be suitable in sampling number using the method for the present invention compared with Shannon framework
Or it is less in the case of so that reconstructed error is smaller.
Table I
The sampling period T that four test signals are obtained by algorithms of different
| Alg. | T1 | T2 | T3 | T4 |
| Shannon theory | 0.00149600 | 0.00149600 | 0.00297619 | 0.00345066 |
| K=1 of the present invention | 0.01423525 | 0.01423525 | 0.01128379 | 0.00021787 |
| K=2 of the present invention | 0.02892037 | 0.02892037 | 0.02058358 | 0.00051808 |
| K=3 of the present invention | 0.04176203 | 0.04176203 | 0.02415283 | 0.00081923 |
| K=4 of the present invention | 0.05207465 | 0.05207465 | 0.02905929 | 0.00106607 |
One, it was demonstrated that as follows：
K+1 sampled point be present on sampling interval [nT, (n+k) T].Utilize k rank rank Lagrange interpolation polynomials, energy
Obtain reconstructing analog signal xr(t) form is following (such as document [19])
Wherein li(t) it is Lagrange's interpolation basic function,
Wherein ti=(n+i-1) T, tj=(n+j-1) T.
Reconstructed error is obtained (such as document [19]) by Lagrange's interpolation remainder
So
We used average inequality (such as document [20]) in (19) formula.
If k+1 is an odd number, Wo MenyouPair and a single sampled point.Between a pair of sampled points it is maximum away from
From being kT, the ultimate range between reconstruction point and this single sampled point isTherefore we can be readily available
If k+1 is an even number, Wo MenyouTo sampled point.Similarly, we can obtain
So reconstructed error can be defined as
Sampled point x (nT), x ((n+1) T) ..., x ((n+k) T) has been used to reconstruct x in (16) formular(t) in section
Content on [nT, (n+1) T].Similar proof procedure can equally prove, if using arbitrarily includes x (nT) and x ((n+
1) T) including k+1 continuous sampling point reconstruct xr(t) content on section [nT, (n+1) T], conclusion (22) formula is still
Set up.
Two, implement algorithm
0) determine to reconstruct polynomial parameter k
1) x (t), t ∈ [0,1], and the point-by-point worst error permissible value ε of time domain are inputted
2) estimate
3) sampling interval is set
4) sampled point x (nT) is obtained,
5) reconstruction signal x is obtained using formula (16) and (17)r(t)。
It is below existing literature involved in the present invention：
[1]Abdul J.Jerri,“The Shannon sampling theorem—Its various
extensions and applications: A tutorial review,”Proceedings of the IEEE,
vol.65,no.11,pp.1565–1596,1977.
[2]John G.Proakis and Dimitris G.Manolakis,Digital Signal Processing:
Principles, Algorithms,and Applications,4th Ed.,Prentice-Hall,2006.
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decades after Nyquist,”IEEE Transactions on Circuits and Systems I:
Fundamental Theory and Applications, vol.48,no.9,pp.1094–1109,2001.
[4]Ahmed I Zayed,Advances in Shannon’s sampling theory,CRC press,
1993.
[5]Robert J II Marks,Advanced topics in Shannon sampling and
interpolation theory, Springer Science&Business Media,2012.
[6]Michael Unser,“Sampling—50years after Shannon,”Proceedings of the
IEEE,vol.88, no.4,pp.569–587,2000.
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bandpass sampling,” IEEE Transactions on signal processing,vol.39,no.9,
pp.1973–1984,1991.
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approach [lecture notes],”IEEE Signal Processing Magazine,vol.29,no.4,pp.90–
96,2012.
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Vandergheynst,Dimitri Van De Ville,and Yves Wiaux,“Sparse signal
reconstruction on the sphere:implications of a new sampling theorem,”IEEE
Transactions on image processing,vol.22,no.6,pp.2275–2285,2013.
[10]Hiromi Ueda and Toshinori Tsuboi,“A sampling theorem for periodic
functions with no minus frequency component and its application,”in
Communications(APCC),2013 19th Asia- Pacific Conference on.IEEE,2013,pp.225–
230.
[11]Stephane Mallat,A wavelet tour of signal processing,2nd Ed.,
Academic press,1999.
[12]George C Stey,“Upper bounds on time jitter and sampling rate
errors,”in IEE Proceedings G-Electronic Circuits and Systems.IET,1983,vol.130
(5),pp.210–212.
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Claims (3)
- A kind of 1. method for the analog signal sampling and reconstruct for ensureing the point-by-point maximum reconstructed error of time domain, it is characterised in that：Wait to adopt Sample analog signal x (t) tries to achieve its amplitude after simulating first differential circuit several times, to output signal by analog circuit Maximum, to calculate sampling period T, and the analog signal x (t) to be sampled after delay is sampled with sampling period T, In reconstruct, each sampled value is formed into a piecewise polynomial come approximate simulation to be sampled using Lagrange interpolation polynomial Signal x (t).
- A kind of 2. side of analog signal sampling and reconstruct for ensureing the point-by-point maximum reconstructed error of time domain according to claim 1 Method, it is characterised in that：Sampling period is arranged toIn formula：ε is maximum allowable reconstructed error, and unit is consistent with the unit of continuous analog signal x (t) to be sampled；K is to reconstruct polynomial top step number, k >=1, (k+1)！It is k+1 factorial；ηk+1For the maximum of the k+1 order derivative amplitudes of continuous analog signal x (t) to be sampled.
- A kind of 3. side of analog signal sampling and reconstruct for ensureing the point-by-point maximum reconstructed error of time domain according to claim 2 Method, it is characterised in that：Sampled using the sampling period T, signal reconfiguring method：In formula：K is to reconstruct polynomial top step number, k >=1；Integer m span is：n-k+1≤m≤n；li(t) it is Lagrange's interpolation basic functionWherein ti=iT, tj=jT.The all-order derivative of analog signal need not be sampled, only utilize the sampled point of analog signal to be sampled.It is every in reconstruct During signal content on one sampling period [nT, (n+1) T], the k+1 including sampled point x (nT) and x ((n+1) T) is utilized Individual continuous sampling point carries out signal reconstruction.
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