DE102004048204A1 - Method for reconstructing a non-equidistant sampled analog signal - Google Patents

Abstract

The present invention provides a method with the following steps for reconstructing a non-equidistantly sampled analog signal that is observable only at limited, periodically recurring times: DOLLAR A a.) Provide an approximation matrix of dimension MxN which is a finite number N has predeterminable approximation functions for each one sampling instant of a finite number M of predeterminable sampling instants; DOLLAR A b.) Providing a function approach, which is formed by a multiplication of the approximation matrix with a weighting vector to be determined; DOLLAR A c.) Non-equidistant sampling of the signal at the predetermined sampling instants to form a sample vector; DOLLAR A d.) Performing a least-squares method to minimize a deviation of the function approach from the sample vector such that the weighting vector resulting in a minimum deviation is determined; DOLLAR A e.) Substituting predeterminable reconstruction instants into the approximation matrix to form a reconstruction matrix and DOLLAR A f.) Multiplying the reconstruction matrix with the determined weighting vector to form the reconstruction vector having a reconstruction value at each predetermined reconstruction time.

Description

The The present invention relates to a method of reconstruction a non-equidistant sampled, analog signal.

In the telecommunications industry is the sampling of analog signals usually to equidistant Sampling times with a constant sampling rate or sampling frequency carried out. The sampling theorem, also called the first Nyquist condition, which the (in the sense of information) lossless transition from value- and time-continuous signals to continuous value ensures discrete-time signals. The sampling theorem states that exactly then signals are reconstructed exactly can be if the frequencies contained in the signal are at most half as large as the used sampling rate or sampling frequency are. Will the sampling condition injured, i. the signal has shares with greater frequencies than half Sampling rate, this causes interference, so-called overlap error (Aliasing). In the process, high frequency components become low frequencies postponed and distorted so the reconstructed signal. An exact reconstruction of the transmitted Signal is therefore no longer possible.

For example, in multiplex systems, especially in time-division multiplex systems, ie in systems where two or more signals are alternately transmitted and received, signals occur which are observable only at time-limited and periodically recurring time slots. A single one of these signals is exemplary in FIG 1 shown. 1 shows a schematic diagram of an analog signal s (t) in the time domain, which is observable at the receiver only to time slots defined for this analog signal s (t). The time slots defined for the signal s (t) correspond to the observation period T _B. At all other times, the analog signal s (t) is indeterminate and can be assumed to be zero. Especially in multiplex systems, the analog signal s (t) is often reconstructed by low-pass filtering. Due to the multiplexing of the analog signal s (t) inevitably aliasing occurs, ie there are disturbances in the reconstruction of the signal to the extent that the analog signal s (t) contains frequencies that are higher than half the multiplexing rate. If the analog signal s (t) according to 1 be reconstructed without errors, so within an observation period T _B and a multiplexing period T _{M to} record multiple samples. The acquisition of several samples per observation period T _B results in a non-equidistant sampling of the analog signal s (t).

It is known from theory that a non-equidistantly sampled signal can theoretically be restored to its fullest using special reconstruction functions as well as a sample that has met the sampling condition. 2 shows a schematic diagram for a non-equidistant sampling of the analog signal. Within the observation period T _B , according to this example, seven samples s _{m are taken} at the sampling times tA _m . Outside the observation period T _B no sample s _{m is} recorded. The special reconstruction functions to be used for reconstruction are infinitely extended in time and also generally show very large amplitude values. The very large amplitude values in the practical implementation lead both to numerical instabilities and to a disproportionate amplification of the noise. For these reasons, the ideal reconstruction of the analog signal s (t) by means of the special reconstruction functions is practically barely usable.

ADVANTAGES OF INVENTION

The inventive method for the reconstruction of a non-equidistant sampled, analog signal having the features of claim 1 points opposite the known approaches to the Advantage on that the only too limited, periodically recurring Times observable analog signal is robustly reconstructable, taking the analog signal may have frequencies higher than are half the multiplexing rate. Another advantage is that that inventive method numerically stable, since the large amplitudes, which at the use of known reconstruction functions can occur avoided become. Furthermore, lets the process of the invention using digital signal processing so that the transmitted, analog signal can be processed in real time.

• a.) Bereitstellen einer Approximationsmatrix mit einer Dimension MxN, welche eine endliche Anzahl N an vorbestimmbaren Approximationsfunktionen für jeweils einen Abtastzeitpunkt einer endlichen Anzahl M an vorbestimmbaren Abtastzeitpunkten aufweist;
• b.) Bereitstellen eines Funktionsansatzes, welcher durch eine Multiplikation der Approximationsmatrix mit einem zu bestimmenden Gewichtungsvektor ausgebildet wird;
• c.) nicht-äquidistantes Abtasten des Signals an den vorbestimmten Abtastzeitpunkten zur Ausbildung eines Abtastvektors;
• d.) Durchführen eines Kleinste-Fehlerquadrate-Verfahrens zur Minimierung einer Abweichung des Funktionsansatzes von dem Abtastwertvektor, so dass der Gewichtungsvektor, welcher zu einer minimalen Abweichung führt, bestimmt wird;
• e.) Einsetzen von vorbestimmbaren Rekonstruktionszeitpunkten in die Approximationsmatrix zur Ausbildung einer Rekonstruktionsmatrix; und
• f.) Multiplizieren der Rekonstruktionsmatrix mit dem bestimmten Gewichtungsvektor zur Ausbildung des Rekonstruktionsvektors, welcher zu jedem vorbestimmten Rekonstruktionszeitpunkt einen Rekonstruktionswert aufweist.

Essentially, the idea underlying the present invention is to provide a method comprising the following steps for reconstructing a non-equidistantly sampled analog signal to deliver:

• a.) providing an approximation matrix having a dimension MxN which has a finite number N of predeterminable approximation functions for each one sampling instant of a finite number M of predeterminable sampling instants;
• b.) providing a functional approach, which is formed by a multiplication of the approximation matrix with a weighting vector to be determined;
• c.) non-equidistant sampling of the signal at the predetermined sampling instants to form a sample vector;
• d.) performing a least-squares method to minimize a deviation of the functional approach from the sample vector such that the weighting vector resulting in a minimum deviation is determined;
• e.) inserting predeterminable reconstruction instants into the approximation matrix to form a reconstruction matrix; and
• f.) multiplying the reconstruction matrix with the determined weighting vector to form the reconstruction vector having a reconstruction value at each predetermined reconstruction time.

advantageously, The signal may be due to non-equidistant sampling frequencies have higher than half the multiplexing rate. The inventive method is numerically stable, since the approximation functions are selectable that the resulting amplitudes do not exceed certain limits. Furthermore the approximation functions are selectable such that they are finite in time are extended, allowing the processing of the signal in real time possible becomes. The inventive method is also possible in real time, since the approximation matrix and the functional approach already exist the non-equidistant Be provided so that after sampling at Carry out the method according to the invention only addition, subtraction and multiplication steps with fixed coefficients are necessary for the reconstruction of the signal.

• – Abtastzeitpunkte für das Kleinste-Fehlerquadrate-Verfahren,
• – Rekonstruktionszeitpunkte für den Rekonstruktionsvektor,
• – Approximationsstützstellen für die Approximationsfunktion.

Another advantage of the method according to the invention is that the following functions or parameters are freely selectable depending on the specific problem at hand:

• Sampling times for the least squares method,
• Reconstruction times for the reconstruction vector,
• - Approximation support points for the approximation function.

In the dependent claims find advantageous developments and refinements of in claim 1 specified method for reconstructing a non-equidistant sampled, analog signal.

According to one preferred embodiment is the non-equidistant scanning of the signal for forming the sample vector such that in each multiplex period or in each observation period, which for the analog signal is provided, at least two samples be scanned. Advantageously, this is the permissible bandwidth of the to be transferred, analog signal in multiplex systems not to half the multiplexing rate or sampling rate limited. Advantageously, then the Aliasing errors avoided.

According to one Another preferred embodiment is for each approximation function provided exactly a predeterminable approximation support point. advantageously, can the approximation support points independently chosen from the sampling times become. This independence contributes decisively for flexibility the method according to the invention at. Furthermore, the respective approximation support point is dependent on the present one Problem or the signal to be received by the developer freely determinable.

According to one Another preferred embodiment is to determine the weighting vector using the least-squares method, a pseudoinverse to the approximation matrix and the calculated pseudoinverse is multiplied by the sample vector. advantageously, is the inventive method insensitive by using the least-squares approximation across from Noise.

According to one Another preferred development is the approximation function as B-splines, as a Hermitean function or as one of one Si function derived function is formed.

According to a further preferred development, the reconstruction times are provided with a reconstruction period, so that the reconstruction period is shorter than the multiplexing period. In front It is thus advantageous for the reconstruction vector to have reconstruction values at times which are outside the observation period.

According to one Another preferred embodiment, the sampling times are within of an observation period equidistant selected. Advantageously, thereby the computational effort for reconstruction of the analog signal is minimized.

DRAWINGS

embodiments The invention is illustrated in the drawings and in the following Description closer explained.

It demonstrate:

1 a schematic diagram of an only too limited, periodically recurring times observable analog signal in the time domain;

2 a schematic diagram for a non-equidistant sampling of the analog signal;

3 a schematic flow diagram of a first embodiment of the inventive method for reconstructing a non-equidistantly sampled signal;

4 a schematic block diagram of a second embodiment of the inventive method for reconstructing a non-equidistant sampled signal;

5 a schematic diagram for non-equidistant sampling of the analog signal according to the present invention; and

6 a schematic diagram for reconstructing the non-equidistant sampled signal according to 5 ,

DESCRIPTION THE EMBODIMENTS

In the same reference numerals designate the same or functionally identical Ingredients.

3 shows a schematic flow diagram of a first embodiment of the inventive method for reconstructing a non-equidistantly sampled signal. The inventive method for reconstructing the non-equidistantly sampled, analog signal s (t) has the following method steps:

Process step a:

Providing an approximation matrix Θ with a dimension NxM which has a finite number N of predeterminable approximation functions φ _n (t) for one sampling time tA _{m of} a finite number M of predeterminable sampling times tA _m . For each approximation function φ _n (t), it is preferable to provide exactly one predeterminable approximation support point tS _n . Advantageously, the approximation support points tS _n can be selected independently of the sampling times tA _m . This independence contributes significantly to the flexibility of the method according to the invention. Preferably, the sampling times tA _{m are} chosen to be equidistant within an observation period T _B. An approximation function φ _n (t) is formed for example as B-splines, as a Hermitian function or as a function derived from an Si function.

Process step b:

Providing a functional approach (Z), which by multiplying the approximation matrix Θ with a weighting vector to be determined a is trained.

Process step c:

Non-equidistant sampling of the signal s (t) at the predetermined sampling instants tA _m to form a sample vector s , Preferably, the non-equidistant sampling of the signal s (t) becomes the sample vector s performed such that in each observation period T _B , which is provided for the analog signal s (t), at least two samples s _{m are} sampled. In particular, the number of samples s _m sampled in a single observation period T _{B may} be selected by the corresponding developer depending on the present problem.

Process step d:

Performing a least-squares approximation method to minimize a deviation of the functional approach Z from the sample vector s so that the weighting vector a , which leads to a minimal deviation, is determined. Preferably, to determine the weighting vector a a pseudoinverse P of the approximation matrix Θ is calculated by means of the least squares method and the calculated pseudoinverse P is compared with the sample vector s multiplied.

Process step e:

Insertion of predeterminable reconstruction times tR _k into the approximation matrix Θ for forming a reconstruction matrix Θ ~. Preferably, the reconstruction times tR _{k are provided} with a reconstruction period T _R , so that the reconstruction period T _{R is} smaller than the multiplexing period T _M.

Process step f:

Multiply the reconstruction matrix Θ ~ by the determined weighting vector a for the formation of the reconstruction vector z which has a reconstruction value z _k for every predetermined reconstruction time tR _k . The reconstruction vector z with its reconstruction values _k z to the predetermined reconstruction points in time t R _k corresponds to the reconstructed, non-equidistantly sampled analog signal s (t), where the analog signal s (t) frequencies can have the over half the multiplexing rate (f _M / 2) go out.

4 shows a schematic block diagram of a second embodiment of the inventive method for reconstructing a non-equidistant sampled signal s (t). In the method according to the invention, before the sampling of the analog signal s (t) in the receiver, both an approximation matrix Θ and a functional approach Z are provided for the reconstruction of the transmitted analog signal s (t). An approximation matrix Θ (see equation (1)) with a dimension NxM is provided. The approximation matrix Θ has a finite number N of predeterminable approximation functions φ _n (t) for one sampling time tA _{m of} a finite number M of predeterminable sampling times tA _m (m ∈ [1, ..., M]). The respective developer can adapt both the approximation functions φ _n (t) and the sampling times tA _m to the present problem or to the signal to be received and reconstructed. Also, for each approximation function φ _n (t), the designer can predetermine exactly one approximation support point tS _n, depending on the corresponding problem. The approximation function φ _n (t) may be formed as B splines, as a Hermitian function, or as a function derived from an Si function.

Furthermore, a function approach Z (see equation (3)) is provided, which is achieved by multiplying the approximation matrix Θ by a weighting vector to be determined a is trained. The weighting vector a (see Equation (2)) provides a weighting parameter a _n for each of the N predetermined approximation functions φ _n (t) (n ∈ [1, ..., N]).

In the receiver, the received, analog signal s (t) of a scanning device 1 is supplied, which receives the signal s (t) at the predetermined sampling times tA _m to form a sample vector s non-equidistant scanning.

The sample vector s (see equation (4)) is fed to a computing device 2, which determines the weighting vector a (see equation (7)).

To minimize the deviation of the functional approach Z from the sample vector s a least squares method is performed. The deviation of the functional approach Z from the sample vector s is then minimal if the error square e ^T e the error vector e is minimal (see equation (5)). e = Θ · a - s (5)

In the least squares method, the error square becomes e ^T · e depending on the weighting vector a minimized (compare equation (6)).

The least squares approach is solved by the remaining error vector e on the approximation matrix Θ is perpendicular (see equation (7)).

• [1] Haykin, Simon S., Adaptive Filter Theory, Prentice Hall, Englewood Cliffs, New Jersey, 1986.
• [2] Freeman, H., Discrete-Time Systems, John Wiley & Sons, New York, 1965.

The weighting vector a is uniquely computable by means of the pseudoinverse P of the approximation matrix Θ. For the transformation according to equation (7) reference is made to the following literature:

• [1] Haykin, Simon S., Adaptive Filter Theory, Prentice Hall, Englewood Cliffs, New Jersey, 1986.
• [2] Freeman, H., Discrete-Time Systems, John Wiley & Sons, New York, 1965.

The pseudoinverse P depends only on the approximation functions φ _n (t) and the sampling times tA _m , but not on the sampled values s _m or the sample vector s and therefore only needs to be determined once when designing the system.

The determined weighting vector a is supplied to a reconstruction device 3, which by means of the weighting vector a a reconstruction vector z with reconstruction values z _k of the predetermined reconstruction points in time t R _k calculated. For the time being, desired reconstruction times tR _{k are} predetermined. The predetermined reconstruction times tR _k are used in the approximation matrix Θ to form the reconstruction matrix Θ ~. The reconstruction matrix Θ ~ becomes with the determined weighting vector a for the formation of the reconstruction vector z multiplied.

To determine the reconstruction vector z from the sample vector s only a simple matrix multiplication with a fixed matrix determined in the design process is needed. This matrix can be passed through Using symmetric approximation functions φ _n (t) with limited temporal extent greatly simplify. The possible simplifications depend on the chosen function approach Z for estimating the signal s (t) as a function of the weighting vector a from.

5 shows a schematic diagram for non-equidistant sampling of the analog signal according to the present invention. The analog signal s (t) is transmitted eg via a time-division multiplex channel to a receiver. At the receiver, the analog signal s (t) is observable only within the observation period T _B , which is provided for the analog signal s (t). Outside the observation period T _B , the analog signal s (t) is not observable at the receiver. At each predetermined sampling time tA _m , a sample s _{m is} sampled. The samples s _m at the sampling times tA _m form the sample vector s out.

6 shows a schematic diagram for the reconstruction of the non-equidistant sampled signal according to 5 , From the sample vector s according to 5 is by means of the method according to the invention 3 and 4 the reconstruction vector z which generates a reconstruction value z _k for every predetermined reconstruction time t R _k . For this purpose, the reconstruction times tR _k with a reconstruction period T _{R are} provided such that the reconstruction period T _{R is} smaller than the multiplex period T _M. Advantageously, thus the reconstruction vector z Reconstruction values z _k at times tR _k , which are outside the observation period T _B.

Although the present invention has been described above with reference to preferred embodiments, it is not limited thereto, but modifiable in many ways. Thus, in particular, the selection of the approximation functions by the developer with regard to the present problem can be selected. Likewise, reference is made in the above description to various points to a multiplex system in which analog signals s (t) are observable only at limited, periodically recurring times. However, these analog signals also occur in other systems. Here, the terms "multiplex rate" and "multiplex period" are synonymous with the rate of recurrence or the period of the periodically recurring observation periods T _B.

Claims (7)

A method of reconstructing a non-equidistantly sampled analog signal (s (t)) that is observable only at limited, periodically recurring times, comprising the steps of: a) providing an approximation matrix (Θ) having a dimension MxN which is a finite Number N of predeterminable approximation functions (φ _n (t)) for each one sampling instant (tA _m ) of a finite number M of predeterminable sampling times (tA _m ); b) providing a function approach (Z) which is obtained by multiplying the approximation matrix (Θ) by a weighting vector ( a ) is formed; c) non-equidistant sampling of the signal (s (t)) at the predetermined sampling instants (tA _m ) to form a sample vector ( s ); d) carrying out a least-squares method for minimizing a deviation of the functional approach (Z) from the sample vector ( s ), so that the weighting vector ( a ), which leads to a minimal deviation, is determined; e) substituting predefinable reconstruction times (tR _k ) in the approximation matrix (Θ) to form a reconstruction matrix (Θ ~); and f) multiplying the reconstruction matrix (Θ~) by the determined weighting vector ( a ) for forming the reconstruction vector ( z ), which has a reconstruction value (z _k ) at every predetermined reconstruction time (tR _k ). A method according to claim 1, characterized in that the non-equidistant sampling of the signal (s (t)) for forming the sample vector ( s ) is performed such that in each observation period (T _B ), which is provided for the analog signal (s (t)), at least two samples (s _m ) are sampled. Method according to Claim 1, characterized in that precisely one predeterminable approximation support point (tS _n ) is provided for each approximation function (φ _n (t)). Method according to claim 1, characterized in that for determining the weighting vector ( a ) a pseudoinverse (P) is calculated to the approximation matrix (Θ) by means of the least-squares method, and the calculated pseudoinverse (P) is compared with the sample vector (P). s ) is multiplied. Method according to claim 1 or 3, characterized in that the approximation function (φ _n (t)) is formed as B-splines, as a hermetic function or as a function derived from an Si-function. A method according to claim 1, characterized in that the reconstruction times (tR _k ) are provided with a reconstruction period (T _R ) such that the reconstruction period (T _R ) is less than the multiplexing period (T _M ). A method according to claim 1, characterized in that the sampling times (tA _m ) within an observation period (T _B ) are selected equidistant.