EP1183646A1

EP1183646A1 - Method and device for determining interpolated intermediate values of a sampled signal

Info

Publication number: EP1183646A1
Application number: EP00930996A
Authority: EP
Inventors: Andreas Falkenberg; Ulf Niemeyer; Christoph Rohe
Original assignee: Infineon Technologies AG
Current assignee: Infineon Technologies AG
Priority date: 1999-04-22
Filing date: 2000-04-03
Publication date: 2002-03-06
Also published as: US6526428B2; KR20020021787A; CN1203450C; JP2002543644A; US20020152249A1; CN1353847A; WO2000065537A1

Abstract

To determine interpolated intermediate values of a sampled signal, for example a digital mobile telephone signal, the parameters a, b and c of the second-degree polynomial y = ax2 + bx + c are determined from three known sampling values (x¿1?, y2), (x2, y2) and (x3, y3), so that intermediate values can be calculated on the basis of said polynomial. It is assumed that the signal is sampled at constant intervals d, notably at intervals normalized to d = 1, and the sampling time x2 = 0 is assigned to the know middle sampling value, so that calculation requires little hardware.

Description

description

Method and device for determining interpolated intermediate values of a sampled signal

The present invention relates to a method and a device for determining interpolated intermediate values of a sampled signal, in particular a mobile radio signal, according to the preamble of claim 1 and the preamble of claim 11.

The determination of interpolated intermediate values of a sampled signal is necessary wherever due to certain restrictions only a limited number of samples of the signal are available. For example, the calculation of interpolated values is required at various points in the field of computer graphics or in the field of digital mobile radio. In the field of mobile radio in particular, it is necessary to estimate intermediate values by interpolation, since only a small number of samples are available at certain points for evaluating a received signal.

In the interpolation, an attempt is essentially made to determine a curve or line which runs through a certain number of known points or sampled values and which is continuous. The desired intermediate values can then be determined using the now known line.

In the following it is assumed that three samples of the signal to be evaluated are known, which can be represented by three pairs of values (xi, yi), (x ₂ , y ₂ ) and (x ₃ , y ₃ ), with - as in FIG. 2 is shown - the x values correspond to the sampling time and the y values correspond to the respective signal value. The time intervals between the individual samples can be different. In the case of three known samples or pairs of samples, interpolation can be carried out using the second degree polynomial y = ax ² + bx + c, since the three unknown parameters a, b and c can be determined using the three known pairs of values. The following system of equations must therefore be solved as efficiently as possible:

y ₂ = ax ₂ + bx ₂ + cy ₃ = ax ₃ ² + bx ₃ + c

Using these three equations, the sought parameters a, b and c can be determined so that any intermediate value can then be calculated using the general relationship y = ax ² + bx + c.

Since mostly more than three sampling points are known, such as the value pairs (x ₄ , y ₄ ) and (x ₅ , y ₅ ) according to FIG. 2, partial curves are calculated for each three known sampling values, which are then combined. One sub-curve is based on the samples (Xi, yi), (x ₂ , y ₂ ) and (X ₃ Λ y ₃ ) and the other sub-curve is based on the samples (x ₃ , y ₃ ), (x ₄ , y ₄ ) and (x ₅ , y ₅ ) are calculated.

So that there is no kink at the interface (x ₃ , y ₃ ) between the two sub-curves, the slope of the curve, which is defined by the first derivative at the corresponding point, can be included in the calculation. The slope m is generally for the second degree polynomial mentioned above

m = 2ax + b

The procedure described above is relatively complex.

The object of the present invention is therefore to develop a method and a device for determining interpo- To provide the measured intermediate values of a sampled signal, which enables a determination or calculation of intermediate values in a simple manner and with less hardware expenditure.

This object is achieved by a method having the features of claim 1 and an apparatus having the features of claim 11. The subclaims each describe preferred and advantageous embodiments of the present invention.

The present invention relates to the determination or calculation of interpolated intermediate values of a signal in the presence of a special case, which occurs particularly frequently in the mobile radio sector, so that the solution according to the invention can be used almost without restricting generality.

On the one hand, it is assumed according to the present invention that the signal is sampled at constant intervals. In addition, the mean sampling time is defined as x ₂ = 0, so that for the unknown parameters a, b and c of the second degree polynomial to be determined, y = ax ² + bx + c expressions result, which result from the known samples y _x , y ₂ and y ₃ can be calculated.

A further simplification can be achieved by normalizing the constant distance between the known sampling times to 1, so that the intermediate values in the range [-1, 1] around x ₂ = 0 can be determined. It is particularly advantageous if the x values of the desired intermediate values are selected such that expressions result for the corresponding y values on the basis of the polynomial determined, in which only quotients with denominators which can be represented as powers of 2 result. In this case, a simple, optimized hardware implementation of the interpolator is possible, where used several times in several circuit units and divisions are not carried out explicitly, but are realized by moving the data lines or by renaming the corresponding bits, so that no additional hardware is required for this. Multiplications are also not carried out as such, but are replaced by much simpler and cleverly chosen additions.

The present invention can be used in particular for the analysis of a digital received signal in digital mobile radio systems, e.g. be used in UMTS terminals (Universal Mobile Telecommunication System).

The invention is explained below on the basis of preferred exemplary embodiments with reference to the drawing.

1 shows an illustration to explain the procedure according to the invention,

2 shows an illustration for explaining the conventional sampling and interpolation of a signal,

3 shows a block diagram of an interpolator according to a first exemplary embodiment of the invention,

FIG. 4 shows the structure of a first calculation unit shown in FIG. 3,

FIG. 5 shows the structure of a second calculation unit shown in FIG. 3, and

6 shows a block diagram of an interpolator according to a second exemplary embodiment of the invention.

According to the invention, it is assumed that - as shown in FIG. 1 - the signal to be analyzed is sampled at constant intervals d, ie: d = x ₂ - Xi = ₃ - x ₂

In FIG. 1, the known pairs of measured values or samples thus obtained are represented by solid arrows. These known samples are now to be used to reconstruct only a few discrete intermediate values of the corresponding signal curve, which runs through the known samples, which are shown in FIG. 1 by dashed arrows. This is particularly necessary if the signal sampling is not possible with an arbitrarily precise speed or a finite measuring accuracy is required, which is the case with applications in digital mobile radio.

1 shows the case of a fourfold oversampling, ie from the three neighboring known samples (xi yi), (x ₂ , y ₂ ) and (x ₃ , y ₃ ), six intermediate values in between are calculated by interpolation. In a further step, the known pairs of values (Xi, yi),

(x ₂ , y ₂ ) and (x ₃ , y ₃ ) replaced by (x ₂ , y ₂ ), (x ₃ , y ₃ ) and (x ₄ , y ₄ ), ie three new neighboring samples for further interpolation used, the new samples being shifted by d from the previous samples. Intermediate values between x ₂ and x ₄ can then be calculated in an analogous manner.

Kink points are permitted within the scope of the present invention since no continuous function is to be generated.

As a further simplification, the sampling time of the average sample is set to x ₂ = 0. This simplification, together with the assumption of constant sampling distances, does not limit the accuracy.

Based on the above conditions or assumptions, it follows: Xi = -dx ₂ = 0 x ₃ = d

The system of equations presented in the introduction to determine the parameters a, b and c of the polynomial y = ax ² + bx + c is thus simplified as follows:

y ₂ = cy ₃ = ad ² + bd + c

The parameter c can thus be replaced by y ₂ in the formulas for yi and y ₃ , so that after subtracting the resulting formulas, the following expression results:

The parameter b sought thus has the value by ₃ -yι

2d

By inserting the now known values for b and c into the above equation for calculating y _x :

y _λ = ad * - ^ -d ₊ y ₂ = ad> - ^ - ₊ y ₂

For the remaining parameter a of the polynomial searched

of second degree a =

The parameters obtained in this way can now be used in the equation y = ax ² + bx + c to calculate any intermediate values. Since the distances between adjacent x values are always the same by definition, this distance can be normalized to 1, so that d = 1.

The above formulas for parameters a, b and c are therefore simplified again as follows:

^a = - = - -y.

c = y ₂

The calculation of the range -d and d, i.e. in the range between -1 and 1, around x = 0 existing intermediate values can thus be done using the following formula:

^ J ^^ -Al + ^ -HΛ

A circuit implementation of the calculation of

Intermediate values with the help of the last-mentioned formula become particularly easy if it is assumed that the positions to be calculated are fixed, i.e. the intermediate values are always calculated in the same places. For reasons which will be explained in more detail below, it is particularly advantageous to select the digits in such a way that by inserting the corresponding x values in the above polynomial, expressions result for the y values which only have quotients, the denominator of which are powers of 2 can be represented.

According to a first embodiment of the invention, it is therefore assumed that the intermediate values are always for the x-

3 _J 1 1 1 1 3

Values - —d, - —d, - —d, —d, —d and - d are calculated

4 2 4 4 2 4 should. Based on the requirement d = 1, the Sample values for x = ³ -1 1 1 1 3

- and - calculated, for 4 ' ^~ 2' ^~ 4 '4 the following values result:

, ³ , 9, y_ + y, ³ , I__2-L _Λ _ _L ^{g (} -4 ^{~) =} TE ^(~ 2 ^~ -> -4-T- ^{) +}

An interpolator which carries out the calculations shown above thus receives the three known samples yi, y ₂ and y ₃ at its inputs and outputs six intermediate values as a function thereof, as is also shown in FIG. 1. The circuit structure of the interpolator can be optimized by providing units that are used several times. Furthermore, in the hardware implementations described below, multiplications are not carried out as such, but are replaced by cleverly chosen additions, which are much simpler. Since the quotients of the above formulas are exclusively quotients whose denominators can be represented as powers of two, the divisions to be carried out can also be carried out by moving the corresponding data lines, ie by renaming or re-evaluating the corresponding bits of the value to be divided, be implemented so that in principle no additional hardware is required for the divisions.

1 shows a block diagram of a corresponding interpolator 24, to which a known unit 23, which is used to calculate the intermediate values, is used by a scanning unit 23. Sample values y * ι, y ₂ and y _{3 are} supplied, which have been obtained by the scanning unit 23 in particular by scanning the corresponding signal with constant intervals d = 1.

The interpolator 24 comprises two calculation units 1 and 2, five division or bit renaming units 3-7, eight adders 8-12 and 16-18 and three subtractors 13-15. The interconnection of the individual components can be seen in FIG. 3.

The calculation unit 1 is used to calculate the expression y_ + y__ -y ₂ , which is required to calculate each intermediate value above. The structure of the calculation unit 1 is shown by way of example in FIG. 4 and essentially comprises an adder 19 and a subtractor 21. The division by 2 is carried out by a bit renaming unit 20 which shifts the output lines of the adder 19 so that the output value of the adder 19 is halved by shifting the bits of the output value of the adder 19 by one position.

The calculation unit 2 is used to calculate the expression - -, which is also required to calculate each intermediate value. The structure of the calculation unit 2 is shown in FIG. 5 and essentially comprises a subtractor 22, the output value of which is again halved by a bit renaming unit 23 in that the data output lines of the subtractor 22 are shifted.

The circuit thus proposed is suitable for quick calculations, since all intermediate values are calculated in parallel. The present at the inputs known Abtastwer- ^te Vi, y ₂ and y ₃ are applied at twice the sampling frequency of the scanning unit 23 to the inputs of the interpolator. A further simplification in terms of hardware can be achieved if only the intermediate values for

1 1 1 x = -—, -— and - can be calculated. A corresponding one

2 4 4 Circuit example is shown in Fig. 6, wherein the elements corresponding to the elements shown in Fig. 5 are given the same reference numerals. In this case, however, the scanning unit applies new samples with the sampling frequency to the interpolator 24, so that a calculation must be carried out at each sampling time in order to calculate all the desired intermediate values. The circuit shown in FIG. 6 already manages with the four adders 11-12, 17 and 19 and the four subtractors 14, 15, 21 and 22.

Both circuits shown calculate an interpolation, which increases the virtual sampling rate to four times the value. The circuit shown in FIG. 5 supplies six intermediate values in parallel, the inputs of the interpolator being reassigned in every second step, so that the elements provided in the circuit shown in FIG. 5 can work quite slowly. In the circuit shown in FIG. 6, on the other hand, three intermediate values are calculated, but the inputs have to be reset with each sampling time.

Claims

claims

1. Method for determining interpolated intermediate values of a sampled signal, the signal being sampled by at least three known samples of the signal in the form of pairs of values (x _x , yi), (x ₂ , y ₂ ) or (x ₃ , y ₃ ) to be obtained, the parameters a, b and c of the second degree polynomial y = ax ² + bx + c being determined on the basis of the three known sample values, and the desired intermediate values after determining the parameters of the second degree polynomial using the polynomial are calculated, characterized in that the three known samples are obtained by sampling the signal at constant intervals d, and in that the sample time x ₂ = 0 is assigned to the middle sample, so that the desired intermediate values are based on the second degree polynomial

be calculated.

2. The method of claim 1, d a d u r c h g e k e n n z e i c h n e t that the distance d between the known samples is normalized to d = 1, so that the desired intermediate values using the second degree polynomial with

be calculated.

3. The method according to claim 1 or 2, characterized in that a limited number of discrete intermediate values is calculated using the second degree polynomial.

4. The method according to claim 2 and 3, d a d u r c h g e k e n n z e i c h n e t that the x values of the desired intermediate values are selected such that the y values of the desired intermediate values that can be represented by the second degree polynomial as a result only have quotients with a power of 2 as a denominator.

5. The method according to claim 4, characterized in that the intermediate values using the second degree polynomial for x = -3/4, x = -1/2, x = -1/4, x = 1/2, x = 1/4 and x = 3/4 can be calculated.

6. The method of claim 4, d a d u r c h g e k e n n z e i c h n e t that the intermediate values are calculated using the second degree polynomial for x = -1/2, x = -1/4 and x = 1/4.

7. The method according to any one of claims 4-6, d a d u r c h g e k e n n z e i c h n e t that when calculating the intermediate values, divisions with a divisible by a power of 2 are realized by renaming the bits of the value to be divided.

8. The method according to any one of claims 4-7, d a d u r c h g e k e n n z e i c h n e t that when calculating the intermediate values, multiplications are realized by corresponding additions of quotients, the denominators of which can be represented by powers of 2.

9. The method according to any one of the preceding claims, characterized in that interpolated intermediate values of a sampled digital mobile radio signal are determined.

10. The method according to any one of the preceding claims, d a d u r c h g e k e n n z e i c h n e t that after the determination of the interpolated intermediate values based on the three known samples, the method is repeated again for three known samples whose sampling times are shifted by d compared to the previous three known samples.

11. Device for determining interpolated intermediate values of a sampled signal, the device (24) having at least three known samples of the signal, which are in the form of value pairs (xi, yi), (x ₂ , yz) or (x ₃ , y ₃ ) can be displayed, and the device is designed such that it uses the three known sample values to determine the parameters a, b and c of the second degree polynomial y = ax + bx + c and then desired intermediate values using the polynomial calculated, characterized in that the device (24) is designed such that it uses the second degree polynomial with y, +, - y - ~ —yι -y ₂ y ₃ -ya = -p,, b = and c = yd 2d calculated, where d denotes a constant time interval between the known sample values (xi, yi), (x ₂ , y ₂ ) and (x ₃ , y ₃ ) supplied to the device (24).

12. The apparatus of claim 11, d a d u r c h g e k e n n z e i c h n e t that the device (24) is designed such that it the desired intermediate values using the second degree polynomial for d = 1 with

a "= - y ⁺ -— ^y ι -y ₂ , b = - y ^~ -— ^y and c = y calculated and a first calculation unit (1) for calculating the value τ ⁺ y ₃ y ₂ and a second calculation unit (2)

for calculating the value.

13. The apparatus according to claim 12, characterized in that the first calculation unit (1) comprises an adder (19), a bit renamer (20) and a subtractor (21), the adder (19) as input signals the known samples yi and y ₃ receives and adds, the bit renaming (20) renaming the bits of the output value of the adder (19) in such a way that the output value of the adder (19) is halved, and the subtractor (21) as input signals the output signal of the bit renamer (20) and the known sample y ₂ receives and subtracts.

14. The apparatus of claim 12 or 13, characterized in that the second calculation unit (2) comprises a subtractor (22) and a bit renamer (23), wherein the subtractor (22) receives and subtracts the known samples y ₃ and y ₁ as input signals , and wherein the bit renamer (23) renames the bits of the output value of the subtractor (22) such that the output value of the subtractor (22) is halved.

15. Device according to one of claims 12-14, characterized in that the device (24) is designed such that it uses the second degree polynomial for the intermediate values for x = - 3/4, x = -1/2, x = - 1/4, x = 1/2, x = 1/4 and x = 3/4 calculated.

16. The apparatus according to claim 13, 14 and 15, characterized in that the device (24) comprises a first bit renaming device (3-5) and a second bit renaming device (6, 7) as well as first to eighth adders (8-12, 16-18) and first to third subtractors (13-15) The first bit renaming device (3-5) is designed in such a way that it renames the bits of the output value of the first calculation unit (1) in such a way that the first bit renaming device (3-5) outputs a first, second and third output value , wherein the first output value corresponds to half the output value of the first calculation unit (1), the second output value corresponds to the output value of the first calculation unit (1) divided by 4, and the third output value corresponds to the output value of the first calculation unit (1) divided by 16, the second Bit renaming device (6, 7) is designed such that it renames the bits of the output value of the second calculation unit (2) in such a way that the second A first and a second output value are output in the bit renaming device (6, 7), the first output value corresponding to half the output value of the second calculation unit (2) and the second output value corresponding to the output value of the second calculation unit (2) divided by 4, the first Adder (8) adds the first and third output values of the first bit renaming device (3-5), the second adder (9) adding the first and second output values of the second bit renaming device (6, 7), the third adder (10) adding the output value of the first adder (8) and the known sample value y ₃ , the fourth adder (11) adding the second output value of the first bit renaming device (3-5) and the known sample value y ₃ , the fifth adder (12 ) adds the third output value of the first bit renaming device (3-5) and the known sample value y ₃ , the first subtractor (13) adding the output value of the second add ierers (9) from the output value of the third adder (10) subtracts and outputs the desired intermediate value for x = -3/4, the second subtractor (14) subtracting the first output value of the second bit renaming device (6, 7) from the output value of the fourth adder (11) and the desired intermediate value for x = - Outputs 1/2, the third subtractor (15) subtracting the second output value of the second bit renaming device (6, 7) from the output value of the fifth adder (12) and outputting the desired intermediate value for x = -1/4, the sixth adder (16) adds the first output value of the second bit renaming device (6, 7) and the output value of the fourth adder (11) and outputs the desired intermediate value for x = 1/2, the seventh adder (17) the second output value of the second Bit renaming means (6, 7) and the output value of the fifth adder (12) added and outputs the desired intermediate value for x = 1/4, and wherein the eighth adder (18) the output value of the second Adders (9) and the output value of the third adder (10) are added and the desired intermediate value for x = 3/4 is output.

17. The apparatus of claim 13 or 14, characterized in that the device (24) is designed such that it uses the second degree polynomial for x = - 1/2, x = -1/4 and x = 1/4 calculated.

18. The apparatus of claim 13, 14 and 17, characterized in that the device (24) a first bit renaming device (4, 5) and a second bit renaming device (6, 7) and first to third adders (11, 12, 17) and one comprises first and a second subtractor (14, 15), the first bit renaming device (4, 5) being designed such that it renames the bits of the output value of the first calculation unit (1) in such a way that the first bit renaming device (4, 5) outputs a first and a second output value, the first output value dividing the output value of the first calculation unit (1) by 4 and the second output value dividing the output value of the first calculation unit (1) by 16 The second bit renaming device (6, 7) is designed such that it renames the bits of the output value of the second calculation unit (2) such that the second bit renaming device (6, 7) outputs a first and a second output value , wherein the first output value corresponds to half the output value of the second calculation unit (2) and the second output value corresponds to the output value of the second calculation unit (2) divided by 4, the first adder (11) representing the first output value of the first bit renaming device (4, 5) and adds the known sample value y ₃ , the second adder (12) adding the second output value t of the first bit renaming device (4, 5) and the known sample value y _{3 are} added, the first subtractor (14) subtracting the first output value of the second bit renaming device (6, 7) from the output value of the first adder (11) and the desired intermediate value for outputs x = -1/2, the second subtractor (15) subtracting the second output value of the second bit renaming device (6, 7) from the output value of the second adder (12) and outputting the desired intermediate value for x = -1/4, and wherein the third adder (17) adds the second output value of the second bit renaming device (6, 7) and the output value of the second adder (12) and outputs the desired intermediate value for x = 1/4.

19. Use of a device according to one of claims 11-18 for determining interpolated intermediate values of a sampled digital mobile radio signal.

20. Use according to claim 19, characterized in that a sampling device (23) for generating the known samples samples the digital mobile radio signal at constant time intervals d, the distance d being standardized to d = 1 in order to obtain the known samples, and in that the sampling instant x ₂ = 0 is assigned to the mean sampling value of the known sampling values thus obtained.