JP2009284451A - Method of designing filter, and filter - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To obtain an easy-to-design digital filter, having a linear passband and few ripples, and a small change in impulse response. <P>SOLUTION: A method of designing a filter is provided, in which data symmetric in a frequency axis direction is added to frequency characteristic data of a target filter and then the data are inversely Fourier-transformed, to obtain basic data of a target filter coefficient. The method is configured so that the characteristic of a generated filter does not deviate from the target specifications even when the number of taps is changed by applying a function or a plurality of piecewise functions to the transition portion of the frequency characteristic to be input, specifying the amplitude and frequency at two points in the function part in accordance with the specifications, adjusting the constants constituting the function without fixing the frequency coordinates of the connection part between the function part and the straight line data. There are also provided software and a filter generated by the method. <P>COPYRIGHT: (C)2010,JPO&INPIT

Description

  The present invention relates to a filter design method and a filter, and is particularly suitable for a filter used in the digital communication field.

Conventionally, as a method of designing an FIR filter, a filter coefficient is obtained by performing inverse Fourier transform using data obtained by linearly approximating a frequency characteristic of a target filter and adding a symmetrical characteristic that is turned back at half the sampling frequency. The method of obtaining the basic value of was common.
However, this method has a problem that the number of coefficients becomes very large, and if the coefficients are cut off halfway, a truncation error occurs and does not match the target frequency characteristic.
In order to improve this, there is known a technique of performing so-called windowing that multiplies a specific function to smoothly reduce the value of the truncation part. However, as a result, the coefficient value becomes complicated, and many operations are required for the configuration of the filter, which increases the scale.

In addition to this, there is also proposed a method that uses trigonometric functions and piecewise polynomials in the transition part without approximating the frequency characteristics linearly, but it reduces the passband ripple, especially removing the main overshoot at the end of the passband. Was difficult.
In designing, it is necessary to repeatedly optimize the calculation to obtain the required characteristics. In addition, it is necessary to perform the above-mentioned windowing, and it is usual to use a dedicated design device.

  As digital communication technology has become more sophisticated, the characteristics of the filters used therein have become severely demanded, and it has become increasingly difficult to cope with conventional design techniques. In particular, the linearity of the passband and the reduction of the ripple are important, and these improvements are expected to improve the performance of the equipment such as intersymbol interference and signal-to-noise ratio.

The filter of the present invention does not require a special design device in designing. If a predetermined target specification value and initial n value of a predetermined filter are input by a normal personal computer and attached software, the frequency characteristics can be obtained by calculation. This can be further symmetrized and subjected to inverse Fourier transform to obtain basic data of filter coefficients, which can be rearranged and rounded to obtain the necessary filter coefficients.
The obtained frequency characteristic of the filter coefficient has very little passband ripple, little bounce of the amplitude at the end of the passband, and 10 bits of calculation bits, and it fits within plus or minus 0.1 dB. The phase characteristic is a straight line, and the impulse response hardly changes even when a plurality of filters are cascaded.
Furthermore, if the n value is changed and recalculated, the number of taps changes within the specification value of the characteristic. Therefore, by optimizing the n value by moving the n value in the direction of decreasing the number of taps, It can be minimized.

According to the present invention configured as described above, frequency characteristic data necessary in the design process can be easily created by inputting a target specification value and n.
Next, symmetric data is added to the frequency characteristic data and inverse Fourier transform is performed to obtain a basic coefficient of the filter.
The basic coefficient is rounded by a power of 2 and rearranged around the maximum value of the coefficient to obtain a target filter coefficient.
The frequency characteristics and phase characteristics obtained by Fourier transform of the filter coefficients need not be modified, but if the number of processing bits is increased, the number of taps will increase, but the transition band gradient will remain the same and the passband ripple will be further increased. It is possible to improve and increase the attenuation in the attenuation region.
If the characteristic is within the specification, a filter coefficient satisfying the target standard and minimizing the number of taps can be obtained by optimizing the n value.

Embodiments of filter design according to the present invention and characteristics of a filter to be generated will be described below with reference to the drawings.
FIG. 1 is a diagram showing two functions of the transition region portion of the low-pass filter (1) according to the first embodiment and the frequency characteristics created therefrom.
1 and 2 are the central axes of the functions 1 and 2, and 3 is the point at which the functions 1 and 2 are switched at the center frequency of the transition region of the filter. The function 1 is on the left of 3 and the function 2 is on the right. 4 is an intersection of the passband attenuation y1 and the passband end frequency x1 on the function 1, and 5 is an intersection of the stopband attenuation y2 and the stopband end frequency x2 on the function 2.
The numbers 4 and 5 are given by the target specification.

  FIG. 2 is a table in which an arithmetic expression for creating the frequency characteristic of the low-pass filter (1) is described for each of the x divisional ranges. x is a range of the divided frequency, and y is an expression representing a frequency characteristic applied to the section. f (x) and g (x) are the expressions of f (x) and g (x) in the expression of y.

  FIG. 3 is a table showing the calculation formulas for the constants m, d, and a of the low-pass filter (1). According to this equation, m, d, a can be calculated and substituted for m, d, a in the equation of FIG.

FIG. 4 is a diagram displaying the two functions and the frequency characteristics created from these using the high-pass filter (1) of the first embodiment.
6 and 7 are center axes of the function 1 and the function 2, and 8 is a point where the function 1 and the function 2 are switched with the same gradient on the center frequency of the transition region of the filter, and the frequency is a. 8 is the function 2 on the left, the function 1 on the right, 9 is the intersection of the passband attenuation y1 and the passband end frequency x1 on the function 1, and 10 is the stopband attenuation y2 and the stopband end frequency on the function 2 This is the intersection of x2.
The numbers 9 and 10 are given by the target specification.

  FIG. 5 is a table in which an arithmetic expression for creating the frequency characteristic of the high-pass filter (1) is described for each of the x divisional ranges.

  FIG. 6 is a table showing the calculation formulas for the constants m, d, and a of the high-pass filter (1). The frequency characteristics can be obtained by calculating m, d, and a according to this equation and substituting the results into the equation of FIG.

FIG. 7 is a diagram displaying the functions of the transition region of the low-pass filter (2) of the second embodiment and the frequency characteristics created from these functions.
11 is the central axis of function 1, and 12 is function 1. 11 is a straight line 1 on the left side and a function 1 up to a frequency 1 on the right side. 13 is an intersection of the passband attenuation y1 and the passband end frequency x1 on the function 1, and 14 is an intersection of the stopband attenuation y2 and the stopband end frequency x2 on the function 1.
The values 13 and 14 are given by the target specification.

FIG. 8 is a frequency characteristic calculation formula for each frequency section of the low-pass filter (2).
FIG. 9 shows formulas for calculating m and a necessary for the calculation.

FIG. 10 is a diagram displaying the function of the transition region portion of the high-pass filter (2) of the second embodiment and the frequency characteristics created therefrom.
15 is the central axis of the function 1, and 16 is the function 1. The right side from 15 is a straight line 1, and the left side is a function 1. 17 is an intersection of the passband attenuation y1 and the passband end frequency x1 on the function 1, and 18 is an intersection of the stopband attenuation y2 and the stopband end frequency x2 on the function 1.
The numbers 17 and 18 are given by the target specification.

  FIG. 11 is a frequency characteristic calculation formula for each frequency section of the high-pass filter (2). FIG. 12 shows equations for calculating m and a necessary for the calculation of the high-pass filter (2).

FIG. 13 shows target specifications of the low-pass filter (1) of the embodiment, and FIG. 14 shows calculated values necessary for designing the low-pass filter (1) of the embodiment.
FIG. 15 is a table showing the coefficient values and coefficient numbers of the low-pass filter (1) of the embodiment. This filter is an FIR filter and has 37 taps.
If the coefficients are multiplied by 2 to the 10th power, they become integers as shown in the rightmost column of the figure, so that a multiplier is not required to realize these coefficient values in a circuit. The last division of 2 to the power of 10 is also possible by bit shift.

FIG. 16 shows the frequency characteristics of the 37-tap low-pass filter obtained as a result. As a result, the passband is flat and the attenuation in the stopband satisfies the specification of -45 dB.

  FIG. 17 is an enlarged view of the passband portion of this filter, and the ripple in the passband is within plus or minus 0.1 dB. There is no overshoot at the end of the passband.

  FIG. 18 shows the impulse responses of this filter alone and two and three connected in cascade, with the time axis shifted and superimposed. From this, it can be seen that the impulse responses overlap and the phase characteristics are good.

  FIG. 19 shows the number of taps when the n value and the number of processing bits are changed without changing the gradient of the transition portion of this filter. The white portion is the minimum number of taps in the same processing bit, and this portion may be selected finally.

FIG. 20 shows changes in the m value and the a value when the n value of the filter is changed. When the n value increases in this way, the a value slightly changes in proportion to this, and the m value changes logarithmically and has a function value within the specification.
FIG. 21 shows changes in the transition characteristics when the n value is changed, and FIG. 22 shows changes in the number of bits versus the number of taps when the n value is changed. n = 3.4 and n = 2.6 intersect each other around 10 bits. From this, it can be seen that when increasing the number of bits, it is better to move the n value in the increasing direction.

FIG. 23 is a table showing the coefficient values and coefficient numbers of the low-pass filter (2) of the second embodiment. This filter is an FIR filter and has 47 taps.
If the coefficients are multiplied by 2 to the 10th power, they become integers as shown in the rightmost column of the figure, so that a multiplier is not required to realize these coefficient values in a circuit. The last division of 2 to the power of 10 is also possible by bit shift.

FIG. 24 shows the frequency characteristics of the 47-tap low-pass filter obtained as a result. The passband is flat and the stopband attenuation meets the -45 dB specification.

  FIG. 25 is an enlarged view of the passband portion of this filter, and the ripple in the passband is within plus or minus 0.1 dB. There is no overshoot at the end of the passband.

  FIG. 26 shows the impulse responses of this filter alone and two and three connected in cascade, with the time axis shifted and superimposed. From this, it can be seen that the impulse responses overlap and the phase characteristics are good.

  FIG. 27 shows the number of taps when the n value and the number of processing bits are changed without changing the gradient of the transition portion of this filter. The white portion is the minimum number of taps in the same processing bit, and this portion may be selected finally.

FIG. 28 shows changes in the m value and the a value when the n value of this filter is changed. When the n value increases in this way, the a value slightly changes in proportion to this, and the m value changes logarithmically to keep the function value within the specification.
FIG. 29 shows changes in the transition characteristics when the n value is changed, and FIG. 30 shows changes in the number of bits versus the number of taps when the n value is changed. n = 4.1 and n = 2.8 intersect each other in the vicinity of 13 bits. From this, it can be seen that when increasing the number of bits, it is better to move the n value in the increasing direction.

  Although two types of low-pass filters have been described above, functions can be similarly applied to the transition region for high-pass filters, band-pass filters, and the like.

  The filter according to the present invention can be configured by a combination of a gate and a flip-flop without using a multiplier, and is suitable for applications where power consumption is limited at high frequencies. And useful for digital communication equipment.

  It is a figure explaining the production | generation method of the frequency characteristic of the low-pass filter (1) of this invention.   It is a figure which shows the calculation formula for creating the frequency characteristic for every division frequency range of the low-pass filter (1) of this invention.   It is a figure which shows the calculation formula of the constant of the calculation formula for creating the frequency characteristic of the low-pass filter (1) of this invention.   It is a figure explaining the production | generation method of the frequency characteristic of the high pass filter (1) of this invention.   It is a figure which shows the calculation formula for creating the frequency characteristic for every division | segmentation frequency range of the high pass filter (1) of this invention.   It is a figure which shows the calculation formula of the constant of the calculation formula for creating the frequency characteristic of the high pass filter (1) of this invention.   It is a figure explaining the production | generation method of the frequency characteristic of the low-pass filter (2) of this invention.   It is a figure which shows the calculation formula for creating the frequency characteristic for every division | segmentation frequency range of the low-pass filter (2) of this invention.   It is a figure which shows the calculation formula of the constant of the calculation formula for creating the frequency characteristic of the low-pass filter (2) of this invention.   It is a figure explaining the production | generation method of the frequency characteristic of the high pass filter (2) of this invention.   It is a figure which shows the calculation formula for creating the frequency characteristic for every division | segmentation frequency range of the high pass filter (2) of this invention.   It is a figure which shows the calculation formula of the constant of the calculation formula for producing the frequency characteristic of the high pass filter (2) of this invention.   It is a figure which shows an example of the target specification of the low-pass filter (1) used for the Example of this invention.   It is a figure which shows an example of the calculation result of the constant term of the frequency characteristic type | formula of the low-pass filter (1) used for the Example of this invention.   It is a figure which shows the 10th power value of the 2nd of all the coefficients of an example of the low-pass filter (1) calculated in the Example of the system 1, and all the coefficients.   It is a figure which shows the frequency characteristic of the low-pass filter (1) calculated with the coefficient of FIG.   It is the figure which expanded the frequency characteristic near the pass band of the low-pass filter (1) calculated with the coefficient of FIG.   FIG. 10 is a diagram in which low-pass filters (1) calculated with the coefficients of FIG. 9 are connected individually and in two and three in cascade, and the time axis of each impulse response is shifted and superimposed.   This is the number of taps when the n value and the number of processing bits are changed without changing the gradient of the transition part in the low-pass filter (1) calculated with the coefficients of FIG.   It is a figure which shows the change of m value and a value when n value is changed with the low-pass filter (1) calculated with the coefficient of FIG.   It is a figure which shows the change of the frequency characteristic of a transition region when n value is changed with the low-pass filter (1) calculated with the coefficient of FIG.   FIG. 10 is a diagram illustrating the relationship between the number of processing bits and the number of taps when the n value is changed by the low-pass filter (1) calculated using the coefficients of FIG. 9.   It is a figure which shows the 10th power value of all the coefficients of an example of the low-pass filter (2) calculated by the system 2, and all the coefficients.   It is a figure which shows the frequency characteristic of the low-pass filter (2) calculated with the coefficient of FIG.   It is the figure which expanded the frequency characteristic near the pass band of the low-pass filter (2) calculated with the coefficient of FIG.   It is the figure which displayed the low-pass filter (2) calculated with the coefficient of FIG. 23 individually, two, and three in cascade, and shifted and superimposed the time axis | shaft of each impulse response.   This is the number of taps when the n value and the number of processing bits are changed without changing the gradient of the transition part in the low-pass filter (2) calculated with the coefficients of FIG.   It is a figure which shows the change of m value and a value when n value is changed with the low-pass filter (2) calculated with the coefficient of FIG.   It is a figure which shows the change of the frequency characteristic of a transition region when n value is changed with the low-pass filter (2) calculated with the coefficient of FIG.   FIG. 10 is a diagram showing the relationship between the number of processing bits and the number of taps when the n value is changed by the low-pass filter (2) calculated using the coefficients of FIG. 9.

Explanation of symbols

1,6,11,15 Function 1 center frequency 2,7 Function 2 center frequency 3,8, Center of frequency characteristic transition region 4,9,13,17 Passband end position of target specification filter 5,10 , 14, 18 Attenuation band edge position of target specification filter 12, 16 Function 1

Claims (5)

In the filter design method to obtain the basic data of the target filter coefficient by adding symmetric data in the frequency axis direction to the frequency characteristic data of the target filter and performing inverse Fourier transform on this data, the transition of the input frequency characteristics Apply a function or multiple piecewise functions to the part, fix the amplitude and frequency of the two points of the function part according to the specifications, and adjust the constants that make up the function without fixing the frequency coordinates of the joint between the function part and the linear data Then, a filter design method and software configured so that the generated filter characteristics do not deviate from the target specification even if the number of taps is changed, and a filter generated thereby. In claim 1, when the amplitude value is normalized by 1, two exponential functions that are in contact with the amplitudes 1 and 0 at the top are alternately joined in the transition region of the filter based on the specification of the filter. And a filter design method and software generated by joining the top of each exponential function and a straight line 1 or 0 to form data of a predetermined frequency characteristic. As the exponential function of claim 2, the central frequency of the transition region is a, the central frequencies of two exponential functions connected obliquely according to the frequency specifications before and after that are a−d, a + d, m is a constant of each exponential function, b the logarithm base (e, 2,10, etc.), fractional or integer associated with the number of processing bits n, when the amplitude of y, a _{m m = -log b (0.5)} / d n
And the frequency characteristic of the filter is 0 ≦ x <ad when y = 1 (the direction of the transition region is negative)
y = 0 (the direction of the transition zone is positive)
When a−d <x <a, y = b ^{f (x)} (the direction of the transition region is negative)
y = 1−b ^{g (x)} (the direction of the transition region is positive)
At this time, f (x) = − m | x− (ad) | ⁿ
g (x) = − m | x− (a + d) | ⁿ
When a <x <a + d, y = 1−b ^{g (x)} (transition zone direction is negative)
y = b ^{f (x)} (direction of transition region is positive)
When a + d <x ≦ 1, y = 0 (the direction of the transition zone is negative)
y = 1 (the direction of the transition zone is positive)
Further, when the amplitude value of the passband end frequency x1 of the specification value is y1 in order to eliminate a from the arithmetic expression, a = −log _b (y1) ^{(1 / n)} / m ^{(1 / n)} + x1 +
(-Log _b (0.5)) ^{(1 / n)} / m ^{(1 / n)}
Is substituted into the calculation formula, and the calculation formula is configured so that the function portion changes while maintaining the specification value with respect to the change of n. Thus, a frequency characteristic data is obtained by fitting a real number (for example, a decimal number or an integer between 2 and 4) according to the number of processing bits as an initial n value, and then symmetrized to perform inverse Fourier transform and rounding processing, 2. The filter design method and software according to claim 1, wherein the number of taps of the resulting coefficient sequence is minimized by optimizing n values to obtain a filter with the minimum number of taps satisfying the specifications, and the filter generated thereby . In claim 1, when the amplitude value is normalized by 1, the top of the exponential function that is in contact with the amplitude 1 is joined to the straight line 1 at one end of the transition region, and a constant of the exponential function is calculated from the n value and specification to tap. A filter design method and software for obtaining a coefficient sequence having a predetermined frequency characteristic by minimizing the number, and a filter generated thereby. As an exponential function of claim 4, a is a transition band start frequency, m is a constant of the exponential function, b is a logarithmic base (e, 2, 10, etc.), n is a real number for adjusting the number of taps, and y is an amplitude. When the frequency characteristic is 0 ≦ x <a, y = 1 (the direction of the transition region is negative)
y = b ^{f (x)} (direction of transition region is positive)
Where f (x) = − m | x−a | ⁿ
When a <x ≦ 1, y = b ^{f (x)} (transition zone direction is negative)
Where f (x) = − m | x−a | ⁿ
y = 1 (the direction of the transition zone is positive)
In order to eliminate a from the calculation, the specified passband edge frequency is x1, and the amplitude of x1 is y1. A = −logb (y1) ^{(1 / n)} / m ^{(1 / n)} + x1
Is substituted into the calculation formula, and the function part is configured to change while maintaining the specification value with respect to the change of n, so that initially n is a real number (for example, a decimal number between 2 and 4 or 2 A filter design method and software, and a filter generated thereby, in which an integer) is applied, frequency characteristics are calculated, symmetrized, inverse Fourier transform is performed, and the number of taps of the obtained filter coefficient is minimized by optimization of n.

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