A method for designing a FIR linear phase filter

Abstract

The application discloses a FIR linear phase filter design method, the group delay of the filter is, and the filter unit impulse response is about symmetry. When a frequency sampling method is used to design a new linear phase filter, the frequency sampling value is either a real number or an imaginary number. The filter design method greatly simplifies the filter design process, significantly reduces the number of operation devices required by the filter, and improves the stability of the filter.

Description

Technical Field

[0001] This invention belongs to the field of digital signal processing technology, and in particular, relates to a design method for an FIR linear phase filter. Background Technology

[0002] A filter is a device or circuit used to process signals, primarily allowing useful signals to pass through while blocking unwanted signals as much as possible. With the development of computer technology, integrated circuit technology, and materials technology, digital filters have become widely used in signal processing systems. A digital filter implements a linear time-invariant system according to a predetermined finite-precision algorithm, and its implementation includes both software and hardware implementations. From a software perspective, a digital filter can be understood as a computational program or algorithm. From a hardware perspective, a digital filter can be viewed as a device composed of multipliers, adders, and delay units. Therefore, although digital filters run on hardware platforms (DSP chips, microcontrollers, and FPGA chips), the performance of the filter is not determined by the hardware, but rather by the digital system itself and the filter algorithm.

[0003] Like amplitude-frequency characteristics, phase-frequency characteristics are crucial features of a signal. With the continuous development of the communication market, the demands for communication quality are increasing, making phase-frequency characteristics increasingly important. Filters with undistorted amplitude and phase responses can reduce the system's bit error rate and improve communication quality. In television, network, digital communication, and radar systems, the output signal must meet the condition of no phase distortion to recover the original information carried by the input signal. The performance indicators of a cable television headend system determine the quality of its signal, and filters are a vital component of this system. They can filter or suppress interference signals to obtain the desired television image and sound for the user. However, if the phase-frequency characteristics of the transfer function change, the resulting phase distortion will distort the received television signal. Furthermore, linear phase filters have important applications in radar pulse signal transmission and power amplifier pre-distortion technology. Sometimes, systems need to meet certain phase characteristics, such as pulse compression or spreading, or to compensate for phase distortion caused by other components, such as filters or dispersive structures, which also require linear phase filters. FIR filters have wide applications in communication and video. FIR digital filters are the most basic units in digital signal processing systems. They can achieve the goal of maintaining strict linear phase frequency characteristics while having arbitrary amplitude frequency characteristics. Furthermore, because their unit impulse response is finite, the stability of the filter is guaranteed.

[0004] Frequency sampling method (FSM) is one of the most commonly used methods for designing FIR filters. Traditional FSM samples complex values ​​during frequency sampling, which is not conducive to simplifying and implementing the design of filters using frequency sampling method. The system function of the designed filter is more complex and requires a larger number of computing devices.

[0005] The existing patent publication number CN105958966A discloses an analytical design method for finite impulse response (FIR) filters. This method enables automatic selection of the frequency sampling mode for all-phase filters, requiring only the substitution of boundary frequency parameters into the corresponding analytical expression to achieve filter design. The technical solution adopted in this invention is an analytical design method for FIR filters, which includes constructing a set of candidate passband critical frequencies for two sampling modes, searching for the index corresponding to the minimum distance between the cutoff frequency and the candidate frequency set under the two sampling modes, and determining the selected sampling mode. Specifically, the steps are: 1) Given the cutoff frequency ωc and the length N0 of the frequency vector H, the passband cutoff frequency ωp is obtained according to equations (1) and (2); 2) The sampling mode is determined and selected, and the final filter coefficients are calculated. This patent mainly targets the design of odd-length FIR filters and does not explicitly address the application scenarios of even-length FIR filters. Summary of the Invention

[0006] This invention addresses the issue that traditional FSMs in the prior art use complex sample values ​​during frequency sampling, which is not conducive to simplifying and implementing the frequency sampling method for filters. The system function of the designed filters is relatively complex, and a large number of computing devices are required. Therefore, this invention proposes an FIR linear phase filter design method.

[0007] A design method for an FIR linear phase filter includes the following steps: S1. Determine the design specifications of the filter, and solve for the impulse response of the ideal filter based on the design specifications. h ( n ); S2, regarding the above h ( n Apply a symmetric constraint about N / 2, which limits the group delay of the filter. And the filter's unit impulse response coefficient is related to symmetry; S3. Determine the system function of the filter, the system function The expression is:

[0008] In the above formula, the It means that the Indicates the first kEach frequency sample value, the ; Step S3 further includes: when the frequency sample value sequence Satisfy about ( When the system function is subject to multiple nested symmetric conditions, The process of recursively merging and simplifying continues until the symmetric conditions for the next level can no longer be met.

[0009] Furthermore, in step S3, the filter includes a unit impulse response sequence. h ( n Even symmetry and odd symmetry.

[0010] Furthermore, each major category of filters includes two subcategories of linear phase filters with corresponding filter lengths N being odd and even.

[0011] Furthermore, the group delay α and phase function of the filter and impulse response h ( n It has the following properties:

[0012] or

[0013] In the above formula, the The constant is not equal to 0. Indicates the filter number Each time-domain coefficient.

[0014] Furthermore, step S2 also includes determining whether the frequency domain sampling value of the filter is a pure real number or a pure imaginary number based on the symmetry constraint.

[0015] Furthermore, when the unit impulse response When even symmetric, all All are real numbers; when the unit impulse response When odd symmetry, and For real numbers, the rest are as described. All are imaginary numbers.

[0016] Furthermore, when the stated h ( n Even symmetry, N When it is an odd number, For real numbers, the system function simplifies to: .

[0017] Furthermore, when h( n Even symmetry, N When it is even, For real numbers, the system function simplifies to: .

[0018] Furthermore, when h ( n Odd symmetry N When it is even, and the frequency sample value Satisfy about Under the condition of symmetry, the system function can be further simplified to: ; In the above formula, the , represents the imaginary part of a complex number.

[0019] Furthermore, when h ( n Odd symmetry N When the number is odd, the system function simplifies to: .

[0020] Compared with the prior art, the beneficial effects of the present invention are as follows: The linear phase filter designed in this invention has a group delay of According to the properties of the Fourier transform: if the frequency response length of a filter is equal to its order, then Unit impulse response An even-symmetric linear-phase filter whose frequency response is real and about Even symmetry; Unit impulse response An odd-symmetric linear-phase filter, whose frequency response is divided by the origin and... Apart from these two points being real numbers, all others are imaginary numbers and are related to... It exhibits odd symmetry. Applying this property to the frequency sampling design method of filters yields simpler and more efficient FIR linear-phase filters. Attached Figure Description

[0021] Figure 1 The amplitude function and phase angle diagram of the linear phase filter of this invention; Figure 2 The sampling diagrams show the ideal filter frequencies of the present invention, which are 0.15π~0.35π and 0.65π~0.85π. Figure 3 This is a schematic diagram of frequency sampling for the dual-passband linear phase filter of the present invention; Figure 4 This is a schematic diagram showing the amplitude-frequency and phase characteristics of the dual-bandpass FIR linear-phase linear filter of the present invention; Figure 5 This invention relates to a dual-passband linear phase filter network structure. Figure 6 This is the amplitude-frequency response diagram of the notch filter of the present invention; Figure 7 This is the network structure of the notch filter of the present invention. Detailed Implementation

[0022] To clearly illustrate the technical features of the present invention, the present invention will be described in detail below through specific embodiments and in conjunction with the accompanying drawings.

[0023] Many specific details are set forth in the following description in order to provide a full understanding of the invention. However, the invention may also be practiced in other ways different from those described herein, and therefore the scope of protection of the invention is not limited to the specific embodiments disclosed below.

[0024] In the description of this specification, references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.

[0025] Example 1 A design method for an FIR linear phase filter includes the following steps: S1. Determine the design specifications of the filter, and solve for the impulse response of the ideal filter based on the design specifications. h ( n ); S2, regarding the above h ( n Apply a symmetric constraint about N / 2, which limits the group delay of the filter. And the filter's unit impulse response coefficient is related to Symmetry; based on the symmetry constraint, the frequency domain sampling value of the filter is determined to be either a pure real number or a pure imaginary number; S3. Determine the system function of the filter, the system function The expression is:

[0026] In the above formula, the It means that the Indicates the first k Each frequency sample value, the ; Step S3 further includes: when the frequency sample value sequence Satisfy about ( When the system function is subject to multiple nested symmetric conditions, The process of recursively merging and simplifying continues until the symmetric conditions for the next level can no longer be met.

[0027] Based on the pure real or pure imaginary characteristics of the frequency domain sampled values, the complex first-order resonant units in the system function of the filter are merged into real-coefficient second-order resonant units or real-coefficient fourth-order resonant units.

[0028] Independent FIR linear phase filters and their amplitudes: For a The first-order FIR digital filter has a unit impulse response of , ... Its system function is: (1) Its filter frequency response is: (2) In the above formula, the Represents the Euler number, the , represents the imaginary part of a complex number, the said Indicates frequency, range is .

[0029] Bundle Independent FIR linear phase filter frequency response Represented as: (3) in Let be the amplitude function of the filter. As the phase function, the middle and rear parts of equation (3) are... After elimination, it is represented as: (4) The real and imaginary parts on both the left and right sides of the equation are equal, and the ratio of the real to the imaginary part is also equal. (5) Multiplying both sides of the above equation by cross and then moving the terms on the right side to the left side, we get: (6) The phase function and unit impulse response sequence satisfying the above equation There are two symmetry relationships, namely: (7) (8) Similar to traditional FIR linear phase filters, the above two equations represent the unit pulse response sequence, respectively. There are two main categories of linear phase filters: even-symmetric and odd-symmetric. Each category corresponds to a filter length. N These are linear phase filters in two subclasses: odd and even.

[0030] The amplitude characteristics of the filter in this embodiment are as follows: Even symmetry, N If the number is odd, when the stated h ( n Even symmetry, N When it is an odd number, For real numbers, the system function simplifies to: .

[0031] Will Substituting into equation (3), we get: (9) From the above equation, we can see that the amplitude function of the filter in this case is... With phase function : (10) because The value of is generally small, which affects the passband ripple and stopband attenuation performance of the filter, but has little impact on the filter's throughput. When analyzing the filter's throughput, is not considered. The impact. When hour, , about Odd symmetry makes this type of linear phase unsuitable. hour Filters, such as high-pass and band-stop filters.

[0032] In this embodiment, when Even symmetry, N When the number is even, use the same method as above to... Substituting into equation (3), we get: (11) From the above equation, we can see that the amplitude function of the filter in this case is... and phase function The expression is as follows: (12) In the above formula, since about Even symmetry, then Also about Even symmetry.

[0033] when Odd symmetry N For odd numbers, use the same method as above to... Substituting into equation (3), we get: (13) From the above equation, we can see that the amplitude function of the filter in this case is... With phase function The expression is as follows: (14) In the above formula, when hour, ,Right now exist There is a zero point, and for It exhibits odd symmetry. Therefore, it cannot be used in the design of low-pass and band-stop filters.

[0034] when Odd symmetry N Even number, Using the above method, Substituting into equation (3), we get: (15) From the above equation, we can see that the amplitude function of this filter is... and phase function The expression is: (16) In the above formula, when hour, ,Right now exist There is a zero point, and for They exhibit odd symmetry. Therefore, they cannot be used in the design of low-pass, high-pass, and band-stop filters. A summary of the characteristics of these four types of linear-phase filters is shown in Table 1.

[0035] Table 1. Four new types of linear-phase FIR filters.

[0036]

[0037] Example 2 In this embodiment, Figure 1 for Amplitude and phase characteristics of traditional FIR linear phase filters and new FIR linear phase filters. Figure 1 The two sub-graphs at the top are traditional linear-phase FIR filters. N Even number . Figure 1 The two sub-figures at the bottom show the new linear-phase FIR filter. N Even number Both filters employ a Kaiser window design, in which... design Figure 1 The two subgraphs above are traditional linear phase filters. ,design Figure 1 The following two subgraphs are for new linear-phase FIR filters. When selecting the final filter coefficients, the first 32 are taken and the last coefficient is discarded.

[0038] Designed using frequency sampling method An independent FIR filter system. The frequency sampling method described here starts from the frequency domain and measures the ideal frequency response. Apply equally spaced sampling; its expression is as follows: (17) Then, with Discrete samples of the frequency characteristics of a practical FIR filter .Depend on Finite-length sequences can be obtained using the inverse discrete Fourier transform. The expression is as follows: (18) If the filter has a unit impulse response Satisfying the new linear phase filter pair Symmetry requirements Then the discrete samples of frequency It also satisfies this symmetry requirement. When the unit impulse response When even symmetric, all All are real numbers. When the unit impulse response When even symmetric, remove and Except for real numbers, all others All are imaginary numbers. The expression for the system function of the FIR filter obtained by the frequency sampling method is as follows: (19) when Even symmetry, when N is odd. It is a real number. Apart from the independent symmetric coefficients, it has no symmetric relationship with other coefficients. Utilizing... The symmetry simplifies the system function of equation (19) to: (20) when Even symmetry, when N is even. It is a real number. and It is independent and has no symmetrical relationship with other coefficients. It can be utilized... The symmetry simplifies the system function of equation (19) to: (twenty one) when When the number is even and not divisible by 4, and has Utilizing coefficients The symmetry relation can simplify the system function of equation (21) to: ;(twenty two) when When the number is even and divisible by 4, and has Utilizing coefficients The symmetry relation can simplify the system function of equation (21) to: ;(twenty three) like The system function in equation (23) It can be further simplified, as long as It satisfies the requirement of even symmetry, and the center of symmetry of the system function truly exists during the previous simplification, as shown in equation (23). Then the system function can be further simplified using the methods of equations (22) and (23).

[0039] when Odd symmetry N When it is an odd number, ,Apart from Outside of real numbers (because) It can only take the value 0 or 1. Therefore, The other coefficients only contain the imaginary part, and the coefficients are used to... The symmetry relation will transform the system function of equation (19) Simplified to: ;(twenty four) when Odd symmetry N When it is even, , The other coefficients only contain the imaginary part, using the coefficients The symmetry relation will transform the system function of equation (19) Simplified to: (25) Equation (25) and Equation (24) are exactly the same because the independent terms Equal to zero, the difference between the two is not reflected in equation (25). When When the number is even and not divisible by 4, and The system function of equation (25) can be simplified to: (26) when When it is even and divisible by 4, and Utilizing coefficients The symmetry relation can simplify the system function of equation (25) to: (27) In the above formula, if Then it can be further simplified. As long as... It satisfies the odd symmetry requirement, and the center of symmetry of the system function actually exists during the last simplification, as shown in equation (27). Then the system function can continue to be simplified using the methods of equations (26) and (27).

[0040] Example 3 A design method for an FIR linear phase filter includes the following steps: S1. Determine the design specifications of the filter, and solve for the impulse response of the ideal filter based on the design specifications. h ( n ); S2, regarding the above h ( n Apply a symmetric constraint about N / 2, which limits the group delay of the filter. And the filter's unit impulse response coefficient is related to Symmetry; based on the symmetry constraint, the frequency domain sampling value of the filter is determined to be either a pure real number or a pure imaginary number; S3. Determine the system function of the filter, the system function The expression is:

[0041] In the above formula, the It means that the Indicates the first k Each frequency sample value, the ; Step S3 further includes: when the frequency sample value sequence Satisfy about ( When the system function is subject to multiple nested symmetric conditions, The process of recursively merging and simplifying continues until the symmetric conditions for the next level can no longer be met.

[0042] This embodiment simulates the filter, using a frequency domain sampling method to design the group delay. Multi-passband linear phase filters and multi-bandstop linear phase filters were developed, and the filter systems were verified.

[0043] 1. Multi-bandpass linear phase filter By properly planning the symmetry of the frequency sampling values, the multi-bandpass linear-phase filter can fully utilize the methods of equations (19) to (27) above to simplify the filter. This embodiment designs a two-passband FIR linear-phase multi-bandpass filter with passband frequencies between 0 and π, specifically 0.15π to 0.35π and 0.65π to 0.85π, and a filter order N = 131. Since the filter order is odd and the filter type is a bandpass filter, the first and third type filters in Table 1 above are suitable. Considering that the third type filter requires fewer computational components after simplification, this filter is designed as an odd-symmetric filter, i.e. , Frequency sampling methods sample from an ideal filter, such as... Figure 2 The examples shown are ideal filters with passband frequencies between 0 and π, ranging from 0.15π to 0.35π and from 0.65π to 0.85π.

[0044] The sampled values ​​of the fourth type of FIR linear-phase multi-bandpass filter in the frequency domain, excluding those in Table 1, are as follows: Apart from real numbers, all others must be imaginary numbers, and follow the rules... Odd symmetry. From equations (24) to (27), it can be seen that in the filter frequency system function, All are accompanied by the imaginary part symbol. To simplify calculations, in practical design, the equations (24) to (27) will be simplified. Cancel; all frequency domain sampled values ​​will be real numbers.

[0045] Begin uniformly sampling the amplitude of the ideal filter, taking real numbers for each sample, following... Odd symmetry and Equals 0. During sampling, it is assumed that the signal is unsigned, and the sign of the filter signal does not need to be considered. To reduce passband ripple and stopband attenuation, two additional sampling points are added to the transition band. The frequency sampling results are as follows:

[0046] The frequency sampling of the filter is as follows Figure 3 The upper sub-figure is shown. Considering the phase characteristics of the filter, the ideal filter sample is equal to the amplitude sample value multiplied by... Frequency sampling, including phase characteristics, such as Figure 3 Lower subgraph. From Figure 3 As can also be seen, frequency sampling has... , , The filter can be simplified three times based on these three sets of symmetries. Further simplification using the system function of equation (27) yields: (28) Figure 4 Design a dual bandpass filter using the aforementioned FIR linear phase filter design method. Figure 5 The network structure of the designed dual-passband filter is shown. Figure 5 Equation (28) shows that the filter only requires 19 multipliers, 27 adders, and 19 delayers.

[0047] 2. Notch Filter Design This embodiment also designs an FIR notch filter with a sampling frequency of 250Hz, a center stopband frequency of 50Hz, a stopband width of 0.4Hz, a stopband attenuation greater than 20dB, and passband cutoff frequencies of 49.2Hz and 50.8Hz, respectively. The maximum passband attenuation is 0.02dB. The notch filter of this embodiment can be used for electrocardiogram diagnosis in biomedical engineering.

[0048] Because the stopband attenuation is greater than 20 dB, three sampling points will be inserted into the transition band. The stopband width is equal to 2π × 0.4 / 250 = 0.0032π. If more sampling points are added in the stopband, then 2π / N ≤0.0032π and N ≥625. This embodiment sets the sampling length. N =1250, use the second type of filter in Table 1 to design this notch filter.

[0049] The corresponding understanding of the notch filter is as follows:

[0050] Wherein, the cutoff frequency in the above formula Based on the above understanding, the group delay of the notch filter is... N The frequency sampling of / 2 is as follows: .

[0051] To simplify the filter's transfer function, we will sample the frequency. It is divided into two parts, namely:

[0052] in

[0053] and .

[0054] From frequency sampling points The transfer function of the designed filter is From frequency sampling points The transfer function of the designed filter is: .

[0055] From frequency sampling points The transfer function of the designed filter for: .

[0056] like Figure 6 The figure shows the amplitude response characteristics of the designed notch filter. From... Figure 6 It can be seen that when the passband ripple is less than 0.012 dB and the stopband width is equal to 0.0032π, the stopband attenuation is greater than -20 dB. The designed notch filter meets the required specifications.

[0057] like Figure 7 The network structure of the notch filter is shown. From... Figure 7 As we can see, the filter only requires 28 adders, 11 multipliers, and 15 delayers.

[0058] Compared to traditional linear-phase filters, the new linear-phase filter is more suitable for filter design using the frequency sampling design method. In principle, the linear-phase filter in this embodiment is the same as the traditional linear-phase filter. Theoretical analysis and experimental simulation results show that the linear-phase filter in this embodiment is correct and feasible both theoretically and experimentally. The new linear-phase filter designed using the frequency design method requires significantly fewer arithmetic units, only 6 (or m+4) units (multipliers and adders). The case of an odd-numbered filter order can be avoided by increasing the number of sampling points, thus solving the problem of the inability to simplify the system function of the new filter.

[0059] Obviously, the embodiments described above are merely examples for clearly illustrating the present invention and are not intended to limit the implementation of the present invention. Those skilled in the art can make other variations or modifications based on the above description. It is neither necessary nor possible to exhaustively list all possible implementations. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the claims of the present invention.

Claims

1. A design method for an FIR linear phase filter, characterized in that, Includes the following steps: S1. Determine the design specifications of the filter, and solve for the impulse response of the ideal filter based on the design specifications. h ( n ); S2, regarding the above h ( n Apply a symmetric constraint about N / 2, which limits the group delay of the filter. And the filter's unit impulse response coefficient is related to symmetry; S3. Determine the system function of the filter, the system function The expression is: In the above formula, the express The transformed complex frequency domain variables, the Indicates the first k Each frequency sample value, the express ; Step S3 further includes: when the frequency sample value sequence Satisfy about ( When the system function is subject to multiple nested symmetric conditions, The merging and simplification process is repeated recursively until the symmetric conditions for the next level can no longer be met.

2. The FIR linear phase filter design method according to claim 1, characterized in that, In step S2, the filter includes a unit impulse response sequence. h ( n Even symmetry and odd symmetry.

3. The FIR linear phase filter design method according to claim 2, characterized in that, Each major category of filters includes a corresponding filter length. N These are two subclasses of linear phase filters: odd and even.

4. The FIR linear phase filter design method according to claim 2, characterized in that, The group delay α and phase function of the filter and impulse response h ( n It has the following properties: or In the above formula, the The constant is not equal to 0. Indicates the filter number n Each time-domain coefficient.

5. The FIR linear phase filter design method according to claim 1, characterized in that, Step S2 further includes determining whether the frequency domain sampling value of the filter is a pure real number or a pure imaginary number based on the symmetry constraint.

6. The FIR linear phase filter design method according to claim 1, characterized in that, When the unit impulse response When even symmetric, all All are real numbers; when the unit impulse response When odd symmetry, and For real numbers, the rest are as described. All are imaginary numbers.

7. The FIR linear phase filter design method according to claim 1, characterized in that, When the h ( n Even symmetry, N When it is an odd number, For real numbers, the system function simplifies to: 。 8. The FIR linear phase filter design method according to claim 1, characterized in that, when h ( n Even symmetry, N When it is even, For real numbers, the system function simplifies to: 。 9. The FIR linear phase filter design method according to claim 1, characterized in that, when h ( n Odd symmetry N When it is even, and the frequency sample value Satisfy about Under the condition of symmetry, the system function can be further simplified to: ； In the above formula, the , represents the imaginary part of a complex number.

10. The FIR linear phase filter design method according to claim 1, characterized in that, when h ( n Odd symmetry N When the number is odd, the system function simplifies to: 。

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