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Based on amplitude-frequency characteristics of all phase FIR filter design compensation

Print View , by: iSee ,Total views: 32 ,Word Count: 3346 ,Date: Wed, 22 Jul 2009 Time: 10:03 AM

1 Introduction

Control of the border has been the frequency of FIR filter design problems. Traditional filter design, such as sampling frequency and window function method, because beyond the control of the critical frequency, and their use is restricted. Some modern filter design methods, such as neural network ..., the immune algorithm, these methods are designed to filter coefficient optimization algorithm with a certain amplitude and frequency on the target function approximation process, but did not address the Optimization of the process of how to control the frequency of the border. A literature FRM (FreqLtency Responses Masking, frequency response shield) design method, it is first of all to design two prototype to meet the range of complementary filters, and then the prototype filter uses M delay of each delay device to replace the ( that the interpolation process), and then designed two-way filter mask to filter out as a result of interpolation of the image arising from the frequency characteristics of the final superposition of the two routes in response to that was the final filter output. This has generated sparse coefficient filter characteristics, while the total length of the filter does not significantly increase, because this method can be limited to a very narrow transition zone width has been widely used in, but the methods of the prototype filter and shielding the order of filter, the ripple frequency bands matching the performance of mutual influence and the problems that typically use linear regulation of complex mathematical way to solve the problem.

This paper put forward in the literature of all phase filter design based on the sampling frequency through changing the traditional model of the sampling frequency of symmetric dual models and the introduction of two-phase shift to compensate for the composition and structure of the single-window filter all phase approach, with in the matlab design, FIR filter makes the location of the critical frequency by changing the parameter λ can be solved, it has no multi-step iterative optimization of the characteristics of the design method is simple.

2 dual-symmetric frequency sampling of the entire phase FIR filter

2.1 equivalent full-phase FIR filter design steps

Literature all phase DFT filter design method, sampling frequency and window of the dual nature of function, and pointed out that: filter performance can be added before the window or rear window b and f can be improved, f and b can be divided into the set for three situations: no windows, a single window and double window. N order to design all-phase filter, the frequency of the need to set up a vector H, the ultimate all-phase filter can be equivalent to a length of 2N-1 of the FIR filter, its design can be divided into three steps: (1) H IDFT to generate h, then h defined on the domain extension, the formation of (2N-1) the length of the vector h '= [h (-N +1) ... h (0) ... h (N-1)] T. (2) before the window of f, rear window b for normalized convolution and deconvolution to generate the window after the wc. (3) h ', wc multiply the corresponding elements that have the equivalent FIR filter. Based on the above steps to generate 2N-l length of the FIR filter coefficient g is derived as follows:

Set the frequency vector for H = [H (O) H (1) ... H (N 1 1)] T, the assumption that to meet the traditional symmetric H (k) = H (Nk), (k = 1, ..., N-1 ), the IDFT of H corresponding to k = [h (0) h (1) ... h (N-1)] T. The WN = e-j2π / N, for h (n) of the definition of domain extension available vector h ':

Based on amplitude-frequency characteristics of all phase FIR filter design compensation

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Sampling frequency of 2.2 under the symmetric dual full-phase FIR filter

In fact, H can also be set to dual form of symmetry, that is, to meet the H (k) = H (N-1 a k), (k = 0, ..., N-1). If N = 16 when they set: H = [1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1] T, then type (1) the results of IDFT plural, which type (4) FIR coefficients for the plural. FIR coefficients to be real, you need to type (4) g multiplied by the volume of a phase shift v0 = [v0 (-N +1) ... v0 (a 1) v0 (0) v0 (1) ... v0 (N-1 )] T, which Based on amplitude-frequency characteristics of all phase FIR filter design compensation

Picture not clear? Click here to view the image (larger). Combination of type (4), the FIR filter coefficient becomes:

Based on amplitude-frequency characteristics of all phase FIR filter design compensation

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The frequency sampling interval △ ω = 2π / N, documentation to prove: no windows and a single window all phase of the transmission curve of filter frequency set point through the strict k △ ω, k = 0,1, ..., N-l and type (5 ) on the filter coefficient multiplied by the amount of phase shift after v0, according to Fourier transform the nature of the frequency shift, the frequency set point is also shifted to right the corresponding 0.5 △ ω, that is, strictly through ω = (k +0.5) △ ω , so that the formation of a symmetrical dual mode sampling frequency. For example: when N = 8 when the band adopted the traditional vector H = [1 1 l 0 0 01 1] T-symmetric dual-frequency vector mining He = [1 1 0 0 0 0 1 1] T. Both models are in [O, 2π) distribution of sampling points as shown in Figure 1.

Based on amplitude-frequency characteristics of all phase FIR filter design compensation

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3-phase full compensation under the amplitude-frequency low-pass filter design

To N = 16 as an example, the frequency of sampling will be even symmetric vector H as: H = [1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 l] T, the increase in single-Hamming window, the use of in front of all phase FIR filter design steps, as shown in Figure 2 can be the amplitude-frequency curves.

Based on amplitude-frequency characteristics of all phase FIR filter design compensation

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Figure 2 can be seen from: | H (ejω) | curve in the passband and stop-band than the flat, and strict border through the pass-band frequency sampling points ω1 = 2.5 △ ω and stopband boundary frequency sampling points ω3 = 3.5 △ ω, so the transition zone can be strictly controlled in the frequency of sampling every question within △ ω. In addition, also found that the frequency of sampling points ωl with ω3 curve between the very good linearity (N bigger, the better the linearity, the linear function of the degree can be adjusted through the window), if this approximation curve as a straight line above paragraph, 3dB cut-off frequency can be estimated that the approximate location of ω2 for ω2 *= (3.5-0.7071) △ ω = 2.792 △ ω = 1.097 (rad.s-1) The actual figure 2 ω2 = 2.839 △ ω = 1.115 (rad.s-1), both the existence of 0.018 (rad.s-1) the small difference between the increased N or select a good window function can reduce this difference.

Following the adoption of amplitude-frequency characteristic compensation method can be realized Figure 2 Frequency of the Boundary ωl, ω2, ω3 precise location of translational control:

The frequency vector H is divided into two parts SSB H = Hl + H2, which H1 = [l 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0] T, H2 = [0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1] T. According to H1, H2, the same filter structure in accordance with all phase steps can be the formation of two sub-filters h1, h2, according to type (1) and type (5), corresponding to their respective filter coefficients, respectively (M = 3, N = 16):

Based on amplitude-frequency characteristics of all phase FIR filter design compensation

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Obviously satisfy h1 (n) = h2 * 2 (n), which according to the nature of Fourier transform, are:

Based on amplitude-frequency characteristics of all phase FIR filter design compensation

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As shown in Figure 3: The two curves on the amplitude-frequency ω = π symmetry.

Based on amplitude-frequency characteristics of all phase FIR filter design compensation

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Practical requirements of the cut-off frequency is often not exactly as shown in Figure 2 fall on the frequency of sampling points, the traditional methods only increase the filter length N to realize to adjust the frequency of the border. Ⅳ does not change for the realization of the border to control the frequency of these two sub-filters can pan curves to achieve. Figure 3 as a result of the two sub-filter curve is symmetrical, if the two curves in the opposite direction of their shift the same distance, and then two sons and a filter synthesis filter coefficients can be real low-pass filter, based on the assumption λ translation of △ ω, while the two sons after translation of the FIR filter coefficients as follows:

Based on amplitude-frequency characteristics of all phase FIR filter design compensation

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Filter coefficients to the sum of two sons after the filter coefficient g 'as follows:

Based on amplitude-frequency characteristics of all phase FIR filter design compensation

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Take λ = 0.25 when the two sub-post-translational filter and its transmission after the composite curve superimposed in Figure 4 (a), 4 (b) below.

Based on amplitude-frequency characteristics of all phase FIR filter design compensation

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Compared to Figure 3: Figure 4 (a) of sub-filter transmission curves 1,2, respectively, moved to the right and left 0.25 △ ω. Figure 4 (b) is a complex superposition of two sons and filter results: low-frequency region can be found a dent, that is to say, after the compound does not have the low-pass filter characteristics, and thus all phase compensation needed to be addressed.

With computer-aided design can be G '(ejω) at ω = 0 and 2π / N the value of Office, so that a = | G (ej0) |, b = | G (ej2π / N) |, set up a frequency vector Hc = [1-al of a b 0 0 00 0 0 OOO 0 0 0 0 1-b] T, the use of Hc-phase design using the entire structure of a compensation filter: Figure 5 (a) in Canada for kaiser (N, 1) convolution with the rectangular window of the window to form a single window and the compensation of the amplitude-frequency filter hc curve, Figure 5 (b) after the filter to compensate for the amplitude-frequency curve g. Obviously, the compensation of the amplitude-frequency curve after the elimination of the low-frequency notch area, was flat characteristics.

As a result of the filter compensation coefficient by the two sub-filter coefficients h1 ', h2' and compensation from the filter hc stacking, which are:

Based on amplitude-frequency characteristics of all phase FIR filter design compensation

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Figure 5 (b) frequency of the border: ωl = 2.75 △ ω, ω2 = 3.089 △ ω, ω3 = 3.75 △ ω. Compared to Figure 2, are accurate translation 0.25 △ ω.

As a result of hl ', h2' factor conjugate to each other, and the filter compensation coefficient hc for real, and therefore the coefficient g filter for real.

4 full phase compensation method of other types of filter design

In addition to low-pass filter design, the use of such compensation can design any type of FIR filter, can be roughly divided into the following steps: (1) from the frequency of vector symmetric dual H symmetry derived from two complementary vector Hl, H2 (2) the use of H1, H2 in accordance with the full-phase filter design steps to design the two sub-filters h1, h2 (also available on the formation of sub-filter). (3) and then sub-filters hl, h2, respectively, multiplied by the phase conjugate to each other towards the volume vl, v2, so that after the phase shift of the sub-filter h1 ', h2'. (4) filter h1 ', h2' to be composite g ', the corresponding amplitude-frequency function G' (ejω). (5) through the use of computer-aided design, come to G '(ejω) the need for compensation in the k △ ω the frequency of point value. (6) compensation in accordance with the frequency of point G '(ejω) the value of the compensation structure the frequency response vector Hc, and Hc in accordance with the selection of appropriate compensation for a single window constructed filter hc. (7) h1 ', h2', hc superimposed to give the final composite filter g.

Structural differences between various types of filter is only steps (1) the frequency of vector-derived approach steps (4) of the composite method and the steps (5) compensation for the choice of the location of the frequency points are different. Due to limited space, only for all types of simple description:

4.1 high-pass filter design

The steps (1) the high-pass frequency vector H next to fill the law with forms derived from the frequency of the two symmetrical vectors H1, H2, and then use high-frequency amplitude-frequency curve of the value of the regional structure of the amplitude-frequency compensation filter hc, in accordance with type (10) give the final high-pass filter coefficient.

4.2 Band-Pass Filter Design

The basic idea is using two different cut-off frequency low-pass filter coefficients by the method of band-pass filter. Since each low-pass filter frequency can be divided into two vectors, so the need to break down into four single-sideband frequency vector. To N = 32 for example, two low-pass frequency vector are as follows:

Based on amplitude-frequency characteristics of all phase FIR filter design compensation

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By Ha, Hb can be split into two frequency vectors, which can be four single-sideband frequency vector:

Based on amplitude-frequency characteristics of all phase FIR filter design compensation

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Set of two shift parameters λ1, λ2, makes Ha1, Ha2 filter transmission curve corresponding to the opposite direction towards each other their respective mobile λ1 of 2π / N (rad / s), and Hb1, Hb2 corresponding filter transmission curve the opposite direction towards each other their respective mobile λ2 of 2π / N (rad / s), the assumption that after the four SSB frequency shift corresponds to the filter coefficient vector for the ha1 ', ha2', hb1 ', hb2', after the composite The band-pass filter coefficients can be expressed as:

Based on amplitude-frequency characteristics of all phase FIR filter design compensation

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As the ha1, ha2, Hb1, Hb2 are the corresponding low-pass transmission characteristic, so the need for frequency compensation point also appears in the low-frequency, use of 3.1 next to law can be compensated up filter hc. Again in accordance with the steps (7) to give the final band-pass filter g.

Can be seen that by setting the frequency shift parameters λ1, λ2 can be flexible to adjust the band-pass filter with position and bandwidth, and thus control of the border frequency.

4.3 Notch Filter Design

Specific examples to illustrate all phase-based compensation of amplitude and frequency of the notch filter designed to N = 16 as an example, assume that dual symmetric notch frequency vector H = [l 1 1 0 1 1 ll 1 1 l 1 0 1 1 1] T, the corresponding amplitude-frequency characteristics and its attenuation characteristics as shown in Figure 6: Amplitude-frequency curve as a result of the adoption of a strict (k +0.5) △ ω the frequency of sampling points, so the border frequency ω1 = 2.5 △ ω , ω3 = 3.5 △ ω, which points ω3 Department notch up to a 300dB attenuation of the following; 3 dB angular frequency ω2 = 2.830 △ ω, 3 dB bandwidth △ ω2 = 1.34l △ ω.

H can be derived for the two frequency vectors Hl, H2:

Based on amplitude-frequency characteristics of all phase FIR filter design compensation

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Based on amplitude-frequency characteristics of all phase FIR filter design compensation

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The frequency shift parameter λ = 0.25, the H1, H2, λ into equation (8) available after the corresponding filter pan h1 ', h2', according to their type (12) overlay, that is a composite of depression after filter coefficients:

Based on amplitude-frequency characteristics of all phase FIR filter design compensation

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Type (12) of g (0) to minus l, because the h1 ', h2' superposition process will be introduced to the size of the direct flow of 1, thus less need to direct traffic. DC Notch adjusted curve shown in figure 7, Figure 7 (a) shows that: the frequency of all the border points have been moved precisely 0.25 △ ω, the frequency of the border but the plan into 7 (b) shows that: depression filter attenuation performance variation due to Notch Point is, after moving, ω1 is still sub-filter 1, the amplitude of the frequency set to 0, but deviated from the sub-filter 2 amplitude for the frequency set point l. MATLAB can be measured by G '(ejω1) value, so that μ = | G' (ejω1) |, with-μ values to replace the H1, H2 in the amplitude of 0 the value of the frequency of sampling, re-generation-in (9) and type (11) of the attenuation curve has been shown in Figure 8.

Based on amplitude-frequency characteristics of all phase FIR filter design compensation

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Based on amplitude-frequency characteristics of all phase FIR filter design compensation

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Figure 8 shows that, after a revised value of μ the attenuation characteristics of the following is still up to-300dB. The corresponding notch filter coefficients as shown in Table l.

5 Conclusion

In this paper, all phase-based amplitude-frequency characteristic compensation FIR filter design method, the frequency of symmetry in the dual sample basis, through the introduction of dual-phase-shift filter composition and structure of the method of compensation, it can control the transition zone sampling in the frequency intervals . Increase the number of N-order filter will help control the transition zone within the linear amplitude-frequency curve and reduce the band width of the border. And in the actual applications of digital filtering can not be increased in the case of N, by setting the frequency shift of the parameter λ to solve the low-pass, high pass, band pass, notch filter of the frequency of the location of the border at any point mobile control problem.


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