TWI876412B - Compensated composite filter and signal processing method thereof - Google Patents

Abstract

A compensated composite filter includes a prototype filter and a compensation filter. The prototype filter is configured to multiply an input signal by a prototype transfer function to generate a prototype filtered output signal, and the prototype filter has a first group delay in a passband. The compensation filter is connected to the prototype filter in series, and configured to multiply the prototype filtered output signal by a compensation transfer function to generate an output signal. The compensation filter includes a plurality of shaping filters connected in cascade, and has a second group delay in the passband. The first group delay is added to the second group delay to generate a composite group delay, and the composite group delay is a constant. Therefore, the first group delay is compensated by the second group delay, so that the compensated composite filter can realize linear phase response in the passband and has a low-complexity circuit structure.

Description

Compensation type composite filter and signal processing method thereof

The present disclosure relates to a compensating composite filter and a signal processing method thereof, and in particular to a compensating composite filter with linear phase response and a signal processing method thereof.

Configuring a filter with perfect frequency response (i.e., flat passband, sharp transition band, highly suppressed stopband, and linear phase response) is always the ultimate goal of any digital signal processor (DSP) practitioner. In the prior art, high-performance filtering with linear phase response can be achieved through finite impulse response (FIR) filters.

However, the FIR filter contains many adders and multipliers, which increases the system complexity of the DSP. Compared with FIR filters, infinite impulse response (IIR) filters can achieve high-performance filtering based on low system complexity through their own recursive structure. However, due to the recursive structure, IIR filters cannot provide linear phase response and limit the applicability of IIR filters. It can be seen from this that there is currently a lack of a filter and signal processing method with low complexity and linear phase response on the market, so relevant industries are looking for solutions.

Therefore, the purpose of the present disclosure is to provide a compensating composite filter and a signal processing method thereof. The compensating composite filter not only has a low-complexity circuit structure, but also can achieve a linear phase response in the passband. The signal processing method of the compensating composite filter can completely retain the waveform of the input signal and filter the interference signal at the same time.

According to an implementation method of the structural aspect of the present disclosure, a compensated composite filter is provided, which includes a prototype filter and a compensation filter. The prototype filter is used to multiply an input signal by a prototype transfer function to generate a prototype filter output signal, and the prototype filter has a first group delay in a passband. The compensation filter is connected in series with the prototype filter, and is used to multiply the prototype filter output signal by a compensation transfer function to generate an output signal. The compensation filter includes cascade-connected complex shaping filters and has a second group delay in the passband. The first group delay is added to the second group delay to produce a composite group delay, which is a constant.

Thus, the compensated composite filter of the present disclosure achieves linear phase response and low complexity by connecting the prototype filter in series with the compensating filter including a plurality of cascade-connected shaping filters and using the second group delay of the compensating filter to compensate for the first group delay of the prototype filter.

Other embodiments of the above-mentioned implementation are as follows: The above-mentioned prototype transfer function and the compensation transfer function are multiplied to generate a composite transfer function. The prototype transfer function is represented as , the compensation transfer function is expressed as , the composite transfer function is expressed as , and the composite transfer function conforms to the following formula: ;in is a variable, is the number of these shaping filters, is an integer transfer function of one of these shaping filters, Is a positive integer between 1 and between.

Other embodiments of the above-mentioned implementation are as follows: The above-mentioned first group delay is expressed as , the second group delay is expressed as , the composite group delay is expressed as , and the complex group delay meets the following formula: ;in is the angular frequency, is the number of these shaping filters, is the group delay of one of these shaping filters, Is a positive integer between 1 and between.

Other embodiments of the aforementioned embodiment are as follows: The group delay of one of the aforementioned shaping filters conforms to the following formula: ;in, is a filter parameter for one of these shaping filters, is another filter parameter of one of these shaping filters; wherein a plurality of group delays of these shaping filters are different from each other.

Other embodiments of the aforementioned implementation method are as follows: the aforementioned prototype filter is a Butterworth filter, an elliptic filter, an infinite impulse response filter (IIR filter) or a Chebyshev filter, and each shaping filter is an all-pass filter.

According to an embodiment of the method aspect of the present disclosure, a signal processing method of a compensating composite filter is provided, which includes the following steps: a first filtering step and a second filtering step. The first filtering step is to drive a prototype filter to receive an input signal and multiply the input signal by a prototype transfer function to generate a prototype filter output signal, and the prototype filter has a first group delay in a passband. The second filtering step is to drive a compensation filter to receive the prototype filter output signal and multiply the prototype filter output signal by a compensation transfer function to generate an output signal, and the compensation filter includes cascade-connected complex shaping filters and has a second group delay in the passband. The first group delay is added to the second group delay to generate a complex group delay, and the complex group delay is a constant.

Thus, the signal processing method of the compensated composite filter disclosed in the present invention converts the input signal into the output signal according to the prototype transfer function and the compensated transfer function, thereby effectively suppressing the interference signal outside the passband and retaining the signal in the passband with high fidelity.

Other embodiments of the above-mentioned implementation are as follows: The above-mentioned prototype transfer function and the compensation transfer function are multiplied to generate a composite transfer function. The prototype transfer function is represented as , the compensation transfer function is expressed as , the composite transfer function is expressed as , and the composite transfer function conforms to the following formula: ;in is a variable, is the number of these shaping filters, is an integer transfer function of one of these shaping filters, Is a positive integer between 1 and between.

Other embodiments of the above-mentioned implementation are as follows: The above-mentioned first group delay is expressed as , the second group delay is expressed as , the composite group delay is expressed as , and the complex group delay meets the following formula: ;in is the angular frequency, is the number of these shaping filters, is the group delay of one of these shaping filters, Is a positive integer between 1 and between.

Other embodiments of the aforementioned embodiment are as follows: The group delay of one of the aforementioned shaping filters conforms to the following formula: ;in, is a filter parameter for one of these shaping filters, is another filter parameter of one of these shaping filters; wherein a plurality of group delays of these shaping filters are different from each other.

Other embodiments of the aforementioned implementation method are as follows: the aforementioned prototype filter is a Butterworth filter, an elliptic filter, an infinite impulse response filter (IIR filter) or a Chebyshev filter, and each shaping filter is an all-pass filter.

The following will describe multiple embodiments of the present disclosure with reference to the drawings. For the sake of clarity, many practical details will be described together in the following description. However, it should be understood that these practical details should not be used to limit the present disclosure. In other words, in some embodiments of the present disclosure, these practical details are not necessary. In addition, in order to simplify the drawings, some commonly known structures and components will be depicted in a simple schematic manner in the drawings; and repeated components may be represented by the same number.

In addition, in this article, when a certain component (or unit or module, etc.) is "connected/linked" to another component, it may refer to that the component is directly connected/linked to another component, or it may refer to that a certain component is indirectly connected/linked to another component, that is, there are other components between the component and the other component. When it is clearly stated that a certain component is "directly connected/linked" to another component, it means that there are no other components between the component and the other component. The terms first, second, third, etc. are only used to describe different components, and there is no restriction on the components themselves. Therefore, the first component can also be renamed as the second component. Moreover, the combination of components/units/circuits in this article is not a generally known, conventional or familiar combination in this field. Whether the components/units/circuits themselves are known cannot be used to determine whether their combination relationship is easy to be completed by ordinary knowledgeable people in the technical field.

Please refer to FIG. 1, which is a block diagram of a compensated composite filter 100 according to a first embodiment of the present disclosure. As shown in FIG. 1, the compensated composite filter 100 includes a prototype filter 200 and a compensation filter 300. The prototype filter 200 is used to multiply an input signal S _in by a prototype transfer function to generate a prototype filter output signal S _pro , and the prototype filter 200 has a first group delay in a passband. The compensation filter 300 is connected in series to the prototype filter 200, and is used to multiply the prototype filter output signal S _pro by a compensation transfer function to generate an output signal S _out . The compensating filter 300 includes cascade-connected complex shaping filters 310, 320, and 330, and has a second group delay in the passband. The first group delay of the prototype filter 200 is added to the second group delay of the compensating filter 300 to generate a complex group delay, and the complex group delay is a constant.

Specifically, when the input signal S _in is input to the compensated composite filter 100, the compensated composite filter 100 filters the input signal S _in according to the prototype transfer function of the prototype filter 200 and the compensated transfer function of the compensated filter 300, and generates an output signal S _out after a constant complex group delay, and the output signal S _out can completely retain the same waveform as the input signal S _in . Other bands outside the passband are suppressed by the prototype filter 200, and because the complex group delay is constant in the passband, the phase response of the compensated composite filter 100 can be linear in the passband.

The prototype transfer function is multiplied by the compensation transfer function to generate a composite transfer function. The prototype transfer function is expressed as , the compensation transfer function is expressed as , the composite transfer function is expressed as , and the composite transfer function can satisfy the following formula (1): (1).

In formula (1), is a variable (i.e. a complex number in the Z transform), is the number of shaping filters 310, 320, 330, is a shaping transfer function of one of the shaping filters 310, 320, 330, Is a positive integer between 1 and between.

The first group delay of the prototype filter 200 is expressed as , the second group delay of the compensation filter 300 is expressed as , the composite group delay is expressed as , and the complex group delay can meet the following formula (2): (2).

In formula (2), is the angular frequency, is the number of shaping filters 310, 320, 330, is a group delay of one of the shaping filters 310, 320, 330, Is a positive integer between 1 and From equation (1) and equation (2), it can be seen that the compensated composite filter 100 of the present disclosure achieves linear phase response and low complexity by connecting the prototype filter 200 in series with the compensating filter 300 including a plurality of cascade-connected shaping filters 310, 320, and 330, and using the second group delay of the compensating filter 300 to compensate the first group delay of the prototype filter 200. The following paragraphs will be used in conjunction with the drawings to explain in detail the operating mechanism and configuration specifications of the compensated composite filter 100 of the present disclosure.

Please refer to FIG. 1, FIG. 2 and FIG. 3 together, wherein FIG. 2 is a group delay comparison diagram of the shaping filters 310, 320, and 330 of the compensated composite filter 100 of FIG. 1; and FIG. 3 is a group delay comparison diagram of the compensated composite filter 100 of FIG. 1 and the prototype filter 200. As shown in FIG. 2, the plurality of group delays of the shaping filters 310, 320, and 330 are different from each other, that is, the phase responses of the shaping filters 310, 320, and 330 are different from each other.

The group delay of one of the shaping filters 310, 320, 330 may satisfy the following equation (3): (3).

In formula (3), is a filter parameter of one of the shaping filters 310, 320, 330, is another filter parameter of one of the shaping filters 310, 320, 330. Filter parameters of the shaping filters 310, 320, 330 The default values are , and , where the parameter corresponds to the peak of the group delay curve, parameter Controls the shape of the curve. , so that the shaping filters 310, 320, 330 tend to be stable, so the group delay of the shaping filters 310, 320, 330 in all frequency bands can be positive. Filter parameters The selection of can provide a degree of freedom for the group delay of the shaping compensating composite filter 100. In other embodiments, the compensating filter of the present disclosure can also use more shaping filters (i.e., the number of shaping filters is greater than 4), but it will inevitably increase the number of multipliers and adders inside the compensating composite filter.

Specifically, the prototype filter 200 can be a Butterworth filter, an elliptic filter, an infinite impulse response filter (IIR filter) or a Chebyshev filter, and each shaping filter 310, 320, 330 is an all-pass filter. The prototype filter 200 needs to meet the configuration specification of the amplitude response. The compensation filter 300 is mainly used to reconstruct the phase response of the compensation composite filter 100.

Although conventional FIR filters may have a linear phase response over the entire frequency band, this is not necessary for bandwidth-limited signals. Therefore, the phase response of the compensated composite filter 100 is linear only in the passband PB (shown in FIG. 3 ) because other bands have been suppressed by the prototype filter 200 .

As shown in Figure 3, the passband PB is between 0 and the passband edge frequency. A group delay margin _MGD for compensating the first group delay _D1 of the prototype filter 200 to the complex group delay _DC can be provided by the second group delay _D2 of the compensation filter 300, and the second group delay _D2 can be formed by the group delay _D21 of the shaping filter 310, the group delay _D22 of the shaping filter 320, and the group delay _D23 of the shaping filter 330 superimposed on each other. An average value of the complex group delay _DC of the compensated composite filter 100 in the passband PB is expressed as , and can meet the following formula (4): (4).

In addition, the flatness of the complex group delay _DC over the passband PB is defined as the complex transfer function The root mean square (RMS) group delay error can be expressed as And it meets the following formula (5): (5).

RMS Group Delay Error It can be used as the configuration specification of the compensated composite filter 100. The configuration specification of the compensated composite filter 100 may include the specification of (i) passband edge frequency (rad./sample); Specification (ii) Stopband edge frequency (rad./sample); Specification (iii) Passband ripple peak-to-peak less than 0.05 dB; Specification (iv) Stopband suppression not less than 73 dB; Specification (v) RMS group delay error on passband PB less than 0.2; specification (vi) the order of the compensating composite filter 100 is not greater than 20; and specification (vii) the number of shaping filters 310, 320, 330 is greater than or equal to 1 and less than or equal to 3, that is .

Please refer to Figures 1-3, 4 and 5 together, wherein Figure 4 is a graph showing a comparison of the amplitude response of the prototype filter 200 of the present disclosure and a known FIR filter; and Figure 5 is a graph showing a comparison of the group delay of the prototype filter 200 of the present disclosure and a known FIR filter. Specifications (i)-(iv) are mainly used as configuration specifications of the prototype filter 200. As shown in FIG. 4 , the known FIR filter (i.e., the least-squares finite impulse response filter Least-Squares FIR filter, hereinafter referred to as the least-squares FIR filter), the least p th-norm IIR filter (hereinafter referred to as the least polynomial IIR filter) and the Chebyshev II filter all meet the above specifications (i)-(iv). However, in order to reduce the complexity of the system, the present disclosure selects the least polynomial IIR filter (8th order) and the Chebyshev II filter (12th order) with fewer orders as the prototype filter 200. As shown in FIG. 5 , the minimum polynomial IIR filter has a larger group delay margin than the Chebyshev type II filter, but it can be filled by the shaping filters 310 , 320 , 330 .

Please refer to Figures 1-5, 6, 7 and 8 together, wherein Figure 6 is a curve diagram showing the root mean square group delay error of the prototype filter 200 of the present disclosure; Figure 7 is a phase response comparison diagram of the compensated composite filter 100 of the present disclosure and the known FIR filter; and Figure 8 is a curve diagram showing the group delay of the compensated composite filter 100, the prototype filter 200 and the shaping filters 310, 320, 330 of Figure 1. As shown in FIG. 6 , compared with the minimum polynomial IIR filter, the selection of the Chebyshev II filter as the prototype filter 200 can achieve the minimum root mean square group delay error. This is because the group delay margin required to be compensated for the Chebyshev II filter is smaller than the group delay margin of the minimum polynomial IIR filter. Therefore, the preferred configuration of the compensated composite filter 100 is as follows. The Chebyshev II filter is selected as the prototype filter 200, and the number of shaping filters 310, 320, and 330 is 3 ( ), and the filter parameters of the shaping filters 310, 320, 330 are Can be divided into , and . The root mean square group delay error of the prototype filter 200 is 0.1196. Based on the parameters of the above preferred configuration, the phase response of the compensated composite filter 100 in FIG. 7 can not only be linear in the passband PB like the phase response of the least squares FIR filter, but also have low complexity. The complex group delay _DC of the compensated composite filter 100 in FIG. 8 can be 58.

Please refer to Table 1, which lists the complexity (e.g., the number of multipliers and the number of adders) of filters (i)-(iv), wherein filters (i)-(iv) are respectively a compensated complex filter 100, a least squares FIR filter, an equivalent ripple FIR filter, and an FIR filter using a window design method in a filter design tool (e.g., MATLAB). Table 1 Filter(i) Filter (ii) Filter (iii) Filter (iv) Number of multipliers 36 101 68 97 Number of adders 36 200 134 182

As shown in Table 1, the compensated composite filter 100 of the present disclosure has fewer multipliers and adders than the other three FIR filters. Therefore, the compensated composite filter 100 can not only achieve linear phase response in the passband PB, but also has a low-complexity circuit structure.

Please refer to Figures 1, 9A, 9B, 10A, 10B and 10C together, wherein Figure 9A is a waveform diagram of the input signal S _in of the present disclosure; Figure 9B is a spectrum diagram of the input signal S _in of the present disclosure after discrete-time Fourier transform (DTFT); Figure 10A is a waveform diagram of the output signal S ₁ generated after the input signal S _in of Figure 9A is filtered by a Chebyshev filter; Figure 10B is a waveform diagram of the output signal S _out generated after the input signal S _in of Figure 9A is filtered by a compensated composite filter 100; and Figure 10C is a waveform diagram of the input signal S in of Figure 9A. Schematic diagram of the waveform of the output signal _S2 generated after _in is filtered by the least squares FIR filter.

As shown in FIG. 10A , although the noise has been filtered from the output signal S ₁ , the waveform of the output signal S ₁ obviously produces multiple jump points and is different from the input signal S _in . As shown in FIG. 10B and FIG. 10C , the compensated composite filter 100 with linear phase response and the least squares FIR filter both retain the waveform of the input signal S _in with high fidelity and filter the interference signal at the same time. Since the least squares FIR filter has an order of 200 and has a larger group delay, the compensated composite filter 100 of the present disclosure can still have a lower group delay in the passband under a low-complexity configuration compared to the least squares FIR filter.

Please refer to FIG. 1 and FIG. 11 together, wherein FIG. 11 is a diagram showing the zero-pole distribution of the compensated composite filter 100 of FIG. 1. Since the number of shaping filters 310, 320, and 330 is 3, the shaping filters 310, 320, and 330 have a total of six (2×3) zero-pole pairs _P1 , _P2 , _P3 , _P4 , _P5 , and _P6 . As shown in FIG. 11, the zero-pole pairs _P1 , _P2 , _P3 , _P4 , _P5 , and _P6 are distributed in the frequency band And the corresponding six poles are all located in the unit circle, so the compensated composite filter 100 has high stability.

Please refer to Figures 1-8 and 12 together, wherein Figure 12 is a flow chart showing a signal processing method 400 of a compensating composite filter according to a second embodiment of the present disclosure. As shown in Figure 12, the signal processing method 400 of a compensating composite filter can be applied to the compensating composite filter 100 of the first embodiment, and includes the following steps: a first filtering step S02 and a second filtering step S04.

The first filtering step S02 is to drive the prototype filter 200 to receive the input signal S _in and multiply the input signal S _in by the prototype transfer function to generate the prototype filter output signal S _pro , and the prototype filter 200 has a first group delay D ₁ in the passband PB. The second filtering step S04 is to drive the compensation filter 300 to receive the prototype filter output signal S _pro and multiply the prototype filter output signal S _pro by the compensation transfer function to generate the output signal S _out . The compensation filter 300 includes cascade-connected shaping filters 310, 320, and 330, and has a second group delay D ₂ in the passband PB. In the second filtering step S04, the second group delay _D2 is superimposed on the first group delay _D1 to form a complex group delay _DC , and the complex group delay _DC is a constant in the passband PB. Therefore, the signal processing method 400 of the compensated composite filter of the present disclosure converts the input signal Sin _into the output signal Sout according to the prototype transfer function and the _compensated transfer function, so as to effectively suppress the interference signal outside the passband PB and retain the signal in the passband PB _with high fidelity, so as to make the waveform of the _output signal Sout the same as the input signal Sin.

In summary, the compensated composite filter 100 and the signal processing method 400 of the compensated composite filter provided by the present disclosure have the following advantages: First, in a digital signal processor, the compensated composite filter can replace the conventional FIR filter, thereby reducing the system complexity. Second, under a low-complexity circuit structure, the compensated composite filter can still achieve a linear phase response. Third, the compensated composite filter has high stability.

Although the present invention has been disclosed in the above embodiments, it is not intended to limit the present invention. Anyone skilled in the art can make various changes and modifications without departing from the spirit and scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the scope defined in the attached patent application.

100: Compensated composite filter 200: Prototype filter 300: Compensated filter 310, 320, 330: Shaping filter 400: Signal processing method of compensated composite filter D ₁ : First group delay D ₂ : Second group delay D ₂₁ , D ₂₂ , D ₂₃ : Group delay _DC : Complex group delay _MGD : Group delay margin PB: Passband S02: First filtering step S04: Second filtering step S _in : Input signal S _pro : Prototype filter output signal S _out , S ₁ , S ₂ : Output signal :Passband edge frequency

FIG. 1 is a block diagram of a compensated composite filter according to the first embodiment of the present disclosure; FIG. 2 is a group delay comparison diagram of a shaping filter of the compensated composite filter of FIG. 1; FIG. 3 is a group delay comparison diagram of the compensated composite filter of FIG. 1 and a prototype filter; FIG. 4 is an amplitude response comparison diagram of the prototype filter of the present disclosure and a known finite pulse impulse response filter; FIG. 5 is a group delay comparison diagram of the prototype filter of the present disclosure and a known finite pulse impulse response filter; Figure 6 is a curve diagram showing the root mean square group delay error of the prototype filter of the present disclosure; Figure 7 is a phase response comparison diagram showing the compensated composite filter of the present disclosure and the known finite pulse impulse response filter; Figure 8 is a curve diagram showing the group delay of the compensated composite filter, the prototype filter and the shaping filter of Figure 1; Figure 9A is a waveform diagram showing the input signal of the present disclosure; Figure 9B is a spectrum diagram showing the input signal of the present disclosure after discrete time Fourier transformation; FIG. 10A is a waveform diagram of the output signal generated after the input signal of FIG. 9A is filtered by the Chebyshev filter; FIG. 10B is a waveform diagram of the output signal generated after the input signal of FIG. 9A is filtered by the compensated composite filter; FIG. 10C is a waveform diagram of the output signal generated after the input signal of FIG. 9A is filtered by the least squares finite pulse impulse response filter; FIG. 11 is a zero-pole distribution diagram of the compensated composite filter of FIG. 1; and FIG. 12 is a flow diagram of the signal processing method of the compensated composite filter according to the second embodiment of the present disclosure.

100: Compensatory composite filter

200: Prototype filter

300: Compensation filter

310,320,330: Shaping filter

S _in : input signal

S _pro : prototype filter output signal

S _out : output signal

Claims (8)

A compensated composite filter comprises: a prototype filter for multiplying an input signal by a prototype transfer function to generate a prototype filter output signal, wherein the prototype filter has a first group delay in a passband; and a compensation filter connected in series with the prototype filter and for multiplying the prototype filter output signal by a compensation transfer function to generate an output signal, wherein the compensation filter comprises a complex shaping filter connected in cascade and has a second group delay in the passband; wherein the first group delay is added to the second group delay to generate a complex group delay, and the complex group delay is a constant; wherein the prototype transfer function is multiplied by the compensation transfer function to generate a complex transfer function _, the prototype transfer function is represented by HP ( z ), and the compensation transfer function is represented by

, the composite transfer function is represented by H _C ( z ), and the composite transfer function conforms to the following formula:

, 1

M

3; wherein z is a variable, M is the number of the shaping filters, HS _,m ( z ) is an shaping transfer function of one of the shaping filters, and m is a positive integer between 1 and M. The compensated composite filter as claimed in claim 1, wherein the first group delay is represented _by grd [ HP ( ω )], and the second group delay is represented by

, the complex group delay is expressed as grd [ H _C ( ω )], and the complex group delay meets the following formula:

, 1

M

3; wherein ω is an angular frequency, and grd [ HS _,m ( ω )] is a group delay of one of the shaping filters. The compensated composite filter as described in claim 2, wherein the group delay of one of the shaping filters satisfies the following formula:

Wherein, r is a filter parameter of one of the shaping filters, and θ is another filter parameter of one of the shaping filters; wherein the complex group delays of the shaping filters are different from each other. A compensating composite filter as described in claim 1, wherein the prototype filter is a Butterworth filter, an elliptic filter, an infinite impulse response filter (IIR filter) or a Chebyshev filter, and each shaping filter is an all-pass filter. A signal processing method of a compensated composite filter comprises the following steps: a first filtering step, which is to drive a prototype filter to receive an input signal and multiply the input signal by a prototype transfer function to generate a prototype filter output signal, wherein the prototype filter has a first group delay in a passband; and a second filtering step, which is to drive a compensation filter to receive the prototype filter output signal and multiply the prototype filter output signal by a prototype transfer function. The output signal is multiplied by a compensation transfer function to generate an output signal, wherein the compensation filter includes cascade-connected complex shaping filters and has a second group delay in the passband; wherein the first group delay is added to the second group delay to generate a complex group delay, and the complex group delay is a constant; wherein the prototype transfer function is multiplied by the compensation transfer function _to generate a complex transfer function, the prototype transfer function is represented by HP ( z ), and the compensation transfer function is represented by

, the composite transfer function is represented by H _C ( z ), and the composite transfer function conforms to the following formula:

, 1

M

3; wherein z is a variable, M is the number of the shaping filters, HS _,m ( z ) is an shaping transfer function of one of the shaping filters, and m is a positive integer between 1 and M. The signal processing method of the compensated composite filter as described in claim 5, wherein the first group delay is represented by grd [ HP ₍ ω )], and the second group delay is represented by

, the complex group delay is expressed as grd [ H _C ( ω )], and the complex group delay meets the following formula:

, 1

M

3; wherein ω is an angular frequency, and grd [ HS _,m ( ω )] is a group delay of one of the shaping filters. The signal processing method of the compensated composite filter as described in claim 6, wherein the group delay of one of the shaping filters satisfies the following formula:

Wherein, r is a filter parameter of one of the shaping filters, and θ is another filter parameter of one of the shaping filters; wherein the complex group delays of the shaping filters are different from each other. A signal processing method for a compensating composite filter as described in claim 5, wherein the prototype filter is a Butterworth filter, an elliptic filter, an infinite impulse response filter (IIR filter) or a Chebyshev filter, and each of the shaping filters is an all-pass filter.

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