TWI550945B - Method of designing composite filters with sharp transition bands and cascaded composite filters - Google Patents

Description

Design of composite filter with rapid transition zone Method and its series composite filter

The present invention relates to a method of designing a composite filter, and more particularly to a method of designing a composite filter of a rapid transition zone using a series structure.

According to its impulse response characteristics, the filter can be divided into two types: Infinite Impulse Response (IIR) filter and Finite Impulse Response (Finite Impulse Response) filter.

In the general case, the infinite pulse filter of the same filter order has a more turbulent transition band than the finite pulse filter. However, the conventional problem is that most infinite pulse filters generate an overshoot at the edge of their passband, that is, the Gibbs's phenomenon, so that the waveform is not flat.

Therefore, in order to overcome the Gibbs phenomenon and satisfy the filter specifications to form an ideal filter, the higher the order of the filter, the better. But high order filter The waver inevitably leads to its feedback instability, and the increase of the order represents an increase of the multiplier, which not only leads to an increase in cost, but also increases the volume of the circuit.

The object of the present invention is to provide a method for designing a composite filter having a rapid transition band, which uses a pulse shaping filter or a complementary pulse shaping filter to connect an infinite pulse filter in series, so that the composite filter has an impetus Transition zone.

An embodiment of a method according to the present invention is to provide a method for designing a composite filter having an imminent transition band, the steps of which include:

a. one of the composite filters is filtered according to one of a transfer function of a pulse shaping filter and a complementary transfer shaping function of a complementary pulse shaping filter The filter order (K) and the three selected bands correspond to a selection table, and the filter parameter is a positive integer greater than zero, and the selected frequency bands are a Low Pass Subband (LPS) and a High Pass Subband (High Pass Subband, respectively). HPS) and Band Pass Subband (BPS).

b. Determining a selected frequency band and its corresponding at least one pulse shaping filter or at least one complementary pulse shaping filter from the selection table according to a frequency band required by the composite filter.

c. Connect the corresponding pulse shaping filter or complementary pulse shaping filter in step b with an Infinite Impulse Response (IIR) filter.

d. Calculate one of the pulse shaping filter or the complementary pulse shaping filter from one to zero to zero width according to a De Moivre's formula, a transfer function or a complementary transfer function.

e. Determining a range of filtering parameters based on a frequency response peak of one of the infinite pulse filters and a half-zero to zero width of the pulse shaping filter or the complementary pulse shaping filter. as well as

f. Select the filter parameters from the selection table according to the filter parameter range.

According to one embodiment of the foregoing method embodiment, wherein the transfer function of the pulse shaping filter is: Where z is the complex frequency domain and K is the filtering parameter. According to the selection table: if the selected frequency band is the high-pass band, then K 1, and K is an odd number; and if the selected band is a band pass band, then K 2.

According to another embodiment of the foregoing method embodiment of the present invention, wherein the complementary transfer function of the complementary pulse shaping filter is: Where z is the complex frequency domain and K is the filtering parameter. According to the selection table: if the selected frequency band is a low-pass band, then K 1; if the selected frequency band is the high-pass band, then K 2, and K is an even number; and if the selected band is a band pass band, then K 3.

According to still another embodiment of the method embodiment of the present invention, wherein the half-zero to zero width of the pulse shaping filter or the complementary pulse shaping filter is: ¹ / ₂ Δ ω ₁ = ¹ / ₂ Δ ω ₂ = ^π / _K Where Δ ω ₁ is the zero-to-zero width of the pulse shaping filter, and Δ ω ₂ is the zero-to-zero width of the complementary pulse shaping filter.

According to still another embodiment of the method embodiment of the present invention, the number of the pulse shaping filter or the complementary pulse shaping filter may be a complex number, and the pulse shaping filter or the complementary pulse shaping filter and the infinite pulse filter are The series positions can be interchanged freely.

According to an embodiment of the present invention, there is provided a series composite filter comprising a pulse shaping filter, a complementary pulse shaping filter and an infinite pulse filter, the pulse shaping filter having a first The passband chopping, complementary pulse shaping filter has a second passband chopping, an infinite pulse filter series pulse shaping filter and a complementary pulse shaping filter, and the infinite pulse filter has an overshoot passband edge (overshoot) Passband edge), wherein the first passband chopping or the second passband chopping cancels the edge of the overshoot passband, so that the composite filter has an imminent transition band.

According to an embodiment of the foregoing structural embodiment, the number of the pulse shaping filter and the complementary pulse shaping filter is a complex number, and the pulse shaping filter and the complementary pulse shaping filter and the infinite pulse filter are connected in series. Any interchange. The pulse shaping filter can be a comb filter (comb Filter), the infinite pulse filter can be a Chebyshev filter or an elliptic filter.

Therefore, the design method of the composite filter having the rapid transition band and the series composite filter all have the effects described later, and the selection function is first calculated by using the transfer function of the pulse shaping filter and the complementary transfer function of the complementary pulse shaping filter. The frequency band corresponds to the selection table of the filtering parameters, and the frequency band required by the composite filter determines that the pulse shaping filter or the complementary pulse shaping filter is to be connected in series with the infinite pulse filter, and then according to the Limefu formula and the transfer respectively. The function or complementary transfer function is used to calculate the half-zero to zero width of the pulse shaping filter or the complementary pulse shaping filter. Finally, the composite filter with the sharp transition band is determined according to the frequency response peak of the infinite pulse filter and the half-zero to zero width. The filter parameter range can be used, and the filter parameter is selected from the selection table according to the filter parameter range.

S01~S06‧‧‧Steps

100‧‧‧Series composite filter

200‧‧‧pulse shaping filter

300‧‧‧Complementary pulse shaping filter

400‧‧‧Infinite Impulse Filter

The above and other objects, features, advantages and embodiments of the present invention will become more <RTIgt; </ RTI> <RTIgt; </ RTI> <RTIgt; </ RTI> <RTIgt; Flowchart of the composite filter.

2A is a frequency response diagram in which the filter parameter of the pulse shaping filter and the complementary pulse shaping filter is equal to 5 according to an embodiment of the invention.

2B is a frequency response diagram of a filter parameter equal to 6 of a pulse shaping filter and a complementary pulse shaping filter according to an embodiment of the invention.

Figure 3 is a schematic diagram showing an embodiment of a series hybrid filter in accordance with the present invention.

4 is a schematic diagram showing a signal of a series hybrid filter versus an infinite pulse filter according to an embodiment of the invention.

Please refer to FIG. 1 , 2A and 2B , wherein FIG. 1 is a flow chart showing a design of a composite filter having a rapid transition band according to an embodiment of the invention. 2A and 2B are respectively frequency response diagrams in which the filter parameters of the pulse shaping filter and the complementary pulse shaping filter are equal to 5 and equal to 6 in accordance with an embodiment of the present invention. The flow step of designing the composite filter with the rapid transition band includes: step S01, one of the composite filters according to a transfer function of one pulse shaping filter and one complementary transfer function of a complementary pulse shaping filter The filtering parameter and the three selected frequency bands are corresponding to a selection table, and the filtering parameter is a positive integer greater than zero, and the selected frequency bands are respectively a low pass band, a high pass band and a band pass band; and step S02, according to one of the required bands of the composite filter Determining a selected frequency band and its corresponding at least one pulse shaping filter or at least one complementary pulse shaping filter in the selection table; in step S03, the corresponding pulse shaping filter or complementary pulse shaping filter in step S02 is An infinite pulse filter is connected in series; Step S04, calculating a half-zero to zero-point width of the pulse shaping filter or the complementary pulse shaping filter according to a Lime's formula, a transfer function or a complementary transfer function; and step S05, according to one of the infinite pulse filters, the frequency response peak value And a half-zero to zero width of the pulse shaping filter or the complementary pulse shaping filter determines a filtering parameter range; and in step S06, the filtering parameter is selected from the selection table according to the filtering parameter range.

First, for step S01, the transfer function H ₁ ( z ) of the pulse shaping filter and the complementary transfer function H ₂ ( z ) of the complementary pulse shaping filter are respectively Where z is the complex frequency domain, K is the filtering parameter, and K 1. The frequency response of the pulse shaping filter and the complementary pulse shaping filter can be obtained by substituting z = e ^jω into equations (1) and (2), respectively.

As shown in Fig. 2, taking K = 5 as an example, since the transfer function H ₁ ( ω ) has a null at ω = 0, the pulse shaping filter cannot be used as a low pass filter. The complementary transfer function H ₂ ( ω ) has a zero point when ω=π, and the complementary pulse shaping filter cannot be used as a high-pass filter.

As shown in Fig. 2B, taking K = 6 as an example, H ₁ ( ω ) has a zero point at both ω = 0 and ω = π, and the pulse shaping filter can be used as a band pass filter. While H ₂ ( ω ) has no zero point at ω=0 and ω=π, the complementary pulse shaping filter can be used as a high-pass filter, a low-pass filter and a band-pass filter. The K =1, 2, 3...8 corresponding transfer function H ₁ ( ω ), the complementary transfer function H ₂ ( ω ), and the selectable frequency bands are organized as follows (H represents a high-pass filter, B in the table) Represents a bandpass filter and L stands for a lowpass filter):

That is to say, when the selected frequency band is a low pass band, then the K of the complementary transfer function H ₂ ( ω ) 1; when the selected frequency band is a high-pass band, then the transfer function H ₁ ( ω ) K 1, and K is an odd number, the complementary transfer function H ₂ (ω) of K 2, and K is an even number; when the selected band is a band pass band, then the transfer function H ₁ ( ω ) K 2K of the complementary transfer function H ₂ ( ω ) 3.

Therefore, step S02 and step S03 are performed according to the above conditions, and at least one pulse shaping filter or at least one complementary pulse shaping filter can be selected from the frequency band required by the composite filter to connect the infinite pulse filter in series.

Next, in step S04, in order to find the zero-to-zero width of the pulse shaping filter, substituting the Limefu formula into z ^K =1 is as shown in formula (3): Comparing equation (3) with equation (4), the zero position ω _{1 k} of the transfer function H ₁ ( ω ) can be located and the half-zero to zero width of the pulse shaping filter can be calculated. Δω ₁ : z= e ^jω =cos( ω )+ j sin( ω ) (4); In the same way, the zero position ω _{2 k} and the half-zero to zero width of the complementary pulse shaping filter can be located. △ω ₂ is:

Finally, in step S05 and step S06, the frequency response peak of the infinite pulse filter is first calculated, and a filter parameter range is obtained according to the half-zero to zero width of the selected pulse shaping filter or the complementary pulse shaping filter, and The available filtering parameters can be selected from Table 1.

An example of a series-connected composite filter of a low pass band will be exemplified below in order to explain the above steps in detail. Please refer to FIG. 3 and FIG. 4 simultaneously. FIG. 3 is a schematic diagram of a series composite filter according to another embodiment of the present invention. 4 is a schematic diagram showing a signal of a series hybrid filter versus an infinite pulse filter according to an embodiment of the invention.

The series composite filter 100 of FIG. 3 includes at least one pulse shaping filter 200 or a complementary pulse shaping filter 300 connected in series with an infinite pulse filter 400. The infinite pulse filter 400 is illustrated as Direct form II is not limited to this. Those of ordinary skill in the art will appreciate that a (1), a (2) in Fig. 3 represent the coefficients of the denominator of the transfer function of the infinite pulse filter 400, and b (0), b (1), b ( 2) A coefficient representing the transfer function molecule. The position and number of the pulse shaping filter 200 or the complementary pulse shaping filter 300 are shown, and the user can change the position and number according to the requirements.

First, the present embodiment is a low pass band series hybrid filter 100, and according to the selection table of Table 1, only the complementary pulse shaping filter 300 can be selected in series with the infinite pulse filter 400, and the complementary conversion function H _{2 is selected.} The filter parameter of ( ω ) can be 1, 2, ... 8. In Fig. 4, the vertical axis is the intensity (in decibels, dB), and the horizontal axis is the frequency (unit is the diameter/sampling point, rad./sample). First, according to Fig. 4, the frequency response peak ω _max of the infinite pulse filter 400 can be obtained to be about 0.3 and according to the formula (10) as follows: Therefore, K ₁ may be equal to 1, 2, ... 10, and in this embodiment, K ₁ = 6 may be selected therefrom, as shown in the figure, after the infinite pulse filter 400 is connected in series with a complementary pulse shaping filter 300 The passband edge of the original infinite pulse filter 400 (approximately ω = 0.5 (rad./sample)) is offset by the zero of the complementary pulse shaping filter 300.

If the user wants to suppress the high frequency response at ω=1 (rad./sample), then Therefore, K ₂ can be equal to 3 (2 or 4), as shown in the figure, after the infinite pulse filter 400 is connected in series with the two complementary pulse shaping filter 300, the height of ω = 1 (rad. / sample) The frequency response has been cancelled by the zero of the complementary pulse shaping filter 300 of the second series.

Similarly, the composite filter of the high-pass band or bandpass band can be used to estimate the applicable pulse shaping filter, complementary pulse shaping filter and the available filtering parameters.

Therefore, the composite filter with the rapid transition band of the present invention has the design method of the composite filter with the rapid transition band and the series composite filter has the following advantages: the hardware is simple, and the cost can be effectively saved; there is a rapid transition zone, noise The inhibition effect is remarkable.

Although the present invention has been disclosed in the above embodiments, it is not intended to limit the present invention, and the present invention can be modified and modified without departing from the spirit and scope of the present invention. The scope is subject to the definition of the scope of the patent application attached.

S01~S06‧‧‧Steps

Claims (10)

A method for designing a composite filter with an imminent transition band, the steps comprising: a. according to a transfer function of a pulse shaping filter and a complementary transfer function of a complementary pulse shaping filter, the composite filter The filter parameter and the three selected frequency bands are corresponding to a selection table, wherein the filter parameter is a positive integer greater than zero, and the selected frequency bands are respectively a low pass band, a high pass band and a band pass band; b. according to the composite filter required Determining, from the selection list, a selected frequency band and at least one of the pulse shaping filters or at least one of the complementary pulse shaping filters; c. corresponding to the pulse shaping filter in step b Or the complementary pulse shaping filter is connected in series with an infinite pulse filter; d. calculating the pulse shaping filter or the complementary pulse shaping filter according to a Lime's formula, the transfer function or the complementary transfer function Half of the zero to zero width; e. according to one of the infinite pulse filter frequency response peaks and the half of the pulse shaping filter or the complementary pulse shaping filter to Dot width determining a filter parameter range; and selecting the filter parameter f from the selected table based on the filtering parameters. The method for designing a composite filter having an imminent transition band as described in claim 1, wherein the transfer function of the pulse shaping filter is: H ₁ ( z ) 1- z ^-K , where z represents the complex frequency domain and K is the filtering parameter. A method for designing a composite filter having an emergency transition band as described in claim 2, wherein according to the selection table: if the selected frequency band is the high-pass band, then K 1, and K is an odd number; and if the selected frequency band is the band pass band, then K 2. The method for designing a composite filter having an imminent transition band as described in claim 1, wherein the complementary transfer function of the complementary pulse shaping filter is: H ₂ ( z ) 1+ z ^-K , where z represents the complex frequency domain and K is the filtering parameter. A method for designing a composite filter having a rapid transition band as described in claim 4, wherein according to the selection table: if the selected frequency band is the low pass band, then K 1; if the selected frequency band is the high-pass band, then K 2, and K is an even number; and if the selected frequency band is the band pass band, then K 3. The method for designing a composite filter having a rapid transition zone as described in claim 1, wherein the half-zero to zero width of the pulse shaping filter or the complementary pulse shaping filter is: ¹ / ₂ △ ω ₁ = ¹ / ₂ △ ω ₂ = ^π / _K , where Δ ω _{1 is} the zero-to-zero width of the pulse shaping filter, and Δ ω _{2 is} one of the complementary pulse shaping filters from zero to zero width. The method for designing a composite filter having an imminent transition band as described in claim 1, wherein the number of the pulse shaping filter or the complementary pulse shaping filter is plural, and the pulse shaping filter or The position of the complementary pulse shaping filter and the infinite pulse filter can be interchanged arbitrarily. A series-type composite filter is applied to a design method of a composite filter having an emergency transition band as described in claim 1, wherein the series composite filter comprises: a pulse shaping filter having a first a passband chopping wave; a complementary pulse shaping filter having a second passband chopping; and an infinite pulse filter connected in series with the pulse shaping filter and the complementary pulse shaping filter, and The infinite pulse filter has an overshoot passband edge; wherein the overshoot passband edge is cancelled by the first passband chopping or the second passband chopping, so that the composite filter has an imminent transition band. The tandem composite filter according to claim 8, wherein the number of the pulse shaping filter and the complementary pulse shaping filter is a plurality, and the pulse shaping filter and the complementary pulse shaping filter The series position of the device and the infinite pulse filter can be interchanged arbitrarily. The tandem composite filter according to claim 8, wherein the pulse shaping filter is a comb filter, and the infinite pulse filter is a Chebyshev filter or an ellipse. Filter (elliptic filter).