本發明是有關於一種設計複合濾波器之方法，且特別是有關於一種利用串聯結構來設計急遽過渡帶的複合濾波器之方法。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.
濾波器依照其脈衝響應特性可分為無限脈衝(Infinite Impulse Response，IIR)濾波器及有限脈衝(Finite Impulse Response)濾波器兩種。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.
而在一般的情況下，同樣濾波器階數(filter order)的無限脈衝濾波器比起有限脈衝濾波器，具有更急遽的過渡帶。然而習知的問題在於，大多無限脈衝濾波器在其通帶的邊緣處會產生過衝(overshoot)，即吉布斯現象(Gibbs’s phenomenon)，使得波型並不平坦。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.
所以，為了克服吉布斯現象及滿足濾波器規格而形成理想的濾波器，濾波器的階數越高越好。但階數高的濾波器又必然導致其回授不穩定，且階數的提高代表乘法器的增加，不僅導致成本上升，電路的體積也會隨之擴大。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 filterThe 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.
本發明之目的是在於提供一種具有急遽過渡帶的複合濾波器之設計方法，其利用一脈衝塑形濾波器或一互補脈衝塑形濾波器來串聯一無限脈衝濾波器，使複合濾波器具有一急遽過渡帶。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.
根據本發明一方法態樣的實施方式是在提供一種具有急遽過渡帶的複合濾波器之設計方法，其步驟包含：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.根據一脈衝塑形濾波器(pulse shaping filter)之一轉移函數(transfer function)及一互補脈衝塑形濾波器(complementary pulse shaping filter)之一互補轉移函數，將複合濾波器之一濾波階數(filter order(K))及三選擇頻帶對應成一選擇表，濾波參數為大於零之正整數，選擇頻帶分別為一低通頻帶(Low Pass Subband，LPS)、一高通頻帶(High Pass Subband，HPS)及一帶通頻帶(Band Pass Subband，BPS)。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.根據複合濾波器所需之一頻帶從選擇表中決定一選擇頻帶及其對應之至少一脈衝塑形濾波器或至少一互補脈衝塑形濾波器。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.將步驟b中對應之脈衝塑形濾波器或互補脈衝塑形濾波器與一無限脈衝(Infinite Impulse Response，IIR)濾波器串聯。c. Connect the corresponding pulse shaping filter or complementary pulse shaping filter in step b with an Infinite Impulse Response (IIR) filter.
d.根據一隸美弗公式(De Moivre’s formula)、轉移函數或互補轉移函數分別計算脈衝塑形濾波器或互補脈衝塑形濾波器之一半零點到零點寬度(null to null width)。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.根據無限脈衝濾波器之一頻率響應峰值以及脈衝塑形濾波器或互補脈衝塑形濾波器之半零點到零點寬度決定一濾波參數範圍。以及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.根據濾波參數範圍從選擇表中選擇濾波參數。f. Select the filter parameters from the selection table according to the filter parameter range.
根據前述方法實施方式之一實施例，其中脈衝塑形濾波器之轉移函數為：，其中z表示複頻域，K為濾波參數。而根據選擇表：若選擇頻帶為高通頻帶時，則K1，且K為奇數；以及若選擇頻帶為帶通頻帶時，則K2。According to one embodiment of the foregoing method embodiment, wherein the transfer function of the pulse shaping filter is: Wherez is the complex frequency domain andK is the filtering parameter. According to the selection table: if the selected frequency band is the high-pass band, thenK 1, andK is an odd number; and if the selected band is a band pass band, thenK 2.
根據本發明前述方法實施方式之另一實施例，其中互補脈衝塑形濾波器之互補轉移函數為：，其中z表示複頻域，K為濾波參數。而根據選擇表：若選擇頻帶為低通頻帶時，則K1；若選擇頻帶為高通頻帶時，則K2，且K為偶數；以及若選擇頻帶為帶通頻帶時，則K3。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: Wherez is the complex frequency domain andK is the filtering parameter. According to the selection table: if the selected frequency band is a low-pass band, thenK 1; if the selected frequency band is the high-pass band, thenK 2, andK is an even number; and if the selected band is a band pass band, thenK 3.
根據本發明方法實施方式之又一實施例，其中脈衝塑形濾波器或互補脈衝塑形濾波器之半零點到零點寬度為：1/2△ω1=1/2△ω2=π/K，其中△ω1為脈衝塑形濾波器之零點到零點寬度，△ω2為互補脈衝塑形濾波器之零點到零點寬度。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 Δω1 =1 /2 Δω2 =π /K Where Δω1 is the zero-to-zero width of the pulse shaping filter, and Δω2 is the zero-to-zero width of the complementary pulse shaping filter.
根據本發明方法實施方式之再一實施例，其中脈衝塑形濾波器或互補脈衝塑形濾波器之數量可為複數，且脈衝塑形濾波器或互補脈衝塑形濾波器與無限脈衝濾波器之串聯位置可任意互換。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.
根據本發明一結構態樣的實施方式是在提供一種串聯式複合濾波器，包含一脈衝塑形濾波器、一互補脈衝塑形濾波器以及一無限脈衝濾波器，脈衝塑形濾波器具有一第一通帶漣波，互補脈衝塑形濾波器具有一第二通帶漣波，無限脈衝濾波器串聯脈衝塑形濾波器及互補脈衝塑形濾波器，且無限脈衝濾波器具有一過衝通帶邊緣(overshoot passband edge)，其中，藉由第一通帶漣波或第二通帶漣波抵消過衝通帶邊緣，使複合濾波器具有一急遽過渡帶。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.
根據前述結構實施方式之一實施例，其中脈衝塑形濾波器及互補脈衝塑形濾波器之數量為複數，且脈衝塑形濾波器及互補脈衝塑形濾波器與無限脈衝濾波器之串聯位置可任意互換。脈衝塑形濾波器可為梳狀濾波器(comb filter)，無限脈衝濾波器可為切比雪夫濾波器(Chebyshev filter))或橢圓濾波器(elliptic filter)。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 (combFilter), the infinite pulse filter can be a Chebyshev filter or an elliptic filter.
因此，前述具有急遽過渡帶的複合濾波器之設計方法及串聯式複合濾波器皆具有後述效果，其利用脈衝塑形濾波器之轉移函數及互補脈衝塑形濾波器之互補轉移函數先計算出選擇頻帶對應濾波參數的選擇表，並由複合濾波器所需的頻帶來決定要將脈衝塑形濾波器或互補脈衝塑形濾波器來跟無限脈衝濾波器串聯，接著分別根據隸美弗公式以及轉移函數或互補轉移函數來計算脈衝塑形濾波器或互補脈衝塑形濾波器之半零點到零點寬度，最後依據無限脈衝濾波器之頻率響應峰值以及半零點到零點寬度決定具有急遽過渡帶的複合濾波器可使用之濾波參數範圍，並根據濾波參數範圍從選擇表中選擇濾波參數。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‧‧‧步驟S01~S06‧‧‧Steps
100‧‧‧串聯式複合濾波器100‧‧‧Series composite filter
200‧‧‧脈衝塑形濾波器200‧‧‧pulse shaping filter
300‧‧‧互補脈衝塑形濾波器300‧‧‧Complementary pulse shaping filter
400‧‧‧無限脈衝濾波器400‧‧‧Infinite Impulse Filter
為讓本發明之上述和其他目的、特徵、優點與實施例能更明顯易懂，所附圖式之說明如下：第1圖係繪示依照本發明一方法實施例的一種設計具有急遽過渡帶的複合濾波器之流程圖。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.
第2A圖係繪示依照本發明一實施方式中脈衝塑形濾波器及互補脈衝塑形濾波器的濾波參數等於5之頻率響應圖。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.
第2B圖係繪示依照本發明一實施方式中脈衝塑形濾波器及互補脈衝塑形濾波器的濾波參數等於6之頻率響應圖。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.
第3圖係繪示依照本發明一種串聯式複合濾波器實施例之示意圖。Figure 3 is a schematic diagram showing an embodiment of a series hybrid filter in accordance with the present invention.
第4圖係繪示依照本發明一實施例之串聯式複合濾波器比對無限脈衝濾波器之訊號示意圖。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.
請參照第1、2A、2B圖，其中第1圖係繪示依照本發明一實施方式的一種設計具有急遽過渡帶的複合濾波器之流程圖。第2A圖及第2B圖分別繪示依照本發明一實施方式中脈衝塑形濾波器及互補脈衝塑形濾波器的濾波參數等於5及等於6之頻率響應圖。設計具有急遽過渡帶的複合濾波器之流程步驟依序包含：步驟S01，根據一脈衝塑形濾波器之一轉移函數及一互補脈衝塑形濾波器之一互補轉移函數，將複合濾波器之一濾波參數及三選擇頻帶對應成一選擇表，濾波參數為大於零之正整數，選擇頻帶分別為一低通頻帶、一高通頻帶及一帶通頻帶；步驟S02，根據複合濾波器所需之一頻帶從選擇表中決定一選擇頻帶及其對應之至少一脈衝塑形濾波器或至少一互補脈衝塑形濾波器；步驟S03，將步驟S02中對應之脈衝塑形濾波器或互補脈衝塑形濾波器與一無限脈衝濾波器串聯；步驟S04，根據一隸美弗公式、轉移函數或互補轉移函數分別計算脈衝塑形濾波器或互補脈衝塑形濾波器之一半零點到零點寬度；步驟S05，根據無限脈衝濾波器之一頻率響應峰值以及脈衝塑形濾波器或互補脈衝塑形濾波器之半零點到零點寬度決定一濾波參數範圍；以及步驟S06，根據濾波參數範圍從選擇表中選擇濾波參數。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.
首先針對步驟S01，脈衝塑形濾波器之轉移函數H1(z)及互補脈衝塑形濾波器之互補轉移函數H2(z)分別為
如第2圖所繪示，以K=5為例，由於轉移函數H1(ω)在ω=0時具有零點(null)，此時脈衝塑形濾波器無法作為低通濾波器。而互補轉移函數H2(ω)在ω=π時具有零點，此時互補脈衝塑形濾波器無法作為高通濾波器。As shown in Fig. 2, takingK = 5 as an example, since the transfer functionH1 (ω ) has a null at ω = 0, the pulse shaping filter cannot be used as a low pass filter. The complementary transfer functionH2 (ω ) has a zero point when ω=π, and the complementary pulse shaping filter cannot be used as a high-pass filter.
如第2B圖所繪示，以K=6為例，H1(ω)在ω=0及ω=π同時具有零點，此時脈衝塑形濾波器可以作為帶通濾波器。而H2(ω)在ω=0及ω=π同時不具有零點，此時互補脈衝塑形濾波器可以作為高通濾波器、低通濾波器及帶通濾波器。將K=1,2,3...8對應轉移函數H1(ω)、互補轉移函數H2(ω)以及可選擇的頻帶整理成表1如下(表中以H代表高通濾波器、B代表帶通濾波器、L代表低通濾波器)：
也就是說當選擇頻帶為低通頻帶時，則互補轉移函數H2(ω)之K1；當選擇頻帶為高通頻帶時，則轉移函數H1(ω)之K1，且K為奇數，互補轉移函數H2(ω)之K2，且K為偶數；當選擇頻帶為帶通頻帶時，則轉移函數H1(ω)之K2互補轉移函數H2(ω)之K3。That is to say, when the selected frequency band is a low pass band, then theK of the complementary transfer functionH2 (ω ) 1; when the selected frequency band is a high-pass band, then the transfer functionH1 (ω )K 1, andK is an odd number, the complementary transfer functionH2(ω) ofK 2, andK is an even number; when the selected band is a band pass band, then the transfer functionH1 (ω )K2K of the complementary transfer functionH2 (ω ) 3.
因此，根據上述條件執行步驟S02及步驟S03，可由複合濾波器所需之頻帶從表1選擇至少一脈衝塑形濾波器或至少一互補脈衝塑形濾波器來串聯無限脈衝濾波器。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.
接下來在步驟S04中，為了求得脈衝塑形濾波器之零點到零點寬度，將隸美弗公式代入zK=1如公式(3)：
最後在步驟S05及步驟S06中，先計算無限脈衝濾波器之頻率響應峰值，並根據所選擇脈衝塑形濾波器或互補脈衝塑形濾波器之半零點到零點寬度以獲得一濾波參數範圍，並可從表1中選擇可用之濾波參數。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.
以下將舉例應用於低通頻帶之一串聯式複合濾波器，以便於詳細說明上述步驟。請同時參照第3圖及第4圖，第3圖係本發明另一實施方式的一種串聯式複合濾波器之示意圖。第4圖係繪示依照本發明一實施例之串聯式複合濾波器比對無限脈衝濾波器之訊號示意圖。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.
第3圖中之串聯式複合濾波器100，其包含至少一脈衝塑形濾波器200或一互補脈衝塑形濾波器300串聯一無限脈衝濾波器400，圖中無限脈衝濾波器400雖繪示為直接形式(direct form II)但不限於此。本技術領域之通常知識者當可明白第3圖中的a(1)、a(2)代表無限脈衝濾波器400的轉移函數分母的係數，而b(0)、b(1)、b(2)代表其轉移函數分子的係數。脈衝塑形濾波器200或互補脈衝塑形濾波器300之位置以及數量皆為示意，使用者可根據需求變化其位置及數量。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 thata (1),a (2) in Fig. 3 represent the coefficients of the denominator of the transfer function of the infinite pulse filter 400, andb (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.
首先本實施例為一低通頻帶之串聯式複合濾波器100，而根據表1之選擇表，僅可選擇互補脈衝塑形濾波器300與無限脈衝濾波器400串聯，而選擇互補轉換函數H2(ω)之濾波參數可為1,2,...8。第4圖中縱軸為強度(單位為分貝，dB)，橫軸為頻率(單位為徑度/取樣點，rad./sample)。首先根據第4圖可得到無限脈衝濾波器400之頻率響應峰值ωmax約為0.3並根據公式(10)如下：
如使用者欲需再壓抑ω=1(rad./sample)處之高頻響應，則
同理，高通頻帶或帶通頻帶之複合濾波器亦由此可推算得知適用之脈衝塑形濾波器、互補脈衝塑形濾波器以及可用之濾波參數。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.
因此，本發明之具有急遽過渡帶的複合濾波器具有急遽過渡帶的複合濾波器之設計方法以及串聯式複合濾波器具有以下優點：硬體簡單，可有效節省成本；有急遽過渡帶，雜訊抑制效果顯著。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.
雖然本發明已以實施方式揭露如上，然其並非用以限定本發明，任何熟習此技藝者，在不脫離本發明之精神和範圍內，當可作各種之更動與潤飾，因此本發明之保護範圍當視後附之申請專利範圍所界定者為準。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‧‧‧步驟S01~S06‧‧‧Steps
| Application Number | Priority Date | Filing Date | Title |
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| TW103144832ATWI550945B (en) | 2014-12-22 | 2014-12-22 | Method of designing composite filters with sharp transition bands and cascaded composite filters |
| Application Number | Priority Date | Filing Date | Title |
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| TW103144832ATWI550945B (en) | 2014-12-22 | 2014-12-22 | Method of designing composite filters with sharp transition bands and cascaded composite filters |
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| TW201624821A TW201624821A (en) | 2016-07-01 |
| TWI550945Btrue TWI550945B (en) | 2016-09-21 |
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| TW103144832ATWI550945B (en) | 2014-12-22 | 2014-12-22 | Method of designing composite filters with sharp transition bands and cascaded composite filters |
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