DIRECT SYNTHESIS OF LADDER WAVE DIGITAL FILTERS WITH TUNABLE PARAMETERS

This paper proposes a method for the synthesis of ladder wave digital filters (WDFs) directly from the digital domain. This method avoids the need for the synthesis of analog reference filters conventionally required in WDF design. This direct method allows for the determination of the WDF coefficients from the digital domain transfer function. This is similar to conventional infinite impulse response (IIR) filter coefficient determination but the WDF will give a more efficient realization. Due to the WDFs power complementary properties, a first-order ladder WDF can simultaneously realize both lowpass and highpass responses using the same structure, while a second-order WDF can realize both the bandpass and bandstop responses simultaneously. By appropriately choosing the WDF adaptor configuration and structure, tunable parameters can be determined from the digital domain transfer function that controls the 3dB cut-off frequency of the lowpass and highpass filters, and the centre frequency and 3-dB bandwidth of the bandpass and bandstop filters. This results in the WDFs requiring a minimum number of multipliers for realization.


INTRODUCTION
Digital filters are important in many applications.Next to nonrecursive digital filters and filters composed of first and second-order sections in cascade, wave digital filters (WDFs) represent a viable type of digital filter suitable for implementation.This is due to their excellent sensitivity properties with respect to coefficient value variation and also due to the modular structure of WDFs. 1,2,3Fs represent a class of digital filters that are closely related to lossless resistively terminated classical filter networks.Every WDF has a reference analog filter from which it is derived.. 6 However, unlike conventional IIR filters, a WDF also derived its structure from the reference filter.There are many types of WDFs based on the reference filter types, the most popular being lattice and ladder WDFs. 7,8 lthough simpler to design, lattice WDFs exhibit low sensitivity only at passband frequencies while ladder WDFs have low sensitivity properties in both the passband and stopband frequencies.A ladder WDF can therefore be implemented using a smaller wordlength than a lattice WDF satisfying the same specifications. 9,10e conventional method of synthesizing a WDF involves designing an analog LC filter network and subsequently converting the design to the digital domain. 1 Methods to directly synthesis WDFs purely in the digital domain has been proposed to avoid the analog filter synthesis requirement. 11,12However, the complexity of these techniques is comparable to conventional WDF design as it is essentially the digital version of the reactance reduction technique used for analog network synthesis. 13,14,15.Direct relationship has not been shown between the digital domain transfer function and the coefficients of the WDFs.
In this paper, a method that allows for the synthesis of tunable first and second-order ladder wave digital filters (WDFs) directly from the digital domain transfer function is shown.The power complementary transfer function properties of the WDF result in an efficient realization of the lowpass and highpass responses with the same first-order structure, and the bandpass and bandstop responses with the same second-order transfer function.These simple frequency selective filters are adequate in many applications and a correspondence between the transfer function and the tunable parameters has been shown for all-pass filter realizations. 16This paper will show that by appropriately choosing the WDF adaptor configuration and structure, tunable parameters that controls the 3-dB cut-off frequency for the lowpass and highpass WDFs, and the center frequency and 3-dB bandwidth of the bandpass and bandstop WDFs, can be determined directly from the digital domain transfer function.

FIRST-ORDER IIR FILTERS
Designing IIR filters via the bilinear transformation requires the application of both the bilinear transformation and its inverse to arrive at the transfer function.The sampling period T therefore has no effect on the expression for the transfer function and can be omitted from the original definition of the bilinear transformation written as Hence, the bilinear transformation that maps one-to-one the points on the analog s-plane to the digital z-plane is The relationship between the digital transfer function G(z) and the parent analog transfer function HA(z) using ( 2) is The relationship between the analog frequency, , on the s-plane and the digital frequency, , on the unit circle on the z-plane is tan 2 A first-order lowpass transfer function with a 3-dB cutoff frequency at c is given by Applying the bilinear transformation, the expression for a first-order lowpass transfer function G(z) is given by Rearranging terms, (6) can be written as where or in terms of the digital cutoff frequency c, using (4) and half angle identities, A realization of G LP (z) in a Direct II IIR structure requires three non-unity multipliers. 4,5

SECOND-ORDER IIR FILTERS
Spectral transformation can be used to obtain other filter types from lowpass IIR filters.As such, the lowpass to bandpass transformation is used to obtain the bandpass filter transfer function.The spectral transformation that maps the zero frequency of the lowpass filter to the desired location of the centre frequency, , of the bandpass filter is In addition, the 3-dB bandwidth of the bandpass filter is equal to cut-off frequency of the lowpass filter.The bandwidth is defined as c2 c1 (13)   where c2 and c1 are the upper and lower edgeband frequencies, respectively.
Applying ( 12) to ( 7) the transfer function of a second-order bandpass filter is obtained as where the coefficients are A conventional Direct II realization of G BP (z) requires four non-unity non-zero multipliers.4,5

WAVE DIGITAL FILTER ELEMENTS AND ADAPTORS
A WDF is designed using the bilinear transformation (2) that is used for an IIR filter design.However, in addition to the transfer function, the structure of the WDF itself is derived from the reference analog filter.The recourse of a 2-port network is used to obtain the WDF equivalents of analog filter components.For a classical port designated by a port resistance, R, instantaneous voltage, v, and instantaneous current, i, the incident wave, a, and reflected wave, b, are defined by By representing each analog element as a 2-port network having the appropriate port resistance, and using the incident and reflected waves as the signal variables, a WDF element is constructed.The WDF equivalents are shown in Table 1 for the capacitor, inductor, resistor and resistive source.
In addition, the coefficients are related as For an n-port parallel adaptor, the equation that relates the input and output signals is As such, the adaptors required for the first-order and second WDFs are the 2-port and 3-port adaptors.Realizing an n-port adaptor with a minimum number of multipliers is desirable to reduce cost.For a 2-port adaptor, a parallel realization is essentially equivalent to that of a series.From ( 25) and (30), only one coefficient is required as the other can be derived from it.A symmetrical realization of a 2-port adaptor is obtained by the manipulation of the adaptor equations and expressing the single adaptor coefficient as where The equation that relates the input and output of the 2-port adaptor is For a 3-port adaptor, a single multiplier realization is obtained when two of the ports are equivalent in value hence having equal coefficient value.The other remaining coefficient is derived from this coefficient.Due to the two ports having an equal resistance value, the resulting single coefficient, m, for both the parallel and series adaptor has the range of 0 m 1 (36) The symbols and signal flow diagrams of single multiplier adaptors are shown in Table 2.

FIRST-ORDER LADDER WDF
The transfer function of a ladder wave digital filter is obtained from an analog filter by treating the analog filter as a 2-port network inserted between resistive terminations.Figure 1 shows the analog network and its WDF equivalent.
the WDF transfer function for the network as shown in Figure 1b is Figure 2 shows the components of a first-order ladder analog network.The transfer function of this circuit is The transmittance matrix giving the lowpass filter response is therefore where c is the filter cut-off frequency.
At a normalized frequency of 1 rad/sec the values of the resistors are R 1 =R 2 =1 and L=2 H.
To change the cut-off frequency of this normalized filter to a frequency c, requires the value of the inductor to be scaled according to The WDF digital transfer function obtained using (41) is equal to (6) as both filters have the same response.To obtain the WDF structure, each analog element is converted to its WDF equivalent.Consider the 3-port series adaptor shown in Figure 3 where ports 1 and 2 are the resistive ports and port 3 is the inductive port.The coefficient for the 3-port series adaptor according to ( 24) is where Therefore the coefficient value is with its range given by (36).
By rearranging terms, equation ( 6) can be written in terms of the adaptor coefficient as The WDF coefficient can therefore be extracted from the digital domain transfer function (7)  where In addition to requiring only one multiplier, by varying the WDF coefficient the cut-off frequency of the lowpass filter is varied hence making it a tunable parameter.In terms of the digital frequency c , (47) can be written as The WDF has a doubly complementary highpass response that is not available with a conventional IIR filter realization.Due to the WDF lossless property, the equation that relates the lowpass to highpass response is 1 The filter shown in Figure 2 has a dual realization in which case a capacitor is connected in parallel with the resistors.The WDF equivalent in this case is the dual of that shown in Figure 3 in which case a 3-port parallel adaptor is required with the third port representing the capacitor.However, the transfer function and the coefficient value are equivalent to those for the series adaptor realization.

SECOND-ORDER LADDER WDF
A second-order analog filter can be obtained from a first-order lowpass filter using frequency transformation.For the circuit shown in Figure 2 the resulting second-order analog filter is as shown in Figure 5.The transfer function for this circuit is The values for the reactive elements can be obtained by applying the low-pass to bandpass transformation for analog filter given by where B is the bandwidth while 0 is the centre frequency of the bandpass filter defined as with c1 and c2 the lower and upper passband edge frequencies, respectively.
Applying (53) to (42) the second-order analog filter transfer function can be written in terms of the bandwidth and centre frequency as  Applying the WDF principles to the analog components in Figure 5 will result in several possible configurations.Consider the realization of the ladder WDF consisting of a series adaptor and a 2-port adaptor as shown in Figure 6.
From (24) the coefficient of the first adaptor is with its range given by (36).The 2-port adaptor is used to connect the series resonant elements to the first adaptor.As such, the port resistance values are 1 The coefficient of the 2-port adaptor using ( 32) is with its range given by (33).
To obtain the transfer function of the WDF, (41) is applied to (56), to give By rearranging terms, H(z) can be written in terms of m 1 and m 2 as The WDF coefficients can be extracted from the digital domain transfer function (14) according to In terms of the digital domain parameters, the coefficient m 1 , using (4) and (60), can be simplified to where the digital filter bandwidth is as defined in (13).Using (62) and ( 4), and trigonometric identities, it can be shown that the centre frequency of the digital filter is related to m 2 according to m 2 cos( 0 ) cos (68) where 0 ( The two WDF coefficients control the bandwidth and the centre frequency of the bandpass filter.In addition, due to the inherent properties of the WDF, a stopband response that is power complementary to the bandpass filter is also realized simultaneous at output port b 1 as indicated in Figure 6. Figure 7a shows the magnitude responses of the bandpass/bandstop WDF for the case where the bandwidth is fixed but the centre frequency varied.Figure 7b shows the magnitude responses with a fixed centre frequency but with two different bandwidths.
The analog filter shown in Figure 5 has a dual realization consisting of a parallel connection of the reactive elements and resistive components.The WDF realization is as shown in Figure 6 but with a 3-port parallel adaptor connected to the 2-port adaptor.However, the resulting transfer function and coefficients are equivalent to those for the series adaptor.Hence, for a given transfer function, the coefficients for the ladder WDF can be determined and the WDF realized either as a series or parallel 3-port adaptor in combination with a 2-port adaptor.

CONCLUSION
This paper has shown a method for synthesizing first and second-order WDFs from the digital domain transfer function by selecting the appropriate adaptors that contain the coefficients.A 3-port parallel or series adaptor is required for a first-order WDF, while the second-order WDF requires a 2-port adaptor in conjunction with either a series or parallel 3-port adaptor.One of the advantages of WDFs is that the same structure is used to obtain both the lowpass and highpass response for the first-order filter, and for the case of the second-order filter, both bandpass and bandstop responses are obtained simultaneously.In addition, the WDFs are realized with a minimum number of multipliers.The multipliers control the filter parameters and are tunable to fit the required specifications.For the first-order WDF, the 3-dB cut-off frequency is controlled by a single coefficient while for the second-order WDF, two coefficients determine the 3-dB bandwidth and centre frequency.The coefficients can be extracted from the transfer function describing the IIR digital filter.This method of designing WDFs can be easily incorporated in a digital filter design software to extent its capability to include WDFs since this technique avoids the requirement of synthesizing the reference analog filter usually required in WDF design.Alternatively, the coefficients can be calculated directly from the equations shown in this paper and used to obtain the digital domain transfer function.

Table 1 :
WDF Equivalents of Analog ElementsAdaptors are required to connect the basic elements having different port resistance values in a manner satisfying Kirchhoff's laws.Adaptors connect the signals either in series or parallel.For an n-port adaptor, each port, k, has its incident and reflected wave, a k , and, b k , respectively.Using the column vectors A and B to represent these signals as10 the equation for the n-port series adaptor can be written asB (I M s )A (22)where I is a n by n unity matrix and the coefficient matrix is M s m s1 m s1 .. .. m s1 m s2 m s2 .. .. m s2 : : .. .. : : : .. .. : m sn m sn .. .. m sn (23) For each port k the coefficient is p2 .. .. m pn m p1 m p2 .. .. m pn Direct Synthesis of Ladder Wave Digital FiltersHere, G k is the port conductance, which is the reciprocal of the port resistance.The coefficients are related as a combination of different adaptors to emulate the analog filter structures.

Figure 3 : 1 tan
Figure 3 : First-order WDF with series adaptor is complementary to S 21 .Frequencies in the passband of S 21 fall into the stopband of S 11 , and vice versa.This high-pass response, S 11 , has the input signal a 1 , as for the low-pass filter, but with the output obtained at b 1 as shown in Figure3.Both the lowpass and highpass responses are obtained simultaneously.Figure4shows the composite magnitude response of both responses for two different values of c .

Figure 4 :
Figure 4 : Magnitude responses of lowpass/highpass WDF for different values of c

2 B
the values for the reactive elements are L (57)

14 S
.A. Samad Direct Synthesis of Ladder Wave Digital Filters

Figure 7a :Figure 7b :
Figure 7a : Magnitude responses of bandpass/bandstop WDF for different values of 0 with fixed bandwidth at = 0.04

Table 2 :
WDF Adaptors with Single Multiplier