1 Introduction
FIR filters and wideband beamformers have numerous applications ranging from Sonar, Radar, audio processing, ultrasound imaging, radio astronomy, earthquake prediction, medical diagnosis, to communications, etc Van Trees (2002); Liu Weiss (2010). Many optimization methods have been employed in the past to design FIR filters and wideband beamformers with required specifications. General convex optimization is one of the techniques that has been extensively explored from this perspective ElKeyi . (2005); Liao Raza (2011); Duan . (2008); Zhao . (20111) with the inherent drawback of long computation time required to reach a feasible solution.
Although it can be considered as a special case of the convex optimization approach, least squares based design has been adopted as a simple but effective solution to both design problems, which minimizes the mean squared error between the desired and designed responses Liu Weiss (2010); Zhao . (20112); Doclo Moonen (2003)
. The solution of the standard least squares cost function involves matrix inversion to obtain the required weight vector. Since matrix inversion poses numerical instability with long filters
Tkacenko . (2003), another method was proposed based on the least squares approach by performing eigenvector decomposition of a cost function to extract the required weight vector in the form of an eigenvector. This method is called eigenfilter design and has been explored for designing both filters and beamformers Vaidyanathan Nguyen (1987); Nguyen (1993); Pei Tseng (2001); Zhang Chen (2002); Doclo Moonen (2002); Zhao . (20113). Moreover, the design of linearphase FIR Hilbert transformers and arbitrary order digital differentiators were considered by Pei and Shyu Pei Shyu (1988, 1989), who also investigated the design of nonlinearphase filters with arbitrary complexvalued coefficients Pei Shyu (1992, 19931). Twodimensional (2D) extension to the eigenfilter method was proposed by Nashashibi and Charalambous Nashashibi Charalambous (1988), and later considered by Pei Pei Shyu (1990, 19932). Eigenfilters have also been used to design infinite impulse response (IIR) and allpass filters Laakso . (1993); Shyu Pei (1992).In this work, we revisit the eigenfilter method for designing FIR filters and wideband beamformers and reveal a serious performance issue in the passband of the designed FIR filters and the mainlobe of the designed wideband beamformers in the light of an inherent design formulation flaw. An overall critical analysis of the performance of this approach is presented with the suggested modification for tackling this issue. In particular, an additional constraint is imposed at the passband/mainlode of the system to control the resultant responses.
This paper is organized as follows. The eigenfilter based design formulation for FIR filters and wideband beamformers along with the critical analysis is presented in Section 2. The proposed solution to the highlighted problem is given in Section 3. Design examples for different types of FIR filters and wideband beamformers affected by the problem are provided in Section 4 followed by results using the proposed solution. Conclusions are drawn in Section 5.
2 Least squares based design and critical analysis
2a FIR filter design
Consider an tap FIR filter. Its frequency response is given by
(1) 
where is the th tap/coefficient of the filter. In vector form, it can be expressed as
(2) 
where w is the weight vector holding the coefficients , , and
(3) 
Now consider designing a lowpass filter as an example. The desired response is given by
(4) 
where represents the desired linear phase at the passband with a delay of samples along with the desired stopband response equal to zero.
The design process involves formulating the cost function in the standard eigenfilter form, based on the RayleighRitz principle which states that for any Hermitian matrix R, its RayleighRitz ratio is given by
(5) 
This ratio reaches its maximum/minimum when w
is the eigenvector corresponding to the maximum/minimum eigenvalue of
R. The maximum and minimum values of this ratio are respectively the maximum and minimum eigenvalues. For FIR filter design, a reference frequency point was introduced by Nguyen in the passband region of the cost function to help represent it into the quadratic form as desired by (5) Nguyen (1993). The cost function with the reference frequency point incorporated is given as(6) 
where is the weighting function and and represent the desired and designed responses at reference frequency, respectively. This expression can also be written as
(7)  
For stopband, the desired response . Substituting this value into the expression above, we have
(8) 
Substituting the expression in (2) into (8), the expression further simplifies to
(9) 
Then we can express (9) as
(10) 
where is a symmetric, positive definite matrix of order x given by
(11) 
The passband cost function is derived by incorporating the desired passband response into (7)
(12)  
After simplification, we have
(13)  
This expression can also be written as
(14) 
where is a symmetric, positive definite matrix of order x given by
(15)  
The total cost function is a combination of the passband and stopband cost functions with a tradeoff factor
(16) 
which can be transformed into
(17) 
where
(18) 
Combining (11) and (15) in (18) and taking the real part, we have
(19)  
The solution rests in finding the eigenvector w corresponding to the minimum eigenvalue of P which minimizes . The norm constraint is also incorporated to avoid trivial solution. The final expression of solution for the eigenfilter based FIR filter design problem is given by
(20) 
After investigating the designed filter’s performance, it is found that although the design performs well for most of the cases with varying specifications for short filters, it produces ever increasingly inconsistent results as the number of filter taps increases for the same set of specifications. With those longer filters, the passband performance starts varying and switches from one case with flatness around near unity gain to another case with flatness achieved at almost zero magnitude.
This unstable performance can be attributed to the formulation in (19) where the first part of the cost function measures the difference between the filter’s response at the reference frequency and those at the other frequencies in the passband. The term compensates for different phase shifts of the response at different frequencies. This expression minimizes the relative variation of the filter’s response at different passband frequencies and ensures a flat passband response. However, there is no control over the absolute value of the filter’s response in passband, allowing any type of flat passband response with arbitrary absolute magnitude leading to inconsistent design performance.
2B Wideband beamformer design
Consider a wideband beamformer with tapped delay lines (TDLs) or FIR filters shown in Figure 1, where is the number of delay elements associated with each of the sensors. The wideband beamformer samples the propagating wave field in both space and time. Its response as a function of signal angular frequency and direction of arrival is given by Liu Weiss (2010)
(21) 
where is the delay between adjacent taps of the TDL and is the spatial propagation delay between the sensor and the reference sensor. We can also express (21) as
(22) 
where w is the coefficient vector
(23) 
and is the x steering vector
(24) 
where denotes the Kronecker product. The terms and are defined as
(25) 
(26) 
For a uniform linear array (ULA) with an interelement spacing , and angle measured from the broadside, the spatial propagation delay is given by . With normalized angular frequency, , and , the steering vector is given by
(27) 
(28) 
(29) 
Now we have (22) as a function of and , given by
(30) 
The desired response for the wideband beamformer is represented by . Then, the eigenfilter based cost function can be expressed as
(31)  
where is the reference point. We can change this expression into
(32) 
where
(33)  
Consider a typical design case with desired sidelobe response equal to zero and response at look direction given by equal to a pure delay; and represent the reference frequency and passband frequency range, respectively, and is the weighting factor for the mainlobe. The expression in (33) is modified accordingly for realvalued beamformer coefficients and given by
(34)  
Then, the solution to the wideband beamformer design problem is given by
(35) 
Similar to the FIR filter design case, testing of the designed wideband beamformer through the eigenfilter method showed an inconsistent design performance. The design performed well for some look directions, while attained a very poor response for other look directions.
This variable nature of look direction response for the same set of specifications can again be traced back to the design formulation in (34), where the first part of the expression calculates the difference between the beamformer response at reference point and those at other frequencies in the look direction . The term compensates for the different phase shifts experienced by the wideband signal at different frequencies. The formulation ensures minimzation of the relative error at the look direction for different frequencies, thus providing flat response at . However, just like the FIR filter case, there is a lack of control for exact response in the look direction which can lead to design failure.
3 Proposed Solution with an Additional Constraint
As shown in our analysis of the eigenfilter design for both FIR filters and wideband beamformers in Section 2, the key issue is its lack of control of the achieved response at the passband/look direction compared to the desired one in the formulation. To solve this problem, we add an additional constraint to the formulation to specify the required response explicitly at the reference point. Since the original formulation will minimize the variation of the achieved response in the passband/look direction, the explicit control of the response of the designed filter/beamformer at one reference point of the passband/look direction will guarantee the design reaches the desired response for the whole considered passband/look direction region with a minimum overall error.
Now, constraining the reference frequency response to unity by adding a linear constraint to (20) gives us the following modified design formulation
(36) 
where the constraint matrix C and the response vector f provide the required constraint on the weight vector w so that the resultant design can have the required exact response at the reference frequency. The constraint matrix C in its most basic form corresponds to the real and imaginary parts of the reference frequency vector where we want to constrain the response for this reference frequency vector in the passband of a filter or the look direction of a wideband beamformer to a fixed desired response with its real and imaginary parts contained in the response vector f.
For example, consider the design of a lowpass filter. In order to provide correction for the original formulation flaw, we incorporate a constraint for the filter passband response at the reference frequency to be equal to the desired response with unity gain magnitude and linear phase. For a reference frequency , in (3) changes to
(37) 
Then, the constraint matrix C just becomes a constraint vector with with the response vector f containing the desired unity gain as the response of the filter at represented by
(38) 
which is simply
(39) 
This constraint will make sure that the designed response of the filter at the reference frequency in the passband is equal to the desired response. As the original formulation will minimize the variation in the response achieved at other frequencies in the passband with respect to the reference frequency, the overall designed response in the passband will be equal to the desired response, thus solving the original formulation problem.
Note that we can also add other constraints to the formulation of C and f so that more flexible constraints can be imposed on the design. For example, we can add a constraint to make sure the resultant design has an exact zero response at some stopband frequencies.
The solution to (36) can be obtained by the Lagrange multipliers method and it is given by
(40) 
For the wideband beamformer design, the modified problem is given by
(41) 
where C and f again correspond to the constraint matrix and response vector, respectively. For the wideband beamformer case, just like the filter design scenario, this constraint matrix will correspond to the reference frequency steering vector, where .
By constraining the response of the wideband beamformer at this reference frequency steering vector equal to the desired response as
(42) 
the overall response of the wideband beamformer at the look direction for different frequencies will be equal to the desired response, thus mitigating the initial formulation problem. The solution to (41) is then given by
(43) 
Note that there are matrix inversion operations in (40) and (43), which can be computationally intensive for larger filters and beamformers. However, there are other approaches available in literature e.g. null space based methods to solve (36) and (41) avoiding the need to compute matrix inversion Liu Weiss (2010).
4 Design Examples
In this section, design examples are provided to show the inconsistent performance produced by the original unconstrained eigenfilter design method. The examples are then redesigned through the proposed constrained eigenfilter method to show the improvement.
4a Unconstrained eigenfilter design
First, we consider the lowpass filter design scenario where the whole frequency range from was discretized into 400 points. The design specifications include the passband from [0, 0.5] and stopband from [0.8, ]. A 70tap filter with tradeoff parameter = 0.97 and reference frequency at 0.35 is then designed using the original formulation. The result is shown in Fig. 2 in blue colour (solid curve) with a clearly satisfactory design performance showing a passband to stopband ratio of 140 dB.
In the second case, we just change the number of taps to 76, while keeping all the other specifications the same as the first case. The result is shown in Fig. 2, highlighted in dashed curve with red colour. We can see that the passband response is out of control, with a flat response of around 118 dB, and the resulting ratio between passband and stopband is just around 19 dB (if ignoring the unacceptable response at the transition band), clearly highlighting the problem with the original formulation.
For highpass filters, again two cases are presented. For the first case, we consider an 81tap filter, where the design specifications include a stopband from [0, 0.4] and passband from [0.7, ]. The tradeoff factor = 0.71 and the reference frequency is set to 0.74. The result is depicted in Fig. 3 with solid curve and blue colour, where a very satisfactory design performance can be observed with a passband to stopband ratio of 150 dB.
For the second case, we just change the reference frequency to 0.94 and the result is shown in Fig. 3 with dashed red colour, which is without any doubt unacceptable, with a passband response at around 130 dB leaving a passband to stopbad ratio of only 15 dB. The results for lowpass and highpass filter design examples clearly demonstrate the magnitude of the problem at hand for different arbitrary design scenarios.
Now we extend this observation to the design of bandpass filters to see if the same problem can be observed in those filters as well.
For the bandpass filter design scenario, we again consider two cases for comparison. For the first case, we have 91 taps, where the design specifications include the 1 stopband from [0, 0.15], passband from [0.35, 0.65] and the 2 stopband from [0.85, ]. The tradeoff factor = 0.96 and the reference frequency is set to 0.55. The satisfactory design result is shown in Fig. 4 with solid curve and blue colour, where a suitable passband to stopband ratio of 145 dB can be observed.
For the second case, we change the reference frequency to 0.49, while keeping the remaining specifications similar to the first case and the result is shown with dashed red colour where it can be seen that the flat passband again has dropped to a very low unacceptable magnitude of 80 dB with a passband to stopband ratio of 36 dB, providing further evidence for the kind of inconsistent results caused by the flawed design formulation.
For the wideband beamformer design, we consider an array with 10 sensors and a TDL length of 10 taps. The look direction is chosen as an offbroadside direction of with the desired response equal to . The considered wideband signal has a frequency range of with the reference frequency and chosen as the reference point. The weighting function is set to at the look direction and 0.4 at the sidelobe region, which runs from from to and to . The frequency range is discretized into 20 points, while the angle range is divided into 360 points.
(a) Lowpass  (b) Highpass  (c) Bandpass 
The result is shown in Fig. 5(a), where a satisfactory design performance is achieved with the look direction to sidelobe ratio around 20 dB. The same scenario is again tested by changing the look direction to the broadside of with the sidelobe region ranging from to and to with the remaining specifications unchanged. The result is shown in Fig. 5(b), where it can be observed that the look direction response plunges to 40 dB with a flat response attained, which is even lower than the sidelobes.
We provide another example for a scenario where we consider an array with 11 sensors and a TDL structure of 10 taps. For the first case, the look direction is chosen as the broadside direction with and the desired response equal to . For the design specifications we consider a wideband signal having a frequency range of with the reference frequency and chosen as the reference point. The weighting function is the same as the previous example and the sidelobe region is from to and to . The result is shown in Fig. 6(a), where an excellent design response is achieved with a look direction to sidelobe response ratio of 40 dB. For the second case, we change the look direction to an offbroadside direction of with the sidelobe ranging from to and to with the remaining specifications unchanged. The result is shown in Fig. 6(b), where the look direction response again has no absolute control and achieves flatness around 30 dB with the resulting look direction response even lower than the sidelobes, again demonstrating the presence of this problem in a wide range of design scenarios.
4B Constrained eigenfilter design
We now apply the constrained eigenfilter formulation in (36) to design the lowpass, highpass and bandpass filters presented using unconstrained design formulation. The new results are presented in Fig. 7(a), (b) and (c). Although there is still a noticeable bump in the transition band for the design results in Fig. 7(a) and (b) for lowpass and highpass, respectively, the overall response has improved significantly compared to the results in Figs. 2 and 3. The bandpass filter designed with the new formulation in Fig. 7(c) achieves a very satisfactory response compared to the result in Fig. 4.
For the beamformer design presented in Figs. 5(b) and 6(b), we redesign them using the constrained formulation in (41) and the result is provided in Figs. 8 and 9, where the look direction response has improved significantly with a decent look direction to sidelobe ratio achieved as per the desired specifications.
We have tried various designs for different types of filters and wideband beamformers with varying design specifications and the proposed method has been found to perform consistently well in different scenarios.
5 Conclusion
The classic eigenfilter approach has been revisited and critically analyzed, where a formulation problem is highlighted in the passband/look direction part of the cost function which leads to an inconsistent design performance. A solution was then proposed by adding a linear constraint, explicitly setting the designed passband response at the reference frequency point to the desired one. Results have been provided for different design scenarios based on FIR filter and wideband beamformer design to demonstrate the crucial issue of the original formulation and the satisfactory performance by the proposed one.
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