Frequency Domain Limitations in the Design of ... - Semantic Scholar
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Frequency Domain Limitations in the Design of Nonnegative Impulse Response Filters Yuzhe Liu, Student Member, IEEE, and Peter H. Bauer, Fellow, IEEE
Abstract—This paper provides a fundamental characterization of nonnegative impulse response (NNIR) filters. The analysis is performed in terms of both geometrically and equidistantly spaced frequencies. Various inferences on the frequency domain limitations in NNIR filter design are analyzed. It is found that a performance boundary exists for NNIR lowpass filters independent of the system order. A specification test procedure is also provided for NNIR lowpass filters based on these analyses. The results are presented for systems in the discrete-time (D-T) domain, although equivalent results exist for systems in the continuous-time domain. Index Terms—Filter, frequency response, frequency-domain bounds, nonnegative impulse response (NNIR), performance boundary.
NOTATIONS AND CONDITIONS We list the notations that will be used throughout the remainder of this paper. Angular frequency in D-T domain. Frequency response of a D-T NNIR filter. : impulse response function of a real-valued D-T NNIR filter. Convolution operator. Set of natural numbers. Fourier-transform. -degree Chebyshev polynomial of the first kind in variable . -degree Chebyshev polynomial of the second kind in variable . Throughout this paper, we assume that Condition 1 holds. 1) Condition 1: The impulse response of a digital filter is nonnegative, i.e. (1) Manuscript received August 15, 2009; accepted April 24, 2010. Date of publication May 17, 2010; date of current version August 11, 2010. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ivan Selesnick. This work was supported by the Defense Threat Reduction Agency and by CRANE NAVAL by Grant N00164-07-C-8570. The authors are with the Mobile Sensor Systems (MOSES) Lab, Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2010.2050142
I. INTRODUCTION ONNEGATIVITY of the impulse response is a widely required feature in control systems, electronic amplifiers, and many other industrial applications [1]. For control systems such as machine-tool axis control and trajectory-following in robotics, local extrema of the step response are not acceptable [2]. In ferro-magnetic systems, when controlling the pulsed operations such as focusing and deflecting electromagnets for particle accelerators, overshoot of current in an electromagnet could result in ill-defined operating points, which is due to the hysteresis effect. In voltage control for automatic focusing of an electron beam of a cathode ray tube, overshoot of the controlling voltage could reduce the spot size and cause tube to burn [1]. Recently, one emerging class of applications comes from the design of so-called Evidence Filters [3], which are based on Dempster-Shafer Theory [4], [5]. An evidence filter has the capability of selectively fusing evidence (pieces of possibly imperfect information) from heterogeneous sources, while still weighing previously acquired information, and making inferences on various events of interest over time. The fusion process requires the NNIR feature in addition to other constraints. Unfortunately, the relationship between a nonnegative impulse response and its corresponding frequency response is still not clear. Due to this incomplete knowledge, optimization approaches usually deliver unsatisfactory designs, and significant approximation errors are incurred from theoretically unachievable specifications in the frequency domain. In order to guide the NNIR filter design, we explore some fundamental limitations in the frequency domain, when the impulse response is constrained to be nonnegative in the time domain. Such a system characterization is approached for both geometrically spaced and equidistantly spaced frequencies. The formulated frequency response properties are of particular importance in various types of NNIR filter designs, whose performance boundaries are currently not clear. Analysis of these properties explain the difficulties associated with NNIR filter design for systems other than lowpass filters, and show the limitations in the frequency selectivity of various types of NNIR filters, regardless of the filter order. Based on these properties, a specification test procedure (including guidance on tuning the specifications) and the derivation of the upper performance boundary are provided for NNIR lowpass filters. Therefore, this study sheds some light on the concrete design problem of NNIR filters. This paper is organized as follows: Section II states some of the related work reported in the literature over several decades.
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Section III explores the relationship of the upper-bounds on the increase and decrease of the squared magnitude function over geometrically spaced frequency regions. This analysis is used to determine the type of NNIR filters that are easiest to achieve in theory. Section IV analyzes the limitations on the decrease of the squared magnitude function over equidistantly spaced frequency regions, as well as their respective impact on the design of a variety of NNIR filter types. A specification test and tuning procedure and the derivation of the upper performance boundary are also provided for NNIR lowpass filters. Section V offers conclusions. II. RELATED WORK The work by Papoulis [6] shows that if the impulse response is nonnegative, the power attenuation over subsequent octaves on the frequency axis is severely limited when the attenuation over the first octave is given. The work in [7] shows that a high-pass filter with more than 6 dB/octave roll-off near the origin in the frequency response will have a step response that is nonnegative or nonpositive. Regarding fixed, lumped, linear, and stable networks, the work in [8] and [9] provides several bounds on the impulse/step response of various classes of transfer functions whose real/imaginary parts are restricted to be nonnegative/nonpositive or monotonically decreasing on the positive real-frequency axis. Bounds are also provided on the frequency response when the impulse response is restricted. The work in [10] gives the bounds on the attenuation of the real part of when the impulse rethe transfer or immittance function sponse is restricted to be nonnegative. Tighter bounds are prois further restricted to be nonnegative and vided when monotonically decreasing. Dual bounds on the impulse response is restricted to be nonnegative. The work are found when in [11] provides a set of limitations to the even-order transient response of a positive real network, when the real part of the transfer function is limited to be positive real. Our work in [12], [13] contains a preliminary analysis of the limitations on the frequency response of an NNIR system, which is performed in terms of geometrically and equidistantly spaced frequencies. Some of these results are extended or generalized in this paper and are employed in the derivation of a specification test procedure for NNIR lowpass filters. The work in [14]–[17] is motivated by the applications in the field of optics, where inherent requirements of nonnegativity and band-limitation of luminance point-spread functions exist. The work in [14] provides the best possible bound on the passband of a nonnegative integrable time-domain function whose frequency response vanishes for on the continuous frequency axis. The work in [15] provides point-wise and integral frequency-domain bounds for nonnegative, strictly band-limited 1-D, radially symmetric 2-D functions. The results in [16] and [17] show a technique for proving bounds of the Boas-Kac-Lukosz type [14] for smoothly restricted functions with nonnegative Fourier transforms. Unfortunately, these results only work for a passband analysis of lowpass filters, and are not applicable to other filter types or to frequency regions that are not in the passband, while the bounds given in this paper have no such limitations.
Another closely related system is called a positive linear system. A positive linear system has nonnegative states and output, if its initial state and input are nonnegative [18]. Typical applications include Charge Routing Networks [19] and industrial processes involving chemical reactors, heat exchangers and distillation columns. The work in [20] gives an essentially complete characterization of a nonnegatively realizable rational transfer function with a nonnegative impulse response sequence in terms of the location of poles. The work in [21] provides necessary and sufficient conditions for positive realizability by means of convex analysis using a unitary framework for continuous and discrete-time systems. Generally, a positive linear system implies an NNIR system while the opposite is not true, i.e., the internal states of an NNIR system cannot always be made positive through any change of the basis of the state space. III. LIMITATIONS OF THE FREQUENCY RESPONSE AT GEOMETRICALLY SPACED FREQUENCIES This section addresses frequency response limitations of NNIR filters at geometrically spaced frequencies. It shows that NNIR lowpass filters are inherently easier to realize than other NNIR filter types. A. Decrease of the Squared Magnitude Function In this section, an upper-bound on the maximum attenuation over a frequency region is derived under the assumption that the attenuation near frequency zero is given. (Note that any NNIR filter that is not of lowpass type will always generate a passband around frequency zero!) Theorem 1: If (1) is satisfied, and the squared magnitude function at frequency is bounded by (2) , where squared magnitude function from by
, then the decrease of the to is bounded
(3) regardless of the filter order. Proof: See Appendix I. Theorem 1 is demonstrated in Fig. 1. From the proof of Theorem 1, it is important to note that the is very upper-bound is more meaningful when the chosen . small since the bound is obtained when B. Increase of the Squared Magnitude Function Assuming the same conditions as in Section III-A, this section derives an upper-bound on the maximum gain increase of the squared magnitude function over a given frequency interval. Theorem 2: If (1) is satisfied, and the squared magnitude function at is bounded by
(4)
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Fig. 1. The relationship between the attenuation at ! and the decrease of the squared magnitude function from m ! to m !.
Fig. 3. Tightness of analytical bounds B
and B (m = 2;
k = 1).
C. The Relationship Between the Maximal Squared Magnitude Function Increase and Decrease Based on the observations in Theorem 1 and Theorem 2, this section explores the relationship between the maximal decrease and the maximal increase in the squared magnitude funcand , when the attenuation at is tion between bounded. To do so, it is helpful to have knowledge of the tightness of the bounds that were derived so far. Definition: When the attenuation at the frequency point is bounded by , i.e. (6) Fig. 2. The relationship between the attenuation at ! and the increase in the squared magnitude function from m ! to m !.
the tightness of a bound on the decrease or increase in the to , denoted squared magnitude function from by and , respectively, can be defined as (7)
where , squared magnitude function from by
, then the increase in the to , is bounded
(5) Proof: See Appendix II. Theorem 2 is demonstrated in Fig. 2. Remark: It should be noted that the bound derived in Theorem 2 represents a valid but only a conservative upper-bound. However, it is sufficiently tight (in the sense of norm of the bounding error) to discuss the effect of the the bounds on NNIR filter design, as will be shown in detail in Section III-C shortly. Obtaining a nonconservative upper-bound with a small bounding error by applying a similar derivation procedure to that of Theorem 1 can be done but it is unnecessary as well as nontrivial. In that case, it is equivalent to obtaining the global maximum of the function , . This is a nonconvex optimization problem, which can be solved by converting it into a convex optimization problem in each of the subfrequency-regions partitioned by ’s such that . The maximum in each of these regions can be found. The global numerical upper-bound can then be obtained as the maximum of all maxima.
where and denote the maximally possible decrease and increase in the squared magnitude function, respectively, and denote the bounds derived in Sections III-A and III-B, respectively, i.e. (8a) (8b) (8c) (8d) and and tively:
denote the bounding error of
and
respec(9)
Consider the case when is small (near the vicinity of frequency zero), we have the following expression from Appendix III: (10) where
is defined in (71).
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Fig. 4. Impossible designs of BP/LP/HP NNIR filters that have 0 dB gain at multiple frequencies within only the passband. (a) Lowpass. (b) Bandpass. (c) Highpass.
From Appendix IV, we have (11) is defined in (76). where and is shown in Fig. 3 for the case The tightness of and , by plotting and versus . Fig. 3 shows is tight, with the bounding error , while is that relatively loose, with the bounding error approximately 70% . of . Similar observation can be found for other Therefore, we have
response properties in terms of equidistantly spaced frequencies is of particular importance to the characterization and the design of NNIR filters, such as evidence filters[3], which often deal with frequency regions of interest that are not geometrically spaced. A. General Considerations of the Properties of NNIR Filters In our previous work[12], we have shown the following preliminary result, as restated in Lemma 4 for convenience. Lemma 4: If (1) is satisfied, and the squared magnitude funcis bounded by tion at (13)
(12) Based on (12), we have the relationship described in Theorem 3. Theorem 3: Let denote the amount of attenuation at , i.e., , . Let , be defined as in (8a) and (8b), respectively. For a fixed at the lower edge [as denoted in (8)], the following of the transition band relationship holds:
Proof: See Appendix V. 1) Comments: Theorem 3 demonstrates that the maximally achievable increase of the squared magnitude function is much less than the maximally achievable decrease of the squared magnitude function over the transition band of an NNIR filter when the attenuation near frequency zero does not exceed a prescribed value. This means that NNIR lowpass filters are theoretically easier to achieve than NNIR highand pass or bandpass filters. Since , the frequency response at satisfies . Hence the presented result is consistent with the presence of a passband around frequency zero for any NNIR filter. IV. LIMITATIONS OF THE FREQUENCY RESPONSE AT EQUIDISTANTLY SPACED FREQUENCIES In Section III, upper-bounds on the decrease/increase in the squared magnitude function over the interval are given to provide some general considerations on the feasibility of designing NNIR filters of different types. In this section, we approach the frequency domain limitations for equidistantly spaced frequencies. The formulation of frequency
where is bounded by
,
, then the attenuation at
(14) , then It was also shown that if . When , become equidistantly spaced frequencies. Then one of its immediate implications on NNIR filter design is that an NNIR filter of any type (except allpass) cannot have a gain of 0 dB over its finite passband only, as demonstrated in Fig. 4(a)–(c). For an digital NNIR filter of any type (except allpass), 0 dB gain at multiple frequencies within only the passband [as shown in Fig. 4(a)–(c)]) would imply 0 dB gain at multiple frequencies across the whole normalized frequency axis. This is due to the even symmetry of the magnitude response of an NNIR filter and the periodicity of the frequency distribution within (with the period being ). To further reveal the frequency response properties of NNIR filters, Theorem 5 analyzes the inherent interrelation between the decreases/increases in the squared magnitude function over two consecutive equidistantly spaced frequency regions. and denote the decrease Theorem 5: Let and in the squared magnitude function over respectively, where , ,
true for and magnitude function at
(15) . If (1) is satisfied, and the squared is bounded by (16)
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Fig. 6. Frequency domain specifications for a typical NNIR lowpass filter. Fig. 5. Difference between the decreases in the squared magnitude function 1) ] and [ ( + 1) ]. over [(
m 0 ! ;m!
m! ; m
!
then the difference between the decreases in the squared magnitude function over two consecutive equidistantly spaced frequency regions is always bounded by a constant, regardless of , i.e., (17) Similarly, let and denote the increase in the squared magnitude function over the regions and , respectively, where
(18) true for
and
, we have (19)
Proof: See Appendix VI. Equation (17) in Theorem 5 is illustrated in Fig. 5. 1) Comments: Theorem 5 describes the way in which the magnitude response is allowed to decrease or increase across the frequency axis, regardless of filter order: There is a linear increase in the achievable decrease/increase in the squared magnitude function in consecutive equi-length frequency regions along the frequency axis. It is consistent with the result in Theorem 1, i.e., a geometric increase in the achievable attenuation over an geometrically spaced frequency region. Further, Theorem 5 implies that the magnitude responses at different frequencies are inherently interrelated, and cannot be chosen arbitrarily. B. Upper Performance Boundary of NNIR Lowpass Filters Since the magnitude responses of an NNIR filter at different frequencies are inherently interrelated, as explicated in Theorem 5, successfully designing an NNIR filter that fulfills the given specifications depends on whether the most critical specification in the specification set is achievable or not. For an NNIR lowpass filter, the specification on the transition band usually represents the most critical requirement on the design, while the specification on other bands are relatively mild, due to smoother changes of the magnitude response. Based on such an understanding, a procedure is provided to obtain the upper performance boundary for NNIR lowpass fil-
ters. Before we show this procedure, we extend the result in [13, Theorem 2.2] and state it in Lemma 6 as follows, then illustrate the typical NNIR lowpass filter specifications in the frequency domain. Lemma 6: If (1) is satisfied, and the squared magnitude funcis bounded by tion at (20) where , , then the decrease in to is the squared magnitude function from bounded by (21) Comments: Lemma 6 is useful for studying low-pass NNIR filters. It implies that the maximum roll-off that can be achieved to is determined in the transition band from . by the magnitude response at the passband edge Slightly different from conventional lowpass filters, the set of the frequency domain specifications for NNIR lowpass filters is described as follows: (target specification) • passband ripple (target specification) • stopband ripple • minimal transition band attenuation
(22) (target specification) • passband edge frequency • stopband edge frequency (target specification). minNote we assume that the specification of a imal transition band attenuation must be maintained and cannot be negotiated. This is because the set of specifications for conventional lowpass filters may not be feasible when the impulse response is constrained to be nonnegative. It is possible that some tuning of the specifications is required. However, regardless of how the system is specified, it is always meaningful to ensure that a system satisfies some minimum signal-to-noise-ratio (SNR). The set of specifications for NNIR lowpass filters is illustrated in Fig. 6. It should also be noted that since it is impossible to achieve the flat nominal passband starting from frequency zero [as shown in Fig. 4(a)], and since an NNIR lowpass design could have a continuously decreasing passband gain from 0 dB, the conventional concept of “cutoff frequency” where the passband gain deviates
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by a nominal amount from 0 dB does not always apply in the realm of NNIR lowpass design. Given a set of frequency domain specifications for an NNIR lowpass filter, a test procedure (including guidance on tuning the specifications) and the procedure to obtain the upper performance boundary in the passband and the transition band are provided in the following five steps (without loss of generality, is assumed to be 1 here). 1) Testing if the Set of Specifications is Feasible: Let denote the set of design solutions such that the set of specifications is satisfied, i.e.
(23) Define the function (24) Let (25) where , and is chosen arbitrarily small by properly choosing the value of and . From the attenuation bound provided in (21) in Lemma 6, we also define the function (26) and represent the gain difference between Note that and in linear scale, while the hard requirement of minimal in represents the ratio of transition band attenuation the gain at to the gain at in logarithmic scale. describes the deviation Also note that the passband ripple from the ideal gain of 1, while the stopband ripple of describes the deviation of from the ideal gain of 0. and are kept at a Therefore, when dB ratio, i.e. fixed (27) is a monotonically increasing function of
, i.e. (28)
and i.e.
is a monotonically decreasing function of
, (29)
Therefore, given a set of specifications , we have the following results from (28) and (29):
Fig. 7. Specifications test and tuning for an NNIR lowpass filter.
2) Tuning the Specifications if Required (Optional): As shown in Fig. 7, if the given specifications are not feasible, AdjustSpec_Proc procedure should be called to produce a feasible set of specifications:
This procedure adjusts the pair so that
and/or the pair
(31) is satisfied. According to (28) and (29), when the hard requirement of a minimal transition band attenuation of is fixed, one can (and decrease correspondingly) to decrease only increase and to increase so that (31) can be satisfied. specFig. 8 illustrates the adjustment procedure of ification pairs, each of which conforms to the hard specification ratio). and (corresponding to the upper solid ( red curve) belong to the original unfeasible specifications. and (corresponding to the dotted red curves) are still not and decrease of feasible, due to the insufficient increase of . and (corresponding to the green curve) belong to the adjusted specifications where (32) The and on any curve that lies below the green curve and (corresponding to the are feasible. For example, bottom curve) consists of a valid set of specifications after the adjustment. in to It is also possible to only adjust the pair and do satisfy (31). From (24), we notice that adjusting . It only affects due to (25) and (33) not affect
(30) Based on (28), (29), and (30), we have the procedure described in Fig. 7 which determines whether the given specification set is feasible.
(33)
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5) Obtaining the Upper Performance Boundary in the Passdenote a curve with a band and the Transition Band: Let minimally required attenuation at each frequency within the passband and the transition band. According to the result in [12, Theorem 3.2], we have (39) From (38) and the definition of the slope, we have Fig. 8. The relationship between R and R that are K dB apart on a double linear gain-frequency plot.
When the pair is fixed, one needs to increase to satisfy (31). Therefore, one needs to increase and decrease , i.e., one should specify a smaller and a larger . and can be adjusted Of course, the pairs simultaneously to produce a feasible set of specifications. 3) Finding the Optimal Gain at the Passband Edge That Satisfies the (Tuned) Specifications: When (31) is satisfied, it only means that the (tuned) specifications are now feasible for a design. Procedure CompH_Proc is used to find the achievable with the highest possible passband and transition band that satisfies the specifications. At the gain passband edge, let (34)
(40) From (39) and (40), the upper performance boundary is approximated as (41) where . It should be noted that in the computation in step 3) should be small in order to have a better approximation. (This results in a large and .) Now observe the expression in (41): It is straightforward to is monosee that the upper performance boundary curve tonically decreasing. We have obtained the performance boundary in the passband and the transition band of an NNIR lowpass filter. Regarding the frequency response in the stopband, there are no severe constraints due to the following reasoning. Using the notations in (15) in Theorem 5, we have
CompH_Proc: According to (21) in Lemma 6, to avoid unnecessary attenuation at , it is desired that the maximum drop within the transhould be exploited. When this occurs, we sition band have (35)
(42) where
(35)
. Mathematically it is perfectly possible that . The same is true for . as the sampling points, it is possible Consider to keep the magnitude response in each of frequency regions flat. That is
(36)
(43)
From (35), we have
Obviously, of
is a monotonically increasing function . Therefore, we have
(37) 4) Finding the Minimally Required Attenuation at Frequency : Once is obtained, the minimally required attenuation at frequency , denoted by , can be obtained, based on Lemma 4 (let )
(38)
This reasoning also implies that there are less constraints on NNIR all-pass filters. High performance NNIR all-pass filter design is therefore possible. The above approach to the upper performance boundary can be readily employed to test the feasibility of the given specifications and evaluate the quality of a design. This is illustrated in the following example. for an Example: The frequency domain specifications NNIR lowpass filter is provided as follows: (target specification) • passband ripple (target specification) • stopband ripple • minimal transition band attenuation 33 dB (hard specification) (target specification) • passband edge frequency (target specification). • stopband edge frequency
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(48) Therefore, it is impossible to design an NNIR lowpass filter that satisfies . and and keeping After tuning the specifications on and unchanged in AdjustSpec_Proc, a new set of specificais formulated as follows: tions (target specification) • passband ripple (target specification) • stopband ripple • minimal transition band attenuation 33 dB (hard specification) (target specification) • passband edge frequency (target specification). • stopband edge frequency It should be noted that this adjustment is not unique, and that still keeps the specification of 33 dB minimal transition band attenuation. Now we have (49) Recomputing
and
, we have: (50)
infers that if a design attenuates to Note that at , then the attenuation within the transition band can be arbitrarily large. Since (51) Fig. 9. The magnitude response and impulse response of an NNIR lowpass filter and its upper performance boundary. (a) The magnitude response of a design and the performance boundary. (b) NNIR of the design.
is a feasible set of specifications. From (37) in step 3), we obtain (52)
Before starting the design of an NNIR lowpass filter, it is important to test the feasibility of the given specifications.
Then in step 4), we obtain
from (38) and (52) (53)
(44) In step 1), we choose
. Therefore, we have
is obtain from (54)
(45) that is (46) From (24), (26), and (30), we have
(47) In the procedure AdjustSpec_Proc in step 2), we find that
Finally, the upper performance boundary (41) in step 5)
Now the quality of a specific design that satisfies can be . evaluated by comparing it to -order NNIR FIR Using the method provided in [3], an filter design that satisfies is obtained. Its magnitude response is shown in Fig. 9(a), accompanied by the upper performance in the passband and transition band. Its nonboundary negative impulse response is shown in Fig. 9(b). For this specific design, we notice that the passband gain attenuates in a monotonically decreasing approach, like the performance boundary curve. However, compared to the performance boundary, it actually attenuates more than necessary. For example, the passband ripple of this specific design is , against the of the performance about
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boundary. Therefore, there is still room in improving the performance in the passband and the transition band. This can be achieved through improving the design procedure. It also shows the conventional concept of cutoff frequency where the passband gain deviates by a nominal amount from 0 dB does not make sense in the NNIR case. It should be noted that studying the performance boundary is important for the design of NNIR filters in order to achieve higher performance with lower order systems.
Since (59) and (60) we have (61)
V. CONCLUSION This paper investigates some fundamental limitations of nonnegative impulse response systems in the frequency domain. The results explain the difficulties associated with NNIR filter design for systems other than lowpass filters, and shows the limitations in the frequency selectivity of various types of NNIR filters, regardless of the filter order. These fundamental properties are found to be inherently interrelated, and serve as important tools for analysis and design of various types of NNIR filters. Based on these properties, a specification test procedure (including guidance on tuning the specifications) and the derivation of the upper performance boundary are provided for NNIR lowpass filters. The obtained performance boundary sheds some light on high-performance NNIR lowpass filter designs. All results are provided for systems in the discrete-time domain. However, equivalent results exist for systems in the continuous-time domain.
Therefore
(62)
APPENDIX II PROOF OF THEOREM 2 Proof: It can be proved via induction that
APPENDIX I PROOF OF THEOREM 1
(63)
Proof: First we define [The proof of (63) is extremely lengthy. Please refer to addendum [22].] Therefore, we have
(55) From the property of
, we have (56)
(64) Similarly, we have (65), shown at the bottom of the next page. Summing up the left- and right-hand side of (65), we have the upper-bound for the gain in the squared magnitude function to : from
Therefore, from (55) and (56), we have
(57) Hence, we obtain
(66) (58)
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APPENDIX III DERIVATION OF Proof: Now we consider the case of a decrease in the squared magnitude function. From Theorem 1, we have:
(73)
(67) Therefore, (68b) becomes
Therefore, we obtain
(74) (68a) APPENDIX IV DERIVATION OF
(68b)
Proof: Consider now the case of an increase in the squared magnitude function. Similarly, we consider the function
Now consider
(75) and define (76) (69)
Then we have
is chosen such that a sufficient amount of energy of where is contained. defined as follows: Now let’s consider the function
(70) Let (71) Therefore, we have (77)
(72) Therefore, we have Since
is small, we have (78)
.. . (65)
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(83)
APPENDIX V DERIVATION OF THEOREM 3
Therefore
at , there must exist a corProof: For a given at [12]. responding minimum attenuation such that . Therefore, there always exists some Then from Theorem 1, we immediately have the following result :
(82) , which simplifies (82) to the following form: Let [see (83) at the top of the page]. Therefore
From (9), we can rewrite
Similarly, from Theorem 2, we have
Rewrite
as (84) Similarly, it can be proved that (19) holds.
From the definitions in (7), we have
REFERENCES (79) Using (12) to simplify(79), we have
APPENDIX VI DERIVATION OF THEOREM 5 Proof: Based on the recurrence relation of the Chebyshev polynomial of the first kind
(80) we have
(81)
[1] N. G. Meadows, “In-line pole-zero conditions to ensure nonnegative impulse response for a class of filter systems,” Int. J. Contr., vol. 15, no. 6, pp. 1033–1039, 1972. [2] M. El-Khouq, O. D. Crisalle, and R. Longchamps, “Influence of zero locations on the number of step-response extrema,” Automatica, vol. 29, no. 6, pp. 1571–1574, Nov. 1993. [3] D. A. Dewasurendra, P. Bauer, and K. Premaratne, “Evidence filtering,” IEEE Trans. Signal Process., vol. 55, pp. 5796–5805, Dec. 2007. [4] G. Shafer, A Mathematical Theory of Evidence. Princeton, NJ: Princeton Univ. Press, 1976. [5] A. P. Dempster, “Upper and lower probabilities induced by a multivalued mapping,” Ann. Math. Stat., vol. 38, no. 1, pp. 325–339, 1976. [6] A. Papoulis, “Attenuation limits for filters with monotonic step response,” IRE Trans. Circuit Theory, vol. 9, no. 1, p. 86, 1962. [7] E. Meyer, “A note on the step response of high-pass filters,” IEEE Trans. Circuit Theory, vol. 15, no. 4, pp. 481–482, Dec. 1968. [8] A. Zemanian, “Bounds existing on the time and frequency responses of various types of networks,” Proc. IRE, vol. 42, no. 5, pp. 835–839, May 1954. [9] A. Zemanian, “Further bounds existing on the transient responses of various types of networks,” Proc. IRE, vol. 43, no. 3, pp. 322–326, Mar. 1955. [10] P. Chirlian, “Some necessary conditions for a nonnegative unit impulse response and for a positive real immittance function,” IRE Trans. Circuit Theory, vol. 8, no. 2, pp. 105–108, Jun. 1961. [11] R. Rohrer, “Limitations on the transient response of positive real function,” IEEE Trans. Circuit Theory, vol. 10, no. 1, pp. 110–111, Mar. 1963. [12] Y. Liu and P. Bauer, “Fundamental properties of nonnegative impulse response filters—Theoretical bounds I,” in Proc. 34th IEEE Int. Conf. Acoust., Speech Signal Process. (ICASSP), Apr. 19–24, 2009, pp. 3209–3212.
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[13] Y. Liu and P. Bauer, “Fundamental properties of nonnegative impulse response filters—Theoretical bounds II,” in Proc. 34th IEEE Int. Conf. Acoust., Speech Signal Process. (ICASSP), Apr. 19–24, 2009, pp. 3213–3216. [14] R. Boas and M. Kac, “Inequalities for Fourier transforms of positive functions,” Duke Math. J., vol. 12, pp. 189–206, 1945. [15] A. J. E. M. Janssen, “Frequency-domain bounds for nonnegative bandlimited functions,” Philips J. Res., vol. 45, pp. 325–366, 1990. [16] A. J. E. M. Janssen, “Frequency-domain bounds for nonnegative, unsharply band-limited functions,” J. Fourier Anal. Appl., vol. 1, no. 1, pp. 39–65, Feb. 1994. [17] A. J. E. M. Janssen, “More epsilonized bounds of the Boas-Kac-Lukosz type,” J. Fourier Anal. Appl., vol. 1, no. 2, pp. 171–191, 1994. [18] L. Farina and S. Rinaldi, Positive Linear Systems: Theory and Applications. New York: Wiley Intersci., 2000. [19] L. Benvenuti and L. Farina, “On the class of linear filters attainable with charge routing networks,” IEEE Trans. Circuits Systems-II: Analog Digit. Signal Process., vol. 43, no. 8, pp. 618–622, 1996. [20] B. Anderson, M. Deistler, L. Farina, and L. Benvenuti, “Nonnegative realization of a linear system with nonnegative impulse response,” IEEE Trans. Circuits Syst. I: Fund. Theory Appl., vol. 43, no. 2, pp. 134–142, Feb. 1996. [21] L. Farina and L. Benvenuti, “Positive realizations of linear systems,” Syst. Contr. Lett., vol. 26, pp. 1–9, 1995. [22] Y. Liu and P. H. Bauer, “Addendum to ‘Fundamental Properties of Nonnegative Impulse Response Filters—Theoretical Bounds I,’” Moses Group, Dept. EE, Univ. Notre Dame, Jul. 2008 [Online]. Available: http://www.nd.edu/~yliu5/Addendum1 Y. Liu (S’02) received the Bachelor of Engineering degree (Honors Degree) from the Northwestern Polytechnic University, Xi’An, China, in 2000, and the Master of Engineering degree in electrical and computer engineering from the National University of Singapore, Singapore, and the Institute of Infocomm Research (I R), A-STAR, Singapore, in 2004. He is currently pursuing the Ph.D. degree at the University of Notre Dame, Notre Dame, IN. He is now a Research Assistant with the University of
Notre Dame. His research interests include circuit and system theory, digital signal processing, and multidimensional systems.
Peter H. Bauer (F’05) received the Diplom degree from the Technical University of Munich, Munich, Germany, in 1984, and the Ph.D. degree from the University of Miami, Coral Gables, FL, in 1987, both in electrical engineering. In 1999, he served as the Director of the University of Notre Dame, Notre Dame, IN, EE/CSE London Program, London, U.K. He is currently a Professor with the Department of Electrical Engineering, University of Notre Dame, where he is the Head of the Mobile Sensor Systems (MOSES) Laboratory. He is the author or coauthor of more than 150 technical papers and six book chapters. His research interests include sensor and actuator networks, mobile wireless sensing, congestion control, evidential filtering and fusion, and stability theory of discrete-time systems. Dr. Bauer served as an Associate Editor for the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II (1989–1991, 2003–2005) and has served as an Associate Editor for the journal Multidimensional Systems and Signal Processing (2000–2006). He received the E. I. Jury Award from the University of Miami in 1987, the NASA Technical Innovator Award in 1992, the Alexander von Humboldt Fellowship in 1997, and the University of Notre Dame Kaneb Center Best Teacher Award in 2005.
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Frequency Domain Limitations in the Design of Nonnegative Impulse Response Filters Yuzhe Liu, Student Member, IEEE, and Peter H. Bauer, Fellow, IEEE
Abstract—This paper provides a fundamental characterization of nonnegative impulse response (NNIR) filters. The analysis is performed in terms of both geometrically and equidistantly spaced frequencies. Various inferences on the frequency domain limitations in NNIR filter design are analyzed. It is found that a performance boundary exists for NNIR lowpass filters independent of the system order. A specification test procedure is also provided for NNIR lowpass filters based on these analyses. The results are presented for systems in the discrete-time (D-T) domain, although equivalent results exist for systems in the continuous-time domain. Index Terms—Filter, frequency response, frequency-domain bounds, nonnegative impulse response (NNIR), performance boundary.
NOTATIONS AND CONDITIONS We list the notations that will be used throughout the remainder of this paper. Angular frequency in D-T domain. Frequency response of a D-T NNIR filter. : impulse response function of a real-valued D-T NNIR filter. Convolution operator. Set of natural numbers. Fourier-transform. -degree Chebyshev polynomial of the first kind in variable . -degree Chebyshev polynomial of the second kind in variable . Throughout this paper, we assume that Condition 1 holds. 1) Condition 1: The impulse response of a digital filter is nonnegative, i.e. (1) Manuscript received August 15, 2009; accepted April 24, 2010. Date of publication May 17, 2010; date of current version August 11, 2010. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ivan Selesnick. This work was supported by the Defense Threat Reduction Agency and by CRANE NAVAL by Grant N00164-07-C-8570. The authors are with the Mobile Sensor Systems (MOSES) Lab, Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2010.2050142
I. INTRODUCTION ONNEGATIVITY of the impulse response is a widely required feature in control systems, electronic amplifiers, and many other industrial applications [1]. For control systems such as machine-tool axis control and trajectory-following in robotics, local extrema of the step response are not acceptable [2]. In ferro-magnetic systems, when controlling the pulsed operations such as focusing and deflecting electromagnets for particle accelerators, overshoot of current in an electromagnet could result in ill-defined operating points, which is due to the hysteresis effect. In voltage control for automatic focusing of an electron beam of a cathode ray tube, overshoot of the controlling voltage could reduce the spot size and cause tube to burn [1]. Recently, one emerging class of applications comes from the design of so-called Evidence Filters [3], which are based on Dempster-Shafer Theory [4], [5]. An evidence filter has the capability of selectively fusing evidence (pieces of possibly imperfect information) from heterogeneous sources, while still weighing previously acquired information, and making inferences on various events of interest over time. The fusion process requires the NNIR feature in addition to other constraints. Unfortunately, the relationship between a nonnegative impulse response and its corresponding frequency response is still not clear. Due to this incomplete knowledge, optimization approaches usually deliver unsatisfactory designs, and significant approximation errors are incurred from theoretically unachievable specifications in the frequency domain. In order to guide the NNIR filter design, we explore some fundamental limitations in the frequency domain, when the impulse response is constrained to be nonnegative in the time domain. Such a system characterization is approached for both geometrically spaced and equidistantly spaced frequencies. The formulated frequency response properties are of particular importance in various types of NNIR filter designs, whose performance boundaries are currently not clear. Analysis of these properties explain the difficulties associated with NNIR filter design for systems other than lowpass filters, and show the limitations in the frequency selectivity of various types of NNIR filters, regardless of the filter order. Based on these properties, a specification test procedure (including guidance on tuning the specifications) and the derivation of the upper performance boundary are provided for NNIR lowpass filters. Therefore, this study sheds some light on the concrete design problem of NNIR filters. This paper is organized as follows: Section II states some of the related work reported in the literature over several decades.
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Section III explores the relationship of the upper-bounds on the increase and decrease of the squared magnitude function over geometrically spaced frequency regions. This analysis is used to determine the type of NNIR filters that are easiest to achieve in theory. Section IV analyzes the limitations on the decrease of the squared magnitude function over equidistantly spaced frequency regions, as well as their respective impact on the design of a variety of NNIR filter types. A specification test and tuning procedure and the derivation of the upper performance boundary are also provided for NNIR lowpass filters. Section V offers conclusions. II. RELATED WORK The work by Papoulis [6] shows that if the impulse response is nonnegative, the power attenuation over subsequent octaves on the frequency axis is severely limited when the attenuation over the first octave is given. The work in [7] shows that a high-pass filter with more than 6 dB/octave roll-off near the origin in the frequency response will have a step response that is nonnegative or nonpositive. Regarding fixed, lumped, linear, and stable networks, the work in [8] and [9] provides several bounds on the impulse/step response of various classes of transfer functions whose real/imaginary parts are restricted to be nonnegative/nonpositive or monotonically decreasing on the positive real-frequency axis. Bounds are also provided on the frequency response when the impulse response is restricted. The work in [10] gives the bounds on the attenuation of the real part of when the impulse rethe transfer or immittance function sponse is restricted to be nonnegative. Tighter bounds are prois further restricted to be nonnegative and vided when monotonically decreasing. Dual bounds on the impulse response is restricted to be nonnegative. The work are found when in [11] provides a set of limitations to the even-order transient response of a positive real network, when the real part of the transfer function is limited to be positive real. Our work in [12], [13] contains a preliminary analysis of the limitations on the frequency response of an NNIR system, which is performed in terms of geometrically and equidistantly spaced frequencies. Some of these results are extended or generalized in this paper and are employed in the derivation of a specification test procedure for NNIR lowpass filters. The work in [14]–[17] is motivated by the applications in the field of optics, where inherent requirements of nonnegativity and band-limitation of luminance point-spread functions exist. The work in [14] provides the best possible bound on the passband of a nonnegative integrable time-domain function whose frequency response vanishes for on the continuous frequency axis. The work in [15] provides point-wise and integral frequency-domain bounds for nonnegative, strictly band-limited 1-D, radially symmetric 2-D functions. The results in [16] and [17] show a technique for proving bounds of the Boas-Kac-Lukosz type [14] for smoothly restricted functions with nonnegative Fourier transforms. Unfortunately, these results only work for a passband analysis of lowpass filters, and are not applicable to other filter types or to frequency regions that are not in the passband, while the bounds given in this paper have no such limitations.
Another closely related system is called a positive linear system. A positive linear system has nonnegative states and output, if its initial state and input are nonnegative [18]. Typical applications include Charge Routing Networks [19] and industrial processes involving chemical reactors, heat exchangers and distillation columns. The work in [20] gives an essentially complete characterization of a nonnegatively realizable rational transfer function with a nonnegative impulse response sequence in terms of the location of poles. The work in [21] provides necessary and sufficient conditions for positive realizability by means of convex analysis using a unitary framework for continuous and discrete-time systems. Generally, a positive linear system implies an NNIR system while the opposite is not true, i.e., the internal states of an NNIR system cannot always be made positive through any change of the basis of the state space. III. LIMITATIONS OF THE FREQUENCY RESPONSE AT GEOMETRICALLY SPACED FREQUENCIES This section addresses frequency response limitations of NNIR filters at geometrically spaced frequencies. It shows that NNIR lowpass filters are inherently easier to realize than other NNIR filter types. A. Decrease of the Squared Magnitude Function In this section, an upper-bound on the maximum attenuation over a frequency region is derived under the assumption that the attenuation near frequency zero is given. (Note that any NNIR filter that is not of lowpass type will always generate a passband around frequency zero!) Theorem 1: If (1) is satisfied, and the squared magnitude function at frequency is bounded by (2) , where squared magnitude function from by
, then the decrease of the to is bounded
(3) regardless of the filter order. Proof: See Appendix I. Theorem 1 is demonstrated in Fig. 1. From the proof of Theorem 1, it is important to note that the is very upper-bound is more meaningful when the chosen . small since the bound is obtained when B. Increase of the Squared Magnitude Function Assuming the same conditions as in Section III-A, this section derives an upper-bound on the maximum gain increase of the squared magnitude function over a given frequency interval. Theorem 2: If (1) is satisfied, and the squared magnitude function at is bounded by
(4)
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Fig. 1. The relationship between the attenuation at ! and the decrease of the squared magnitude function from m ! to m !.
Fig. 3. Tightness of analytical bounds B
and B (m = 2;
k = 1).
C. The Relationship Between the Maximal Squared Magnitude Function Increase and Decrease Based on the observations in Theorem 1 and Theorem 2, this section explores the relationship between the maximal decrease and the maximal increase in the squared magnitude funcand , when the attenuation at is tion between bounded. To do so, it is helpful to have knowledge of the tightness of the bounds that were derived so far. Definition: When the attenuation at the frequency point is bounded by , i.e. (6) Fig. 2. The relationship between the attenuation at ! and the increase in the squared magnitude function from m ! to m !.
the tightness of a bound on the decrease or increase in the to , denoted squared magnitude function from by and , respectively, can be defined as (7)
where , squared magnitude function from by
, then the increase in the to , is bounded
(5) Proof: See Appendix II. Theorem 2 is demonstrated in Fig. 2. Remark: It should be noted that the bound derived in Theorem 2 represents a valid but only a conservative upper-bound. However, it is sufficiently tight (in the sense of norm of the bounding error) to discuss the effect of the the bounds on NNIR filter design, as will be shown in detail in Section III-C shortly. Obtaining a nonconservative upper-bound with a small bounding error by applying a similar derivation procedure to that of Theorem 1 can be done but it is unnecessary as well as nontrivial. In that case, it is equivalent to obtaining the global maximum of the function , . This is a nonconvex optimization problem, which can be solved by converting it into a convex optimization problem in each of the subfrequency-regions partitioned by ’s such that . The maximum in each of these regions can be found. The global numerical upper-bound can then be obtained as the maximum of all maxima.
where and denote the maximally possible decrease and increase in the squared magnitude function, respectively, and denote the bounds derived in Sections III-A and III-B, respectively, i.e. (8a) (8b) (8c) (8d) and and tively:
denote the bounding error of
and
respec(9)
Consider the case when is small (near the vicinity of frequency zero), we have the following expression from Appendix III: (10) where
is defined in (71).
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Fig. 4. Impossible designs of BP/LP/HP NNIR filters that have 0 dB gain at multiple frequencies within only the passband. (a) Lowpass. (b) Bandpass. (c) Highpass.
From Appendix IV, we have (11) is defined in (76). where and is shown in Fig. 3 for the case The tightness of and , by plotting and versus . Fig. 3 shows is tight, with the bounding error , while is that relatively loose, with the bounding error approximately 70% . of . Similar observation can be found for other Therefore, we have
response properties in terms of equidistantly spaced frequencies is of particular importance to the characterization and the design of NNIR filters, such as evidence filters[3], which often deal with frequency regions of interest that are not geometrically spaced. A. General Considerations of the Properties of NNIR Filters In our previous work[12], we have shown the following preliminary result, as restated in Lemma 4 for convenience. Lemma 4: If (1) is satisfied, and the squared magnitude funcis bounded by tion at (13)
(12) Based on (12), we have the relationship described in Theorem 3. Theorem 3: Let denote the amount of attenuation at , i.e., , . Let , be defined as in (8a) and (8b), respectively. For a fixed at the lower edge [as denoted in (8)], the following of the transition band relationship holds:
Proof: See Appendix V. 1) Comments: Theorem 3 demonstrates that the maximally achievable increase of the squared magnitude function is much less than the maximally achievable decrease of the squared magnitude function over the transition band of an NNIR filter when the attenuation near frequency zero does not exceed a prescribed value. This means that NNIR lowpass filters are theoretically easier to achieve than NNIR highand pass or bandpass filters. Since , the frequency response at satisfies . Hence the presented result is consistent with the presence of a passband around frequency zero for any NNIR filter. IV. LIMITATIONS OF THE FREQUENCY RESPONSE AT EQUIDISTANTLY SPACED FREQUENCIES In Section III, upper-bounds on the decrease/increase in the squared magnitude function over the interval are given to provide some general considerations on the feasibility of designing NNIR filters of different types. In this section, we approach the frequency domain limitations for equidistantly spaced frequencies. The formulation of frequency
where is bounded by
,
, then the attenuation at
(14) , then It was also shown that if . When , become equidistantly spaced frequencies. Then one of its immediate implications on NNIR filter design is that an NNIR filter of any type (except allpass) cannot have a gain of 0 dB over its finite passband only, as demonstrated in Fig. 4(a)–(c). For an digital NNIR filter of any type (except allpass), 0 dB gain at multiple frequencies within only the passband [as shown in Fig. 4(a)–(c)]) would imply 0 dB gain at multiple frequencies across the whole normalized frequency axis. This is due to the even symmetry of the magnitude response of an NNIR filter and the periodicity of the frequency distribution within (with the period being ). To further reveal the frequency response properties of NNIR filters, Theorem 5 analyzes the inherent interrelation between the decreases/increases in the squared magnitude function over two consecutive equidistantly spaced frequency regions. and denote the decrease Theorem 5: Let and in the squared magnitude function over respectively, where , ,
true for and magnitude function at
(15) . If (1) is satisfied, and the squared is bounded by (16)
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Fig. 6. Frequency domain specifications for a typical NNIR lowpass filter. Fig. 5. Difference between the decreases in the squared magnitude function 1) ] and [ ( + 1) ]. over [(
m 0 ! ;m!
m! ; m
!
then the difference between the decreases in the squared magnitude function over two consecutive equidistantly spaced frequency regions is always bounded by a constant, regardless of , i.e., (17) Similarly, let and denote the increase in the squared magnitude function over the regions and , respectively, where
(18) true for
and
, we have (19)
Proof: See Appendix VI. Equation (17) in Theorem 5 is illustrated in Fig. 5. 1) Comments: Theorem 5 describes the way in which the magnitude response is allowed to decrease or increase across the frequency axis, regardless of filter order: There is a linear increase in the achievable decrease/increase in the squared magnitude function in consecutive equi-length frequency regions along the frequency axis. It is consistent with the result in Theorem 1, i.e., a geometric increase in the achievable attenuation over an geometrically spaced frequency region. Further, Theorem 5 implies that the magnitude responses at different frequencies are inherently interrelated, and cannot be chosen arbitrarily. B. Upper Performance Boundary of NNIR Lowpass Filters Since the magnitude responses of an NNIR filter at different frequencies are inherently interrelated, as explicated in Theorem 5, successfully designing an NNIR filter that fulfills the given specifications depends on whether the most critical specification in the specification set is achievable or not. For an NNIR lowpass filter, the specification on the transition band usually represents the most critical requirement on the design, while the specification on other bands are relatively mild, due to smoother changes of the magnitude response. Based on such an understanding, a procedure is provided to obtain the upper performance boundary for NNIR lowpass fil-
ters. Before we show this procedure, we extend the result in [13, Theorem 2.2] and state it in Lemma 6 as follows, then illustrate the typical NNIR lowpass filter specifications in the frequency domain. Lemma 6: If (1) is satisfied, and the squared magnitude funcis bounded by tion at (20) where , , then the decrease in to is the squared magnitude function from bounded by (21) Comments: Lemma 6 is useful for studying low-pass NNIR filters. It implies that the maximum roll-off that can be achieved to is determined in the transition band from . by the magnitude response at the passband edge Slightly different from conventional lowpass filters, the set of the frequency domain specifications for NNIR lowpass filters is described as follows: (target specification) • passband ripple (target specification) • stopband ripple • minimal transition band attenuation
(22) (target specification) • passband edge frequency • stopband edge frequency (target specification). minNote we assume that the specification of a imal transition band attenuation must be maintained and cannot be negotiated. This is because the set of specifications for conventional lowpass filters may not be feasible when the impulse response is constrained to be nonnegative. It is possible that some tuning of the specifications is required. However, regardless of how the system is specified, it is always meaningful to ensure that a system satisfies some minimum signal-to-noise-ratio (SNR). The set of specifications for NNIR lowpass filters is illustrated in Fig. 6. It should also be noted that since it is impossible to achieve the flat nominal passband starting from frequency zero [as shown in Fig. 4(a)], and since an NNIR lowpass design could have a continuously decreasing passband gain from 0 dB, the conventional concept of “cutoff frequency” where the passband gain deviates
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by a nominal amount from 0 dB does not always apply in the realm of NNIR lowpass design. Given a set of frequency domain specifications for an NNIR lowpass filter, a test procedure (including guidance on tuning the specifications) and the procedure to obtain the upper performance boundary in the passband and the transition band are provided in the following five steps (without loss of generality, is assumed to be 1 here). 1) Testing if the Set of Specifications is Feasible: Let denote the set of design solutions such that the set of specifications is satisfied, i.e.
(23) Define the function (24) Let (25) where , and is chosen arbitrarily small by properly choosing the value of and . From the attenuation bound provided in (21) in Lemma 6, we also define the function (26) and represent the gain difference between Note that and in linear scale, while the hard requirement of minimal in represents the ratio of transition band attenuation the gain at to the gain at in logarithmic scale. describes the deviation Also note that the passband ripple from the ideal gain of 1, while the stopband ripple of describes the deviation of from the ideal gain of 0. and are kept at a Therefore, when dB ratio, i.e. fixed (27) is a monotonically increasing function of
, i.e. (28)
and i.e.
is a monotonically decreasing function of
, (29)
Therefore, given a set of specifications , we have the following results from (28) and (29):
Fig. 7. Specifications test and tuning for an NNIR lowpass filter.
2) Tuning the Specifications if Required (Optional): As shown in Fig. 7, if the given specifications are not feasible, AdjustSpec_Proc procedure should be called to produce a feasible set of specifications:
This procedure adjusts the pair so that
and/or the pair
(31) is satisfied. According to (28) and (29), when the hard requirement of a minimal transition band attenuation of is fixed, one can (and decrease correspondingly) to decrease only increase and to increase so that (31) can be satisfied. specFig. 8 illustrates the adjustment procedure of ification pairs, each of which conforms to the hard specification ratio). and (corresponding to the upper solid ( red curve) belong to the original unfeasible specifications. and (corresponding to the dotted red curves) are still not and decrease of feasible, due to the insufficient increase of . and (corresponding to the green curve) belong to the adjusted specifications where (32) The and on any curve that lies below the green curve and (corresponding to the are feasible. For example, bottom curve) consists of a valid set of specifications after the adjustment. in to It is also possible to only adjust the pair and do satisfy (31). From (24), we notice that adjusting . It only affects due to (25) and (33) not affect
(30) Based on (28), (29), and (30), we have the procedure described in Fig. 7 which determines whether the given specification set is feasible.
(33)
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5) Obtaining the Upper Performance Boundary in the Passdenote a curve with a band and the Transition Band: Let minimally required attenuation at each frequency within the passband and the transition band. According to the result in [12, Theorem 3.2], we have (39) From (38) and the definition of the slope, we have Fig. 8. The relationship between R and R that are K dB apart on a double linear gain-frequency plot.
When the pair is fixed, one needs to increase to satisfy (31). Therefore, one needs to increase and decrease , i.e., one should specify a smaller and a larger . and can be adjusted Of course, the pairs simultaneously to produce a feasible set of specifications. 3) Finding the Optimal Gain at the Passband Edge That Satisfies the (Tuned) Specifications: When (31) is satisfied, it only means that the (tuned) specifications are now feasible for a design. Procedure CompH_Proc is used to find the achievable with the highest possible passband and transition band that satisfies the specifications. At the gain passband edge, let (34)
(40) From (39) and (40), the upper performance boundary is approximated as (41) where . It should be noted that in the computation in step 3) should be small in order to have a better approximation. (This results in a large and .) Now observe the expression in (41): It is straightforward to is monosee that the upper performance boundary curve tonically decreasing. We have obtained the performance boundary in the passband and the transition band of an NNIR lowpass filter. Regarding the frequency response in the stopband, there are no severe constraints due to the following reasoning. Using the notations in (15) in Theorem 5, we have
CompH_Proc: According to (21) in Lemma 6, to avoid unnecessary attenuation at , it is desired that the maximum drop within the transhould be exploited. When this occurs, we sition band have (35)
(42) where
(35)
. Mathematically it is perfectly possible that . The same is true for . as the sampling points, it is possible Consider to keep the magnitude response in each of frequency regions flat. That is
(36)
(43)
From (35), we have
Obviously, of
is a monotonically increasing function . Therefore, we have
(37) 4) Finding the Minimally Required Attenuation at Frequency : Once is obtained, the minimally required attenuation at frequency , denoted by , can be obtained, based on Lemma 4 (let )
(38)
This reasoning also implies that there are less constraints on NNIR all-pass filters. High performance NNIR all-pass filter design is therefore possible. The above approach to the upper performance boundary can be readily employed to test the feasibility of the given specifications and evaluate the quality of a design. This is illustrated in the following example. for an Example: The frequency domain specifications NNIR lowpass filter is provided as follows: (target specification) • passband ripple (target specification) • stopband ripple • minimal transition band attenuation 33 dB (hard specification) (target specification) • passband edge frequency (target specification). • stopband edge frequency
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(48) Therefore, it is impossible to design an NNIR lowpass filter that satisfies . and and keeping After tuning the specifications on and unchanged in AdjustSpec_Proc, a new set of specificais formulated as follows: tions (target specification) • passband ripple (target specification) • stopband ripple • minimal transition band attenuation 33 dB (hard specification) (target specification) • passband edge frequency (target specification). • stopband edge frequency It should be noted that this adjustment is not unique, and that still keeps the specification of 33 dB minimal transition band attenuation. Now we have (49) Recomputing
and
, we have: (50)
infers that if a design attenuates to Note that at , then the attenuation within the transition band can be arbitrarily large. Since (51) Fig. 9. The magnitude response and impulse response of an NNIR lowpass filter and its upper performance boundary. (a) The magnitude response of a design and the performance boundary. (b) NNIR of the design.
is a feasible set of specifications. From (37) in step 3), we obtain (52)
Before starting the design of an NNIR lowpass filter, it is important to test the feasibility of the given specifications.
Then in step 4), we obtain
from (38) and (52) (53)
(44) In step 1), we choose
. Therefore, we have
is obtain from (54)
(45) that is (46) From (24), (26), and (30), we have
(47) In the procedure AdjustSpec_Proc in step 2), we find that
Finally, the upper performance boundary (41) in step 5)
Now the quality of a specific design that satisfies can be . evaluated by comparing it to -order NNIR FIR Using the method provided in [3], an filter design that satisfies is obtained. Its magnitude response is shown in Fig. 9(a), accompanied by the upper performance in the passband and transition band. Its nonboundary negative impulse response is shown in Fig. 9(b). For this specific design, we notice that the passband gain attenuates in a monotonically decreasing approach, like the performance boundary curve. However, compared to the performance boundary, it actually attenuates more than necessary. For example, the passband ripple of this specific design is , against the of the performance about
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boundary. Therefore, there is still room in improving the performance in the passband and the transition band. This can be achieved through improving the design procedure. It also shows the conventional concept of cutoff frequency where the passband gain deviates by a nominal amount from 0 dB does not make sense in the NNIR case. It should be noted that studying the performance boundary is important for the design of NNIR filters in order to achieve higher performance with lower order systems.
Since (59) and (60) we have (61)
V. CONCLUSION This paper investigates some fundamental limitations of nonnegative impulse response systems in the frequency domain. The results explain the difficulties associated with NNIR filter design for systems other than lowpass filters, and shows the limitations in the frequency selectivity of various types of NNIR filters, regardless of the filter order. These fundamental properties are found to be inherently interrelated, and serve as important tools for analysis and design of various types of NNIR filters. Based on these properties, a specification test procedure (including guidance on tuning the specifications) and the derivation of the upper performance boundary are provided for NNIR lowpass filters. The obtained performance boundary sheds some light on high-performance NNIR lowpass filter designs. All results are provided for systems in the discrete-time domain. However, equivalent results exist for systems in the continuous-time domain.
Therefore
(62)
APPENDIX II PROOF OF THEOREM 2 Proof: It can be proved via induction that
APPENDIX I PROOF OF THEOREM 1
(63)
Proof: First we define [The proof of (63) is extremely lengthy. Please refer to addendum [22].] Therefore, we have
(55) From the property of
, we have (56)
(64) Similarly, we have (65), shown at the bottom of the next page. Summing up the left- and right-hand side of (65), we have the upper-bound for the gain in the squared magnitude function to : from
Therefore, from (55) and (56), we have
(57) Hence, we obtain
(66) (58)
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APPENDIX III DERIVATION OF Proof: Now we consider the case of a decrease in the squared magnitude function. From Theorem 1, we have:
(73)
(67) Therefore, (68b) becomes
Therefore, we obtain
(74) (68a) APPENDIX IV DERIVATION OF
(68b)
Proof: Consider now the case of an increase in the squared magnitude function. Similarly, we consider the function
Now consider
(75) and define (76) (69)
Then we have
is chosen such that a sufficient amount of energy of where is contained. defined as follows: Now let’s consider the function
(70) Let (71) Therefore, we have (77)
(72) Therefore, we have Since
is small, we have (78)
.. . (65)
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(83)
APPENDIX V DERIVATION OF THEOREM 3
Therefore
at , there must exist a corProof: For a given at [12]. responding minimum attenuation such that . Therefore, there always exists some Then from Theorem 1, we immediately have the following result :
(82) , which simplifies (82) to the following form: Let [see (83) at the top of the page]. Therefore
From (9), we can rewrite
Similarly, from Theorem 2, we have
Rewrite
as (84) Similarly, it can be proved that (19) holds.
From the definitions in (7), we have
REFERENCES (79) Using (12) to simplify(79), we have
APPENDIX VI DERIVATION OF THEOREM 5 Proof: Based on the recurrence relation of the Chebyshev polynomial of the first kind
(80) we have
(81)
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[13] Y. Liu and P. Bauer, “Fundamental properties of nonnegative impulse response filters—Theoretical bounds II,” in Proc. 34th IEEE Int. Conf. Acoust., Speech Signal Process. (ICASSP), Apr. 19–24, 2009, pp. 3213–3216. [14] R. Boas and M. Kac, “Inequalities for Fourier transforms of positive functions,” Duke Math. J., vol. 12, pp. 189–206, 1945. [15] A. J. E. M. Janssen, “Frequency-domain bounds for nonnegative bandlimited functions,” Philips J. Res., vol. 45, pp. 325–366, 1990. [16] A. J. E. M. Janssen, “Frequency-domain bounds for nonnegative, unsharply band-limited functions,” J. Fourier Anal. Appl., vol. 1, no. 1, pp. 39–65, Feb. 1994. [17] A. J. E. M. Janssen, “More epsilonized bounds of the Boas-Kac-Lukosz type,” J. Fourier Anal. Appl., vol. 1, no. 2, pp. 171–191, 1994. [18] L. Farina and S. Rinaldi, Positive Linear Systems: Theory and Applications. New York: Wiley Intersci., 2000. [19] L. Benvenuti and L. Farina, “On the class of linear filters attainable with charge routing networks,” IEEE Trans. Circuits Systems-II: Analog Digit. Signal Process., vol. 43, no. 8, pp. 618–622, 1996. [20] B. Anderson, M. Deistler, L. Farina, and L. Benvenuti, “Nonnegative realization of a linear system with nonnegative impulse response,” IEEE Trans. Circuits Syst. I: Fund. Theory Appl., vol. 43, no. 2, pp. 134–142, Feb. 1996. [21] L. Farina and L. Benvenuti, “Positive realizations of linear systems,” Syst. Contr. Lett., vol. 26, pp. 1–9, 1995. [22] Y. Liu and P. H. Bauer, “Addendum to ‘Fundamental Properties of Nonnegative Impulse Response Filters—Theoretical Bounds I,’” Moses Group, Dept. EE, Univ. Notre Dame, Jul. 2008 [Online]. Available: http://www.nd.edu/~yliu5/Addendum1 Y. Liu (S’02) received the Bachelor of Engineering degree (Honors Degree) from the Northwestern Polytechnic University, Xi’An, China, in 2000, and the Master of Engineering degree in electrical and computer engineering from the National University of Singapore, Singapore, and the Institute of Infocomm Research (I R), A-STAR, Singapore, in 2004. He is currently pursuing the Ph.D. degree at the University of Notre Dame, Notre Dame, IN. He is now a Research Assistant with the University of
Notre Dame. His research interests include circuit and system theory, digital signal processing, and multidimensional systems.
Peter H. Bauer (F’05) received the Diplom degree from the Technical University of Munich, Munich, Germany, in 1984, and the Ph.D. degree from the University of Miami, Coral Gables, FL, in 1987, both in electrical engineering. In 1999, he served as the Director of the University of Notre Dame, Notre Dame, IN, EE/CSE London Program, London, U.K. He is currently a Professor with the Department of Electrical Engineering, University of Notre Dame, where he is the Head of the Mobile Sensor Systems (MOSES) Laboratory. He is the author or coauthor of more than 150 technical papers and six book chapters. His research interests include sensor and actuator networks, mobile wireless sensing, congestion control, evidential filtering and fusion, and stability theory of discrete-time systems. Dr. Bauer served as an Associate Editor for the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II (1989–1991, 2003–2005) and has served as an Associate Editor for the journal Multidimensional Systems and Signal Processing (2000–2006). He received the E. I. Jury Award from the University of Miami in 1987, the NASA Technical Innovator Award in 1992, the Alexander von Humboldt Fellowship in 1997, and the University of Notre Dame Kaneb Center Best Teacher Award in 2005.
