Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

Two-Dimensional FIR Filters 22-322.2.1 Filter Specifications and Approximation CriteriaThe problem of designing a 2-D FIR filter consists of determining the impulse response sequence,h(n 1 , n 2 ), or its system function, H(z 1 , z 2 ), in order to satisfy given requirements on the filter response.The filter requirements are usually specified in the frequency domain, and only this case is consideredhere. The frequency response,* H(v 1 , v 2 ), corresponding to the impulse response h(n 1 , n 2 ), with asupport, I, is expressed asH(v 1 , v 2 ) ¼X(n 1,n 2)2Ih(n 1 , n 2 )e j(n1v1þn2v2) : (22:2)F pNote that H(v 1 , v 2 ) ¼ H(v 1 þ 2p, v 2 ) ¼ H(v 1 , v 2 þ 2p) for all (v 1 , v 2 ). In other words, H(v 1 , v 2 )isaperiodic function with a period 2p in both v 1 and v 2 . This implies that by defining H(v 1 , v 2 ) in theregion { p < v 1 p, p < v 2 p}, the frequency response of the filter for all (v 1 , v 2 ) is determined.For 2-D FIR filters the specifications are usually given in terms of the magnitude response, jH(v 1 , v 2 )j.Attention in this chapter is confined to the case of a two-level magnitude design, where the desiredmagnitude levels are either 1.0 (in the passband) or 0.0 (in the stopband). Some of the procedures can beeasily modified to accommodate multilevel magnitude specifications, as, for instance, in a case thatrequires the magnitude to increase linearly with distance from the origin in the frequency domain.Consider the design of a 2-D FIR low-pass filter whose specifications are shown in Figure 22.1. Themagnitude of the low-pass filter ideally takes the value 1.0 in the passband region, F p , which is centeredaround the origin, (v 1 , v 2 ) ¼ (0, 0), and 0.0 in the stopband region, F s . As a magnitude discontinuity isnot possible with a finite filter support, I, it is necessary to interpose a transition region, F t , between F pand F s . Also, magnitude bounds jH(v 1 , v 2 ) 1jd p in the passband and jH(v 1 , v 2 )jd s in thestopband are specified, where the parameters d p and d s are positive real numbers, typically much lessthan 1.0. The frequency response H(v 1 , v 2 ) is assumed to be real. Consequently, the low-pass filter isspecified in the frequency domain by the regions, F p , F s , and the tolerance parameters, d p and d s .A variety of stopband and passband shapes canbe specified in a similar manner.ω 2 In order to meet given specifications, anadequate filter order (defined here to be the numberof nonzero impulse response samples) needs toπF be determined. If the specifications are stringent,swith tight tolerance parameters and small transitionregions, then the filter support region, I, mustF t–πbe large. In other words, there is a trade-offbetween the filter support region, I, and the frequencydomain specifications. In the general caseωπ 11±δ pthe filter order is not known a priori, and may bedetermined either through an iterative process orusing estimation rules if available. If the filter order±δ sis given, then in order to determine an optimum–πsolution to the design problem, an appropriateoptimality criterion is needed. Commonly usedcriteria in 2-D filter design are minimization ofFIGURE 22.1 Frequency response specifications for a the L p norm, p finite, of the approximation error,2D low-pass filter (jH(v 1 , v 2 ) 1jd p for (v 1 , v 2 ) 2 F p or the L 1 norm. If desired, a maximal flatnessand jH(v 1 , v 2 )jd s for (v 1 , v 2 ) 2 F s ).requirement at desired frequencies can be imposed* Here v 1 ¼ 2pf 1 and v 2 ¼ 2pf 2 are the horizontal and vertical angular frequencies, respectively.