# US 10,374,579 B2 · Method of two-stage FRM filter > Agents: this data is available as JSON from the patentgrep API and MCP server — see https://patentgrep.com/llms.txt - Number: US 10,374,579 B2 - Document ID: US10374579B2 - Status: Granted - Kind: B2 - Application number: 15/320,251 - Filed: 2014-06-19 - Published: 2017-05-25 - Granted: 2019-08-06 - Earliest claimed: 2014-06-05 - Assignees: Shandong University - Applicants: Shandong University - Inventors: Ying Wei; Shaoguang Huang - Web page: https://patentgrep.com/patent/US10374579B2 ## CPC - H03H 17/02: Frequency selective networks {(digital computers for complex mathematical operations G06F17/10)} - H03H 17/06: Non-recursive filters - H03H: IMPEDANCE NETWORKS, e.g. RESONANT CIRCUITS; RESONATORS (waveguides, resonators, lines or other devices of the waveguide type H01P) ## Abstract An improved design method of a two-stage FRM filter includes the following steps: constructing an improved two-stage FRM filter; calculating passband and stopband edge parameters of a prototype filter, passband and stopband edge parameters of a second-stage masking filter and passband and stopband edge parameters of a first-stage masking filter in Case A and Case B, respectively; calculating the complexity of the FRM filter according to the obtained parameters, and finding out one or more sets [M, P, Q] having the lowest complexity within a search range; and optimizing the improved FRM filter. The improved design method of a two-stage FRM filter has the following beneficial effect: as compared to a conventional design method of a two-stage FRM filter, the complexity of a narrow-band FIR (Finite Impulse Response) filter can be reduced through design using the improved method, and power consumption is thus reduced in hardware implementation. ## Claims ### Claim 1 A method of constructing a two-stage FRM filter, comprising the following steps of: determining, with a processor, a transfer function H(z) of a two-stage FRM filter as follows: H(z)=G(z)Hma(1)(z)+(1−G(z))Hmc(1)(z), wherein G(z)=Ha(2)(zM)Hma(2)(zP)+(1−Ha(2)(zM))Hmc(2)(zQ), without any constraint among interpolation factors M, P, Q; Ha(2)(zM) represents a prototype filter, while Hma(1)(z) and Hmc(1)(z) represent first-stage masking filters, respectively, and Hma(2)(zP) and Hmc(2)(zQ) represent second-stage masking filters, respectively; storing the interpolation factors M, P, Q in a memory; searching, with a processor, within a search range for [M, P, Q], and for a certain set [M, P, Q], calculating passband and stopband edge parameters of the prototype filter Ha(2)(z), passband and stopband edge parameters of the second-stage masking filters and passband and stopband edge parameters of the first-stage masking filters in Case A and Case B, respectively, on the basis that a transition band of the whole filter is provided by Ha(2)(zM) or a complement of Ha(2)(zM); calculating calculating, with a processor, the complexity of the FRM filter according to the calculated passband and stopband edge parameters, and finding out one or more sets [M, P, Q] having the lowest complexity within the search range; optimizing, with a processor, an improved FRM filter according to the calculated filter parameters, by determining a number of multipliers implementing the improved FRM filter and a group time delay performed by the improved FRM filter, and constructing the improved FRM filter by arranging the multipliers, based on the calculated filter parameters. ### Claim 2 (depends on 1) The method of constructing a two-stage FRM filter according to claim 1, wherein a way of calculating the passband edge θa and the stopband edge φa of the prototype filter Ha(2)(zM) in the step (2) is as follows: when the transition band of the whole filter is Ha(2)(zM), i.e., in the Case A: m=└ωpM/2π┘, θa=ωpM−2mπ, φa=ωsM−2mπ; when the transition band of the whole filter is the complement of Ha(2)(zM), i.e., in the Case B: m=┌ωsM/2π┐, θa=2mπ−ωsM, φa=2mπ−ωpM; wherein └x┘ represents a largest integer not more than x; ┌x┐ represents a smallest integer not less than x; whether the result satisfies 0<θa<φa<π is determined in the two cases respectively, and if not, the results are abandoned. ### Claim 3 (depends on 1) The method of constructing a two-stage FRM filter according to claim 1, wherein passbands of Ha(2)(Mω) from 0 to π in the step (2) are orderly represented by 0, 2, . . . , 2 × ⌊ M 2 ⌋ , and passbands of 1−Ha(2)(Mω) from 0 to π are orderly represented by 1, 3, . . . , 2 × ⌊ M - 1 2 ⌋ ; assuming that the passband labeled as 2m of Ha(2)(Mω) provides the transition band, the passband of the masking filter from which the transition band is extracted is defined as an effective passband; and in order to reduce the complexity of the two first-stage masking filters, the following restrictive conditions are established: the passband 2m should be at least completely extracted; the passband 2(m+1) should be completely fall outside an effective passband range. ### Claim 4 (depends on 1) The method of constructing a two-stage FRM filter according to claim 1, wherein when the transition band of the whole filter is provided by Ha(2)(zM), i.e., in the Case A, a way of calculating the passband edge ωpma(2) and the stopband edge ωsma(2) of the second-stage masking filter Hma(2)(z), is as follows: (1) when masking is provided by Hma(2)(zP), the case is denoted as Casep=A: { ω pma ( 2 ) = max ⁡ ( 2 ⁢ π ⁢ ⁢ p - ⁢ ω 1 ⁢ P , ω 2 ⁢ P - 2 ⁢ π ⁢ ⁢ p ) ω sma ( 2 ) = ω 3 ⁢ P - 2 ⁢ π ⁢ ⁢ p ; (2) when masking is provided by the complement of Hma(2)(zP), the case is denoted as Casep=B: { ω pma ( 2 ) = 2 ⁢ π ⁢ ⁢ p - ω 3 ⁢ P ω sma ( 2 ) = min ⁡ ( ω 1 ⁢ P - 2 ⁢ π ⁡ ( p - 1 ) , 2 ⁢ π ⁢ ⁢ p - ω 2 ⁢ P ) ; wherein ω1 is a left edge of the passband 2m of the prototype filter, while ω2 is a right edge of the passband 2m of the prototype filter, and ω3 is a left edge of the passband 2(m+1) of the prototype filter; P is the interpolation factor, which is a given value; p is an integer, and should satisfy the following condition: 0 ≤ p ≤ ⌊ P 2 ⌋ ; if no p satisfying the condition exists, the set of parameters is abandoned. ### Claim 5 (depends on 1) The method of constructing a two-stage FRM filter according to claim 1, wherein when the transition band of the whole filter is provided by Ha(2)(zM), i.e., in the Case A, a way of calculating the passband edge ωpmc(2) and the stopband edge ωsmc(2) of the second-stage masking filter Hmc(2)(z) is as follows: (1) when masking is provided by Hmc(2)(zQ), the case is denoted as Caseq=A: { ω pmc ( 2 ) = max ⁡ ( 2 ⁢ π ⁢ ⁢ q - ω 4 ⁢ Q , ω 5 ⁢ Q - 2 ⁢ π ⁢ ⁢ q ) ω smc ( 2 ) = min ⁡ ( ω 6 ⁢ Q - 2 ⁢ π ⁢ ⁢ q , 2 ⁢ π ⁡ ( q + 1 ) - ω 7 ⁢ Q ) ; (2) when masking is provided by the complement of Hmc(2)(zQ), the case is denoted as Caseq=B: { ω pmc ( 2 ) = max ⁡ ( 2 ⁢ π ⁢ ⁢ q - ω 6 ⁢ Q , ω 7 ⁢ Q - 2 ⁢ π ⁢ ⁢ q ) ω smc ( 2 ) = min ⁡ ( ω 4 ⁢ Q - 2 ⁢ π ⁡ ( q - 1 ) , 2 ⁢ π ⁢ ⁢ q - ω 5 ⁢ Q ) ; wherein ω4 is a left passband edge of the complementary filter passband 2m−1 of the prototype filter; ω5 is a right stopband edge of the complementary filter passband 2m−1 of the prototype filter; ω6 is a left passband edge of the complementary filter passband 2m+1 of the prototype filter; ω7 is a right stopband edge of the complementary filter passband 2m+1 of the prototype filter; q is an integer, and should satisfy the following condition: 0 ≤ q ≤ ⌊ q 2 ⌋ ; if no q satisfying the condition exists, the set of parameters is abandoned. ### Claim 6 (depends on 1) The method of constructing a two-stage FRM filter according to claim 1, wherein when the transition band of the whole filter is provided by the complement of Ha(2)(zM), i.e., in the Case B, a way of calculating the passband edge ωpma(2) and the stopband edge ωsma(2) of the second-stage masking filter Hma(2)(z) is as follows: (1) when masking is provided by Hma(2)(zP), the case is denoted as Casep=A: { ω pma ( 2 ) = max ⁡ ( 2 ⁢ π ⁢ ⁢ p - P ⁢ ⁢ ω 4 , ω 5 ⁢ P - 2 ⁢ π ⁢ ⁢ p ) ω sma ( 2 ) = min ⁡ ( ω 6 ⁢ P - 2 ⁢ π ⁢ ⁢ p , 2 ⁢ π ⁡ ( p + 1 ) - ω 7 ⁢ P ) ; (2) when masking is provided by the complement of Hma(2)(zP), the case is denoted as Casep=B: { ω pma ( 2 ) = max ⁡ ( 2 ⁢ π ⁢ ⁢ p - ω 6 ⁢ P , ω 7 ⁢ P - 2 ⁢ π ⁢ ⁢ p ) ω sma ( 2 ) = min ⁡ ( ω 4 ⁢ P - 2 ⁢ π ⁡ ( p - 1 ) , 2 ⁢ π ⁢ ⁢ p - ω 5 ⁢ P ) ; wherein ω4 is a left passband edge of the passband 2(m−1) of the prototype filter; ω5 is a right stopband edge of the passband 2(m−1) of the prototype filter; ω6 is a left passband edge of the passband 2m of the prototype filter; ω7 is a right stopband edge of the passband 2m of the prototype filter; p is an integer, and should satisfy the following condition: 0 ≤ p ≤ ⌊ P 2 ⌋ ; if no p satisfying the condition exists, the set of parameters is abandoned. ### Claim 7 (depends on 1) The method of constructing a two-stage FRM filter according to claim 1, wherein when the transition band of the whole filter is provided by the complement of Ha(2)(zM), i.e., the Case B, a way of calculating the passband edge ωpmc(2) and the stopband edge ωsmc(2) of the second-stage masking filter Hmc(2)(z) is as follows: (1) when masking is provided by Hmc(2)(zQ), the case is denoted as Caseq=A: { ω pmc ( 2 ) = max ⁡ ( 2 ⁢ π ⁢ ⁢ q - ω 1 ⁢ Q , ω 2 ⁢ Q - 2 ⁢ π ⁢ ⁢ q ) ω smc ( 2 ) = ω 3 ⁢ Q - 2 ⁢ π ⁢ ⁢ q ; (2) when masking is provided by the complement of Hmc(2)(zQ), the case is denoted as Caseq=B: { ω pmc ( 2 ) = 2 ⁢ π ⁢ ⁢ q - ω 3 ⁢ Q ω smc ( 2 ) = min ⁡ ( ω 1 ⁢ Q - 2 ⁢ π ⁡ ( q - 1 ) , 2 ⁢ π ⁢ ⁢ q - ω 2 ⁢ Q ) ; wherein ω1 is a left stopband edge of the complementary filter passband 2m−1 of the prototype filter; ω2 is a right passband edge of the complementary filter passband 2m−1 of the prototype filter; ω3 is a left stopband edge of the complementary filter passband 2m+1 of the prototype filter; q is an integer, and should satisfy the following condition: 0 ≤ q ≤ ⌊ q 2 ⌋ ; if no q satisfying the condition exists, the set of parameters is abandoned. ### Claim 8 (depends on 1) The method of constructing a two-stage FRM filter according to claim 1, wherein when the transition band of the whole filter is provided by Ha(2)(zM), i.e., in the Case A: the passband edge ωpma(1) of the first-stage masking filter Hma(1)(z) is equal to ωp; a way of calculating the stopband edge ωsma(1) of the first-stage masking filter Hma(1)(z) is as follows: ω sma ( 1 ) = min ⁡ ( ω sma ⁢ ⁢ _ ⁢ ⁢ temp ⁢ ⁢ 1 ( 1 ) , ω sma ⁢ ⁢ _ ⁢ ⁢ temp ⁢ ⁢ 2 ( 1 ) ) , ⁢ wherein ω sma ⁢ ⁢ _ ⁢ ⁢ temp ⁢ ⁢ 1 ( 1 ) = { t 1 t 1 ∉ R stop ⁡ ( k ) ω 8 t 1 ∈ R stop ⁡ ( k 1 ) , ⁢ ω sma ⁢ ⁢ _ ⁢ ⁢ temp ⁢ ⁢ 2 ( 1 ) = { t 2 t 2 ∉ R pass ⁡ ( k ) ω 9 t 2 ∈ R pass ⁡ ( k 2 ) , ⁢ ω 8 = ( 2 ⁢ π ⁢ ⁢ k 1 = φ a ) / M ; ω 9 = ( 2 ⁢ π ⁢ ⁢ k 2 - θ a ) / M ; ⁢ R pass ⁡ ( k ) = [ 2 ⁢ π ⁢ ⁢ k - θ a M , 2 ⁢ π ⁢ ⁢ k + θ a M ] , k = 0 , … ⁢ , ⌊ M / 2 ⌋ ; ⁢ R stop ⁡ ( k ) = [ 2 ⁢ π ⁢ ⁢ ( k - 1 ) + φ a M , 2 ⁢ π ⁢ ⁢ k + φ a M ] , k = 0 , … ⁢ , ⌊ M / 2 ⌋ ; (1) when Hma(2)(zP) is used for masking unnecessary band of an up-branch, t1=(2π(p+1)−ωsma(2))/P; (2) when the complement of Hma(2)(zP) is used for masking the unnecessary band of the up-branch, t1=(2πp+ωpma(2))/P; (3) when Hmc(2)(ZQ) is used for masking unnecessary band of a down-branch, t2=(2π(q+1)−ωsmc(2))/Q; (4) when a down-branch complement Hmc(2)(ZQ) is used for masking unnecessary band, t2=(2πq+ωpmc(2))/Q; wherein P, Q, M are interpolation factors; ω8 is a right endpoint of Rstop(k1); k1 is an integer satisfying t1∈Rstop(k1); ω9 is a right endpoint of Rpass(k2); k2 is an integer satisfying t2∈Rstop(k2); θa is a passband edge of Ha(2)(z); Hma(1)(z)φa is a stopband edge of Ha(2)(z). ### Claim 9 (depends on 1) The method of constructing a two-stage FRM filter according to claim 1, wherein when the transition band of the whole filter is provided by the complement of Ha(2)(zM), i.e., in the Case B: the passband edge ωpma(1) of the first-stage masking filter Hma(1)(z) is equal to ωp; a way of calculating the stopband edge ωsma(1) of the first-stage masking filter Hma(1)(z) is as follows: ω sma ( 1 ) = min ⁡ ( ω sma ⁢ ⁢ _ ⁢ ⁢ temp ⁢ ⁢ 1 ( 1 ) , ω sma ⁢ ⁢ _ ⁢ ⁢ temp ⁢ ⁢ 2 ( 1 ) ) , ⁢ wherein ω sma ⁢ ⁢ _ ⁢ ⁢ temp ⁢ ⁢ 1 ( 1 ) = { t 1 t 1 ∉ R stop ⁡ ( k ) ω 8 t 1 ∈ R stop ⁡ ( k 1 ) , ⁢ ω sma ⁢ ⁢ _ ⁢ ⁢ temp ⁢ ⁢ 2 ( 1 ) = { t 2 t 2 ∉ R pass ⁡ ( k ) ω 9 t 2 ∈ R pass ⁡ ( k 2 ) , ⁢ ω 8 = ( 2 ⁢ π ⁢ ⁢ k 1 = θ a ) / M ; ω 9 = ( 2 ⁢ π ⁢ ⁢ k 2 - φ a ) / M ; ⁢ R pass ⁡ ( k ) = [ 2 ⁢ π ⁢ ⁢ k - θ a M , 2 ⁢ π ⁢ ⁢ k + θ a M ] , k = 0 , … ⁢ , ⌊ M / 2 ⌋ ; ⁢ R stop ⁡ ( k ) = [ 2 ⁢ π ⁢ ⁢ ( k - 1 ) + φ a M , 2 ⁢ π ⁢ ⁢ k + φ a M ] , k = 0 , … ⁢ , ⌊ M / 2 ⌋ ; (1) when Hmc(2)(zQ) is used for masking unnecessary band, t1=(2π(q+1)−ωsmc(2))/Q; (2) when the complement of Hmc(2)(zQ) is used for masking unnecessary band, t1=(2πq+ωpmc(2))/Q; (3) when Hma(2)(zP) is used for masking unnecessary band, t2=(2π(p+1)−ωsma(2))/P; (4) when the complement of Hma(2)(zP) is used for masking unnecessary band, t2=(2πp+ωpma(2))/P; wherein ω8 is a right endpoint of Rstop(k1); k1 is an integer satisfying t1∈Rstop(k1); ω9 is a right endpoint of Rpass(k2); k2 is an integer satisfying t2∈Rstop(k2); P, Q, M are interpolation factors; θa is a passband edge of Ha(2)(z); Hma(1)(z)φa is a stopband edge of Ha(2)(z). ### Claim 10 (depends on 1) The method of constructing a two-stage FRM filter according to claim 1, wherein (1) when the transition band of the whole filter is provided by Ha(2)(zM), i.e., in the Case A: the stopband edge ωsmc(1) of the first-stage masking filter Hmc(1)(z) is equal to ωs; a way of determining the passband edge ωpmc(1) of the first-stage masking filter is as follows: if t3≥t4, then ω pmc ( 1 ) = { t 3 t 3 ∉ R stop ⁡ ( k ) max ⁡ ( 2 ⁢ π ⁢ ⁢ ( k 3 - 1 ) + φ a M , t 4 ) t 3 ∈ R stop ⁡ ( k 3 ) ; if t3