Table 1 Overall design flow of a three-stage two-band QΣΔM

   1. Obtain the center frequencies (f_C,1 and f_C,2 ) and the suitable frequency offsets (f_offset,1 and f_offset,2) based on the bandwidths of the desired signals, e.g., from [36]

   2. Based on the spectrum sensing information, find the most harmful blockers (f_int,1 and f_int,2 )

• In case of mirror-frequency rejecting STF design f_int,1 = −f_C,1 and f_int,2 = −f_C,2

Design of a three-stage two-band QΣΔM with mirror-frequency rejecting first-stage STF

Transfer function design for the first-stage (two-band NTF and mirror-frequency rejecting STF)

   1. Place the NTF zeros: ${\phi }_{\mathsf{\text{NTF}}.1}^{\left(1\right)}=e^{j2\pi f_{\mathsf{\text{C,1}}}{T}_{\mathsf{\text{S}}}},{\phi }_{\mathsf{\text{NTF}}.2}^{\left(1\right)}=e^{j2\pi f_{\mathsf{\text{C,2}}}{T}_{\mathsf{\text{S}}}}$

   2. Place the STF zeros: ${\phi }_{\mathsf{\text{STF}}.1}^{\left(1\right)}=e^{j2\pi f_{\mathsf{\text{int}},1}{T}_S},{\phi }_{\mathsf{\text{NTF}}.2}^{\left(1\right)}=e^{j2\pi f_{\mathsf{\text{int}},2}{T}_{\mathsf{\text{S}}}}$

   3. Place the common poles: ${\psi }_{\mathsf{\text{common}}1}^{\left(1\right)}=0.5e^{j2\pi f_{\mathsf{\text{C}},1}{T}_{\mathsf{\text{S}}}}$ and ${\psi }_{\mathsf{\text{common}}.2}^{\left(1\right)}=0.5e^{j2\pi f_{\mathsf{\text{C}},2}{T}_{\mathsf{\text{S}}}}$

   4. Solve the modulator coefficients M⁽¹⁾ N⁽¹⁾ using (51)-(52); A⁽¹⁾, B⁽²⁾ and C⁽¹⁾ using (54)-(56); and R⁽¹⁾ and S⁽¹⁾ using (58)-(59).

Transfer function design for the second-stage (two-band NTF and frequency-flat STF)

   5. Place the NTF zeros: ${\phi }_{\mathsf{\text{NTF}}.1}^{\left(2\right)}=e^{j2\pi \left(f_{\mathsf{\text{C,}}1}+f_{\mathsf{\text{offset,1}}}\right)T_{\mathsf{\text{S}}}},{\phi }_{\mathsf{\text{NTF}}.2}^{\left(2\right)}=e^{j2\pi \left(f_{\mathsf{\text{C,}}2}+{f_{\mathsf{\text{offset}}}}_{,2}\right)T_{\mathsf{\text{S}}}}$

   6. Place the STF zeros: ${\phi }_{\mathsf{\text{STF}}1}^{\left(2\right)}=0,{\phi }_{\mathsf{\text{NTF}}.2}^{\left(2\right)}=0$

   7. Place the common poles: ${\psi }_{\mathsf{\text{common}}.1}^{\left(2\right)}=0$ and ${\psi }_{\mathsf{\text{common}}.2}^{\left(2\right)}=0$

   8. Solve the modulator coefficients M⁽²⁾ and N⁽²⁾ using (51)-(52); A⁽²⁾, B⁽²⁾ and C⁽²⁾ using (54)-(56); and R⁽²⁾and S⁽²⁾ using (58)-(59).

Transfer function design for the third-stage (two-band NTF and frequency-flat STF)

   9. Place the NTF zeros: ${\phi }_{\mathsf{\text{NTF}}\mathsf{\text{.1}}}^{\left(3\right)}=e^{j2\pi \left(f_{\mathsf{\text{C}}.1}-f_{o\mathsf{\text{ffset,1}}}\right)T_{\mathsf{\text{S}}}},{\phi }_{\mathsf{\text{NTF}}\mathsf{\text{.1}}}^{\left(3\right)}=e^{j2\pi \left(f_{\mathsf{\text{C}},2}-f_{\mathsf{\text{offset,2}}}\right)T_{\mathsf{\text{S}}}}$

   10. Place the STF zeros: ${\phi }_{\mathsf{\text{STF}}1}^{\left(3\right)}=0,{\phi }_{\mathsf{\text{NTF}}.2}^{\left(3\right)}=0$

   11. Place the common poles: ${\psi }_{\mathsf{\text{common}}.1}^{\left(3\right)}=0$ and ${\psi }_{\mathsf{\text{common}}.2}^{\left(3\right)}=0$

   12. Solve the modulator coefficients M⁽³⁾ and N⁽³⁾ using (51)-(52); A⁽³⁾, B⁽³⁾ and C⁽³⁾ using (54)-(56); and R⁽³⁾ and S⁽³⁾ (58)-(59).