1) Biquad LPF: The generic transfer function of a second-order LPF is Hs LPF () = ω s ++ Q 2 ω0 2 = s ω 2 ss Q ω ω0 2 2 ++1 where ω0 is the natural frequency (also the cutoff frequency in LPF or center frequency in BPF) and Q is the quality factor of the filter. We demonstrate how this transfer function can be decomposed into multiple forms that lead to different circuit topologies. We first express Eq. (1) in a similar form to the transfer function of the closed-loop gain of a negative feedback system, i.e., AA A CL=+() β is the feedback gain. By setting Hs AsLPFCL and defining As As () and β()s as 1 () = ss Q ω ω0 2 2 + we can construct a corresponding block diagram as shown in Fig. 3(a) where Hs Vs Vs topology can be further decomposed into a cascaded LPFLPF()/( ). This () = IN β() 1 s = /1 β where A is the feedforward gain, () = (), (2) 1 (1) structure forming a Lossy integrator and a Lossless integrator as shown in Fig. 3(b) and is also called a TwoIntegrator-Loop topology [22]. The characteristics of both integrator types are shown in Fig. 3(d). We see that the first-order LPF corresponds to the lossy case and an ideal integrator to the lossless case. The lossy integrator can be further decomposed into a lossless integrator associated with a nested feedback path which controls Q of the second-order filter as shown in Fig. 3(c). This two-integrator-loop topology is used in a popular filter implementation called the Tow-Thomas biquad [23]. 2) Biquad BPF With Poles at the Same Frequency: Fig. 3(c) also shows how a BPF response is obtained at the output of the lossy integrator within this topology. This is because the lossless integrator is excluded in the feedforward gain of Hs LPF() case, which in turn acts as LPF() (i.e., × s/).ω0 The the resulting second-order BPF() compared to Hs a differentiation of Hs transfer function of BPF is expressed as H BP sF() = s 2 ω0 Q ++ ω 0s s ω 2 (3) Figure 2. Example of the output of 16 bandpass filter bank channels with center frequencies ranging from 100 Hz to 8KHz on a log-spacing and with Q = 2. The audio input is an example speech from the GSCD samples. SECOND QUARTER 2023 IEEE CIRCUITS AND SYSTEMS MAGAZINE 31