Introduction

M-phase filters offer high-Q filtering by frequency translation of low-Q baseband impedances to the clock frequency. These filters are placed typically at high-impedance RF nodes, and the quality factor of the resulting bandpass filtering at the RF is proportional approximately to this impedance. Low-impedance RF nodes cannot benefit from these M-phase filters for the following two reasons: (a) for a given quality factor, the required baseband capacitors (and, therefore, the filter size) is inversely proportional to this impedance; (b) the attenuation of the filter at far-out frequencies is equal to the ratio of this impedance to the switch resistance. For low-impedance nodes, the far-out attenuation becomes insignificant.

The well-known duality theorem in circuit theory, however, suggests that the low and high impedances of any given electrical network are mapped to high and low impedances, respectively. This phenomenon suggests that because the M-phase filter is beneficial for high-impedance nodes, its dual must be useful for low-impedance nodes. Therefore, in this chapter, we visit briefly the widely recognized concept of duality in electrical networks. Following that, we will describe the dual of the conventional M-phase filter and its variations and will explain how, for low-impedance nodes, these new filters can offer high-Q filtering with centers controlled by clocks.