T, the synthesizer first creates two types of cycles T A and TB . Their length-in-times are given by (1) where I is an integer. When synthesizing a particular frequency (period) fs , it uses T A and TB in an interwoven fashion. Output period (frequency) is expressed in (2) where F = I + r is the frequency (more precisely, period) control word. Fraction r controls the occurrence probability of T A and TB . Frequency resolution can be derived from (2) and is expressed in (3). The left side of figure 6 is its fs vs F curve, which is obtained from (2). It is clear that monotonicity is guaranteed by the 1/x relationship. In small area, f f the relationship is almost linear. 1/x 1/x Linearity can be studied and defined with the help of its tanTangent Linear gent line. In the right-hand side, Line at x0 Small Area a zoom-in view is provided for a selected point x 0 on the curve. Zoom In f1 δy Selecting another point x located somewhere away from x 0 , the 1/x F f2 F curve and the tangent line have δx x0 x their respective values f1 and f2 . (a) (b) Using some simple algebra, linearity can be derived and defined Figure 6. TAF-DPS transfer function (a) and its linearity (b). as in (4) F/2 Reg CLK2 Reg CLK1 K→1 CLK2 2→1 CLK1 f∆ K→1 K Inputs SoC/ASIC fs Functional Circuits CLK1 fs TAF-DPS Reg CLK2 Reg CLK2 Base Unit Generator f0 Control Word XO F (a) (b) Figure 7. Flying-Adder circuit for realizing TAF-DPS (left) and the scheme of integrating it on-chip (right). 16 IEEE CIRCUITS AND SYSTEMS MAGAZINE SECOND QUARTER 2020