Question 9.2: Compare the fractional accumulator output (m = 1) to the thi...
Compare the fractional accumulator output (m = 1) to the third-order \Sigma \Delta modulator output (m = 3) for the case of N(z) = I(z) +. F(z) = 100 + 1/32, with a reference frequency of f_{r} = 10 MHz.
The simulated accumulator outputs are given in Figures 9.19 and 9.20 for m = 1 and m= 3. As shown, the fractional accumulator without \Sigma \Delta modulator (m = 1) has a carry out in every 32 f_{r}cycles, which forces the loop divider to divide by 100 for 31 cycles and then to divide by 101 for 1 cycle. The periodic phase-correction pulse due to dividing by 101 generates fractional spurs with a uniform spacing of f_{r}/32 = 312.5 kHz, as shown in Figure 9.20(a). If three cascaded loops are used, the \Sigma \Delta accumulator outputs are dithered around the correct value, as shown in Figure 9.19(b). Note that the \Sigma \Delta noise shaper breaks the periodicity of the fractional divisor sequences. In the frequency domain, the discrete spurs become more random with their energy pushed towards the higher frequencies, as shown in Figure 9.20(b). Obviously, the fractional spurs look more like frequency noise than discrete tones in the frequency spectrum after the \Sigma \Delta noise shaping.
