Question 9.3: Consider the synthesizer originally designed in Examples 3.4...
First, recall that the reference frequency of the design considered in Chapter 3 was 40 MHz. Thus, the raw \Sigma \Delta phase noise as predicted by (9.80),
\frac{\varphi ^{2}_{\Sigma \Delta} (f)\left[rad^{2}/Hz \right] }{2}= \frac{(2\pi)^{2}}{24f_{r}}.\left[2\sin (\frac{\pi f}{f_{r} } )\right]^{2(m-1)}PN(f) [dBc/Hz]= 10 log\left\{\frac{(2\pi)^{2}}{24f_{r}}.\left[2\sin (\frac{\pi f}{f_{r} } )\right]^{2(m-1)} \right\}
is given in Figure 9.41.
Note that this example considers only up to the third order at the beginning because, if a higher-order \Sigma \Delta modulator is needed, then a more complicated, higher-order loop filter will also be required to attenuate the out-of-band phase noise effectively. Next, the raw phase noise is applied to the transfer function in (9.82),
Note that all values for this formula are taken from the previous examples in Chapter 3. The phase noise from the \Sigma \Delta is compared to the previous total phase noise predicted by this loop in order to determine the effect on overall noise performance. The results of these calculations are shown in Figure 9.42. From this plot, it is easy to see that a first-order modulator (a fractional-N accumulator without noise shaping) produces far too much noise and will in fact completely dominate the noise performance of this synthesizer. On the other hand, a secondorder MASH \Sigma \Deltamodulator will not degrade the in-band noise. However, in the range of 1 to 3 MHz, it will increase the phase noise of the design by 3 dB as the \Sigma \Delta noise is about equal to the total noise. It is also interesting to study the shape of the filtered ∑D noise curve. In-band, the \Sigma \Delta phase noise rises at 20 dB/dec due to the second-order noise-shaping effect. At the loop corner frequency, it becomes flat due to the attenuation of the first loop filter pole. Once the second pole from the loop filter begins to take effect, the filtered \Sigma \Delta phase noise response falls at 20 dB/dec. The third-order \Sigma \Delta modulator, on the other hand, has its noise well below that of the other components in the loop and, therefore, has a negligible effect on the total PLL noise. However, note that even after the second pole in the loop filter, this noise is only flat out of band. If this noise performance is not acceptable and a fourth-order \Sigma \Delta modulator is required, then a higher-order loop filter will be needed to keep the out of-band \Sigma \Delta noise from growing.
It should be pointed out that the above noise PSD only models the random noise produced by the quantizer; therefore, discrete spurs can be expected to be larger than predicted by this formula. Since there is no \Sigma \Delta noise shaping in the first-order modulator, the output of the accumulator contains discrete spurs rather than randomized noise. The phase noise curve for the first-order modulator given in Figure 9.42 will be valid if dithering is used in the fractional accumulator. As shown in Figure 9.41, the slope of the phase noise PSD is 20(m − 1) dB/dec. Thus, there will be no noise-shaping if only one accumulator is used. On the other hand, the phase noise PSD is shaped with slopes of 20 and 40 dB/dec for two and three loops, respectively. Also notice that for every doubling of the reference frequency, which is equivalent to doubling the sampling frequency of the \Sigma \Delta accumulators, the in-band phase noise PSD due to fractional spurs is reduced by 6(m − 1) dB. Note that the noise-shaping slope for fractional frequency errors in (9.42),
f_{0} (z) = [I(z) + .F(z)]f_{r} + (1 − z ^{−1} )^{m}E_{qm } (z)f_{r}and the noise-shaping slope for fractional phase noise PSD in (9.80) are different. During the frequency to-phase conversion, an integration term(1 − z ^{−1} ) ^{−1} is included. In other words, the phase is obtained by integrating the frequency. Phase integration averages the frequency variation; thus, the phase noise PSD has a lower noise-shaping slope than the frequency-error-shaping curve.
