# Structure of All-Digital Frequency Synthesiser for IoT and IoV Applications
## 2 Phase Noise in All-Digital Frequency Synthesisers  
The main parameter of the frequency synthesiser is phase noise, which describes the spectral purity of an output signal. Therefore, requirements for main blocks of frequency synthesiser will be defined based on phase noise requirements. In this chapter noise sources of all-digital frequency synthesiser will be discussed through the \(s\)-domain model. A model of an all-digital frequency synthesiser with included noise sources in the \(s\)-domain is shown in Figure 1[10; 11]. The main blocks are a phase detector, digitally controlled oscillator (DCO) and filter. The main part of phase detector is a time to digital converter (TDC), which measures time intervals between edges of reference and DCO signals. There are two integrators (DCO and integrating path in the filter) and there are additional stages of infinite impulse response (IIR) filters in this model. Therefore, it is a type-II higher order frequency  
Figure 1: \(s\)-domain model of a higher order type-II all digital frequency synthesiser with noise sources. synthesiser. The main advantage of a type-II structure over type-I is better noise filtering capabilities, but it has worse dynamic characteristics than type-I. Therefore, there should be an ability to change the type of the synthesiser: type-I should be used for fast initial frequency acquisition and type-II is used for precise phase (and frequency) tracking. Type-I structure can be made by eliminating the integrator--setting the filter's proportional gain coefficient \(\rho\) to 0. There are three main phase noise sources: the noise of the reference signal (\(\varphi_{\text{n,REF}}\) in Figure 1), TDC (\(\varphi_{\text{n,TDC}}\) in Figure 1) and DCO. Noise induced by DCO has two components: quantisation noise (\(\varphi_{\text{n,DCOq}}\) in Figure 1) and oscillator's noise (\(\varphi_{\text{n,DCO}}\) in Figure 1). TDC creates quantization noise, which depends on the TDC's resolution \(t_{\text{res}}\), DCO's frequency \(f_{\text{DCO}}\) and the frequency of reference signal \(f_{\text{REF}}\). Its single-sided power spectral density is shown in (1) equation [12]. \[\mathcal{L}_{\text{TDC}}=\frac{(2\pi)^{2}}{12}\cdot\frac{(t_{\text{res}}f_{ \text{DCO}})^{2}}{f_{\text{REF}}}, \tag{1}\]  
Spectral density of noise of reference signal and oscillator's component of DCO's noise can be defined by the common equation of oscillator's phase noise [13]:  
\[\mathcal{L}_{\text{Osc}}(f)=\frac{k_{3}}{f^{3}}+\frac{k_{2}}{f^{2}}+\frac{k_{ 1}}{f}+k_{0}, \tag{2}\]  
where \(k_{0}\), \(k_{1}\), \(k_{2}\), \(k_{3}\), \(k_{4}\) are coefficients, describing the individual oscillator and \(f\)--frequency offset from carrier. Quantization component of noise induced by DCO usually is reduced by using \(\Delta\Sigma\) dithering and noise shaping techniques. Single sided power spectral density of DCO's quantisation noise after \(\Delta\Sigma\) shaping is [14]:  
\[ \begin{array}{ll}\mathcal{L}_{\Delta\Sigma}(f)&=\left.\mathcal{L}_{\text{q} }(f)\right|_{\Delta f_{\text{res}}=\frac{\Delta f_{\text{res}}}{2^{W_{\Delta \Sigma}}}}+\left.\mathcal{L}_{\Delta\Sigma}(f)\right|_{\Delta f_{\text{res}}= \Delta f_{\text{res}}}=\\ &=\frac{1}{12}\cdot\left(\frac{\Delta f_{\text{res}}/2^{W_{\Delta \Sigma}}}{f}\right)^{2}\cdot d\frac{1}{f_{\text{REF}}}\cdot\left(\sin c\frac{ f}{f_{\text{REF}}}\right)^{2}+\\ &+\frac{1}{12}\cdot\left(\frac{\Delta f_{\text{res}}}{f}\right)^{2}\cdot\frac{ 1}{f_{\Delta\Sigma}}\cdot\left(2\cdot\sin\frac{\pi f}{f_{\Delta\Sigma}}\right) ^{2n},\end{array}  \tag{3}\]  
here \(\mathcal{L}_{\text{q}}(f)\) is the quantization noise component, \(\mathcal{L}_{\Delta\Sigma}(f)\) is the \(\Delta\Sigma\) noise shaping component, \(\Delta f_{\text{RES}}\)--the tuning step of DCO's frequency, \(f_{\text{REF}}\)--frequency of reference signal, \(f_{\Delta\Sigma}\)--frequency of \(\Delta\Sigma\) dithering, \(f\)--frequency offset from carrier, \(W_{\Delta\Sigma}\)--bit count of \(\Delta\Sigma\) modulator, and \(n\)--order of \(\Delta\Sigma\) modulator.