Abstract:
A solution for synchronizing a network comprising a plurality of interconnected nodes provides a stable synchronized state, especially for large scale networks. Signal transmission speed and the length of each interconnection of the network is configured to cause a delay of the signals received by a node from the other node of the interconnection which is larger than one millionth of the free-running period of the controllable oscillator of the receiving node such that Network-wide synchronization of oscillators is achieved for all nodes of the network in a continuous self-organized process in interaction with the other node of the network.

Description:
FIELD OF THE INVENTION 
       [0001]    The present invention relates to a synchronizable network comprising a plurality of nodes. The present invention specifically relates to clock distribution and self-organized synchronization in large scale networks, like high performance Multiprocessor System-on-Chips (MPSoCs) architectures, massive Multiple-Input Multiple-Output (MIMO) systems in mobile communications, multicore-processor applications, or GPU synchronization. 
         [0002]    The present invention also relates to a method for synchronizing Network comprising a plurality of nodes. 
       BACKGROUND OF THE INVENTION 
       [0003]    Enabled by sub-micrometer semiconductor device fabrication technologies MPSoCs have become a key component for modern communications and computation systems. Lately, the trend to integrate more and more processing cores in a single silicon die has gained momentum, aided by promising benefits in mechanical footprint, computational performance, energy and cost efficiency. Hence, increasing the number of cores directly translates into high performance through parallel processing and high efficiency compared to single-core solutions. 
         [0004]    Nowadays, hundreds of thousands of cores are integrated on one single chip. To ensure a stable and well defined system a common synchronization strategy is to separate clocking of the processing blocks. The globally asynchronous locally synchronous (GALS) clocking yields a simplified clock tree and allows clock generation on-chip to minimize the number of required I/O pins. Hence, the clock frequencies and supply voltages, within a heterogeneous MPSoC, can be dynamically adjusted per core. However, the flexibility, scalability and other benefits of GALS clocking technique goes along with performance penalties caused by additional communication latencies between disjoint clock domains. This exactly describes the bottleneck of the GALS approach. 
         [0005]    In contrast, for high performance microprocessors a globally synchronous design as shown in  FIG. 1 , where all cores ( 11 ) of a clocking network ( 13 ) share one master clock ( 12 ), is used. The communication latencies between cores are drastically reduced compared to GALS clocking. Considering next generation MPSoCs, a very large chip area has to be clocked synchronously. Implementing a master clock based clock tree, see  FIG. 1 , the clock signals within MPSoCs have to be transmitted over ranges of some millimeters, which is a well-known bottleneck for speed, power and reliability. Furthermore, traditional globally synchronous clocking circuits have become too difficult for large MPSoCs with many cores, constantly growing chip size and wire induced delays. In addition, the clock trees consume a significant amount of power which is critical for mobile communication systems. 
         [0006]    Both clocking techniques, GALS and the globally synchronous design, reach their limits at large scale networks like massive Multiple-Input Multiple-Output (MIMO) systems and MPSoCs. 
         [0007]    Another strategy for network synchronization and clock distribution relates to self-organized synchronization of distributed network nodes in absence of an entraining master clock. 
         [0008]    “Mutually connected phase-locked loop networks: dynamical models and design parameters” by F. M. Orsatti, R. Carareto, J. R. C. Piqueira, IET Circuit Devices Syst., 2008, Vol. 2, No. 6, pp. 495-508 relates to distributing clock signals by using mutually connected architectures instead of master-slave type architectures. A mathematical model of mutually connected digital PLL networks is studied numerically; with the class of phase detectors restricted to JK flip flop phase detectors and charge-pump phase detectors. With the setup described in Orsatti et al., it is impossible to build a mutually connected network with three or more nodes by using XOR PDs. Additionally, signal transmission times are explicitly neglected. Conditions for the existence of synchronized states are derived, depending on individual node parameters and network connectivity, considering that the nodes are nonlinear oscillators with nonlinear coupling conditions. 
         [0009]    “Multiple synchronous states in static delay-free mutually connected PLL networks” by F. M. Orsatti, R. Carareto, J. R. C. Piqueira, Signal Processing 90 (2010) 2072-2082 relates to mutually connected networks of digital phase-locked loops. A mathematical model of mutually connected digital PLL networks is studied numerically, with the class of phase detectors restricted to JK flip flop phase detectors. Even for static networks without delays, different synchronous states may exist for the network. 
         [0010]    However, these papers deal with networks for which a time delay between oscillators is not present or negligible. Moreover, in both papers, the class of phase detectors is restricted to JK flip flop and/or charge-pump phase detectors. Hence, the solution presented there does not include networks with different types of phase detectors and cannot be applied to networks exhibiting a significant time delay between network nodes. 
         [0011]    WO 2013/178237 A1 relates to a communication network of interconnected communication nodes, each node comprising an oscillator that is mutually coupled to oscillators of other communication nodes. The oscillator generates periodic synchronization pulses. The communication node further comprises a transmitter for transmitting the synchronization pulses to other communication nodes; a receiver for receiving synchronization pulses from other communication nodes; and a synchronization unit for synchronizing the phase of the synchronization pulses generated by the oscillator with the phase of the synchronization pulses received from other communication nodes by adjusting the phase of the synchronization pulses generated by the oscillator upon receipt of synchronization pulses from other communication nodes. The synchronization unit adjusts the phase of the synchronization pulses generated by the oscillator in such a way that a guaranteed network-wide synchronization is achieved for all communication nodes of the communication network. 
         [0012]    However, WO 2013/178237 A1 explicitly limits a transmission time delay of the synchronization pulses between the communication nodes to one eighth of the period of the oscillator. Hence, this disclosure does not provide a suitable solution for networks exhibiting a transmission time delay exceeding one eighth of the period of the oscillator, e.g., highly integrated chip networks. Moreover, this solution assumes pulse coupling. Stochastic synchronization pulse emission is required to guarantee synchronization. Hence, this solution is not suitable for clock distributions with time-continuous coupling. 
         [0013]    US 2009/183019 A1 relates to a system with multiple clock islands, each clock island clocked by a common clock generator. A predetermined amount of the clock skew may be introduced by programmable delay elements to smear out, over time, instantaneous power supply current demands of respective logic. Moreover, additional delayers are used to compensate for the clock skew between different clock islands for the purpose of information transmission. 
         [0014]    Hence, US 2009/183019 A1 aims at establishing a clock skew in a system with a single clock generator using programmable delay elements. 
       DISCLOSURE OF THE INVENTION 
       [0015]    It is an objective of the present invention to provide a solution for synchronizing a network comprising a plurality of interconnected nodes that provides a stable synchronized state, especially for large scale networks. 
         [0016]    Here synchronized states relate to any state of the network with time-independent phase differences between the nodes of the network. In such a network, each node of the network receives at least one input from another node and transmits its output to at least another node. 
         [0017]    This objective is achieved with a network of nodes according to the independent apparatus claim and a method for synchronizing a network according to the independent method claim. 
         [0018]    The present invention relates to a network comprising a plurality of interconnected nodes. The node comprises a controllable oscillator generating a time-continuous synchronization signal for synchronizing the plurality of interconnected nodes of the network. The node further comprises a controller for comparing and synchronizing a phase of the time-continuous synchronization signal generated by the controllable oscillator with the phase of an external time-continuous synchronization signal received from another node of the network by adjusting a frequency of the time-continuous synchronization signal generated by the controllable oscillator. The external time-continuous synchronization signal received from the other node of the network is delayed with respect to the time-continuous synchronization signal transmitted by the other node by a time delay. Such delays can fulfill the function to enable synchronized states in such systems. The time delay may be a transmission time delay resulting from a transmission time between the transmission of the external time-continuous synchronization signal by the other node and the following receipt of the external time-continuous synchronization signal by the node. The transmission time-delay can be tuned adjusting the length of the connection transmitting the synchronization signal and taking into account the signal transmission speed. The time delay may also comprise any tunable additional time delays in addition to the transmission time delay. 
         [0019]    The controller iteratively adjusts the frequency of the time-continuous synchronization signal generated by the controllable oscillator such that a network-wide synchronization of oscillators is achieved for all nodes of the network. The synchronization is thus achieved in a continuous self-organized process through interactions of nodes in the network. 
         [0020]    The controller may be any control system with feedback of the time-continuous synchronization signal generated by the tunable oscillator. 
         [0021]    Specifically, the controller in combination with the controllable oscillator may form a phase locked loop (PLL). PLLs are electronic components able to synchronize their synchronization signals by evaluating mutual phase differences and adjusting their frequencies accordingly. The controller then comprises a phase detector (PD) and a loop filter (LF). The controllable oscillator may be a voltage controlled oscillator (VCO). The phase detector compares the phase of the external time-continuous synchronization signal with the phase of the time-continuous synchronization signal generated by the controllable oscillator. A tunable signal inverter can be placed in the feedback path between the controllable oscillator and the phase detector and/or in each input path and/or in the output path between the controllable oscillator and the input of at least another node. 
         [0022]    A model for mutually coupled PLLs is described below using an analog PLL as an example. The present invention is not restricted to analog PLLs. 
         [0023]    The VCOs output a sinusoidal with constant amplitude, which can be set to 1 without loss of generality, 
         [0000]        x   k ( t )=sin φ k ( t )  (1)
 
         [0000]    where φ k (t) denotes the phase of the oscillatory signal and k=1, 2 indexes the PLL. The phase detector multiplies an external input signal x l  with the output signal x k  of the VCO. Time delays, for example resulting from transmission time delays and/or tunable additional time delays between the PLLs are accounted for by a delay τ of the received signal. Moreover, a feedback delay between the VCO and the PD is accounted for by a delay τ f  in the VCO signal. However, the feedback delay may be zero. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             x 
                             k 
                             PD 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               x 
                               l 
                             
                              
                             
                               ( 
                               
                                 t 
                                 - 
                                 τ 
                               
                               ) 
                             
                           
                           · 
                           
                             
                               x 
                               k 
                             
                              
                             
                               ( 
                               
                                 t 
                                 - 
                                 
                                   τ 
                                   f 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             1 
                             2 
                           
                            
                           
                             [ 
                             
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         φ 
                                         l 
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           - 
                                           τ 
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
                                         φ 
                                         k 
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           - 
                                           
                                             τ 
                                             f 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         φ 
                                         l 
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           - 
                                           τ 
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       
                                         φ 
                                         k 
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           - 
                                           
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                                             f 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0024]    This signal x k   PD  is filtered by the loop filter 
         [0000]        x   k   C ( t )=∫ 0   ∞   dup ( u ) x   k   PD ( t−u )  (3)
 
         [0000]    according to the impulse response p(u) of the LF. The output x k   C  of the LF yields the control signal for the VCO. The dynamic frequency of the VCO is given by its intrinsic frequency ω, which is modulated by the control signal x k   C , 
         [0000]      {dot over (φ)} k ( t )=ω+ K   VCO   x   k   C ( t )  (4)
 
         [0000]    where {dot over (φ)} k  denotes the time-derivative of φ k  and K VCO  is the sensitivity of the VCO. In Eq. (2), the first term containing a phase difference describes low frequency components of the signal, while the second term containing a sum of phases describes high frequency components. Approximating the LF as ideal, we omit the high frequency components in Eq. (2). Hence, the dynamic frequency of the VCO is given by 
         [0000]      {dot over (φ)} k ( t )=ω+ K∫   0   ∞   dup ( u )cos(φ l ( t−τu )−φ k ( t−τ   f   −u )  (5)
 
         [0000]    where K=K VCO /2 is the coupling strength and has the dimension of a frequency. The cosine function containing the phase difference is called the coupling function. This is a closed phase equation for two mutually delay-coupled PLLs. 
         [0025]    Eq. (5) can be extended to a phase model for N delay coupled PLLs with delays between coupled oscillators. A standard state-of-the-art PLL handles only a single input signal. The controller then compares and synchronizes a phase of the time-continuous synchronization signal generated by the controllable oscillator with the phases of external time-continuous synchronization signals received from a plurality of other nodes of the network by adjusting a frequency of the time-continuous synchronization signal generated by the controllable oscillator. 
         [0026]    One aspect of the present invention relates to a combiner for combining external time-continuous synchronization signals received from other nodes of the network to generate a combined external time-continuous synchronization signal. The phase detector compares the phase of the time-continuous synchronization signal generated by the controllable oscillator with the phase of the combined external time-continuous synchronization signal. The combiner may be part of the phase detector (PD). The combiner may be a non-inverting adder. The phase detector (PD) may be a multiplier for analog signals or an XOR gate for digital signals. Alternatively, the phase detector may compare the phase of the time-continuous synchronization signal generated by the controllable oscillator with the phase of each external time-continuous synchronization signal individually to generate a plurality of phase detector signals. The combiner then combines the phase detector signals to control the controllable oscillator. 
         [0027]    The phase model for N coupled analog PLLs reads 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         φ 
                         . 
                       
                       k 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     ω 
                     + 
                     
                       
                         K 
                         
                           n 
                           k 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             l 
                             = 
                             1 
                           
                           N 
                         
                          
                         
                             
                         
                          
                         
                           
                             d 
                             kl 
                           
                            
                           
                             
                               ∫ 
                               0 
                               ∞ 
                             
                              
                             
                                 
                             
                              
                             
                               
                                  
                                 
                                   up 
                                    
                                   
                                     ( 
                                     u 
                                     ) 
                                   
                                 
                               
                                
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         φ 
                                         l 
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           - 
                                           τ 
                                           - 
                                           u 
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
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                                         k 
                                       
                                        
                                       
                                         ( 
                                         
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                                             f 
                                           
                                           - 
                                           u 
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The connections between PLLs are described by the coupling matrix D=(d kl ) with d kl ε{0,1}, where d kl =1 indicates a connection between k and l. The coupling strength is normalized by the number of input signals 
         [0000]    
       
         
           
             
               n 
               k 
             
             = 
             
               
                 ∑ 
                 l 
               
                
               
                   
               
                
               
                 
                   d 
                   kl 
                 
                 . 
               
             
           
         
       
     
         [0000]    Two examples of coupling matrices are given for global coupling with N=4 oscillators and for nearest-neighbor coupling on a 2×2 lattice with periodic boundary conditions: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       D 
                       global 
                     
                     = 
                     
                       ( 
                       
                         
                           
                             0 
                           
                           
                             1 
                           
                           
                             1 
                           
                           
                             1 
                           
                         
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             1 
                           
                           
                             1 
                           
                         
                         
                           
                             1 
                           
                           
                             1 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                         
                           
                             1 
                           
                           
                             1 
                           
                           
                             1 
                           
                           
                             0 
                           
                         
                       
                       ) 
                     
                   
                   , 
                   
                       
                   
                    
                   
                     
                       D 
                       
                         nearest 
                         - 
                         neighbor 
                       
                     
                     = 
                     
                       ( 
                       
                         
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    For global in-phase synchronous states, the phase of all oscillators satisfy 
         [0000]      φ k ( t )=ω t∀k   (8)
 
         [0000]    where Ω denotes a collective frequency of a synchronized state. The collective frequency Ω satisfies 
         [0000]      Ω=ω+ K  cos(Ω[τ−τ f ])  (9)
 
         [0000]    with ∫ 0   ∞ du p(u)=1, and 
         [0000]    
       
         
           
             
               
                 ∑ 
                 l 
               
                
               
                   
               
                
               
                 d 
                 kl 
               
             
             = 
             
               
                 n 
                 k 
               
               . 
             
           
         
       
     
         [0000]    This result is valid for any coupling topology if there is no disjoint set of nodes. 
         [0028]    One aspect of the present invention relates to tuning the time delay between coupled PLLs. A stable in-phase synchronized solution, Eq. (8), with global frequency Ω cannot be achieved for arbitrary time delays between coupled synchronizers. The time delay is a design parameter and can be tuned by an additional delayer and the design of the network. A node may comprise a delayer for inducing an additional time delay to a transmission time delay. The time delay effectively induces a frequency-dependent phase shift to a synchronization signal and, if appropriately tuned, changes the coupling properties such that stable synchronized states become possible. A delayer may be any means suitable to induce such phase shifts. The delayer may need to be specifically tuned for each input path. 
         [0029]    The time delay may be of the order of a period of the controllable oscillator. Specifically, it may exceed one eighth of a period of the controllable oscillator. Thus, networks with large delays can be synchronized. 
         [0030]    The node may further comprise a feedback delayer for inducing a feedback delay τ f  in a feedback path between the controllable oscillator and the phase detector. The feedback delay may compensate for the time delay. The node may further comprise a tunable signal inverter in each input path and/or in a feedback path between the controllable oscillator and the phase detector and/or a tunable signal inverter in an output path between the controllable oscillators for inducing a signal inversion. The collective frequency then depends on the difference between the time delay τ and the feedback delay τ f . 
         [0031]    Furthermore, the time delay τ may be tuned such that it minimizes a perturbation response rate λ in order to arrive at an in-phase synchronized state with maximum stability as will be explained. 
         [0032]    For a phase perturbed by a perturbation q k (t), 
         [0000]      φ k ( t )=Ω t+εq   k ( t )∀ k   (10)
 
         [0000]    where ε is small, Taylor expansion of Eq. (6) to first order in ε at ε=0 yields the linear dynamics of the perturbation, 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         q 
                         . 
                       
                       k 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       α 
                       
                         n 
                         k 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           1 
                         
                         N 
                       
                        
                       
                           
                       
                        
                       
                         
                           d 
                           kl 
                         
                          
                         
                           
                             ∫ 
                             0 
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                              
                             
                               
                                 up 
                                  
                                 
                                   ( 
                                   u 
                                   ) 
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       q 
                                       l 
                                     
                                      
                                     
                                       ( 
                                       
                                         t 
                                         - 
                                         τ 
                                         - 
                                         u 
                                       
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       q 
                                       k 
                                     
                                      
                                     
                                       ( 
                                       
                                         t 
                                         - 
                                         
                                           τ 
                                           f 
                                         
                                         - 
                                         u 
                                       
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where 
         [0000]      α= K  sin(Ω[τ−τ f ])  (12)
 
         [0000]    Substituting the exponential ansatz q k (t)=c k e λt , where λ is complex, into Eq. (11) the characteristic equation is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     λ 
                      
                     
                         
                     
                      
                     
                       c 
                       k 
                     
                   
                   = 
                   
                     
                       α 
                       
                         n 
                         k 
                       
                     
                      
                     
                       
                         p 
                         ^ 
                       
                        
                       
                         ( 
                         λ 
                         ) 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           1 
                         
                         N 
                       
                        
                       
                           
                       
                        
                       
                         
                           d 
                           kl 
                         
                          
                         
                           ( 
                           
                             
                               
                                 c 
                                 l 
                               
                                
                               
                                  
                                 
                                   - 
                                   λτ 
                                 
                               
                             
                             - 
                             
                               
                                 c 
                                 k 
                               
                                
                               
                                  
                                 
                                   - 
                                   
                                     λτ 
                                     f 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where {circumflex over (p)}(λ)=∫ 0   ∞ du p(u)e −λu  is the Laplace transform of the impulse response p(u) of the LF. The in-phase synchronized state Eq. (8) is linearly stable if and only if Re(λ)&lt;0 for all solutions to Eq. (13). In the absence of a time delay τ and a feedback delay τ f , no stable synchronized can exist: for τ−τ f =0, Eq. (12) implies α=0 and Eq. (13) only permits the solution λ=0. This indicates neutral stability, where any small perturbation persists. Hence, only a non-zero difference of time delay τ and a feedback delay τ f  allows for stable in-phase synchronized states. It shall be remarked that two unfavorable effects for in-phase synchronization (non-attractive coupling, time delay induced by transmission delay) combined yield a desirable technical effect. 
         [0033]    Solutions for λ can be obtained by rewriting Eq. (13) in vector form 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                          
                         λτ 
                       
                        
                       
                         ( 
                         
                           
                             λ 
                             
                               α 
                                
                               
                                   
                               
                                
                               
                                 
                                   p 
                                   ^ 
                                 
                                  
                                 
                                   ( 
                                   λ 
                                   ) 
                                 
                               
                             
                           
                           + 
                           
                              
                             
                               - 
                               
                                 λτ 
                                 f 
                               
                             
                           
                         
                         ) 
                       
                     
                      
                     c 
                   
                   = 
                   
                     
                       D 
                       ~ 
                     
                      
                     c 
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    with c=(c 1 , . . . c N ) T  and the normalized coupling matrix {tilde over (D)}=({tilde over (d)} kl ) with {tilde over (d)} kl d kl /n k . For any solution λ, the scalar coefficient on the left hand side of Eq. (14) is an eigenvalue of {tilde over (D)}. A strategy to solve Eq. (14) is thus to solve the equation e λτ [λ/(α{circumflex over (p)}(λ))+e −λτ     f   ]=ζ. The corresponding eigenvectors c are related to the collective perturbation modes for which the linearized dynamics decouple. 
         [0034]    A generalization of the phase model for N coupled PLLs with not necessarily identical specifications reads 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         φ 
                         . 
                       
                       k 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       ω 
                       k 
                     
                     + 
                     
                       
                         
                           K 
                           k 
                         
                         
                           n 
                           k 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             l 
                             = 
                             1 
                           
                           N 
                         
                          
                         
                             
                         
                          
                         
                           
                             d 
                             kl 
                           
                            
                           
                             
                               ∫ 
                               0 
                               ∞ 
                             
                              
                             
                                 
                             
                              
                             
                               
                                  
                                 
                                   
                                     up 
                                     k 
                                   
                                    
                                   
                                     ( 
                                     u 
                                     ) 
                                   
                                 
                               
                                
                               
                                 
                                   h 
                                   k 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         φ 
                                         l 
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           - 
                                           
                                             τ 
                                             kl 
                                           
                                           - 
                                           u 
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
                                         φ 
                                         k 
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           - 
                                           
                                             τ 
                                             
                                               f 
                                               , 
                                               k 
                                             
                                           
                                           - 
                                           u 
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0000]    here ω k  is the intrinsic frequency, K k  is the coupling strength, p k  is the impulse response of the LF, h k  is the coupling function (which is 2π-periodic), τ f,k  is the feedback delay of PLL k and τ kl  is the time delay between PLLs k and l. 
         [0035]    A synchronized state with time-independent phase differences between PLLs is given by 
         [0000]      φ k ( t )=Ω t+β   k   (16)
 
         [0000]    where Ω denotes a collective frequency and β k  is a phase offset of PLL k. If such a state exists, the collective frequency Ω and the phase offsets β k  satisfy the N equations 
         [0000]    
       
         
           
             
               
                 
                   Ω 
                   = 
                   
                     
                       ω 
                       k 
                     
                     + 
                     
                       
                         K 
                         k 
                       
                        
                       
                         
                           ∑ 
                           
                             l 
                             = 
                             1 
                           
                           N 
                         
                          
                         
                             
                         
                          
                         
                           
                             
                               d 
                               kl 
                             
                             
                               n 
                               k 
                             
                           
                            
                           
                             
                               h 
                               k 
                             
                              
                             
                               ( 
                               
                                 
                                   - 
                                   
                                     Ω 
                                      
                                     
                                       [ 
                                       
                                         
                                           τ 
                                           kl 
                                         
                                         - 
                                         
                                           τ 
                                           
                                             f 
                                             , 
                                             k 
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 - 
                                 
                                   Δ 
                                   kl 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where k=1, . . . , N and Δ kl =β k −β l . For a phase perturbed by a perturbation q k (t), 
         [0000]      φ k ( t )=Ω t+β   k   +εq   k ( t )  (18)
 
         [0000]    where ε is small, Taylor expansion of Eq. (15) to first order in ε at ε=0 yields the linear dynamics of the perturbation. As shown before, the characteristic equation for the perturbation response rate λ can be obtained as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           λ 
                           
                             
                               
                                 p 
                                 ^ 
                               
                               k 
                             
                              
                             
                               ( 
                               λ 
                               ) 
                             
                           
                         
                         + 
                         
                           
                              
                             
                               - 
                               
                                 λτ 
                                 
                                   f 
                                   , 
                                   k 
                                 
                               
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 l 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 
                                   d 
                                   kl 
                                 
                                 
                                   n 
                                   k 
                                 
                               
                                
                               
                                 K 
                                 k 
                               
                                
                               
                                 
                                   h 
                                   k 
                                   ′ 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       - 
                                       
                                         Ω 
                                          
                                         
                                           [ 
                                           
                                             
                                               τ 
                                               kl 
                                             
                                             - 
                                             
                                               τ 
                                               
                                                 f 
                                                 , 
                                                 k 
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                     
                                     - 
                                     
                                       Δ 
                                       kl 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       c 
                       k 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         l 
                         = 
                         1 
                       
                       N 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           d 
                           kl 
                         
                         
                           n 
                           k 
                         
                       
                        
                       
                         K 
                         k 
                       
                        
                       
                         
                           h 
                           k 
                           ′ 
                         
                          
                         
                           ( 
                           
                             
                               - 
                               
                                 Ω 
                                  
                                 
                                   [ 
                                   
                                     
                                       τ 
                                       kl 
                                     
                                     - 
                                     
                                       τ 
                                       
                                         f 
                                         , 
                                         k 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             - 
                             
                               Δ 
                               kl 
                             
                           
                           ) 
                         
                       
                        
                       
                          
                         
                           - 
                           
                             λτ 
                             kl 
                           
                         
                       
                        
                       
                         c 
                         l 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where h′ k  denotes the derivative of h k  with respect to its argument. 
         [0036]    For the synchronized state to be stable, the same condition for λ stated above has to be fulfilled. 
         [0037]    Thus, for a desired collective frequency Ω, the time delays τ kl  in combination with the feedback delays τ f,k  may be tuned such that a maximum perturbation decay can be achieved by optimizing the perturbation response rate λ, see Eq. (19). Moreover, further design parameters for optimizing the perturbation response rate λ are the free running frequencies ω k  of the controllable oscillators, the coupling strengths K k , the coupling functions h k , and the impulse responses p k (u) of a filter within the controllers, i.e., the loop filter. 
         [0038]    For systems, in which the specifications of individual PLLs only vary slightly, the system behavior can be well approximated by setting the intrinsic frequencies ω k , the coupling strengths K k , the coupling functions h k , the impulse responses p k , and the feedback delays τ f,k  to k-independent values. For systems, in which the specifications of the interconnections between the PLLs only vary slightly, the system behavior can be well approximated by setting the time delays τ kl  to a k- and l-independent value. 
         [0039]    One aspect of the present invention relates to optimizing the perturbation response by tuning the cut-off frequency of the loop filter. A large class of loop filters can be described by an impulse response p(t) given by the Gamma distribution, 
         [0000]    
       
         
           
             
               
                 
                   
                     p 
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       t 
                       
                         a 
                         - 
                         1 
                       
                     
                      
                     
                       
                          
                         
                           
                             - 
                             t 
                           
                           / 
                           b 
                         
                       
                       
                         
                           b 
                           a 
                         
                          
                         
                           Γ 
                            
                           
                             ( 
                             a 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Γ is the Gamma function, a corresponds to the order of the utilized loop filter and, together with b, determines the cut-off frequency ω c  according to ω c =(ab) −1 . The transfer function of the filter is given by 
         [0000]      {circumflex over ( p )}(λ)=(1+λ b ) −α   (21)
 
         [0000]    The time-continuous synchronization signal may be a digital signal or an analog signal. The node may be a clocking node and the time-continuous synchronization signal may be a clock signal for clocking a device. The present invention further relates to a network comprising a plurality of interconnected nodes that are continuously coupled. The network may be designed to yield a desired perturbation response rate and/or collective frequency. A design parameter of the network is the distance between a node and another node that contributes to the time delay. The time delay corresponding to an optimum perturbation response rate may be achieved either by adjusting the distance alone and/or the additional time delay induced by a delayer. 
         [0040]    The present invention further relates to a method for synchronizing a network comprising a plurality of interconnected nodes. The method comprises generating a time-continuous synchronization signal in each node; transmitting the time-continuous synchronization signal of each node to at least one other respective node of the network; receiving in each node a delayed external time-continuous synchronization signal from the at least another node of the network; and synchronizing in each node a phase of the time-continuous synchronization signal with a phase of the external time-continuous synchronization signal received from the at least one other node by iteratively adjusting the frequency of the time-continuous synchronization signal such that a network-wide synchronization is achieved for all nodes of the network in a continuous self-organized process. 
         [0041]    For the case of digital PLLs with XOR PDs, the coupling function h is given h(Φ)=Λ(Φ), where Λ is a triangular function whose Fourier representation is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     Λ 
                      
                     
                       ( 
                       Φ 
                       ) 
                     
                   
                   = 
                   
                     
                       - 
                       
                         8 
                         
                           π 
                           2 
                         
                       
                     
                      
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         ∞ 
                       
                        
                       
                           
                       
                        
                       
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 [ 
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     i 
                                   
                                   + 
                                   1 
                                 
                                 ] 
                               
                                
                               Φ 
                             
                             ) 
                           
                         
                         
                           
                             ( 
                             
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 i 
                               
                               + 
                               1 
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
         [0042]    For the case of identical specifications of the individual PLLs and interconnections between the PLLs and no delay in the feedback path, the collective frequency Ω of the in-phase synchronized state satisfies 
         [0000]      Ω=ω+ K ∇(Ωτ)  (23)
 
         [0043]    The characteristic equation for the perturbation response rate λ for this case is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     λ 
                      
                     
                         
                     
                      
                     
                       c 
                       k 
                     
                   
                   = 
                   
                     
                       
                         K 
                          
                         
                             
                         
                          
                         
                           
                             Λ 
                             ′ 
                           
                            
                           
                             ( 
                             
                               - 
                               Ωτ 
                             
                             ) 
                           
                         
                       
                       
                         n 
                         k 
                       
                     
                      
                     
                       
                         p 
                         ^ 
                       
                        
                       
                         ( 
                         λ 
                         ) 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           1 
                         
                         N 
                       
                        
                       
                           
                       
                        
                       
                         
                           d 
                           kl 
                         
                          
                         
                           ( 
                           
                             
                               
                                 c 
                                 l 
                               
                                
                               
                                  
                                 
                                   - 
                                   λτ 
                                 
                               
                             
                             - 
                             
                               c 
                               k 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0044]    Network, node and related method according to the invention are described in more detail herein below by way of exemplary embodiments and with reference to the attached drawings, in which: 
           [0045]      FIG. 1  shows a prior art approach for clock distribution on a globally synchronous network involving a master clock; 
           [0046]      FIG. 2  shows a general approach of a dynamic clocking network involving mutually delay-coupled nodes, which are able to globally synchronize in a self-organized manner according to the invention; 
           [0047]      FIG. 3  shows a block diagram of a network node comprising a PLL with a delayed input according to a first embodiment; 
           [0048]      FIG. 4  shows a block diagram of a network node comprising a delay-coupled PLL with a delayed input and an additional delayer in the input path according to a second embodiment; 
           [0049]      FIG. 5  shows a block diagram of a network node comprising a delay-coupled PLL with delayed inputs, an additional delayer for each input and a combiner to combine a plurality of phase detector signals according to a third embodiment; 
           [0050]      FIG. 6  shows a block diagram of a network node comprising a delay-coupled PLL with delayed inputs, an additional delayer for each input and a combiner to combine a plurality of input signals according to a fourth embodiment; 
           [0051]      FIG. 7  shows a block diagram of a network node comprising a delay-coupled PLL with a delayed input and a feedback delayer and tunable inverters according to fifth embodiment; 
           [0052]      FIG. 8  shows a diagram showing a global frequency of the in-phase and anti-phase synchronized state versus the transmission delay for a system of delay-coupled analog PLLs; and 
           [0053]      FIG. 9  shows a diagram showing global synchrony measured using the Kuramoto order parameter as a function of time; 
           [0054]      FIG. 10  shows a diagram showing the perturbation response rate versus the transmission delay for a system of delay-coupled analog PLLs; and 
           [0055]      FIG. 11  shows a diagram showing the perturbation response rate versus the transmission delay for different cut-off frequencies of the loop filter for a system of delay-coupled analog PLLs. 
           [0056]      FIG. 12  shows a diagram showing a global frequency of the in-phase and/or phase-locked anti-phase synchronized state versus the transmission delay in a system of two delay-coupled digital PLLs; 
           [0057]      FIG. 13  shows a diagram showing a global frequency of the in-phase and phase-locked anti-phase synchronized state versus the transmission delay in a system of two delay-coupled digital PLLs with an active inverter in the feedback path between controllable oscillator and phase detector; 
           [0058]      FIG. 14  shows a diagram showing the perturbation response rate versus the transmission delay for a system of two delay-coupled digital PLLs; 
           [0059]      FIG. 15  shows a diagram showing the perturbation response rate versus the transmission delay for a system of two delay-coupled digital PLLs with an active inverter in the feedback path between controllable oscillator and phase detector; 
           [0060]      FIG. 16  shows a diagram showing a global frequency of the in-phase and a phase-locked synchronized state versus the transmission delay in a system of nine delay-coupled digital PLLs on a 3×3 square lattice with periodic boundaries; 
           [0061]      FIG. 17  shows a diagram showing a global frequency of the in-phase and a phase-locked synchronized state versus the transmission delay in a system of nine delay-coupled digital PLLs on a 3×3 square lattice with open boundaries; 
           [0062]      FIG. 18  shows a block diagram of a network node comprising a PLL with a delayed input and a tunable inverter between the output of the controllable oscillator and the phase detector of at least another node according to a sixth embodiment; 
           [0063]      FIG. 19  shows a block diagram of a network node comprising a PLL with a delayed input and tunable inverters in the feedback path between the controllable oscillator and the phase detector and in the output path according to a seventh embodiment; 
           [0064]      FIG. 20  shows an approach of a dynamic clocking network involving delay-coupled nodes with both unidirectional and bidirectional interconnections, which are able to globally synchronize in a self-organized manner according to an eighth embodiment; 
           [0065]      FIG. 21  shows a block diagram of a network node comprising a PLL with a delayed input and an additional delayer in the output path according to a ninth second embodiment; 
       
    
    
       [0066]    Table 1 shows the specifications of the digital PLLs for which measurements are shown in  FIGS. 12 to 15 . 
         [0067]    Table 2 shows the specifications of the digital PLLs for which measurements are shown in  FIGS. 16 and 17 . 
       DETAILED DESCRIPTION OF THE INVENTION 
       [0068]      FIG. 2  shows a dynamic clocking network  22  comprising a plurality of interconnected clocking nodes  21  that are continuously delay-coupled. Each clocking node is implemented as a PLL. Thus, the clocking network  22  is network of mutually delay-coupled PLLs with continuous coupling. 
         [0069]    Referring to  FIG. 3 , the PLL comprises a phase detector  31 , a loop filter  32  and a voltage controlled oscillator  33  that generates a time-continuous clocking signal x k (t). The PLL synchronizes the phase of the clocking signal generated by the VCO  33  with the phase of the external clocking signal x l (t−τ s ) which is delayed by the transmission time delay τ s , indicated by transmission delayer  34 , by adjusting the frequency of the clocking signal of the VCO such that a network-wide synchronization of the VCOs is achieved for all clocking nodes of the dynamic clocking network. To do so, the phase detector  31  compares the phase of the external clocking signal x l (t−τ s ) with the phase of the clocking signal x k (t) generated by the VCO  33  to generate phase detector signal x k   PD (t). After filtering with the loop filter  32  this yields the control signal x k   C (t) for the VCO  33 . 
         [0070]      FIG. 4  shows the node of  FIG. 3  comprising an additional delayer  45  in the input path to adjust the time delay. Transmission time delay τ s  and additional time delay τ d  yield the time delay τ. The phase detector  41  compares the phase of the additionally delayed external clocking signal x l (t−τ) with the phase of the clocking signal x k (t) generated by the VCO  43  to generate the phase detector signal x k   PD (t). After filtering with the loop filter  42  this yields the control signal x k   C (t) for the VCO  43 . By properly inducing an additional time delay, stable solutions for the collective frequency of the network can be achieved. 
         [0071]      FIG. 5  shows a clocking node with a plurality of external clocking signals x 1 (t), x 2 (t), x 3 (t), . . . , x n (t). Each input path comprises an individual delayer  551 ,  552 ,  553 ,  554  that induces an additional time delay to the transmission time delay indicated by transmission delayers  541 ,  542 ,  543 ,  544 . Each phase detector  511 ,  512 ,  513 ,  514  compares the phase of the clocking signal x k (t) generated by the controllable oscillator  53  with the phase of each external clock signal individually to generate a plurality of phase detector signals. The combiner  56  combines the phase detector signals to generate a combined phase detector signal to control the controllable oscillator. The combined phase detector signal is filtered by the loop filter  52  to yield the control signal x k   C (t) for the VCO. The PLL of each clocking node thus adjusts the frequency of the clocking signal of each VCO such that a network-wide synchronization of the VCOs is achieved for all clocking nodes of the dynamic clocking network. By properly inducing individual additional time delays to each input path, stable solutions for the collective frequency of the network can be achieved. 
         [0072]      FIG. 6  shows a clocking node with a plurality of external clocking signals x 1 (t), x 2 (t), x 3 (t), . . . , x n (t). Each input path comprises an individual delayer  651 ,  652 ,  653 ,  654  that induces an additional time delay to the transmission time delay indicated by transmission delayers  641 ,  642 ,  643 ,  644 . In contrast to the embodiment as shown in  FIG. 5 , where a combiner combines a plurality of phase detector signals, in this embodiment the combiner  66  combines the plurality of external clocking signals to generate a combined external clocking signal. The phase detector  61  compares the phase of the clocking signal generated by the VCO  63  with the phase of the combined external clocking signal to generate the phase detector signal x k   PD (t). After filtering with the loop filter  62  this yields the control signal x k   C (t) for the VCO  63 . 
         [0073]      FIG. 7  shows the clocking node of  FIG. 4  comprising a feedback delayer  77  for introducing a time delay in the feedback loop, a tunable inverter  78  for introducing a signal inversion in the feedback path between the controllable oscillator and the phase detector, and a tunable inverter  79  in the input path of the PLL comprising the phase detector  71 , loop filter  72  and VCO  73 . The feedback delay may be induced to compensate for a time delay. 
         [0074]    The individual time delays τ in any of the described embodiments are design parameters. Only if chosen properly, a stable synchronous state can be achieved as will be explained with reference to  FIG. 8  which shows the global frequency Ω of the in-phase and anti-phase synchronized state as a function of the time delay τ for a clocking network comprising two analog PLLs as clocking nodes. The anti-phase synchronized state is characterized by φ 1 (t)=φ 2 (t)−π. Full lines denote stable solutions and dashed lines denote unstable solutions. Hence, for a desired global frequency of the clocking network, the time delay can be chosen for a given free running frequency of the VCO in order to achieve a desired synchronous state and global frequency of the network. If no additional time delay is induced, the transmission time delay corresponds to the time delay. Thus, by choosing the distances between the coupled nodes of the network accordingly, a transmission time delay can be achieved that yields a stable synchronous state. The curves of  FIG. 8  are shown for the following system parameters: VCO free running frequency ω=2π×3.55 GHz, coupling strength K=2π×1.11 GHz, LF order a=1, LF cut-off frequency ω c =2 π×355 MHz. The frequencies of the different solutions can be obtained by letting the clocking network evolve from different initial phase differences. For example, for τ=0.2×2π/ω, all initial phase differences lead to the in-phase synchronized state, see  FIG. 9 , wherein an order parameter of zero means no synchrony and wherein a value of one implies full synchrony. For values of the time delay for which both solutions are stable, the clock network evolves towards one solution according to its initial condition. Moreover, the time delay, the intrinsic frequencies, the coupling strengths, the filter responses, the feedback delays, and the state of the inverters may be chosen such that a perturbation response rate given by Re(λ) is minimized, see Eq. (14). 
         [0075]      FIG. 10  shows a diagram showing the perturbation response rate versus the time delay of a clocking network comprising two analog PLLs as clocking nodes. The coupling matrix corresponding to two mutually coupled PLLs is given by 
         [0000]    
       
         
           
             
               D 
               ~ 
             
             = 
             
               ( 
               
                 
                   
                     0 
                   
                   
                     1 
                   
                 
                 
                   
                     1 
                   
                   
                     0 
                   
                 
               
               ) 
             
           
         
       
     
         [0000]    and has eigenvalues ζ 1 =1 and ζ 2 =−1. It shows a distinct minimum in the region of stable solutions which corresponds to a time delay that is optimal with respect to the perturbation response rate. It shall be mentioned that for a desired global frequency, a maximum perturbation decay of the clock network can be achieved by simultaneously adjusting the time delay and the free running frequency of the VCO, which shifts the curve of the global frequency, see  FIG. 8 , up or down. The coupling strength and the cut-off frequency of the loop filter also affect the stability of the clock network. The curves are shown for the same parameters as in  FIG. 8 . 
         [0076]      FIG. 11  shows the perturbation response rate versus the time delay of a clocking network comprising two analog PLLs as clocking nodes for different cut-off frequencies of the loop filter. Thus, by properly tuning the time delay in combination with the cut-off frequency a minimum perturbation response rate can be achieved. 
         [0077]    The present invention proposes a novel synchronization strategy, specifically for spatially distributed clocks. These clocks are synchronized by networks of coupled phase-locked loops. An important feature is the time delay in the time-continuous coupling between phase-looked loops which enables synchronized states in the presence of a coupling mechanism that does not permit stable synchronized states for negligible time delays and a non-attractive coupling mechanism. As the transmission time delay is not limited to one eighth of a period of the oscillator as is the case with the solution disclosed in WO 2013/178237 A1, networks with larger time delays between nodes can be synchronized. Important applications are, e.g., high performance MPSoCs architectures, distributed antenna arrays, and other large scale electronic clocking systems communicating by means of time-continuous signals. The present invention specifically provides a simplified clock network compared to the prior art tree structure. The synchronized network thus enables an increased energy efficiency due to shorter connections and less amplification. Moreover, it exhibits increased robustness towards failure of individual components due to a decentralized architecture. Furthermore, the synchronized network is designed for high quality of oscillations. The synchronized network can be realized using readily available hardware components. Thus, this solution works with readily available hardware combined in a novel way and additionally simplifies clock distribution, thereby reducing power consumption and increasing scalability. 
         [0078]      FIG. 12  shows the global frequency Ω of the in-phase and phase-locked (here anti-phase) synchronized state as a function of the time delay τ for a clocking network comprising two digital PLLs as clocking nodes. The curves of  FIG. 12  are obtained using the phase model for digital PLLs. They are shown for the following system parameters: VCO free running frequency ω=2π×1009.5 Hz, coupling strength K VCO =2π×814.5 Hz, LF order a=1, LF cut-off frequency ω c =2π×14 Hz. The symbols show data points measured in an experimental setup with two digital PLLs whose specifications are given in  FIG. 22 . 
         [0079]      FIG. 13  shows the global frequency Ω of the in-phase and phase-locked (here anti-phase) synchronized state as a function of the time delay τ for a clocking network comprising two digital PLLs as clocking nodes with an active inverter in the feedback path between controllable oscillator and phase detector. The curves of  FIG. 13  are obtained using the phase model for digital PLLs. They are shown for the following system parameters: VCO free running frequency ω c =2π×1009.5 Hz, coupling strength K VCO =2π×814.5 Hz, LF order a=1, LF cut-off frequency ω c =2π×14 Hz. The symbols show data points measured in an experimental setup with two digital PLLs whose specifications are given in  FIG. 22 . 
         [0080]      FIG. 14  shows a diagram showing the perturbation response rate versus the time delay for a clocking network comprising two digital PLLs as clocking nodes. The curves of  FIG. 14  are obtained using the phase model for digital PLLs. They are shown for the same parameters as in  FIG. 12 . The symbols show data points measured in an experimental setup with two digital PLLs whose specifications are given in  FIG. 22 . 
         [0081]      FIG. 15  shows a diagram showing the perturbation response rate versus the time delay for a clocking network comprising two digital PLLs as clocking nodes with an active inverter in the feedback path between controllable oscillator and phase detector. The curves of  FIG. 15  are obtained using the phase model for digital PLLs. They are shown for the same parameters as in  FIG. 13 . The symbols show data points measured in an experimental setup with two digital PLLs whose specifications are given in  FIG. 22 . 
         [0082]      FIG. 16  shows the global frequency Ω of the in-phase and a phase-locked synchronized state, in which there is a phase difference of 2π/3 between coupled nodes, as a function of the time delay τ for a clocking network comprising nine digital PLLs on a 3×3 square lattice with periodic boundaries as clocking nodes. The curves of  FIG. 16  are obtained using the phase model for digital PLLs. They are shown for the following system parameters: VCO free running frequency ω=2π×997 Hz, coupling strength K VCO =2π×816 Hz, LF order a=1, LF cut-off frequency ω c =2π×4 Hz. The symbols show data points measured in an experimental setup with nine digital PLLs on a 3×3 square lattice with periodic boundaries whose specifications are given in  FIG. 23 . 
         [0083]      FIG. 17  shows the global frequency Ω of the in-phase and a phase-locked synchronized state, in which there is a phase difference of π between coupled nodes, as a function of the time delay τ for a clocking network comprising nine digital PLLs on a 3×3 square lattice with open boundaries as clocking nodes. The curves of  FIG. 17  are obtained using the phase model for digital PLLs. They are shown for the following system parameters: VCO free running frequency ω c =2π×997 Hz, coupling strength K VCO =2π×816 Hz, LF order a=1, LF cut-off frequency ω c =2π×14 Hz. The symbols show data points measured in an experimental setup with nine digital PLLs on a 3×3 square lattice with open boundaries whose specifications are given in  FIG. 23 . 
         [0084]      FIG. 18  shows the clocking node of  FIG. 4  comprising a tunable inverter  189  for introducing a signal inversion in the output path between the controllable oscillator of the PLL and the phase detector of at least another node. 
         [0085]      FIG. 19  shows the clocking node of  FIG. 4  comprising a tunable inverter  199  for introducing a signal inversion in the output path between the controllable oscillator of the PLL and the phase detector of at least another node and a tunable inverter  198  for introducing a signal inversion in the in the feedback path between the controllable oscillator and the phase detector. 
         [0086]      FIG. 20  shows a dynamic clocking network comprising a plurality of interconnected clocking nodes  201 ,  202 ,  203 ,  204  that are continuously delay-coupled. Each clocking node is implemented as a PLL. The interconnections between the clocking nodes  201 ,  202 ,  203  are unidirectional, whereas the interconnection between the clocking nodes  202  and  204  is bidirectional. Hence, a clocking network of delay-coupled PLLs with continuous coupling can contain both unidirectional and bidirectional interconnections. 
         [0087]      FIG. 21  shows the node of  FIG. 3  comprising an additional delayer  215  in the output path between the controllable oscillator and the phase detector of at least another node to adjust the time delay. 
         [0088]      FIG. 22  shows the specifications of the digital PLLs for which measurements are shown in  FIGS. 12 to 15 . 
         [0089]      FIG. 23  shows the specifications of the digital PLLs for which measurements are shown in  FIGS. 16 and 17 .