cbadc.analog_system.topology.chain

cbadc.analog_system.topology.chain(analog_systems: List[AnalogSystem]) → AnalogSystem

Construct an analog system by chaining several analog systems.

The chaining is achieved by chainging \(\hat{N}\) systems, parameterized by \(\mathbf{A}_1, \mathbf{B}_1, \mathbf{C}^\mathsf{T}_1, \mathbf{D}_1, \mathbf{\Gamma}_1, \tilde{\mathbf{\Gamma}}_1, \dots, \mathbf{A}_{\hat{N}}, \mathbf{B}_{\hat{N}}, \mathbf{C}^\mathsf{T}_{\hat{N}}, \mathbf{D}_{\hat{N}}, \mathbf{\Gamma}_{\hat{N}}, \tilde{\mathbf{\Gamma}}_{\hat{N}}\), as

\(\mathbf{A} = \begin{pmatrix} \mathbf{A}_1 \\ \mathbf{B}_2 \mathbf{C}_1^\mathsf{T} & \mathbf{A}_2 \\ \mathbf{B}_3 \mathbf{D}_2 \mathbf{C}_1^\mathsf{T} & \mathbf{B}_3 \mathbf{C}_2^\mathsf{T} & \mathbf{A}_3 \\ \mathbf{B}_4 \mathbf{D}_3 \mathbf{D}_2 \mathbf{C}_1^\mathsf{T} & \mathbf{B}_4\mathbf{D}_3\mathbf{C}_2^\mathsf{T} & \mathbf{B}_4 \mathbf{C}_3^\mathsf{T} & \mathbf{A}_4 \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix}\)

\(\mathbf{B} = \begin{pmatrix} \mathbf{B}_1 \\ \mathbf{B}_2 \mathbf{D}_1 \\ \mathbf{B}_3 \mathbf{D}_2 \mathbf{D}_1 \\ \mathbf{B}_4 \mathbf{D}_3 \mathbf{D}_2 \mathbf{D}_1 \\ \vdots \end{pmatrix}\)

\(\mathbf{C}^\mathsf{T} = \begin{pmatrix} \mathbf{D}_{\hat{N}}\cdots \mathbf{D}_2 \mathbf{C}_1^\mathsf{T} & \mathbf{D}_{\hat{N}}\cdots \mathbf{D}_3 \mathbf{C}_2^\mathsf{T} & \mathbf{D}_{\hat{N}} \cdots \mathbf{D}_4 \mathbf{C}_3^{\mathsf{T}} & \dots & \mathbf{D}_{\hat{N}} \mathbf{C}_{\hat{N}-1}^\mathsf{T} & \mathbf{C}_{\hat{N}}^\mathsf{T} \end{pmatrix}\)

\(\mathbf{\Gamma} = \begin{pmatrix} \ddots \\ & \mathbf{\Gamma}_\ell \\ & & \mathbf{\Gamma}_{\ell + 1} \\ & & & \ddots \end{pmatrix}\)

\(\tilde{\mathbf{\Gamma}}^\mathsf{T} = \begin{pmatrix} \ddots \\ & \tilde{\mathbf{\Gamma}}^\mathsf{T}_\ell \\ & & \tilde{\mathbf{\Gamma}}^\mathsf{T}_{\ell + 1} \\ & & & \ddots \end{pmatrix}\)

\(\mathbf{D} = \mathbf{D}_{\hat{N}} \mathbf{D}_{\hat{N}-1} \dots \mathbf{D}_{1}\)

where \(N = \sum_\ell N_\ell\), \(L = L_1\), and \(\tilde{N} = N_{-1}\). Above the index \(-1\) refers to the last index of the list.

The systems in the list are chained in the order they are listed.

Parameters

analog_systems (List[cbadc.analog_system.AnalogSystem]) – a list of analog systems to be chained.

Returns

a new analog system

Return type

cbadc.analog_system.AnalogSystem