Tellegen theorem

The Tellegen theorem (developed by BDH Tellegen ) is mainly used in digital signal processing for the design of filters. In its pure form, the theorem is a kind of conservation law, but several relationships between signal flow graphs can be derived from it.

The theorem

Two systems S and S 'that can be compared with the Tellegen theorem

There are two systems S and S ', which are described by signal flow graphs . These do not necessarily have to be linear at first, but have the same number of nodes, namely N. The node signals are denoted by , or , the signals of the paths between nodes i and j are denoted by or, and the input signals are denoted by or . Tellegen's theorem then says: ${\displaystyle w_{k}}$ ${\displaystyle w'_{k}}$ ${\displaystyle s_{ij}}$ ${\displaystyle s'_{ij}}$ ${\displaystyle x_{k}}$ ${\displaystyle x'_{k}}$

${\displaystyle \sum _{k=1}^{N}\sum _{j=1}^{N}(w'_{k}.s_{jk}-w_{k}.s'_{jk})+\sum _{k=1}^{N}(w_{k}.x'_{k}-w'_{k}.x_{k})=0}$

The sum on the left contains only "internal" processes, while the sum on the right only deals with the input signals. No statement can yet be derived from this form; specific cases must be considered.

Derivation

We initially only consider the node signals in the identity, which at first sight seems pointless and trivial

${\displaystyle \sum _{k=1}^{N}(w_{k}\cdot w_{k}'-w_{k}'\cdot w_{k})=0}$

The following can be used for the node signals:

${\displaystyle w_{k}=\sum _{j=1}^{N}s_{jk}+x_{k}}$ or.

${\displaystyle w'_{k}=\sum _{j=1}^{N}s'_{jk}+x'_{k}}$

Inserting and dividing the sum leads exactly to the above form.

LTI case

If the transfer functions of the paths in both systems are linear and time-invariant, then the theorem can be rewritten in a simpler form. First the time signals are replaced by their z-transforms. Each path signal can now be represented as a signal of the root node multiplied by the transfer function of the path . ${\displaystyle F_{ij}}$

${\displaystyle w_{k}[n]\rightarrow W_{k}(z)}$

${\displaystyle x_{k}[n]\rightarrow X_{k}(z)}$

${\displaystyle s_{ij}[n]\rightarrow W_{i}(z).F_{ij}(z)}$

The theorem can now be rewritten as

${\displaystyle \sum _{k=1}^{N}\sum _{j=1}^{N}W'_{j}.W_{k}.(F'_{jk}-F_{kj})+\sum _{k=1}^{N}(W_{k}.X'_{k}-W'_{k}.X_{k})=0}$

From this it is now relatively easy to derive relationships between the systems.

Transposition

If the system to be compared S 'is the system transposed to S , and the systems only have one input and one output each, then they have the same transfer function . This shall now be proven for linear systems using the Tellegen theorem. ${\displaystyle S^{T}}$

The transposed system arises from S, in that the input nodes become the output nodes and vice versa. In addition, all paths are reversed (with the path transfer function remaining the same), i.e. H.

${\displaystyle F_{ij}^{T}=F_{ji}}$.

Inserting this condition into the theorem removes the left sum and it remains ${\displaystyle \sum _{k=1}^{N}\sum _{j=1}^{N}W_{j}^{T}.W_{k}.(F_{jk}^{T}-F_{kj})+\sum _{k=1}^{N}(W_{k}.X_{k}^{T}-W_{k}^{T}.X_{k})=0}$

${\displaystyle \sum _{k=1}^{N}(W_{k}.X_{k}^{T}-W_{k}^{T}.X_{k})=0}$

stand. It is now further assumed that the system S has an input node ( ) and an output node ( ). The transposed system then has the input node at and the output node at . The remaining amount is then reduced to ${\displaystyle w_{a}}$ ${\displaystyle w_{b}}$ ${\displaystyle w_{b}^{T}}$ ${\displaystyle w_{a}^{T}}$

${\displaystyle W_{b}.X_{b}^{T}-W_{a}^{T}.X_{a}=0}$

There follows ${\displaystyle X_{b}^{T}=X_{a}=X}$

${\displaystyle W_{b}=W_{a}^{T}}$

Which means nothing else than that the output signals match with the same input signal, so the transfer function is the same.

Sensitivity analysis

A linear system S is to be considered again, which has only one input and one output signal (can be generalized to any number of inputs and outputs with the same argumentation). It will now be investigated how the transfer function of S changes when exactly one path, e.g. B. that between nodes h and l is changed. ${\displaystyle H(z)}$

So a new system is created

${\displaystyle S\rightarrow S^{\Delta }:F_{hl}(z)\rightarrow F_{hl}^{\Delta }(z)=F_{hl}(z)+\Delta F_{hl}(z)}$

The other system components will also be transferred to the new system

${\displaystyle W_{k}(z)\rightarrow W_{k}^{\Delta }(z)}$; ; ; ${\displaystyle X(z)\rightarrow X^{\Delta }(z)=X(z)}$ ${\displaystyle F_{ij}(z)\rightarrow F_{ij}^{\Delta }(z)=F_{ij}(z)|_{i\neq h\land j\neq l}}$ ${\displaystyle H(z)\rightarrow H^{\Delta }}$

This system is now compared with the transposed starting system using the Tellegen theorem . ${\displaystyle S^{T}}$ ${\displaystyle \sum _{k=1}^{N}\sum _{j=1}^{N}W_{j}^{\Delta }.W_{k}^{T}.(F_{jk}^{\Delta }-F_{kj}^{T})+\sum _{k=1}^{N}(W_{k}^{T}.X_{k}^{\Delta }-W_{k}^{\Delta }.X_{k}^{T})=0}$

In the sum on the left, all summands are again zero, except for those for j = h and k = l. With the requirement of an input signal (node ​​a) and an output signal (node ​​b), the right-hand sum can also be reduced again.

${\displaystyle W_{h}^{\Delta }.W_{l}^{T}.(F_{hl}^{\Delta }-F_{lh}^{T})+W_{a}^{T}.X-W_{b}^{\Delta }.X=0}$

There and the expression can be further simplified to ${\displaystyle F_{lh}^{T}(z)=F_{hl}(z)}$ ${\displaystyle F_{hl}^{\Delta }=F_{hl}(z)+\Delta F_{hl}(z)}$

${\displaystyle W_{h}^{\Delta }.W_{l}^{T}.\Delta F_{hl}+W_{a}^{T}.X-W_{b}^{\Delta }.X=0}$

Whereby now and is. ${\displaystyle W_{a}^{T}(z)=H(z).X(z)}$ ${\displaystyle W_{b}^{\Delta }(z)=H^{\Delta }(z).X(z)}$

The node signals can also be linked to the input signal using (internal) transfer functions. So will and ${\displaystyle W_{h}^{\Delta }(z)=H_{ah}^{\Delta }(z)X(z)}$ ${\displaystyle W_{l}^{T}(z)=H_{bl}^{T}(z).X(z)}$

By reshaping one then obtains

${\displaystyle H^{\Delta }-H=\Delta H=H_{ah}^{\Delta }.H_{bl}^{T}.\Delta F_{hl}=H_{ah}^{\Delta }.H_{lb}.\Delta F_{hl}}$

The only remaining unknown in this equation is . It can be calculated with exactly this equation by using the node h instead of b as the starting node. ${\displaystyle H_{ah}^{\Delta }}$

${\displaystyle H_{ah}^{\Delta }-H_{ah}=H_{ah}^{\Delta }.H_{lh}.\Delta F_{hl}}$.

This can be transformed into

${\displaystyle H_{ah}^{\Delta }={\frac {H_{ah}}{1-H_{lh}.\Delta F_{hl}}}}$.

Inserting it back then gives the equation

${\displaystyle \Delta H={\frac {H_{ah}}{1-H_{lh}.\Delta F_{hl}}}.H_{lb}.\Delta F_{hl}}$,

which only contains functions from the original system.

literature

• Alan V. Oppenheim, Ronald W. Schafer: Digital Signal Processing . Prentice-Hall, 1975, ISBN 0-13-214635-5 .