Tellegen's theorem

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

The theorem

There are two systems S and S ', which are described by the signal flow graph given. This need to not necessarily be linear, but have the same number of nodes, N. The node signals are, respectively, the signals of the paths between nodes i and j, respectively, and the input signals denoted by or. The Tellegen'sche theorem states then:

The left-hand sum contains only "internal" operations, while the right sum covers only the input signals. From this form, no statement can be derived, we should incorporate specific cases are considered.

Derivation

We first consider only the node signals in the time being pointless and seemingly trivial identity

For the nodes signals can be dropped:

Or

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

LTI case

If the transfer functions of the paths in the two systems linear and time-invariant, then the theorem can be rewritten in a simpler form. Will be replaced by their z transforms first timing signals. Each signal path is now as a signal of the root node multiplied represented by the transfer function of the path.

The theorem can now be rewritten as

From this it can be derived between the systems now relatively simple contexts.

Transposition

Is that too comparative system S ' the transposed to S system, and have the systems, only one input and one output, then they have the same transfer function. This will now be proved by means of the Tellegen 's theorem for linear systems.

The transposed results from the system S, and vice versa by the input to the output node. In addition, all paths are reversed ( with constant path transfer function ), i.e.

.

Inserting this condition in the theorem will fall on the left-hand sum and it remains

. stand It is now further assumed that the system S includes an input node () and an output node (). The transpose system is then at the input node and the output node with. The remaining sum then reduces to

As is follows

This means nothing less than that the output signals coincide with the same input signal, the transfer function is thus equal.

Sensitivity analysis

It is again a linear system S are considered, which has only one input and one output signal ( can with the same argument on any number of inputs and outputs can be generalized ). We will now examine how the transfer function of S changes if exactly one path, such as the node is between h and l, as amended.

It creates a new system

The other system components are transferred to the new system

;; ;

This system is then compared using the Tellegen 's theorem with the transposed output system.

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

Since the expression and can be further simplified to

Where and is now.

The node can be made by signals (internal ) transmission functions in conjunction with the input signal. Thus, and

By rearranging, we obtain

The only remaining unknown in this equation is. It can be calculated with this equation exactly by the node h is used as an output node instead of b.

.

This can be rewritten as

.

By default insertion then yields the equation

,

Contains only the functions of the original system.