Phase response

The phase response, and phase frequency response or Phasenmaß (English phase response) is usually considered in conjunction with the amplitude response amplitude or frequency response.

From the phase shift can be a derivative with respect to frequency calculate the group delay, the spoken vividly describes the frequency-dependent signal delay.

Amplitude and phase characteristic point in the frequency domain representation of the signal or in a frequency- sensitive system, the dependence of the amplitude and the phase of the frequency ( amplitude and phase diagram).

Both sizes shown as a graph, also referred to as amplitude ( magnitude frequency response ) and phase response (phase frequency response), which in combination also Bode diagram. If both pieces of information together to form a complex function, one also speaks of the complex frequency response.

Metrological limitations

In metrology, for receiving the phase transition to a continuous sinusoidal signal is normally used, which means that phase shifts can be measured only in the range of ± 180 ° or ± π. From a metrological recorded phase transition, therefore, the group delay can only partly be derived.

Theory

First separating the transfer function of a causal linear time-invariant system according to the real and imaginary parts:

In a second step is required, the transmission factor

Is related to the transfer function by the following equation:

The second factor, in this case, the phase term, therefore corresponds to the phase function of the frequency and the phase response represents

If we introduce now the phase on the original transfer function returns results

The non- uniqueness of the inverse tangent function leads to the restrictions described in the preceding sections (range only up ).

The problem are those points where the transfer function has zeros or poles, as by

For there then arise singularities.

In order to determine the phase now, it makes sense to switch from the Fourier domain into the Laplace domain ( s-plane ) (see Laplace transform ), not just the imaginary axis, but the entire complex frequency plane to look at. A first requirement, which is needed in order to determine the phase response can be

For a start value is set to bypass the non- uniqueness of the phase (). To the phase curve now actually determine you walk in the s- plane along the imaginary axis from the origin to the positive frequencies and from the origin toward the negative frequencies, bypassing the poles and zeros by semicircular " indentations " in the right half-plane.

Statement is an example: n-fold zero of at.

Taylor expansion in the vicinity of zero, termination after the first member:

Wherein the value of the n th derivation of the location means.

Semi-circular indentation: radius, angle

Follows:

And therefore:

For the phase now applies:

Since changes to along this indentation, the phase changes by a total of.

With a pole yields the opposite sign conditions, the phase takes to.