# Phase response

The phase response , also phase frequency response or phase measure (English phase response ), is mostly considered in connection with the amplitude response or amplitude frequency response .

Example of a low-pass phase response

From the phase shift can be a derivative according to the frequency , the group delay calculate the visual terms, describes the frequency-dependent signal delay.

In the representation of the frequency level in a signal or frequency-sensitive system, the amplitude and phase response show the dependence of the amplitude and the phase on the frequency (amplitude and phase diagram).

Put simply, the phase response indicates the frequency-dependent phase shift between the input and output signal. A simple example is a high-pass filter to which a sinusoidal signal is applied. Depending on the frequency, the output signal is out of phase with the input signal.

Both variables shown as a graph are also referred to as the amplitude response (magnitude frequency response) or phase response (phase frequency response), in combination also called the Bode diagram . If both pieces of information are combined into a complex function, one also speaks of a complex frequency response.

## Metrological restrictions

In measurement technology, a continuous sinusoidal signal is usually used to record the phase response, which means that phase shifts can only be measured in the range of ± 180 ° or ± π . The group delay can therefore only be derived to a limited extent from a phase response recorded by measurement.

## theory

First, one separates the transfer function of a causal, linear, time-invariant system into real and imaginary parts :

${\ displaystyle {\ mathcal {}} H (\ mathrm {j} \ omega) = M (\ omega) + \ mathrm {j} N (\ omega)}$

In a second step you need the transfer rate

${\ displaystyle {\ mathcal {}} \ Gamma (\ omega) = A (\ omega) + \ mathrm {j} B (\ omega)}$,

which is related to the transfer function by the following equation:

${\ displaystyle H (j \ omega) = e ^ {- \ Gamma} = e ^ {- (A (\ omega) + \ mathrm {j} B (\ omega))} = e ^ {- A (\ omega )} \ cdot e ^ {- \ mathrm {j} B (\ omega)}}$

The second factor ,, is the phase term, accordingly it corresponds to the phase as a function of the frequency and represents the phase response. ${\ displaystyle {\ mathcal {}} e ^ {- jB (\ omega)}}$${\ displaystyle {\ mathcal {}} B (\ omega)}$

If one now leads the phase back to the original transfer function, the result is ${\ displaystyle {\ mathcal {}} B (\ omega)}$

${\ displaystyle B (\ omega) = - \ arctan {\ frac {N (\ omega)} {M (\ omega)}}}$

The ambiguity of the arctangent function leads to the restrictions described in the sections above (range of values ​​only up to ). ${\ displaystyle {\ mathcal {}} - \ pi}$${\ displaystyle {\ mathcal {}} \ pi}$

The points at which the transfer function has zero or pole points are problematic , as they result ${\ displaystyle {\ mathcal {}} H (j \ omega)}$

${\ displaystyle {\ mathcal {}} \ Gamma = - \ ln {H (\ mathrm {j} \ omega)}}$

for there then singularities result. ${\ displaystyle {\ mathcal {}} \ Gamma}$

In order to be able to determine the phase now, it makes sense to switch from the Fourier range to the Laplace range (s-plane) (see Laplace transformation ), i.e. not only to consider the imaginary axis, but the entire complex frequency plane . A first requirement that is needed in order to be able to determine the phase progression is

${\ displaystyle {\ mathcal {}} \ Gamma (0) = 0}$

This defines a starting value in order to circumvent the ambiguity of the phase ( ). In order to actually be able to determine the phase course, one runs in the s-plane along the imaginary axis starting from the origin to the positive frequencies and from the origin in the direction of the negative frequencies and bypasses the poles and zeros through semicircular "indentations" the right half plane. ${\ displaystyle {\ mathcal {}} \ pm 2 \ pi}$

Explanation based on an example: n-fold zero of at .${\ displaystyle {\ mathcal {}} H (s)}$${\ displaystyle {\ mathcal {}} s = \ mathrm {j} \ omega _ {0}}$

Taylor expansion near the zero point, termination after the first term:

${\ displaystyle {\ mathcal {}} H (s) = (s- \ mathrm {j} \ omega _ {0}) ^ {n} H ^ {(n)}}$

where means the value of the nth derivative at the point . ${\ displaystyle {\ mathcal {}} H ^ {(n)}}$${\ displaystyle {\ mathcal {}} \ mathrm {j} \ omega _ {0}}$

Semicircular indentation: radius , angle${\ displaystyle {\ mathcal {}} \ rho}$${\ displaystyle {\ mathcal {}} \ theta = [- {\ tfrac {\ pi} {2}} \ ldots {\ tfrac {\ pi} {2}}]}$

${\ displaystyle {\ mathcal {}} s = \ mathrm {j} \ omega _ {0} + \ rho e ^ {\ mathrm {j} \ theta}}$

follows:

${\ displaystyle {\ mathcal {}} H (s) (s- \ mathrm {j} \ omega _ {0}) ^ {n} H ^ {(n)} = (\ mathrm {j} \ omega _ { 0} + \ rho e ^ {\ mathrm {j} \ theta} - \ mathrm {j} \ omega _ {0}) ^ {n} H ^ {(n)} = \ rho ^ {n} H ^ { (n)} e ^ {\ mathrm {j} n \ theta}}$

and therefore:

${\ displaystyle {\ mathcal {}} \ Gamma (\ omega) = - \ ln {H (s)} = - \ ln (\ rho ^ {n} H ^ {(n)} e ^ {\ mathrm {j } n \ theta}) = - n \ ln {\ rho} - \ ln {H ^ {(n)}} - \ mathrm {j} n \ theta}$

the following applies to the phase:

${\ displaystyle {\ mathcal {}} B (\ omega) = \ arg (H ^ {(n)}) - n \ theta}$

Since changes to along this indentation , the phase changes overall to . ${\ displaystyle {\ mathcal {}} \ theta}$${\ displaystyle {\ mathcal {}} \ pi}$${\ displaystyle {\ mathcal {}} - n \ cdot \ pi}$

In the case of a pole, the sign ratios are reversed, the phase increases by . ${\ displaystyle {\ mathcal {}} n \ cdot \ pi}$

## literature

• Alfred Fettweis: Elements of communications systems. 2nd Edition. J.Schlembach Fachverlag, Wilburgstetten 2004, ISBN 3-935340-41-9 .
• Gert Hagmann: Fundamentals of electrical engineering. 6th edition. AULA-Verlag GmbH, Wiesbaden 1997, ISBN 3-89104-614-6 .
• Curt Rint : Handbook for high frequency and electrical technicians Volume 2. 13th edition. Hüthig and Pflaum Verlag GmbH, Heidelberg 1981, ISBN 3-7785-0699-4 .