Phase delay

The phase delay , engl. phase delay is the time delay of a carrier of a certain frequency or the time shift between two sinusoidal oscillations with the frequency . The phase delay is the time in which a stationary signal at the output reaches the phase state of the stationary signal at the input. The signal at the output is therefore phase-shifted compared to the signal at the input . ${\displaystyle \tau }$ ${\displaystyle f}$

calculation

Comparison of phase and group delay on a bandpass

Mathematically, the phase delay results from the negative argument of the transfer function of the system, divided by the angular frequency : ${\displaystyle \tau _{\mathrm {ph} }}$ ${\displaystyle H(\mathrm {j} \omega )}$ ${\displaystyle \omega =2\pi f}$

${\displaystyle \tau _{\mathrm {ph} }(\omega )=-{\frac {\varphi (\omega )}{\omega }}\,}$

with . ${\displaystyle \varphi (\omega )=\arg\{H(\mathrm {j} \omega )\}}$

${\displaystyle H(\mathrm {j} \omega )}$is the transfer function of the system, its phase and the phase delay of the system. ${\displaystyle \varphi (\omega )}$ ${\displaystyle \tau _{\mathrm {ph} }(\omega )}$

The argument of the transfer function can be determined from the relationship

${\displaystyle \arctan \left({\frac {\operatorname {Im} \{H(j\omega )\}}{\operatorname {Re} \{H(j\omega )\}}}\right)+k2\pi \,}$.

The real integer factor can be determined from the pole / zero diagram of the transfer function. Because of the ambiguity of the phase around 2π, the group delay is used to describe the system delay , with which there is no ambiguity of the phase. ${\displaystyle k}$

Purely graphically, the phase delay means the time by which a frequency component of the frequency in the output signal is delayed (shifted) compared to the input signal . If the input signal is mono- frequency , the output signal is delayed by the phase delay: ${\displaystyle \tau _{\mathrm {ph} }}$ ${\displaystyle f}$ ${\displaystyle y(t)}$ ${\displaystyle x(t)}$ ${\displaystyle x(t)}$ ${\displaystyle y(t)}$ ${\displaystyle \tau _{\mathrm {ph} }}$

${\displaystyle y(t)=x(t+\tau _{\mathrm {ph} })\,}$.

If the attenuation is constant in the frequency band under consideration , as is the case with a bandpass , the phase delay corresponds to the delay time of the carrier frequency , shown on the right in purple, while the group delay corresponds to the delay time of the envelope curve. The envelope curve is shown as a blue dashed line and represents the information contained in the signal. ${\displaystyle A(\omega )}$ ${\displaystyle \Delta \omega }$ ${\displaystyle \tau _{\mathrm {ph} }}$ ${\displaystyle \tau _{\mathrm {gr} }}$

literature

• Curt Rint : Handbook for Radio Frequency and Electrical Technicians. 2nd volume. 13th revised edition. Hüthig and Pflaum Verlag GmbH, Heidelberg 1981, ISBN 3-7785-0699-4 .
• Helmut Röder, Heinz Ruckriegel, Heinz Häberle: Electronics. Part 3: communications electronics. 5th edition. Verlag Europa-Lehrmittel Nourney, Wuppertal 1980, ISBN 3-8085-3225-4 .