Phase & Group Velocity - Revision history

Carlos.Lopez at 11:35, 23 February 2012

2012-02-23T11:35:00Z

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	\|Level=Intermediate		\|Level=Intermediate
	\|YearOfPublication=2011		\|YearOfPublication=2011
	~~\|Logo=gAGE~~
	\|Title={{PAGENAME}}		\|Title={{PAGENAME}}
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Jaume.Sanz at 11:36, 6 February 2012

2012-02-06T11:36:30Z

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	In the context of GNSS signals, the carrier waves are propagating with the phase velocity and thence, this is the velocity associated to the carrier measurements. The group velocity is what has to be considered for the code measurements, i.e, pseudoranges obtained from the codes modulated in the carriers <ref group="footnotes"> Notice that the information and energy is propagated in a wave as a change in the wave, e.g, by modulating the codes. Thence, the information propagates with the group velocity.</ref>.		In the context of GNSS signals, the carrier waves are propagating with the phase velocity and thence, this is the velocity associated to the carrier measurements. The group velocity is what has to be considered for the code measurements, i.e, pseudoranges obtained from the codes modulated in the carriers <ref group="footnotes"> Notice that the information and energy is propagated in a wave as a change in the wave, e.g, by modulating the codes. Thence, the information propagates with the group velocity.</ref>.

	As it is explained in the next sections, the troposphere is a non dispersive ~~media~~ for radio waves at the GNSS frequencies. Thence, code and carrier phase measurements are associated to the same velocity in the troposphere. By contrast, codes and phases propagate at different speeds in the ionosphere, resulting in a delay in the code and an advance in the carrier measurements, see [[Ionospheric Delay]].		As it is explained in the next sections, the troposphere is a non dispersive medium for radio waves at the GNSS frequencies. Thence, code and carrier phase measurements are associated to the same velocity in the troposphere. By contrast, codes and phases propagate at different speeds in the ionosphere, resulting into a delay in the code and an advance in the carrier measurements, see [[Ionospheric Delay]].

Jaume.Sanz at 11:33, 6 February 2012

2012-02-06T11:33:30Z

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	The ''refractive index'' (<math>n</math>) in a medium is the ratio between the speed ~~propagation~~ of signal in a vacuum (<math>c</math>) to the speed in the medium (<math>v</math>), thence, two different refractive indexes can be defined for the group and phase:		The ''refractive index'' (<math>n</math>) in a medium is the ratio between the propagation speed of signal in a vacuum (<math>c</math>) to the speed in the medium (<math>v</math>), thence, two different refractive indexes can be defined for the group and phase:

	::<math>		::<math>

Jaume.Sanz at 11:31, 6 February 2012

2012-02-06T11:31:44Z

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	Notice that, when the angular frequency and the wave number <math>k</math> are proportional, thence the phase and group velocities are the same, i.e. <math>v_{gr}=v_{ph}</math>. On the other hand, from (2) it is obvious that, in this case, the phase velocity <math>v_{ph}</math> does not depend on the frequency. Media in which this condition is fulfilled are said to be ''non-dispersive''.		Notice that, when the angular frequency and the wave number <math>k</math> are proportional, thence the phase and group velocities are the same, i.e. <math>v_{gr}=v_{ph}</math>. On the other hand, from (2) it is obvious that, in this case, the phase velocity <math>v_{ph}</math> does not depend on the frequency. Media in which this condition is fulfilled are said to be ''non-dispersive''.

	On the contrary, a media which the waves propagation speed varies with the frequency is said to be ''dispersive'' (from the dispersion of light by a prism). Thus, the envelope travels at different speed <math>v_{gr}</math> than the internal oscillations <math>v_{ph}</math>. In this case, if we were riding along with the envelope in figure 1, we would observe the internal oscillations moving forward from one group to the next.		On the contrary, media which the waves propagation speed varies with the frequency are said to be ''dispersive'' (from the dispersion of light by a prism). Thus, the envelope travels at different speed <math>v_{gr}</math> than the internal oscillations <math>v_{ph}</math>. In this case, if we were riding along with the envelope in figure 1, we would observe the internal oscillations moving forward from one group to the next.

Jaume.Sanz at 11:27, 6 February 2012

2012-02-06T11:27:32Z

← Older revision		Revision as of 11:27, 6 February 2012
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	The equation (5) defines the ''Group Velocity.''		The equation (5) defines the ''Group Velocity.''

	Notice that, when the angular frequency and the wave number <math>k</math> are proportional, thence the phase and group velocities are the same, i.e. <math>v_{gr}=v_{ph}</math>. On the other hand, from (2) it is obvious that, in this case, the phase velocity <math>v_{ph}</math> do not depend on the frequency. ~~A media~~ in which this condition is fulfilled is said to be ''non-dispersive''.		Notice that, when the angular frequency and the wave number <math>k</math> are proportional, thence the phase and group velocities are the same, i.e. <math>v_{gr}=v_{ph}</math>. On the other hand, from (2) it is obvious that, in this case, the phase velocity <math>v_{ph}</math> does not depend on the frequency. Media in which this condition is fulfilled are said to be ''non-dispersive''.

	On the contrary, a media which the waves propagation speed varies with the frequency is said to be ''dispersive'' (from the dispersion of light by a prism). Thus, the envelope travels at different speed <math>v_{gr}</math> than the internal oscillations <math>v_{ph}</math>. In this case, if we were riding along with the envelope in figure 1, we would observe the internal oscillations moving forward from one group to the next.		On the contrary, a media which the waves propagation speed varies with the frequency is said to be ''dispersive'' (from the dispersion of light by a prism). Thus, the envelope travels at different speed <math>v_{gr}</math> than the internal oscillations <math>v_{ph}</math>. In this case, if we were riding along with the envelope in figure 1, we would observe the internal oscillations moving forward from one group to the next.

Jaume.Sanz at 11:18, 6 February 2012

2012-02-06T11:18:19Z

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	From equation (4), it follows that, two signals can appear travelling at two different velocities as a result of the superposition of signals <math>s_1(x,t)</math> and <math>s_2(x,t)</math>. That is, in the bottom plot of figure 1, the internal oscillations (in blue) ~~travels~~ at velocity <math>v=\frac{\omega}{k}</math>, whereas the amplitude envelope (in red) travels at velocity <math>\frac{\Delta \omega}{\Delta k}</math>.		From equation (4), it follows that, two signals can appear travelling at two different velocities as a result of the superposition of signals <math>s_1(x,t)</math> and <math>s_2(x,t)</math>. That is, in the bottom plot of figure 1, the internal oscillations (in blue) travel at velocity <math>v=\frac{\omega}{k}</math>, whereas the amplitude envelope (in red) travels at velocity <math>\frac{\Delta \omega}{\Delta k}</math>.

Jaume.Sanz at 11:17, 6 February 2012

2012-02-06T11:17:03Z

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	::[[File:Phase_Group_Vel_vel_grp_ph.png\|none\|thumb\|520px\|'''''Figure 1:''''' The signals of first and second ~~rows~~ plots, with slightly different frequencies are added generating the bottom plot signal.]]		::[[File:Phase_Group_Vel_vel_grp_ph.png\|none\|thumb\|520px\|'''''Figure 1:''''' The signals of first and second row plots, with slightly different frequencies are added generating the bottom plot signal.]]

Jaume.Sanz at 11:07, 6 February 2012

2012-02-06T11:07:10Z

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	{{Article Infobox2		{{Article Infobox2
	\|Category=Fundamentals		\|Category=Fundamentals
	\|Authors=J. Sanz Subirana, JM. Juan Zornoza and M. ~~Hernandez~~-Pajares, University of ~~Catalunia~~, Spain.		\|Authors=J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
	\|Level=Intermediate		\|Level=Intermediate
	\|YearOfPublication=2011		\|YearOfPublication=2011

Carlos.Lopez at 08:43, 23 January 2012

2012-01-23T08:43:45Z

← Older revision		Revision as of 08:43, 23 January 2012
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	{{Article Infobox2		{{Article Infobox2
	\|Category=Fundamentals		\|Category=Fundamentals
	~~\|Title={{PAGENAME}}~~
	\|Authors=J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain.		\|Authors=J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain.
	\|Level=Intermediate		\|Level=Intermediate
	\|YearOfPublication=2011		\|YearOfPublication=2011
			\|Logo=gAGE
			\|Title={{PAGENAME}}
	}}		}}
	In order to introduce the ''Phase and Group Velocity'' definitions, a brief review on wave propagation concepts is given as follows.		In order to introduce the ''Phase and Group Velocity'' definitions, a brief review on wave propagation concepts is given as follows.

Timo.Kouwenhoven: Text replace - "|Level=Medium" to "|Level=Intermediate"

2011-11-06T07:46:11Z

Text replace - "|Level=Medium" to "|Level=Intermediate"

← Older revision		Revision as of 07:46, 6 November 2011
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	\|Title={{PAGENAME}}		\|Title={{PAGENAME}}
	\|Authors=J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain.		\|Authors=J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain.
	\|Level=~~Medium~~		\|Level=Intermediate
	\|YearOfPublication=2011		\|YearOfPublication=2011
	}}		}}