WEBVTT
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welcome friends welcome to the todays class
on linear combinations as you know that g
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p s observables are fraught with errors and
to improve the accuracy in g p s positioning
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we need to improve the quality of the g p
s observables that means we want to reduce
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the errors associated with the g p s observables
in the last class we have seen that the errors
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associated with g p s observables can be minimized
or reduced by going linear difference methods
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today in that linear differences method or
method of differences we do take the difference
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of different same observables from different
satellites or from different receivers now
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today we will discuss another type of combinations
of g p s observables which is called linear
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combinations in which we will take the combination
of g p s observables of different types but
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they will be from same satellites
as you know g p s signal contains observables
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of different types like carrier phase observables
as well as code pseudo range observables so
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in any g p s signal nowadays we get three
types of carrier phase observables and five
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types of pseudo range code observables so
these observables may be combined in such
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a way so that we can minimize the different
types of errors associated with the independent
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observables so in order to carry out these
linear combinations we do take the differences
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undifference observables or double difference
observables as we have find in the last class
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now todays class regarding combinations or
linear combinations of g p s observables we
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will talk on introduction and then different
types of linear combinations we will i will
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take three carrier phase linear combinations
then melbourne wubbena linear combinations
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and some dual carrier phase linear combinations
now as i told you that the linear combinations
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are the observables derived from other observables
of different types from the same satellites
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and we make use of the undifference observables
as well as double difference observables to
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find out the linear combinations observables
now in doing that we will effectively change
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the measurement model from linear or single
or original measurements to combined measurements
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and in doing that we will eliminate the error
most of the code related errors alleviate
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processing computations as well as reduce
bandwidths
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now let us take the three carrier phase linear
combinations that means we will make use of
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the carrier phase observables from a sing
g p s signal of l one l two and l five carrier
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phases which are we can represent it by these
are the three carrier phases that is available
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from a g p s signal now if we combine together
these three then suppose we get a carrier
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signal of linear combination type having phase
for l c so that can be represented by a into
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phi l one plus b phi l two plus c phi l five
which may be represented mathematically like
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so this is the phase of the combined signal
and this a b c are the different coefficients
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which will learn afterwards more so by multiplying
these different we can arrive at some characteristic
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phase of our deserved nature
now if next the frequency of the combined
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signal will be as we know that the frequency
is the time rate of change of phase so if
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we take the the the frequency of the linear
combines signal will be like this which is
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also we can write like so the the frequency
of the combined signal will be the multiplication
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by the parameters a b c of the corresponding
frequency of the signal
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now the wavelength of the combined signal
as we know wavelength is equal to c by f that
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means the wavelength of the combines signal
will be equal to this now through algebraic
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manipulation we can get it now frequency of
this is equal to a into f l one plus c into
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f l five so our will be equal to we can write
it like this c now you know this c is different
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from this is the constant c and this is the
velocity of the light so that distinction
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is to be done now from this we get a into
lambda l two lambda l five plus b into lambda
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l one lambda l five plus c into lambda l one
lambda l two and well get lambda l one lambda
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l two lambda l five so this is what is the
wavelength of the combined signal
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and next all what will be the integer ambiguity
of the combined signal now we know integer
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ambiguity is nothing but phase by lambda so
integer ambiguity of the combined signal will
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be equal to phase of the combined signal divided
by the wavelength of the combined signal though
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we know the phase of the combined signal is
summation of the individual signal multiplied
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by the constants
divided by lambda l c so if we now we have
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to make some algebraic manipulation
so for this i can write these thing by algebraic
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manipulation similarly b into phi l two lambda
l two plus c into
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so this is what we can get from here
now you can see this is a constant because
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this is the wavelength of the l one signal
this is the wavelength of the linearly combined
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signal and a is also constant so all together
this is a constant say it is i and here this
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factor is nothing but it is the integer ambiguity
of the signal l one so phi by l one so integer
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ambiguity of the l one signal similarly this
is another constant j suppose and this is
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the integer ambiguity of the l two signal
and this is constant k the integer ambiguity
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of the l five signal
so we see that the integer ambiguity of the
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combined signal is a summation of integer
ambiguity of the individual signal multiplied
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by some constants these constants are again
depend upon the wavelength of these signal
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as well as wavelength of the combined signal
and a factor a so in this way i can find out
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we can get the cat different characteristics
of the combined signal which really provides
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us an insight into the nature of the combined
signal
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and finally as we are mostly interested in
the errors that will be propagated out of
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the combination so if we consider ionospheric
error and tropospheric error as well as noise
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and multipath are the primary sources of errors
in observables let us see what will be the
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error propagated so the amount of error that
will be propagated to the combined signal
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that is the theory of error propagation we
can write like this
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so this is the error root mean square error
of the combined signal due to the ionosphere
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troposphere and noise in multipath so these
are the errors root mean square error of the
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individual signal l one l one signal in the
ionospheric error tropospheric error this
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is the noise of multipath error
so our errors will be defined in terms of
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the errors that is associated with the l one
signal which is the most prominent signal
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and this is multiplied by
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and this that means this multiplied by this
will be the amount of ionospheric error in
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the combined signal and this multiplied by
this that means this is what is the error
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in the tropospheric signal and your so and
this multiplied by this is the error due to
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noise now we can see that we can found that
the errors associated with the linearly combined
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signal is the weight these are the weights
weighted some of the errors that is associated
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with the ionospheric error tropospheric error
and noise and multipath and these weights
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you can see these weights are primarily dependent
on the coefficients a b c and the nominal
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frequency f l one f l two f l five so these
are the prominent thing on which the error
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will depend and since the amount of error
is the weighted some of these weights weighted
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some so these weights are called the amplification
factor of errors because depending upon these
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weights the error gets amplified or reduced
as the combined signal
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now if we can make this as zero then our this
multiplied by this error with zero so our
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ionospheric error will be zero so the condition
for ionospheric error to be reduced to zero
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or nullify is to make these zero now and to
make the tropospheric error to nullify tropospheric
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error we get to make this character zero similarly
now we can see here all these parameters a
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square plus b square plus c square under all
circumstances whatever we take the value of
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a b c this summation will be always a positive
number this indicates that the error due to
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noise or multipath will always increase in
case of linear combination
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now as we know that the observables are different
in different campaigns so and so the errors
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that will be associated with the g p s observables
thus the constants a b c we take also well
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vary from campaign to campaign to avoid
at a particular characteristics of the combined
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signal now how to decide the value of a b
c the criteria for deciding the value of a
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b c is to make this combined error should
be minimum because this is the primary objective
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of our linear combination is to remove or
minimize the error associated with the linearly
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combined signal so we have to make this expression
to be minimum or in some cases because many
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times we may not make the minimize so it should
be guided that the errors due to ionosphere
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and troposphere as well as noise should be
minimum it should be less in linearly combined
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signal than that is available in only l one
signal
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actually the amount of error that will be
associated with the linearly combined signal
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will depend upon the baseline length of the
g p s observable and the amount of errors
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that will be associated or the mutual errors
that means what is the errors in ionosphere
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and troposphere multipath mutually available
inside the observables and as well as it will
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also depend upon what is a degree of accuracy
a user want to be associated with the observables
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so on these factors the quality or the amount
of errors that will be available in linearly
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combined will depend that means we have to
choose the values of a b c
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with this i will like to go for the next type
of linear combination which is widely prevalent
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in todays scenario nowadays the linear combination
known as melbourne wubbena liner combination
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is most widely used for minimizing or for
operation towards g p s processing to minimize
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the error associated with the g p s observable
now melbourne wubbena linear combination actually
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in this linear combination we make use of
the phase observables of both the carrier
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signal l one l two as well as the single range
of signal p one and p two so in this case
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both the carrier phase observable as well
as pseudo range observables gets combined
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as the objective is to reduce and as a result
of which we do get the observable which are
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free from errors arises out of ionosphere
troposphere satellite geometry as well as
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clocks
now the wavelength of the melbourne wubbena
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is like this one by f l one minus f l two
f l one multiplied by the
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i will use the signal symbol l one that means
this is the this is the carrier phase observable
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range of the signal l one plus one by f l
one plus one by f l two f l one pseudo
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range of the code p one p r means pseudo range
and f l two pseudo range p two so this is
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what is the wavelength and correspondingly
another parameters as we have done in the
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last derivation we can do and in that way
we can get the characteristics these parameters
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will provide us to get an insight into the
characteristics of the signal which will help
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us to see what are the errors and what is
the errors have been removed or minimized
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and where of the other already present still
present and accordingly we can make use of
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this
and apart from these actually nowadays may
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a a some prominent dual frequency or dual
carrier phase combination signals or observables
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are available or linear combinations are available
so dual frequency linear combinations so of
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these there are a big list among this which
are most widely prevalent that is wide lane
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then narrow lane then ionosphere free and
geometry free these are the linear combinations
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dual frequency linear combinations which are
most widely used and in all these cases it
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is the l one and l two carrier phase observables
are being used and corresponding to our a
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and b as we have seen in the three phase carrier
frequency
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in case of wide lane it is the four point
five three and for b it is three point five
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three so that means the signal is we get the
phase of the linear combined wide lane
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we get four point five three phase of l one
minus three point five three phase of l two
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now we know that the frequency of l one frequency
of l one is far far that of than l two so
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the frequency of the linear combined that
means wide lane linear combination will be
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more than what it is available for l one or
l two
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so when the frequency is more means ionospheric
error will be reduced so we can reduce the
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ionospheric error further in wide lane the
cycle slip can be detected restricted and
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integer ambiguity can be resolved easily but
this is having a high noise high noise so
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noise is a embeddedness and as a result we
will see you see that the errors are very
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big five point seven six meter
so now in case of narrow lane we will make
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use of a is equal to point five six b equal
to point four four so here use the so you
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can see the linear combination narrow length
ultimately because l one frequency is le more
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l two frequency is less so ultimately we will
see that the combined frequency will be less
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than the frequency of this will be less than
the frequency of l one single so frequency
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is less means your ionospheric error will
be more but less noise so if we want to create
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a signal which will be having less noise then
we should make use of narrow length concept
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or an these constants we may use
then ionosphere free linear combination were
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two so this is the constants which we have
to multiply with the phase of the l one signal
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and phase of the l two signals and again well
find out well get frequency of ionosphere
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free will be more than the frequency of l
one signal which will provide us a good signal
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for getting the ionosphere free well be able
to compute the ionospheric error easily and
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we can make it free and this type of signal
we design for baselines which are very long
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and geometry free where we get the signals
having constant multiplied by one and minus
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one so here also will get a frequency less
than the frequency of the l one and this will
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provide us a geometry free signal but it will
have so many other errors so depending upon
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the purpose what for we are looking for we
want to preprocess the signal we can go for
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different types of linear sig combinations
and we can reduce some of the parameters we
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can enhance some of the parameters
with this i want to conclude todays class
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but before that i want to summarize todays
class derived observables can also be obtained
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from different types of observable that is
available within a g p s signal we make use
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of different carrier phase observables and
the code phase pseudo ranges available in
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the g p s signal arising from the same satellite
so this is the condition we should take into
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consideration in linear combination we make
use of signal from the same satellite that
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have seen removing or minimizing the correlated
errors
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then as the errors in g p s observable depends
on the nature of campaign and vary from campaign
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to campaign so also the constants that
will multiply to get the linear combinations
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also vary however to make it optimum we should
make we should see that the combine errors
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in the combined signal should be less than
what is available in a single l one signal
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and the optimal value of the constant a b
c will depend upon the baseline length the
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amount of different types of error and their
mutual dependence in the observables and the
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amount of accuracy we do look for our combined
signal
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one of the most widely used linear combination
is the melbourne wubbena which is very good
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in removing or minimizing the ionospheric
error tropospheric error and other geometric
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errors and there is some other dual frequency
linear combinations are available like wide
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lane narrow lane ionosphere free geometry
free etcetera which has their own characteristics
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and depending upon the need we can make use
of these different types of linear combinations
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to serve our purpose particularly
with this i would like to conclude thank you
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and see you again in the next class which
will be on g p s processing we will be talking
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on g p s processing in the next class
thank you