Discussion:
TX Aerial phasing question
(too old to reply)
Liz Tuddenham
2023-07-30 09:14:06 UTC
Permalink
A local VHF radio station has its transmitting aerials in the form of
four slanted dipoles:
www.poppyrecords.co.uk/other/SVFMtx_0432.JPG

I would have assumed this was intended to give an omnidirectional
slant-polarised radiation pattern, but I can't see how the aerials can
be connects so as to achieve both aims.

If we designate the four dipoles North, South, East and West; taking the
N-S pair, they appear to be half a wavelength apart, so if they were in
phase their vertical components in the North and South directions would
cancel. If they were 180-degrees out of phase, their horizontal
components would cancel. The only way there would be both components
would be with a 90-degree (or 270-degree) phase shift. The same would
apply to the E-W pair.

If we now consider the phasing between all four aerials, to obtain an
omnidirectional coverage there needs to be 90-degrees phase difference
between each adjacent pair of aerials - but this is in conflict with the
requirements for opposite pairs to be 90-degrees apart.

I thought I must have misunderstood how this worked, so I did a test on
the radiation. From the top of a hill on the NW fringe of the reception
area, the signal is distinctly horizontally polarised. I have also
done a few random tests inside the service area, which is hilly
countryside and prone to reflections, the general impression I get is
that the horizontal component predominates and the vertical component is
absent or very faint most of the time.

This is in agreement with the theory above, but I find it difficult to
believe that the aerial was never intended to have a vertical componemt.
Has it been mis-wired or is it impossible to get omnidirectional slant
polarisation from this type of aerial?
--
~ Liz Tuddenham ~
(Remove the ".invalid"s and add ".co.uk" to reply)
www.poppyrecords.co.uk
Brian Gaff
2023-07-30 12:51:05 UTC
Permalink
I think its impossible. My only frame of ref though is back in the 70s, when
tv in many countries was on low vhf, we all tried to make an omnidirectional
aerial which did not have dead spots. It never did work in three dimensions.
The best actual results were crossed dipoles phased by I think 90 deg,
buteven then it could be turned through many axis and get better reception
as the reflected signals from the I or f2 lays had various polarities
changing all the time.
So in a hilly area, with only line of sight or direct reflections, they
probably just built watt actually worked for most people after tests. I can
recall in the early days of commercial radio, this was done a lot. Nowadays
of course with 90 percent of stations being networked most of the time they
seem to just do what they please.
You used to get a good fm Signal here before commercial radio was born from
Rowridge for radio Solent pointed straight down the road horizontal. Now you
cannot even here it for all the other stations.
I eventually took down my big fm array, as it was pointless.
Brian
--
--:
This newsgroup posting comes to you directly from...
The Sofa of Brian Gaff...
***@blueyonder.co.uk
Blind user, so no pictures please
Note this Signature is meaningless.!
Post by Liz Tuddenham
A local VHF radio station has its transmitting aerials in the form of
www.poppyrecords.co.uk/other/SVFMtx_0432.JPG
I would have assumed this was intended to give an omnidirectional
slant-polarised radiation pattern, but I can't see how the aerials can
be connects so as to achieve both aims.
If we designate the four dipoles North, South, East and West; taking the
N-S pair, they appear to be half a wavelength apart, so if they were in
phase their vertical components in the North and South directions would
cancel. If they were 180-degrees out of phase, their horizontal
components would cancel. The only way there would be both components
would be with a 90-degree (or 270-degree) phase shift. The same would
apply to the E-W pair.
If we now consider the phasing between all four aerials, to obtain an
omnidirectional coverage there needs to be 90-degrees phase difference
between each adjacent pair of aerials - but this is in conflict with the
requirements for opposite pairs to be 90-degrees apart.
I thought I must have misunderstood how this worked, so I did a test on
the radiation. From the top of a hill on the NW fringe of the reception
area, the signal is distinctly horizontally polarised. I have also
done a few random tests inside the service area, which is hilly
countryside and prone to reflections, the general impression I get is
that the horizontal component predominates and the vertical component is
absent or very faint most of the time.
This is in agreement with the theory above, but I find it difficult to
believe that the aerial was never intended to have a vertical componemt.
Has it been mis-wired or is it impossible to get omnidirectional slant
polarisation from this type of aerial?
--
~ Liz Tuddenham ~
(Remove the ".invalid"s and add ".co.uk" to reply)
www.poppyrecords.co.uk
J. P. Gilliver
2023-07-30 13:59:37 UTC
Permalink
In message <1qeodze.osdv7840rwzgN%***@poppyrecords.invalid.invalid> at
Sun, 30 Jul 2023 10:14:06, Liz Tuddenham
Post by Liz Tuddenham
A local VHF radio station has its transmitting aerials in the form of
www.poppyrecords.co.uk/other/SVFMtx_0432.JPG
I would have assumed this was intended to give an omnidirectional
slant-polarised radiation pattern, but I can't see how the aerials can
be connects so as to achieve both aims.
[]
I thought some local stations aimed at circular polarization, though I
don't think that can be omnidirectional.
--
J. P. Gilliver. UMRA: 1960/<1985 MB++G()AL-IS-Ch++(p)***@T+H+Sh0!:`)DNAf

Philosophy is questions that may never be answered. Religion is answers that
may never be questioned.
Mark Carver
2023-07-30 14:27:47 UTC
Permalink
Post by Liz Tuddenham
A local VHF radio station has its transmitting aerials in the form of
www.poppyrecords.co.uk/other/SVFMtx_0432.JPG
That's a Lindenblad Array

Further reading:-
http://on5au.be/content/storart/lin.pdf
Liz Tuddenham
2023-07-30 16:53:57 UTC
Permalink
Post by Mark Carver
Post by Liz Tuddenham
A local VHF radio station has its transmitting aerials in the form of
www.poppyrecords.co.uk/other/SVFMtx_0432.JPG
That's a Lindenblad Array
Further reading:-
http://on5au.be/content/storart/lin.pdf
Thanks, I suspected it must have a name but without the name I was
unable to look it up.

The reference says all the dipoles should be in phase, but I'm not sure
that it makes sense. Assume your receiver is due North of the array,
there will be virtually no signal from the E and W dipoles because they
are end-on; the small vertical components due to them being tilted will
add, but they won't be very strong.

The N and S dipoles will be spaced half a wavelength apart, so the
vertical component in the Northerly and Southerly directions will be
cancelled - that only leaves the horizontal component - which is what I
found when I tried receiving it.

The concept of circular polarisation isn't very helpful at VHF, as an
end-fire receiving helix of the appropriate size might generate
complaints from the neighbours.
--
~ Liz Tuddenham ~
(Remove the ".invalid"s and add ".co.uk" to reply)
www.poppyrecords.co.uk
J. P. Gilliver
2023-07-30 18:48:43 UTC
Permalink
In message <1qeozui.1pe9qh01o0q6n0N%***@poppyrecords.invalid.invalid> at
Sun, 30 Jul 2023 17:53:57, Liz Tuddenham
<***@poppyrecords.invalid.invalid> writes
[]
Post by Liz Tuddenham
The concept of circular polarisation isn't very helpful at VHF, as an
end-fire receiving helix of the appropriate size might generate
complaints from the neighbours.
I thought circular was sometimes used as a way of ensuring there's
always a significant component of both horizontal and vertical (or
anything else the receiver uses). But I don't think it goes with
omnidirectionality at the transmitter.
--
J. P. Gilliver. UMRA: 1960/<1985 MB++G()AL-IS-Ch++(p)***@T+H+Sh0!:`)DNAf

Can a blue man sing the whites?
Liz Tuddenham
2023-07-31 08:29:41 UTC
Permalink
Post by J. P. Gilliver
Sun, 30 Jul 2023 17:53:57, Liz Tuddenham
[]
Post by Liz Tuddenham
The concept of circular polarisation isn't very helpful at VHF, as an
end-fire receiving helix of the appropriate size might generate
complaints from the neighbours.
I thought circular was sometimes used as a way of ensuring there's
always a significant component of both horizontal and vertical (or
anything else the receiver uses). But I don't think it goes with
omnidirectionality at the transmitter.
I've thought of a mental model that might explain what is supposed to
happen:

Suppose each dipole is replace by a pair of dipoles, one vertical and
one horizontal, each radiating exactly half the power of the original
slant dipole; the effect will be identical with a single slant dipole.

VERTICAL
Seen from the North, the N and S verticals will be spaced half a
wavelength apart in distance and their signals will cancel, but the E
and W verticals will be equidistant so they will appear to be in phase
and therefore will add. Seen from the West, the E and W dipoles will
cancel and the N and S dipoles will add. From the Northeast, the N and
E dipoles will add, but the S and W dipoles will be lagging them by t
(where t is half a wavelength divided by root two). This will reduce
the vertical signal at the quarter points but won't extinguish it
altogether.

HORIZONTAL
Seen from the North, the N and S horizontal dipoles will be half a
wavelength apart but in opposite polarity, so they will add. The E and
W dipoles will be equidistant but in opposite polarity, so they will
cancel. From the Northeast there will appear to be two horizontal
dipoles broadside-on but in phase. They will be angled at 45 degrees to
the receiver, so the net effect will be P (where P is two over root
two).

Thus the vertical and horizontal signals will be equal (circular
polariation) at the cadinal points but the horizontal signal will be
double the vertical signal at the quarter points (elliptical
polarisation).
--
~ Liz Tuddenham ~
(Remove the ".invalid"s and add ".co.uk" to reply)
www.poppyrecords.co.uk
J. P. Gilliver
2023-07-31 09:05:59 UTC
Permalink
In message <1qeq6sp.11ex49vwp4fp2N%***@poppyrecords.invalid.invalid> at
Mon, 31 Jul 2023 09:29:41, Liz Tuddenham
Post by Liz Tuddenham
Post by J. P. Gilliver
Sun, 30 Jul 2023 17:53:57, Liz Tuddenham
[]
Post by Liz Tuddenham
The concept of circular polarisation isn't very helpful at VHF, as an
end-fire receiving helix of the appropriate size might generate
complaints from the neighbours.
I thought circular was sometimes used as a way of ensuring there's
always a significant component of both horizontal and vertical (or
anything else the receiver uses). But I don't think it goes with
omnidirectionality at the transmitter.
I've thought of a mental model that might explain what is supposed to
[good explanation snipped]
Post by Liz Tuddenham
Thus the vertical and horizontal signals will be equal (circular
polariation) at the cadinal points but the horizontal signal will be
double the vertical signal at the quarter points (elliptical
polarisation).
So for this particular arrangement, it's going to radiate circular at
the cardinal points, grading to elliptical between two of them and
vertical otherwise:

ECE
C C
ECE

At first, I thought you meant

HCV VCH
C C or C C
VCH HCV.

Or some variation depending on the overall orientation.

Either seem an odd thing to _want_, though.
--
J. P. Gilliver. UMRA: 1960/<1985 MB++G()AL-IS-Ch++(p)***@T+H+Sh0!:`)DNAf

science is not intended to be foolproof. Science is about crawling toward the
truth over time. - Scott Adams, 2015-2-2
Liz Tuddenham
2023-07-31 16:33:37 UTC
Permalink
Post by J. P. Gilliver
Mon, 31 Jul 2023 09:29:41, Liz Tuddenham
Post by Liz Tuddenham
Post by J. P. Gilliver
Sun, 30 Jul 2023 17:53:57, Liz Tuddenham
[]
Post by Liz Tuddenham
The concept of circular polarisation isn't very helpful at VHF, as an
end-fire receiving helix of the appropriate size might generate
complaints from the neighbours.
I thought circular was sometimes used as a way of ensuring there's
always a significant component of both horizontal and vertical (or
anything else the receiver uses). But I don't think it goes with
omnidirectionality at the transmitter.
I've thought of a mental model that might explain what is supposed to
[good explanation snipped]
Post by Liz Tuddenham
Thus the vertical and horizontal signals will be equal (circular
polariation) at the cadinal points but the horizontal signal will be
double the vertical signal at the quarter points (elliptical
polarisation).
So for this particular arrangement, it's going to radiate circular at
the cardinal points, grading to elliptical between two of them and
ECE
C C
ECE
[...]

I've now had a chance to make a few spot measurements: It is vertical
to the Southeast but horizontal to the Northeast. It gradually changes
from vertical to slant as I travel along a road which runs from
Northwest of the transmitter to West of it. (These are geographical
directions, I don't know how they relate to the orientation of the
array.)

It would appear that they are probably transmitting horizontal
polarisation in the Northeast-Southwest direction and vertical in the
Northwest-Southeast direction. That doesn't appear to be what the array
is intended to do, so something is obviously wrong with the dipoles or
the feeders. The station director is aware that they have a problem and
is going to arrange for their transmitter engineer to make contact with
me.
MB
2023-07-31 13:37:06 UTC
Permalink
There was the case where two Band II folded dipoles (?) were used, one
with most of power feeding the main direction and a lower power feed to
cover another area. Unfortunately they were connected the wrong way
around. :-)
Mark Carver
2023-07-31 13:41:19 UTC
Permalink
Post by MB
There was the case where two Band II folded dipoles (?) were used, one
with most of power feeding the main direction and a lower power feed
to cover another area.  Unfortunately they were connected the wrong
way around.  :-)
 It's not an unusual occurrence. Two UHF panels for Ch 5 at a site
(that will remain nameless) were either wired or mechanically arranged
in such a way the resulting radiation was nulled
charles
2023-07-31 16:00:03 UTC
Permalink
Post by MB
There was the case where two Band II folded dipoles (?) were used, one
with most of power feeding the main direction and a lower power feed to
cover another area. Unfortunately they were connected the wrong way
around. :-)
Yep. Ãrdgour - I found that one, And there was the one on Loch Fyne
(Strachur?) which was transmitting at right angles to its intended
directions
--
from KT24 in Surrey, England - sent from my RISC OS 4té
"I'd rather die of exhaustion than die of boredom" Thomas Carlyle
tony sayer
2023-08-07 19:01:08 UTC
Permalink
Post by Liz Tuddenham
A local VHF radio station has its transmitting aerials in the form of
www.poppyrecords.co.uk/other/SVFMtx_0432.JPG
I would have assumed this was intended to give an omnidirectional
slant-polarised radiation pattern, but I can't see how the aerials can
be connects so as to achieve both aims.
If we designate the four dipoles North, South, East and West; taking the
N-S pair, they appear to be half a wavelength apart, so if they were in
phase their vertical components in the North and South directions would
cancel. If they were 180-degrees out of phase, their horizontal
components would cancel. The only way there would be both components
would be with a 90-degree (or 270-degree) phase shift. The same would
apply to the E-W pair.
If we now consider the phasing between all four aerials, to obtain an
omnidirectional coverage there needs to be 90-degrees phase difference
between each adjacent pair of aerials - but this is in conflict with the
requirements for opposite pairs to be 90-degrees apart.
I thought I must have misunderstood how this worked, so I did a test on
the radiation. From the top of a hill on the NW fringe of the reception
area, the signal is distinctly horizontally polarised.
Should be equal!
Post by Liz Tuddenham
I have also
done a few random tests inside the service area, which is hilly
countryside and prone to reflections, the general impression I get is
that the horizontal component predominates and the vertical component is
absent or very faint most of the time.
Shouldn't be like that normally..
Post by Liz Tuddenham
This is in agreement with the theory above, but I find it difficult to
believe that the aerial was never intended to have a vertical componemt.
Has it been mis-wired or is it impossible to get omnidirectional slant
polarisation from this type of aerial?
Bit unwell to give chapter and verse right now but thats a Lindenblad
invented by a bloke of the same name in the 1940's in America was used
for aircraft comms and sat comms as its gives a low angle of radiation
but mixed as such radiation, the angles and phase can vary slightly but
its essentially Ommni.

Their very good for FM broadcast used a few of them now mixed V and H
pattern and Ommni. Only downside if they don't have that much gain and
you need a top of the mast position to put them they aren't side mount
animals! One here at Madingley near Cambridge sec pic down.


For a lot of stations smaller ones mainly Vertical pol is very easy to
implement Horiz Ommni more difficult but in Urban areas n because of
multipath and the like your RX aerial gets both V and H so if the V is
poor of polarisation skewed or shifted the whole as such is rotated so a
signal is still received if you follow that!

http://tx.mb21.co.uk/gallery/gallerypage.php?txid=1138&pageid=2326
--
Tony Sayer


Man is least himself when he talks in his own person.

Give him a keyboard, and he will reveal himself.
charles
2023-08-07 19:30:03 UTC
Permalink
Post by tony sayer
Post by Liz Tuddenham
A local VHF radio station has its transmitting aerials in the form of
www.poppyrecords.co.uk/other/SVFMtx_0432.JPG
I would have assumed this was intended to give an omnidirectional
slant-polarised radiation pattern, but I can't see how the aerials can
be connects so as to achieve both aims.
If we designate the four dipoles North, South, East and West; taking the
N-S pair, they appear to be half a wavelength apart, so if they were in
phase their vertical components in the North and South directions would
cancel. If they were 180-degrees out of phase, their horizontal
components would cancel. The only way there would be both components
would be with a 90-degree (or 270-degree) phase shift. The same would
apply to the E-W pair.
If we now consider the phasing between all four aerials, to obtain an
omnidirectional coverage there needs to be 90-degrees phase difference
between each adjacent pair of aerials - but this is in conflict with the
requirements for opposite pairs to be 90-degrees apart.
I thought I must have misunderstood how this worked, so I did a test on
the radiation. From the top of a hill on the NW fringe of the reception
area, the signal is distinctly horizontally polarised.
Should be equal!
Post by Liz Tuddenham
I have also
done a few random tests inside the service area, which is hilly
countryside and prone to reflections, the general impression I get is
that the horizontal component predominates and the vertical component is
absent or very faint most of the time.
Shouldn't be like that normally..
Certainly the tests I did on GLR from CP showed that the VP component came
over hills better.
Post by tony sayer
Post by Liz Tuddenham
This is in agreement with the theory above, but I find it difficult to
believe that the aerial was never intended to have a vertical componemt.
Has it been mis-wired or is it impossible to get omnidirectional slant
polarisation from this type of aerial?
Bit unwell to give chapter and verse right now but thats a Lindenblad
invented by a bloke of the same name in the 1940's in America was used
for aircraft comms and sat comms as its gives a low angle of radiation
but mixed as such radiation, the angles and phase can vary slightly but
its essentially Ommni.
Their very good for FM broadcast used a few of them now mixed V and H
pattern and Ommni. Only downside if they don't have that much gain and
you need a top of the mast position to put them they aren't side mount
animals! One here at Madingley near Cambridge sec pic down.
For a lot of stations smaller ones mainly Vertical pol is very easy to
implement Horiz Ommni more difficult but in Urban areas n because of
multipath and the like your RX aerial gets both V and H so if the V is
poor of polarisation skewed or shifted the whole as such is rotated so a
signal is still received if you follow that!
http://tx.mb21.co.uk/gallery/gallerypage.php?txid=1138&pageid=2326
--
from KT24 in Surrey, England - sent from my RISC OS 4té
"I'd rather die of exhaustion than die of boredom" Thomas Carlyle
tony sayer
2023-08-08 16:17:21 UTC
Permalink
Post by charles
Post by tony sayer
Post by Liz Tuddenham
A local VHF radio station has its transmitting aerials in the form of
www.poppyrecords.co.uk/other/SVFMtx_0432.JPG
I would have assumed this was intended to give an omnidirectional
slant-polarised radiation pattern, but I can't see how the aerials can
be connects so as to achieve both aims.
If we designate the four dipoles North, South, East and West; taking the
N-S pair, they appear to be half a wavelength apart, so if they were in
phase their vertical components in the North and South directions would
cancel. If they were 180-degrees out of phase, their horizontal
components would cancel. The only way there would be both components
would be with a 90-degree (or 270-degree) phase shift. The same would
apply to the E-W pair.
If we now consider the phasing between all four aerials, to obtain an
omnidirectional coverage there needs to be 90-degrees phase difference
between each adjacent pair of aerials - but this is in conflict with the
requirements for opposite pairs to be 90-degrees apart.
I thought I must have misunderstood how this worked, so I did a test on
the radiation. From the top of a hill on the NW fringe of the reception
area, the signal is distinctly horizontally polarised.
Should be equal!
Post by Liz Tuddenham
I have also
done a few random tests inside the service area, which is hilly
countryside and prone to reflections, the general impression I get is
that the horizontal component predominates and the vertical component is
absent or very faint most of the time.
Shouldn't be like that normally..
Certainly the tests I did on GLR from CP showed that the VP component came
over hills better.
But was that a Lindenblad array Charles?.

We provide some relays off the Cambridge Madingley TX and at those relay
site locations we rotated the RX aerial and each plane was within .5 dB
of the other!...
Post by charles
Post by tony sayer
Post by Liz Tuddenham
This is in agreement with the theory above, but I find it difficult to
believe that the aerial was never intended to have a vertical componemt.
Has it been mis-wired or is it impossible to get omnidirectional slant
polarisation from this type of aerial?
Bit unwell to give chapter and verse right now but thats a Lindenblad
invented by a bloke of the same name in the 1940's in America was used
for aircraft comms and sat comms as its gives a low angle of radiation
but mixed as such radiation, the angles and phase can vary slightly but
its essentially Ommni.
Their very good for FM broadcast used a few of them now mixed V and H
pattern and Ommni. Only downside if they don't have that much gain and
you need a top of the mast position to put them they aren't side mount
animals! One here at Madingley near Cambridge sec pic down.
For a lot of stations smaller ones mainly Vertical pol is very easy to
implement Horiz Ommni more difficult but in Urban areas n because of
multipath and the like your RX aerial gets both V and H so if the V is
poor of polarisation skewed or shifted the whole as such is rotated so a
signal is still received if you follow that!
http://tx.mb21.co.uk/gallery/gallerypage.php?txid=1138&pageid=2326
--
Tony Sayer


Man is least himself when he talks in his own person.

Give him a keyboard, and he will reveal himself.
charles
2023-08-08 16:45:01 UTC
Permalink
Post by tony sayer
Post by charles
Post by tony sayer
Post by Liz Tuddenham
A local VHF radio station has its transmitting aerials in the form of
four slanted dipoles: www.poppyrecords.co.uk/other/SVFMtx_0432.JPG
I would have assumed this was intended to give an omnidirectional
slant-polarised radiation pattern, but I can't see how the aerials
can be connects so as to achieve both aims.
If we designate the four dipoles North, South, East and West; taking
the N-S pair, they appear to be half a wavelength apart, so if they
were in phase their vertical components in the North and South
directions would cancel. If they were 180-degrees out of phase,
their horizontal components would cancel. The only way there would
be both components would be with a 90-degree (or 270-degree) phase
shift. The same would apply to the E-W pair.
If we now consider the phasing between all four aerials, to obtain an
omnidirectional coverage there needs to be 90-degrees phase
difference between each adjacent pair of aerials - but this is in
conflict with the requirements for opposite pairs to be 90-degrees
apart.
I thought I must have misunderstood how this worked, so I did a test
on the radiation. From the top of a hill on the NW fringe of the
reception area, the signal is distinctly horizontally polarised.
Should be equal!
Post by Liz Tuddenham
I have also done a few random tests inside the service area, which
is hilly countryside and prone to reflections, the general impression
I get is that the horizontal component predominates and the vertical
component is absent or very faint most of the time.
Shouldn't be like that normally..
Certainly the tests I did on GLR from CP showed that the VP component
came over hills better.
But was that a Lindenblad array Charles?.
I don't know if I ever knew about the tx antenna, Using a Yagi for
reception there was equal H & V in open sight of the tx, but VP cleared
hills better.
Post by tony sayer
We provide some relays off the Cambridge Madingley TX and at those relay
site locations we rotated the RX aerial and each plane was within .5 dB
of the other!...
--
from KT24 in Surrey, England - sent from my RISC OS 4té
"I'd rather die of exhaustion than die of boredom" Thomas Carlyle
tony sayer
2023-08-22 16:05:22 UTC
Permalink
Post by charles
I don't know if I ever knew about the tx antenna, Using a Yagi for
reception there was equal H & V in open sight of the tx, but VP cleared
hills better.
Now didn't someone do some research to prove that HP went greater
distances and was that at the birth of Band Two or the advent of band 3
maybe?...
--
Tony Sayer


Man is least himself when he talks in his own person.

Give him a keyboard, and he will reveal himself.
charles
2023-08-22 17:00:02 UTC
Permalink
Post by tony sayer
Post by charles
I don't know if I ever knew about the tx antenna, Using a Yagi for
reception there was equal H & V in open sight of the tx, but VP cleared
hills better.
Now didn't someone do some research to prove that HP went greater
distances and was that at the birth of Band Two or the advent of band 3
maybe?...
The choice of HP was less ignition interference and less liable to
multipath effects. Pawley p 338
--
from KT24 in Surrey, England - sent from my RISC OS 4té
"I'd rather die of exhaustion than die of boredom" Thomas Carlyle
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