Why does NASA use higher frequencies even though they have worse Free Space Path Loss (FSPL)?How to calculate data rate of Voyager 1?Quantitatively, why will optical communication be better than X-band for deep-space communications?Did New Horizons use its smaller medium-gain antenna for most/all downlinking of Pluto and Ultima Thule flyby data?Can I transmit signals from a wristwatch to geostationary orbit?Deep Space Network's transmitter power versus frequency?Will there be “Near Space” Ka-band allocations for TESS?

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Why does NASA use higher frequencies even though they have worse Free Space Path Loss (FSPL)?


How to calculate data rate of Voyager 1?Quantitatively, why will optical communication be better than X-band for deep-space communications?Did New Horizons use its smaller medium-gain antenna for most/all downlinking of Pluto and Ultima Thule flyby data?Can I transmit signals from a wristwatch to geostationary orbit?Deep Space Network's transmitter power versus frequency?Will there be “Near Space” Ka-band allocations for TESS?













6












$begingroup$


Looking at the Formula for FSPL we see that it increases with Frequency.



Why is X-Band used for Deep Space Communications instead of lower bands like S-Band? Is it a question of data-rates? Is the noise floor lower in this band? Is it just a question of infrastructure, as the DSN uses X-Band?



Lower bands would lower the FSPL quite a bit, so I want to know the reason for this decision.










share|improve this question









New contributor



Clex is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$







  • 2




    $begingroup$
    You should rephrase the title of your question (maybe "why using low frequencies for deep space communication?") so that it is easier to browse the website without opening each question about deep space communication
    $endgroup$
    – Manu H
    Jun 2 at 8:28










  • $begingroup$
    The wavelength dependence of the definition of free space path loss (FPSL) is an artifact of he way the receiver's antenna gain is defined in the same link budget calculation. It's referenced to an ideal isotropic antenna with a receive area of roughly 1 square wavelength, which for high frequency gets very small. If you do them together (transmit gain, path loss, receive gain) you'll see that the higher frequency wins.
    $endgroup$
    – uhoh
    Jun 2 at 10:28










  • $begingroup$
    @ManuH how does that look?
    $endgroup$
    – uhoh
    Jun 2 at 15:38















6












$begingroup$


Looking at the Formula for FSPL we see that it increases with Frequency.



Why is X-Band used for Deep Space Communications instead of lower bands like S-Band? Is it a question of data-rates? Is the noise floor lower in this band? Is it just a question of infrastructure, as the DSN uses X-Band?



Lower bands would lower the FSPL quite a bit, so I want to know the reason for this decision.










share|improve this question









New contributor



Clex is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$







  • 2




    $begingroup$
    You should rephrase the title of your question (maybe "why using low frequencies for deep space communication?") so that it is easier to browse the website without opening each question about deep space communication
    $endgroup$
    – Manu H
    Jun 2 at 8:28










  • $begingroup$
    The wavelength dependence of the definition of free space path loss (FPSL) is an artifact of he way the receiver's antenna gain is defined in the same link budget calculation. It's referenced to an ideal isotropic antenna with a receive area of roughly 1 square wavelength, which for high frequency gets very small. If you do them together (transmit gain, path loss, receive gain) you'll see that the higher frequency wins.
    $endgroup$
    – uhoh
    Jun 2 at 10:28










  • $begingroup$
    @ManuH how does that look?
    $endgroup$
    – uhoh
    Jun 2 at 15:38













6












6








6


0



$begingroup$


Looking at the Formula for FSPL we see that it increases with Frequency.



Why is X-Band used for Deep Space Communications instead of lower bands like S-Band? Is it a question of data-rates? Is the noise floor lower in this band? Is it just a question of infrastructure, as the DSN uses X-Band?



Lower bands would lower the FSPL quite a bit, so I want to know the reason for this decision.










share|improve this question









New contributor



Clex is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$




Looking at the Formula for FSPL we see that it increases with Frequency.



Why is X-Band used for Deep Space Communications instead of lower bands like S-Band? Is it a question of data-rates? Is the noise floor lower in this band? Is it just a question of infrastructure, as the DSN uses X-Band?



Lower bands would lower the FSPL quite a bit, so I want to know the reason for this decision.







radio-communication deep-space-network link-budget






share|improve this question









New contributor



Clex is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.










share|improve this question









New contributor



Clex is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.








share|improve this question




share|improve this question








edited Jun 3 at 6:25









RonJohn

248111




248111






New contributor



Clex is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.








asked Jun 2 at 8:11









ClexClex

383




383




New contributor



Clex is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




New contributor




Clex is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









  • 2




    $begingroup$
    You should rephrase the title of your question (maybe "why using low frequencies for deep space communication?") so that it is easier to browse the website without opening each question about deep space communication
    $endgroup$
    – Manu H
    Jun 2 at 8:28










  • $begingroup$
    The wavelength dependence of the definition of free space path loss (FPSL) is an artifact of he way the receiver's antenna gain is defined in the same link budget calculation. It's referenced to an ideal isotropic antenna with a receive area of roughly 1 square wavelength, which for high frequency gets very small. If you do them together (transmit gain, path loss, receive gain) you'll see that the higher frequency wins.
    $endgroup$
    – uhoh
    Jun 2 at 10:28










  • $begingroup$
    @ManuH how does that look?
    $endgroup$
    – uhoh
    Jun 2 at 15:38












  • 2




    $begingroup$
    You should rephrase the title of your question (maybe "why using low frequencies for deep space communication?") so that it is easier to browse the website without opening each question about deep space communication
    $endgroup$
    – Manu H
    Jun 2 at 8:28










  • $begingroup$
    The wavelength dependence of the definition of free space path loss (FPSL) is an artifact of he way the receiver's antenna gain is defined in the same link budget calculation. It's referenced to an ideal isotropic antenna with a receive area of roughly 1 square wavelength, which for high frequency gets very small. If you do them together (transmit gain, path loss, receive gain) you'll see that the higher frequency wins.
    $endgroup$
    – uhoh
    Jun 2 at 10:28










  • $begingroup$
    @ManuH how does that look?
    $endgroup$
    – uhoh
    Jun 2 at 15:38







2




2




$begingroup$
You should rephrase the title of your question (maybe "why using low frequencies for deep space communication?") so that it is easier to browse the website without opening each question about deep space communication
$endgroup$
– Manu H
Jun 2 at 8:28




$begingroup$
You should rephrase the title of your question (maybe "why using low frequencies for deep space communication?") so that it is easier to browse the website without opening each question about deep space communication
$endgroup$
– Manu H
Jun 2 at 8:28












$begingroup$
The wavelength dependence of the definition of free space path loss (FPSL) is an artifact of he way the receiver's antenna gain is defined in the same link budget calculation. It's referenced to an ideal isotropic antenna with a receive area of roughly 1 square wavelength, which for high frequency gets very small. If you do them together (transmit gain, path loss, receive gain) you'll see that the higher frequency wins.
$endgroup$
– uhoh
Jun 2 at 10:28




$begingroup$
The wavelength dependence of the definition of free space path loss (FPSL) is an artifact of he way the receiver's antenna gain is defined in the same link budget calculation. It's referenced to an ideal isotropic antenna with a receive area of roughly 1 square wavelength, which for high frequency gets very small. If you do them together (transmit gain, path loss, receive gain) you'll see that the higher frequency wins.
$endgroup$
– uhoh
Jun 2 at 10:28












$begingroup$
@ManuH how does that look?
$endgroup$
– uhoh
Jun 2 at 15:38




$begingroup$
@ManuH how does that look?
$endgroup$
– uhoh
Jun 2 at 15:38










1 Answer
1






active

oldest

votes


















9












$begingroup$

The wavelength dependence of the definition of free space path loss (FSPL) is an artifact of the way the receiver's antenna gain is defined in the same link budget calculation. It's referenced to an ideal isotropic antenna with a receive area of roughly 1 square wavelength, which for high frequency gets very small. If you do them together (transmit gain, path loss, receive gain) you'll see that the higher frequency wins because the gain of the transmit antenna go up, and the combination of the FSPL and the receive antenna together remain the same.



So the reason higher frequencies are used is because the total link budget is better.



You can't just take the FSPL and ignore the other terms in the budget, it won't make sense because of the way things are defined.



You can see real-world examples of link budgets in this answer (short one) and in this answer (longer one) and in this question. You may find this one interesting as well.




From here"



Link Budget



From this answer which is from this answer:



$$ P_RX = P_TX + G_TX - L_FS + G_RX $$




  • $P_RX$: received power by spacecraft


  • $P_TX$: transmitted power by wristwatch


  • $G_TX$: Gain of wristwatch's transmitting antenna (compared to isotropic)


  • $L_FS$: Free space Loss, what we usually call $1/r^2$


  • $G_RX$: Gain of spacecraft's receiving antenna (compared to isotropic)

$$G sim 20 times log_10left( fracpi dlambda right)$$



$$L_FS = 20 times log_10left( 4 pi fracRlambda right).$$




$$ P_RX - P_TX = G_TX - L_FS + G_RX $$



Change from dB to linear scale:



$$ fracP_RXP_TX = fracG_TXG_RXL_FS = fracpi^4 d_RX^2 d_TX^2lambda^4fraclambda^216 pi^2 R^2 = fracpi^2 d_RX^2 d_TX^2lambda^2frac14^2 R^2 = fracpi^2 d_RX^2 d_TX^24 lambda^2 R^2$$



So the fraction of the transmitted power that is received depends on $lambda^-2$ ; it improves as the frequency goes up and the wavelength goes down.






share|improve this answer











$endgroup$








  • 1




    $begingroup$
    thank you so much! I see that now, did not consider the gain formula, but you are right it is the key to the problem!
    $endgroup$
    – Clex
    Jun 2 at 15:43






  • 1




    $begingroup$
    @Clex ya it was confusing for me at first too. Running through a whole example ike I did here takes some time the first time (it took me hours!), but it helps a lot in the long run.
    $endgroup$
    – uhoh
    Jun 2 at 15:48






  • 2




    $begingroup$
    Yeah, I've always thought that the definition of FSPL (@uhoh, note the acronym isn't FPSL) was kludgy. But once a peer-reviewed paper using it was accepted, saying it should be redefined to be more physically intuitive has the telecom specialists tilting their heads back some while still looking at me, pointing, and uttering, "BLASPHEMY!!".
    $endgroup$
    – Tom Spilker
    Jun 3 at 19:53










  • $begingroup$
    @TomSpilker ha! I can picture it. Thanks for catching my misacronymization also.
    $endgroup$
    – uhoh
    Jun 3 at 21:52











Your Answer








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1 Answer
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active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









9












$begingroup$

The wavelength dependence of the definition of free space path loss (FSPL) is an artifact of the way the receiver's antenna gain is defined in the same link budget calculation. It's referenced to an ideal isotropic antenna with a receive area of roughly 1 square wavelength, which for high frequency gets very small. If you do them together (transmit gain, path loss, receive gain) you'll see that the higher frequency wins because the gain of the transmit antenna go up, and the combination of the FSPL and the receive antenna together remain the same.



So the reason higher frequencies are used is because the total link budget is better.



You can't just take the FSPL and ignore the other terms in the budget, it won't make sense because of the way things are defined.



You can see real-world examples of link budgets in this answer (short one) and in this answer (longer one) and in this question. You may find this one interesting as well.




From here"



Link Budget



From this answer which is from this answer:



$$ P_RX = P_TX + G_TX - L_FS + G_RX $$




  • $P_RX$: received power by spacecraft


  • $P_TX$: transmitted power by wristwatch


  • $G_TX$: Gain of wristwatch's transmitting antenna (compared to isotropic)


  • $L_FS$: Free space Loss, what we usually call $1/r^2$


  • $G_RX$: Gain of spacecraft's receiving antenna (compared to isotropic)

$$G sim 20 times log_10left( fracpi dlambda right)$$



$$L_FS = 20 times log_10left( 4 pi fracRlambda right).$$




$$ P_RX - P_TX = G_TX - L_FS + G_RX $$



Change from dB to linear scale:



$$ fracP_RXP_TX = fracG_TXG_RXL_FS = fracpi^4 d_RX^2 d_TX^2lambda^4fraclambda^216 pi^2 R^2 = fracpi^2 d_RX^2 d_TX^2lambda^2frac14^2 R^2 = fracpi^2 d_RX^2 d_TX^24 lambda^2 R^2$$



So the fraction of the transmitted power that is received depends on $lambda^-2$ ; it improves as the frequency goes up and the wavelength goes down.






share|improve this answer











$endgroup$








  • 1




    $begingroup$
    thank you so much! I see that now, did not consider the gain formula, but you are right it is the key to the problem!
    $endgroup$
    – Clex
    Jun 2 at 15:43






  • 1




    $begingroup$
    @Clex ya it was confusing for me at first too. Running through a whole example ike I did here takes some time the first time (it took me hours!), but it helps a lot in the long run.
    $endgroup$
    – uhoh
    Jun 2 at 15:48






  • 2




    $begingroup$
    Yeah, I've always thought that the definition of FSPL (@uhoh, note the acronym isn't FPSL) was kludgy. But once a peer-reviewed paper using it was accepted, saying it should be redefined to be more physically intuitive has the telecom specialists tilting their heads back some while still looking at me, pointing, and uttering, "BLASPHEMY!!".
    $endgroup$
    – Tom Spilker
    Jun 3 at 19:53










  • $begingroup$
    @TomSpilker ha! I can picture it. Thanks for catching my misacronymization also.
    $endgroup$
    – uhoh
    Jun 3 at 21:52















9












$begingroup$

The wavelength dependence of the definition of free space path loss (FSPL) is an artifact of the way the receiver's antenna gain is defined in the same link budget calculation. It's referenced to an ideal isotropic antenna with a receive area of roughly 1 square wavelength, which for high frequency gets very small. If you do them together (transmit gain, path loss, receive gain) you'll see that the higher frequency wins because the gain of the transmit antenna go up, and the combination of the FSPL and the receive antenna together remain the same.



So the reason higher frequencies are used is because the total link budget is better.



You can't just take the FSPL and ignore the other terms in the budget, it won't make sense because of the way things are defined.



You can see real-world examples of link budgets in this answer (short one) and in this answer (longer one) and in this question. You may find this one interesting as well.




From here"



Link Budget



From this answer which is from this answer:



$$ P_RX = P_TX + G_TX - L_FS + G_RX $$




  • $P_RX$: received power by spacecraft


  • $P_TX$: transmitted power by wristwatch


  • $G_TX$: Gain of wristwatch's transmitting antenna (compared to isotropic)


  • $L_FS$: Free space Loss, what we usually call $1/r^2$


  • $G_RX$: Gain of spacecraft's receiving antenna (compared to isotropic)

$$G sim 20 times log_10left( fracpi dlambda right)$$



$$L_FS = 20 times log_10left( 4 pi fracRlambda right).$$




$$ P_RX - P_TX = G_TX - L_FS + G_RX $$



Change from dB to linear scale:



$$ fracP_RXP_TX = fracG_TXG_RXL_FS = fracpi^4 d_RX^2 d_TX^2lambda^4fraclambda^216 pi^2 R^2 = fracpi^2 d_RX^2 d_TX^2lambda^2frac14^2 R^2 = fracpi^2 d_RX^2 d_TX^24 lambda^2 R^2$$



So the fraction of the transmitted power that is received depends on $lambda^-2$ ; it improves as the frequency goes up and the wavelength goes down.






share|improve this answer











$endgroup$








  • 1




    $begingroup$
    thank you so much! I see that now, did not consider the gain formula, but you are right it is the key to the problem!
    $endgroup$
    – Clex
    Jun 2 at 15:43






  • 1




    $begingroup$
    @Clex ya it was confusing for me at first too. Running through a whole example ike I did here takes some time the first time (it took me hours!), but it helps a lot in the long run.
    $endgroup$
    – uhoh
    Jun 2 at 15:48






  • 2




    $begingroup$
    Yeah, I've always thought that the definition of FSPL (@uhoh, note the acronym isn't FPSL) was kludgy. But once a peer-reviewed paper using it was accepted, saying it should be redefined to be more physically intuitive has the telecom specialists tilting their heads back some while still looking at me, pointing, and uttering, "BLASPHEMY!!".
    $endgroup$
    – Tom Spilker
    Jun 3 at 19:53










  • $begingroup$
    @TomSpilker ha! I can picture it. Thanks for catching my misacronymization also.
    $endgroup$
    – uhoh
    Jun 3 at 21:52













9












9








9





$begingroup$

The wavelength dependence of the definition of free space path loss (FSPL) is an artifact of the way the receiver's antenna gain is defined in the same link budget calculation. It's referenced to an ideal isotropic antenna with a receive area of roughly 1 square wavelength, which for high frequency gets very small. If you do them together (transmit gain, path loss, receive gain) you'll see that the higher frequency wins because the gain of the transmit antenna go up, and the combination of the FSPL and the receive antenna together remain the same.



So the reason higher frequencies are used is because the total link budget is better.



You can't just take the FSPL and ignore the other terms in the budget, it won't make sense because of the way things are defined.



You can see real-world examples of link budgets in this answer (short one) and in this answer (longer one) and in this question. You may find this one interesting as well.




From here"



Link Budget



From this answer which is from this answer:



$$ P_RX = P_TX + G_TX - L_FS + G_RX $$




  • $P_RX$: received power by spacecraft


  • $P_TX$: transmitted power by wristwatch


  • $G_TX$: Gain of wristwatch's transmitting antenna (compared to isotropic)


  • $L_FS$: Free space Loss, what we usually call $1/r^2$


  • $G_RX$: Gain of spacecraft's receiving antenna (compared to isotropic)

$$G sim 20 times log_10left( fracpi dlambda right)$$



$$L_FS = 20 times log_10left( 4 pi fracRlambda right).$$




$$ P_RX - P_TX = G_TX - L_FS + G_RX $$



Change from dB to linear scale:



$$ fracP_RXP_TX = fracG_TXG_RXL_FS = fracpi^4 d_RX^2 d_TX^2lambda^4fraclambda^216 pi^2 R^2 = fracpi^2 d_RX^2 d_TX^2lambda^2frac14^2 R^2 = fracpi^2 d_RX^2 d_TX^24 lambda^2 R^2$$



So the fraction of the transmitted power that is received depends on $lambda^-2$ ; it improves as the frequency goes up and the wavelength goes down.






share|improve this answer











$endgroup$



The wavelength dependence of the definition of free space path loss (FSPL) is an artifact of the way the receiver's antenna gain is defined in the same link budget calculation. It's referenced to an ideal isotropic antenna with a receive area of roughly 1 square wavelength, which for high frequency gets very small. If you do them together (transmit gain, path loss, receive gain) you'll see that the higher frequency wins because the gain of the transmit antenna go up, and the combination of the FSPL and the receive antenna together remain the same.



So the reason higher frequencies are used is because the total link budget is better.



You can't just take the FSPL and ignore the other terms in the budget, it won't make sense because of the way things are defined.



You can see real-world examples of link budgets in this answer (short one) and in this answer (longer one) and in this question. You may find this one interesting as well.




From here"



Link Budget



From this answer which is from this answer:



$$ P_RX = P_TX + G_TX - L_FS + G_RX $$




  • $P_RX$: received power by spacecraft


  • $P_TX$: transmitted power by wristwatch


  • $G_TX$: Gain of wristwatch's transmitting antenna (compared to isotropic)


  • $L_FS$: Free space Loss, what we usually call $1/r^2$


  • $G_RX$: Gain of spacecraft's receiving antenna (compared to isotropic)

$$G sim 20 times log_10left( fracpi dlambda right)$$



$$L_FS = 20 times log_10left( 4 pi fracRlambda right).$$




$$ P_RX - P_TX = G_TX - L_FS + G_RX $$



Change from dB to linear scale:



$$ fracP_RXP_TX = fracG_TXG_RXL_FS = fracpi^4 d_RX^2 d_TX^2lambda^4fraclambda^216 pi^2 R^2 = fracpi^2 d_RX^2 d_TX^2lambda^2frac14^2 R^2 = fracpi^2 d_RX^2 d_TX^24 lambda^2 R^2$$



So the fraction of the transmitted power that is received depends on $lambda^-2$ ; it improves as the frequency goes up and the wavelength goes down.







share|improve this answer














share|improve this answer



share|improve this answer








edited Jun 3 at 21:48

























answered Jun 2 at 10:33









uhohuhoh

44k21172569




44k21172569







  • 1




    $begingroup$
    thank you so much! I see that now, did not consider the gain formula, but you are right it is the key to the problem!
    $endgroup$
    – Clex
    Jun 2 at 15:43






  • 1




    $begingroup$
    @Clex ya it was confusing for me at first too. Running through a whole example ike I did here takes some time the first time (it took me hours!), but it helps a lot in the long run.
    $endgroup$
    – uhoh
    Jun 2 at 15:48






  • 2




    $begingroup$
    Yeah, I've always thought that the definition of FSPL (@uhoh, note the acronym isn't FPSL) was kludgy. But once a peer-reviewed paper using it was accepted, saying it should be redefined to be more physically intuitive has the telecom specialists tilting their heads back some while still looking at me, pointing, and uttering, "BLASPHEMY!!".
    $endgroup$
    – Tom Spilker
    Jun 3 at 19:53










  • $begingroup$
    @TomSpilker ha! I can picture it. Thanks for catching my misacronymization also.
    $endgroup$
    – uhoh
    Jun 3 at 21:52












  • 1




    $begingroup$
    thank you so much! I see that now, did not consider the gain formula, but you are right it is the key to the problem!
    $endgroup$
    – Clex
    Jun 2 at 15:43






  • 1




    $begingroup$
    @Clex ya it was confusing for me at first too. Running through a whole example ike I did here takes some time the first time (it took me hours!), but it helps a lot in the long run.
    $endgroup$
    – uhoh
    Jun 2 at 15:48






  • 2




    $begingroup$
    Yeah, I've always thought that the definition of FSPL (@uhoh, note the acronym isn't FPSL) was kludgy. But once a peer-reviewed paper using it was accepted, saying it should be redefined to be more physically intuitive has the telecom specialists tilting their heads back some while still looking at me, pointing, and uttering, "BLASPHEMY!!".
    $endgroup$
    – Tom Spilker
    Jun 3 at 19:53










  • $begingroup$
    @TomSpilker ha! I can picture it. Thanks for catching my misacronymization also.
    $endgroup$
    – uhoh
    Jun 3 at 21:52







1




1




$begingroup$
thank you so much! I see that now, did not consider the gain formula, but you are right it is the key to the problem!
$endgroup$
– Clex
Jun 2 at 15:43




$begingroup$
thank you so much! I see that now, did not consider the gain formula, but you are right it is the key to the problem!
$endgroup$
– Clex
Jun 2 at 15:43




1




1




$begingroup$
@Clex ya it was confusing for me at first too. Running through a whole example ike I did here takes some time the first time (it took me hours!), but it helps a lot in the long run.
$endgroup$
– uhoh
Jun 2 at 15:48




$begingroup$
@Clex ya it was confusing for me at first too. Running through a whole example ike I did here takes some time the first time (it took me hours!), but it helps a lot in the long run.
$endgroup$
– uhoh
Jun 2 at 15:48




2




2




$begingroup$
Yeah, I've always thought that the definition of FSPL (@uhoh, note the acronym isn't FPSL) was kludgy. But once a peer-reviewed paper using it was accepted, saying it should be redefined to be more physically intuitive has the telecom specialists tilting their heads back some while still looking at me, pointing, and uttering, "BLASPHEMY!!".
$endgroup$
– Tom Spilker
Jun 3 at 19:53




$begingroup$
Yeah, I've always thought that the definition of FSPL (@uhoh, note the acronym isn't FPSL) was kludgy. But once a peer-reviewed paper using it was accepted, saying it should be redefined to be more physically intuitive has the telecom specialists tilting their heads back some while still looking at me, pointing, and uttering, "BLASPHEMY!!".
$endgroup$
– Tom Spilker
Jun 3 at 19:53












$begingroup$
@TomSpilker ha! I can picture it. Thanks for catching my misacronymization also.
$endgroup$
– uhoh
Jun 3 at 21:52




$begingroup$
@TomSpilker ha! I can picture it. Thanks for catching my misacronymization also.
$endgroup$
– uhoh
Jun 3 at 21:52










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