Communications_Engineer
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ofdm time invariant eigenfuction
Hello, can you please help me in understanding the following lines taken
from the book "Fundamentals of Wireless Communication" by David Tse &
Pramod Viswanath.
I quote section 3.4.4, page number 95
-------------------------------------------------------------------------------------------------
Both the single-carrier system with ISI equalization and the DS spread
spectrum system with Rake reception are based on a time-domain view of the
channel.
But we know that if the channel is linear time-invariant, sinusoids are
eigenfunctions and they get transformed in a particularly simple way.
ISI occurs in a single-carrier system because the transmitted signals are
not sinusoids.
This suggests that if the channel is underspread (i.e., the coherence time
is much larger than the delay spread) and is therefore approximately
time-invariant for a sufficiently long time-scale, then transformation into
the frequency domain can be a fruitful approach to communication over
frequency-selective channels. This is the basic idea behind OFDM
"Sinusoids are eigenfunctions of LTI systems, but they are of infinite
duration. If we transmit over only a finite duration, say Nc symbols, then
the sinusoids are no longer eigenfunctions. One way to restore the
eigenfunction property is by adding a cyclic prefix to the symbols"
--------------------------------------------------------------------------------------------------
My questions are,
1) How do we know that if the channel is LTI, 'sinusoids are
eigenfunction' (maybe I dont know what eigenfunctions are), and they get
transformed in a particularly simple way? What simple way? Can someone
please put some light on it
2) Wouldn't it be too trivial to assume the channel to be LTI?
3) What is meant by underspread channel? Is there an overspread channel?
(what I understand, is that the CIR remains constant for the duration of
the excess delay for a symbol, so maybe when we try to equalize we only
have to worry for the effects of flat fading and not for fast fading, would
I be correct when I say this?, still I have got no idea why it would be
underspread)
Thanks for reading through this. I hope you can help me
Hello, can you please help me in understanding the following lines taken
from the book "Fundamentals of Wireless Communication" by David Tse &
Pramod Viswanath.
I quote section 3.4.4, page number 95
-------------------------------------------------------------------------------------------------
Both the single-carrier system with ISI equalization and the DS spread
spectrum system with Rake reception are based on a time-domain view of the
channel.
But we know that if the channel is linear time-invariant, sinusoids are
eigenfunctions and they get transformed in a particularly simple way.
ISI occurs in a single-carrier system because the transmitted signals are
not sinusoids.
This suggests that if the channel is underspread (i.e., the coherence time
is much larger than the delay spread) and is therefore approximately
time-invariant for a sufficiently long time-scale, then transformation into
the frequency domain can be a fruitful approach to communication over
frequency-selective channels. This is the basic idea behind OFDM
"Sinusoids are eigenfunctions of LTI systems, but they are of infinite
duration. If we transmit over only a finite duration, say Nc symbols, then
the sinusoids are no longer eigenfunctions. One way to restore the
eigenfunction property is by adding a cyclic prefix to the symbols"
--------------------------------------------------------------------------------------------------
My questions are,
1) How do we know that if the channel is LTI, 'sinusoids are
eigenfunction' (maybe I dont know what eigenfunctions are), and they get
transformed in a particularly simple way? What simple way? Can someone
please put some light on it
2) Wouldn't it be too trivial to assume the channel to be LTI?
3) What is meant by underspread channel? Is there an overspread channel?
(what I understand, is that the CIR remains constant for the duration of
the excess delay for a symbol, so maybe when we try to equalize we only
have to worry for the effects of flat fading and not for fast fading, would
I be correct when I say this?, still I have got no idea why it would be
underspread)
Thanks for reading through this. I hope you can help me