FPGA Central

How in OFDM the size and type of a cyclic prefix gets out? Wha
characteristics of a liaison channel and a signal influence a prefix?

On Nov 10, 6:13 am, "alex65111" <alex65...@list.ru> wrote:
> How in OFDM the size and type of a cyclic prefix gets out? What
> characteristics of a liaison channel and a signal influence a prefix?

The OFDM cyclic prefix duration provides a guard interval between
consecutive OFDM symbols. Its purpose is to prevent energy from the
previous OFDM being smeared into the current one. It depends how
dispersive the communications channel is. The more dispersive the
channel, the longer the CP needs to be.

The CP also has another purpose. It helps to shape the frequency
spectrum of the subcarriers, such that their main lobes are widened
and more importantly, the side lobe heights are reduced. This means
that any loss of sub-carrier orthogonality at the receiver(will always
happen to some extent due to imperfect synchronisation, etc ), has
less of an effect of inter-subcarrier interference. I think of it as
the classic windowing scenario. Without the CP, the OFDM symbol has
been multiplied by a rectangular window in the time domain. This, of
course, is a convolution with sinc kernel in the frequency domain. By
choosing a CP that rolls off to zero(?) it has the effect of applying
a "nicer" window to the OFDM symbol.

What is a liaison channel?

>By choosing a CP that rolls off to zero(?) it has the effect of applying
>a "nicer" window to the OFDM symbol.
>


As the size of a prefix influences on rolls off?




>What is a liaison channel?
>

It is simply communication channel.

> As the size of a prefix influences on rolls off?

I don't understand the question. Why don't you try creating an OFDM
symbol in matlab and experiment with adding different length prefixes.

In matlab I modelled. I have received results about what you spoke. Abou
dispersive the communications channel it is clear. And about shape th
frequency spectrum of the subcarriers - I see that dependence is, but
can not mathematicly describe it.

For example, at me is 12 subcarriers, on everyone subcarrier - PSK4
everyone has speed of transfer of 50 bod.

What length of a prefix should be if the interval between subcarriers i
100Hz?.
And what length of a prefix should be if the interval between subcarrier
is 60Hz?
How it can be mathematicly calculated?

>>And about shape the
>>frequency spectrum of the subcarriers - I see that dependence is, but I
>>can not mathematicly describe it.

Well if there is no cyclic prefix, then it is a rectangular window =
rect(t). The fourier transform of this is sinc(f). So a single sub-
carrier of frequency f1 in the time domain multiplied by a rectangular
function, is a pair of impulses at +/- f1 in the frequency domain,
convolved with a sinc function.

> What length of a prefix should be if the interval between subcarriers is
> 100Hz?.
> And what length of a prefix should be if the interval between subcarriers
> is 60Hz?

I don't know the answer, but thinking out loud: If the time duration
(including cyclic prefix) of the OFDM symbol is T, and assuming a
rectangular window, then the main-lobe width of a single sub-carrier
in the frequency domain is 1/T. So I suppose the OFDM subcarriers
cannot be spaced any closer than 1/T Hz. So presumably it follows that
a longer cyclic prefix, allows the sub-carriers to be spaced closer
together.

But, my understanding is the cyclic prefix is shaped such that it
gives something other than a rectangular window to the OFDM symbol.
This means a sub-carrier main lobe width is now > 1/T, but the
advantage is the side lobes drop off with frequency at a faster rate,
and/or the first side lobe is reduced. So in this case the sub-
carriers must be spaced > 1/T.

Windowing trades off main lobe width vs side lobe height and roll off
rate. There are many different windows out there. Maybe there is a
particular type that often gets used for OFDM applications?

Disclaimer: I'm still learning about OFDM myself

Cheers
Andrew

On Nov 11, 9:21 am, Andrew FPGA <andrew.newsgr...@gmail.com> wrote:
> >>And about shape the
> >>frequency spectrum of the subcarriers - I see that dependence is, but I
> >>can not mathematicly describe it.
>
> Well if there is no cyclic prefix, then it is a rectangular window =
> rect(t). The fourier transform of this is sinc(f). So a single sub-
> carrier of frequency f1 in the time domain multiplied by a rectangular
> function, is a pair of impulses at +/- f1 in the frequency domain,
> convolved with a sinc function.
>
> > What length of a prefix should be if the interval between subcarriers is
> > 100Hz?.
> > And what length of a prefix should be if the interval between subcarriers
> > is 60Hz?
>
> I don't know the answer, but thinking out loud: If the time duration
> (including cyclic prefix) of the OFDM symbol is T, and assuming a
> rectangular window, then the main-lobe width of a single sub-carrier
> in the frequency domain is 1/T. So I suppose the OFDM subcarriers
> cannot be spaced any closer than 1/T Hz. So presumably it follows that
> a longer cyclic prefix, allows the sub-carriers to be spaced closer
> together.

No. T is the length of the OFDM symbol before the cyclic prefix has
been added.

In general, the cyclic prefix should be at least as long as the delay
spread that you expect from your channel. In practice, it must be
longer than that to allow for symbol timing uncertainty, etc.


--
Oli

> No. T is the length of the OFDM symbol before the cyclic prefix has
> been added.

Umm, what part of what I wrote is no? I can define T how I like, and I
defined it to include the cyclic prefix. If there is no window applied
to the cyclic prefix, then the channel and the maths does not care
about the relative size of the original OFDM symbol and cyclic prefix?
A rect in the time domain is a sinc in the frequency domain. Where do
you think my logic is wrong?

But, I suppose that in the receiver the cyclic prefix gets removed
before the FFT demodulation. So as far as the FFT is concerned the
cyclic prefix was never there, so now the sub-carriers are spaced
closer together. Which raises the question, how does a
shaped(windowed) cyclic prefix help reduce inter-carrier interference
if it gets removed prior to the FFT demod? (I know the cyclic prefix
is added to allow for a guard interval, to deal with spread from one
OFDM symbol to the next. But I have also read papers describing how
the CP is also windowed such that it brings an advantage to the shape
of sub-carrier spectrum).

Regards
Andrew

On Nov 11, 7:21 pm, Andrew FPGA <andrew.newsgr...@gmail.com> wrote:
> > No. T is the length of the OFDM symbol before the cyclic prefix has
> > been added.
>
> Umm, what part of what I wrote is no? I can define T how I like, and I
> defined it to include the cyclic prefix. If there is no window applied
> to the cyclic prefix, then the channel and the maths does not care
> about the relative size of the original OFDM symbol and cyclic prefix?
> A rect in the time domain is a sinc in the frequency domain. Where do
> you think my logic is wrong?

Whilst you're free to define T any way you like, the maths only works
(in the sense of retaining sub-carrier orthogonality) if T is defined
as the length excluding the CP.

>
> But, I suppose that in the receiver the cyclic prefix gets removed
> before the FFT demodulation. So as far as the FFT is concerned the
> cyclic prefix was never there, so now the sub-carriers are spaced
> closer together. Which raises the question, how does a
> shaped(windowed) cyclic prefix help reduce inter-carrier interference
> if it gets removed prior to the FFT demod? (I know the cyclic prefix
> is added to allow for a guard interval, to deal with spread from one
> OFDM symbol to the next. But I have also read papers describing how
> the CP is also windowed such that it brings an advantage to the shape
> of sub-carrier spectrum).

The shaping of the CP is irrelevant when it comes to eliminating ICI.
The standard explanation is along the lines of:

The CP effectively converts the linear convolution of the channel into
circular convolution over the non-CP section of the OFDM symbol. For
a DFT, the convolution theorem states that circular convolution in one
domain is equivalent to pointwise multiplication in the other domain.
In other words, after the FFT in the receiver, the original data may
be recovered simply by multiplying each sub-carrier symbol by a single
complex value. There is no interaction between the sub-carriers.


--
Oli

> The shaping of the CP is irrelevant when it comes to eliminating ICI.
L. Cimini in "Analysis and Simulation of a Digital Mobile Channel
Using Orthogonal Frequency Division Multiplexing" say "The act of
truncating the signal [ofdm symbol] to the interval (0, Ndt) [where N
is number of subcarriers, dt is the data bit time interval] imposes a
sinx/x frequency response on each subchannel with zeros at multiples
of 1/T [where T is defined as T = Ndt]. This spectral shape has large
sidelobes, and gives rise to significant interchannel interference in
the presence of multi-path."

They later go on to say "due to the use of the DFT, each subchannel
pssesses a sinx/x spectrum which has fairly large sidelobes, and which
may cause problems if the orthogonality of the subcarriers is
impaired. The sidelobes can be reduced ... by extending the frame of
data in time [i.e cyclic prefix] and by requiring gradual rather than
abrupt rolloffs of the transmitted waveform... shaped using a raised
cosine"

Which makes sense doesn't it? There will always be some loss of
orthogonality, so the lower the sidelobes are, the less they will
effect other sub-channels. So to me, the CP shaping does have an
effect on ICI.

Cheers
Andrew

There are two different definitions of spectrum. Both are relevant in thei
own respect, but not the same.

As far as the FFT is concerned: It "sees" one cycle of a signal that migh
be periodic from minus to plus infinity. There is ideally no energy leakag
from one subcarrier to the other, including for example adjacent but unuse
frequency regions inside the same FFT.

The other definition becomes relevant when I connect a radio frequenc
spectrum analyzer (or its baseband equivalent).
Other than the FFT, it knows nothing about the cyclic assumption, and i
is not synchronized with the symbol structure.
Suddenly the spectrum looks quite messy, energy seems to leak everywhere

That is caused by the DISCONTINUITIES between symbols: At the symbo
boundary, the signal "jumps" from one instantaneous amplitude to another.
This can be expressed as multiplication of each symbol with a rectangula
window => sinc spectrum. It is completely transparent to the FFT, bu
hurts other users that aren't synchronized and can't take advantage of th
single-cycle processing.
In an implementation, an "ideal" symbol-to-symbol discontinuity from
rectangular window jams too much energy into adjacent bands, and we nee
to tame it somehow (for example filtering)

Hope that clarifies somewhat.

-Markus