FDM vs OFDM: Why Can OFDM Overlap Without ICI?

[quantumlight]

I was working on my thesis when this question popped into my head, why can't there be overlap in FDM wihout ICI but you are allowed to overlap in OFDM without ICI?

Also why is the maximum overlap of subchannels 50% in OFDM before ICI occurs?

From math point of view as long as the carriers are orthogonal and you take the integral from 0 to T it always reduces to a delta function so it doesn't seem to matter how much bandwidth is on each side of the center frequency.

 

[the_emi_guy]

FDM and OFDM both have the same overlap!. In FDM the overlap is in the time domain. In OFDM the overlap is in the frequency domain.
First (you may already know this) the relationship between the rectangular pulse and the sin(x)/x (sinc) function:
A rectangular pulse in the time domain transforms to and from the sinc function in the frequency domain. A sinc function in the time domain transforms to and from a rectangular "brickwall" function in the frequency domain. In other words these two functions transform to each other by either FFT or IFFT.

In both FDM and OFDM we are taking multiple carrier frequencies, modulating them, then combining them for transmission.
For simplicity let's assume each carrier is on/off modulated. In idealized FDM, we modulate each carrier then
send each though a brickwall filter before combining to the antenna. Say the carriers are separated by 500KHz, (say at 1GHz + 500KHz, 1GHz + 1MHz ...)
Each carrier's 500KHz brickwall filter in the frequency domain cause a time domain spreading of its on/off pulses into time domain
sinc function with zero crossings every 1us.
Now, if we make the baud rate 1Mbps, each bit's ideal sampling point (center if eye) occurs at the zero crossing point of
all of the potentially interfering sinc functions from previously received bits. In other words there is lots of ISI, but none at the critical moment when the bits are sampled. This is called "signalling at the Nyquist rate" and is related to but not the same as
Nyquist sampling which you hear a lot about. (see en.wikipedia.org/wiki/Nyquist_ISI_criterion, apparently I am not allowed to include link because I am new to physicsforums).
Of course brick wall filters are hard to make, so we use things like raised cosine filters that create the same beneficial sinc zero crossings.

OFDM is analogous to FDM but with time and frequency domain reversed.
We on/off modulate our carriers, but they are combined as unfiltered rectangular pulses and sent straight to the antenna (simplification of course).
These time domain rectangular pulses become spread in the *frequency* domain as sinc functions.
If we on/off modulate each carrier at 1Mbps (1us symbol time), and simultaneously maintain 1MHz carrier spacing ("orthogonal"),
then the zero crossings of the sinc functions occur every 1MHz. Their positions are such that at each carrier frequency, all other carrier's smearing sinc functions have zero crossings. Thus each carrier frequency is free from interference.
Again there is plenty of interference between these signals, but none at the critical frequencies where the carriers are located.
Note the factor of 2 difference between the OFDM bandwidth and the FDM bandwith in my example. This is due to the convention of including negative frequency in the bandwidth in the OFDM case.
Naturally, there is much more to it than this, but this is the basics.
Hopefully you can figure out from this where the 50% comes from (look at the superimposed sinc functions).

Hope this helps,
Cheers.

 

[quantumlight]

Thanks. I get it from that perspective but how does it work when you demodulate with cos(2πf_ct) where f_c matches the carrier. when you take the combined signal and integrate from 0 to T, you theoretically could obtain the original signal with just that carrier back perfectly because integral from 0 to T of cos(2πf₁t)*cos(2πf₂t) is zero if f₁≠ f₂. What is limitations there?

 

[the_emi_guy]

We recover the signal from carrier f1, as you mentioned, by coorelating (multiplying then averaging) the full signal with a tone at the carrier f1
frequency over one symbol time.
The carrier f2 signal (and all others) will correlate to 0 over one symbol time only when its sinc function has a zero crossing at f1.
This requires f1 and f2 to have a specific relationship to each other.
Specifically the carriers must each have an integer number of cycles within the symbol time.
Of course the magic of OFDM is that all of the coorelation over all carriers is done by a simple FFT. Also the correlation is not just done
with cos(wt) but with sin(wt) as well so that we can extract both dimensions (I & Q) of modulation.

Not sure if this answers your question, let me know.

 

[alphysicist]

FDM (Frequency Division Multiplexing) and OFDM (Orthogonal Frequency Division Multiplexing) are both techniques used in communication systems to divide a single channel into multiple subchannels, allowing for the transmission of multiple signals simultaneously.

One key difference between FDM and OFDM is the use of overlap between subchannels. In FDM, overlap between subchannels is not allowed, as it can lead to Inter-Carrier Interference (ICI). This is because in FDM, each subchannel is allocated a specific frequency band and any overlap would result in interference between the signals.

On the other hand, OFDM allows for overlap between subchannels without ICI occurring. This is because OFDM uses orthogonal subcarriers, meaning they are mathematically orthogonal to each other. This allows for the signals to be overlapped without causing interference.

The maximum overlap of subchannels in OFDM is typically limited to 50% before ICI occurs. This is because as the overlap increases, the orthogonality between the subcarriers decreases, leading to ICI. This can be seen mathematically through the integral from 0 to T, which reduces to a delta function when the subcarriers are orthogonal. As the overlap increases, the integral becomes broader, leading to interference between the subcarriers.

In conclusion, the difference in allowing overlap between subchannels in FDM and OFDM lies in the use of orthogonal subcarriers in OFDM. While FDM does not allow for overlap to avoid ICI, OFDM can tolerate overlap up to a certain limit due to its use of orthogonal subcarriers.