DSL and faxing?

[#1]

Hi all,
Not familiar with DSL and so just wondering about faxing possibilities.
I seem to be reading that, in order to fax from a pc, I have to connect via a modem to the phone side of my connection, which is a pain, because my connection is downstairs and I am bed-bound.
However, as I understood it, the 'DSL filter' is actually filtering out the digital interference from the phone line, not the other way around, so that what is going to the router should be the full body of signals, both analogue and digital, that are on my twisted pair line.
In this case, I ought to be able to send analogue signals through the router, wirelessly from my pc upstairs, and hence,to fax with a faxmodem on my pc.
Is this actually possible? 
If not: why not?
If it's possible, do I need any dedicated software or particular fax-modem to do it?
Alternatively: my downstairs phone is on a wireless base station.  Might it be possible to transmit from the pc to the phone base station, or to join the phone itself to the pc and fax through the phone?
It seems that, I must have enough connectivity options here, to be able to send a fax without getting another peripheral.
Many thanks in advance for additional enlightenment on this curious inability to send two types of signal down one line from one machine.

[#2]

I'm not aware of any router which has fax capabilities, or even the ability to generate the tones that would normally be found on the 'phone' side of the filter, probably because this would be noticeable when the phone was in use if the router was sending tones at those frequencies.
Your best bet would be to run a telephone extension cable from the socket downstairs to the PC, length on these probably won't affect the fax signal as it would a DSL signal so you should be fine with that, and the extension cables are usually that thin they can be run discreetly over skirting boards and under the the carpet grips under a door frame.
Whilst technically it should be possible to do it wireless to the base station you'd probably need a base station and a transceiver for the PC so they can talk to each other but, again, I'm not aware of anything like that existing.
Sorry for the lack of positive information, but it was an impressive idea, if a little ambitious
P.S. Welcome to the forums.
Cheers,
A.

[#3]

I would have expected that most organisations have email as a point of contact avoiding the need to fax.  However if you really need to send a fax, have a look at this - http://www.freepopfax.com/ - as a solution.

[#4]

Router wise they are not cheap but the Fritzboxes have built in Fax
http://www.amazon.co.uk/FRITZ-Box-7490-International-Integrated/dp/B00H3IC3FC/ref=pd_rhf_gw_p_img_1

[#5]

Thanks all: that was quick!
Avatastic:  I have a long reel lead somewhere, but I don't think I've used it for about 15 years since I last had a phone line, and I don't really want to be up and down tidying wires away for the odd times that I might have to fax.
I wasn't looking for a router with fax/modem capabilities: I just can't see why I cannot just generate the tones on the pc with a modem card and send them straight through the router like anything else from the pc.  If I can Skype by this method, I don't see why I can't fax by it.  I suspect it is more because there is not so much call for faxing these days, more than for it being technically infeasible.  It does seem a bit odd though, that there are so many ways to handle data and mix and match devices, and people keep coming up with new options all the time, that nobody seems to have done this already.
Thanks Townman:  Some legalistic Government departments will only give out fax numbers rather than email addresses.  I can often work out what the email addresses are, but it seems to me that a pc ought really to come equipped with the means to dial straight through to a fax number so long as there is a line.  In fact, I don't really see why programmes or devices cannot even send to fax numbers via my mobile, if I want to.  I know about on-line fax services, but of the two main ones, I think one only allows 5 pages a month on the free version, and the other has advertising on the top sheet.  Neither of these would be suitable for a lengthy legal document, even though I might only need to send one a few times per year.  [I'm actually trying to force the Gov departments to allow emailing, under the Equality Act, as it happens, but this will probably be a lengthy process!]
Kevinduke:  That Fritzbox certainly looks like a clever device, but I don't think I can justify the expense for the use it would actually get.  (I suspect that I might find it pretty confusing too!
Thanks for the ideas folks: very impressed at all the keenness! (y)

[#6]

Quote from: Spamlet

I just can't see why I cannot just generate the tones on the pc with a modem card and send them straight through the router like anything else from the pc.  If I can Skype by this method, I don't see why I can't fax by it.

The difference is that Skype works as data over your internet connection, where as a fax is a phone call between two photocopiers.
It's an over simplification but I think it works.

[#7]

You could use an email to fax, or online fax service.
This one is free (and probably add supported - I'd guess ads added to the fax.
http://faxzero.com/
There are paid ones available also.
Here is a search link
https://www.google.com/search?q=email+to+fax+service

[#8]

Something like this may work?

[#9]

Interesting - I had forgotten that these existed.
hmm wonder if they will co-exist with Ethernet over power...  

[#10]

Did Townman just invent POTS over power?

[#11]

Thanks again all.  This is really a hotbed of ideas!
(Incidentally, site design people: I can't see a way to reply under each suggestion?  Maybe I'm blocking something?)
Avatastic: I'm still not seeing why a phone can't be 'pc shaped'. And send through my router to those twisted wires instead of needing a bypass! The filter is only supposed to be to make things easier on my ears, but for this use, I'm not listening (I've not even told anyone my phone number!) 
TonyAtplus:  I only need to fax when sending legal forms/submissions, but I don't do it enough to sign up for a full service.  As it's a legal thing, I need to steer clear of advertising, which I think, lets out most of the free ones, except for one which only lets you do 5 pages a month, and legal stuff tends to be longer than that.  (I'm hoping to force them to give out email addresses--Tribunal was today, but I'm too ill to attend: hence the desire to have faxed--fingers crossed!)
Pwatson/Townman: I too had forgotten about ethernet over power, and these phone sockets.  I'm running out of plug holes, but that ought to work, I think.  With the emphasis on 'ought'!   One of the reviews mentions it doesn't like ADSL, but that might be just for the computer aspect of things.  Might be worth buying just to see what happens!  (y)
Many thanks,
S