Steve wrote:
> Hi,
>
> So I have a really simple question about matched filters in
> communication systems that, quite frankly, I'm a bit embarressed about
> asking. Here goes:
>
> So I understand that the point of the matched filter in a
> communications system is to maximise the frequency spectrum of the
> channel when transmitting pulses. So, for example as described in
> Proakis, instead of sending a unit impluse (which would require an
> infinite spectrum to communicate), a sinc (or in practise a root-raised
> cosine) pulse is used, since that translated into a rectangle (max
> bandwidth efficiency) in the frequency domain. This is superior to the
> square pulse in the time domain since, in the frequency domain, this
> translates to a sinc pulse (which make inefficient use of the available
> bandwidth) Simple.

The use of the pulse shaping is to provide some restriction on the
amount of bandwidth used. If you just transmit a finite length square
pulse, then this will cause a very large amount of bandwidth to be used.
If you extend the duration of the pulse over a much large interval (ie
over many bit windows) then you will cut down on the bandwidth used.
You can use the sinc function to perform this spreading out of the pulse
whilst maintaining symbol independence. The output waveform is then the
linear superposition of each sinc waveform for each previous, current
and future pulse... the sinc has nulls at each bit period, and so if
timing recovery is performed correctly each received bit sample will
still correspond to only the value for that bit.

Never realized how difficult it is to explain this...
Wish I could draw you a picture for it, but I'm sure any good text book
will have sufficient details for further understanding.

You are essentially mapping a series of +1/-1 impulses into a series of
sinc functions. Each sinc function is orthogonal at bit periods, there
is no inter-symbol interference. When one bit undergoes its peak
(either positive or negative depending on the bit value) then all the
other bits have zero contribution to the overall signal value.

ie Signal = Bit0_contrib + Bit1_contrib + Bit2_contrib + ...
Signal is a time varying waveform, as are all bit contributions. When
BitX_contrib is a peak all other bit contributions are zero, simply
because of the nature of the sinc waveform.

> What I don't understand is precisely how this fits into the bigger
> picture. Where it falls apart for me is I transmit the binary sequence
> 1010001010101, according to proakis (or at least my understanding of
> what he is saying) for every '1' in the stream I send a sinc-like
> pulse, for every '0' I send nothing. (these pulses and non-pulses
> overlap to produce an analog waveform that can then be efficiently
> transmitted). Basically, I'm not too sure how this fits in with
> modulation. If i'm using a 256-QAM modulation scheme, does this mean
> that that each sample of this new "analog" waveform (assuming 8-bit
> samping), maps to a specific location on the (is it constellation? I
> can't remember). This doesn't sound very efficient to me: 1bit mapping
> to, in this case, 8-bits? Also, what about spread spectrum? In this
> case, as I understand it, each bit is mapped to n bits according to the
> state of the LFSR even before the filter which is even more
> inefficient.

If we assume something simpler than 256-QAM, such as the classic QPSK.
You should have two orthogonal bit streams, each one is filtered by the
pulse shaping filter (RRC), and then quantised (perhaps it's inherently
quantised by the filter) and mapped into a constellation point.

If you're dealing with something having more resolution in each
orthogonal stream than simply single bits then you still perform the
pulse shaping, however you now use actual numeric values rather than
simply +1 or -1. You should think in terms of antipodal signalling to
start with, using 1 and 0 to transmit signals is not as helpful as using
+1 and -1 and can also be harder to understand.


Sorry about the bad wording etc used in this, it's pretty late here and
the mind isn't ready for explaining things well. If it hasn't helped
I'll try again in the morning...