📄 dsp trick filtering in qam transmitters and receivers.htm
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<H2 align=center>DSP Trick: Filtering in QAM transmitters and receivers</H2>
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<P><TT>Subject: Re: DSP Tricks<BR>From: Allan Herriman<BR>Date:
1999/04/22<BR>Newsgroups: comp.dsp<BR><BR>THIS WORK IS PLACED IN THE
PUBLIC DOMAIN<BR></TT></P>
<P><TT><B>Name:</B> Filtering in QAM transmitters and receivers. When
*NOT* to do what the textbooks tell you to do. </TT></P>
<P><TT><B>Category:</B> Hardware architecture, or implementation</TT></P>
<P><TT><B>Application:</B> QAM receivers (with hardware emphasis)</TT></P>
<P><TT><B>Advantages:</B> The textbook descriptions of QAM receivers
sometimes miss practical details. "Optimal" solutions may not be the best
ones... These tricks are simply a list of possible reasons for deviating
from "normal" QAM filter design.</TT></P>
<P><TT><B>Introduction:</B> (what the textbooks tell you to do)</TT></P>
<P><TT>Any modem which modulates a linear channel (AM, ASK, FM, FSK, PM,
PSK, BPSK, QAM, QPSK, or even baseband signalling) subject to noise will
use filtering to improve the error rate in the receiver.</TT></P>
<P><TT>In general, there will be filters on the output signals (tx) and
the input signals (rx), and also the bit in the middle (the
channel).</TT></P>
<P><TT>There are two parts to this:</TT></P>
<OL>
<LI><TT>Eliminate ISI (Inter Symbol Interference) due to limitations in
the frequency or phase response of the channel.</TT>
<LI><TT>Use matched filtering to produce a maximum likelihood
receiver.</TT> </LI></OL>
<P><TT><B>1.</B> ISI can be eliminated if the channel (including tx and rx
filters) frequency response is (1) linear phase, and (2) has symmetry
about a point at half the symbol rate (Fs/2).</TT></P>
<BLOCKQUOTE><PRE> ^ H(f)
|
1.0 |--------\
| \
| \
| \
| \
0.0 +----------------------------->f
^ ^ ^
| | (1+a)稦s/2
| |
| Fs/2
|
(1-a)稦s/2
</PRE></BLOCKQUOTE>
<P><TT>'a' here is actually 'alpha' - the rolloff factor. (Although I have
seen (1+alpha) used instead of alpha.) Alpha can be between 0 and 1, but
commonly this will be 0.3 to 0.5 (or 1.3 to 1.5 using the other
definition.) for 30% to 50% excess bandwidth</TT></P>
<P><TT>For some reason (because the maths isn't too hard?), most
implemenations use a "raised cosine" response. The rolloff section between
(1-a)稦s/2 and (1+a)稦s/2 is actually a half cycle of a cosine
wave.</TT></P>
<P><TT>Note: it is also possible to use an adaptive equaliser in the
receiver (either before or after the symbol decisions are made) which can
reduce ISI. But this is "simply" a filter which adapts its response so the
above requirement is met. Adaptive equalisers are used when the channel
response is unknown or changing. They may either "adapt" to a training
sequence and then remain fixed (like a fax machine), or they continuously
adapt.</TT> </P>
<P><TT><B>2.</B> A matched filter will produce the lowest errors in the
receiver output for a channel which adds white gaussian noise (an AWGN
channel) if the rx filter impulse response is the time inverse of the tx
pulse shape. (The tx pulse shape is determined by the tx filter.) In the
frequency domain, this means that the magnitude responses of the rx and tx
filters are the same, but the phase responses are opposite (and the
combination has zero phase (linear phase in practice)). The matched filter
output is only valid at the symbol sampling instant.</TT></P>
<P><TT>(This was inherent in the maths. If you want to know more, <A
href="http://www.dspguru.com/info/books/ref2.htm" target=_top>look at a
textbook</A>.) For example, if we transmit square pulses, then the rx
filter should have a square impulse response. This would be an
integrate-and-dump filter.</TT></P>
<P><TT><B>3.</B> Combining 1 and 2 results in the following:</TT></P>
<P><TT>An optimal modem will use root-raised cosine filtering in the tx
and rx filters. (A root-raised cosine filter puts "half" the response in
the tx and "half" in the rx filter, so that the product in the frequency
domain is a raised cosine.) The total channel reponse will have zero ISI,
and the tx and rx filters are the same, so we have minimised the
probability of errors.</TT></P>
<H3 align=center>The Tricks</H3>
<P><TT>The above description can be found in any communications textbook.
Now for what the textbooks leave out: some examples of when *not* to use
"optimal" filters.</TT></P>
<H4>Trick #1:</H4>
<P><TT>Must meet transmit spectral mask because:</TT></P>
<OL>
<LI><TT>Certain regulatory bodies place restrictions on the tx spectrum
from a modem. For RF modems, the out-of-band emissions sometimes have to
be < -80dBc.</TT>
<LI><TT>Sometimes, the tx signal will interfere with the rx signal at
the same end of the link in nearby channels. This is known as NEXT (Near
End Cross Talk). In the case of an RF modem, the tx signal can be more
than 100dB stronger than the rx signal, so NEXT can be a big
problem.</TT> </LI></OL>
<P><TT>Both of these place limits on the tx filter. This will
entail:</TT></P>
<OL>
<LI><TT>Using a small alpha.</TT>
<LI><TT>Truncating the tails of the tx filter frequency response.</TT>
<UL>
<LI><TT>This will result in degraded performance.</TT>
<LI><TT>Truncate the rx filter reponse as well.</TT> </LI></UL>
<LI><TT>Using a non-root-raised cosine tx filter. Pick one that allows a
sharper rolloff.</TT>
<LI><TT>Allocating more of the raised cosine filter to the tx, and less
to the rx </TT></LI></OL>
<H4>Trick #2:</H4>
<P><TT>Interfering signal has non-white spectrum. (AWGN assumption was
made in the matched filter derivation.) Known narrowband interferers can
be handled by putting a notch in the rx filter. If the notch is very
narrow, the tx filter needn't be changed. Adjacent channel interference
can be handled by making the rx filter slightly narrower. (See Trick #1
above)</TT></P>
<H4>Trick #3:</H4>
<P><TT>Symbol timing recovery problems. A matched filter produces a
maximum likelihood estimate of the input symbol at a particular instant
only. This assumes that this instant is known. Some simpler symbol-timing
recovery schemes may require sub-optimal filtering to work. For example,
wideband rx and tx filters allow signal transition detection to be used
for symbol timing recovery. (This is how a UART works.) Symbol timing
recovery is usually easier with larger alpha. (Books could be written
about symbol-timing recovery. Any takers?)</TT></P>
<H4>Trick #4:</H4>
<P><TT>When one of the filters cannot be controlled. Perhaps the receiver
uses analog filtering only, possibly in a SAW filter in the IF (passband)
or RLC filter at baseband (BTW, 2nd and 3rd order butterworth have been
used here). This filter will only be rough approximation for a
root-raised-cosine, and will not have a linear phase response. This can be
compensated for in the (FIR) tx filter.</TT></P>
<H4>Trick #5:</H4>
<P><TT>When there are significant non-linearities (in the tx output
amplifier). Usually, the requirement will be to have the smallest amount
of AM in the tx, which allows the average output power to be higher for a
given amount of spectral spreading (due to the non-linearity). This may
require wider tx filters and narrower rx filters. Useful where power
efficiency is important (satellite links, handheld equipment, etc). There
is also a case for using a larger alpha here. In extreme cases, it is
possible to pick a modulation scheme that has a constant-amplitude
constellation. (OQPSK, GMSK, etc.)</TT></P>
<H4>Trick #6:</H4>
<P><TT>When the rx filter is inside a feedback loop controlling carrier
phase or frequency tracking. The group delay of the rx filter limits the
tracking bandwidth of these loops (due to stability considerations). If a
wider loop bandwidth is required (perhaps because of capture range or
perhaps poor phase noise performance in the up- and downconverters), then
the rx filter may need to be changed if it is not possible to move it
outside the loop. In this case, allocate more of the raised cosine filter
to the tx, and less to the rx (or try harder to move it outside the
loop).</TT></P>
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