DSP Trick: Filtering in QAM transmitters and receivers
Subject: Re: DSP Tricks From: Allan Herriman Date: 1999/04/22 Newsgroups: comp.dsp THIS WORK IS PLACED IN THE PUBLIC DOMAIN
Name: Filtering in QAM transmitters and receivers. When *NOT* to do what the textbooks tell you to do.
Category: Hardware architecture, or implementation
Application: QAM receivers (with hardware emphasis)
Advantages: 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.
Introduction: (what the textbooks tell you to do)
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.
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).
There are two parts to this:
1. 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).
^ H(f) | 1.0 |--------\ | \ | \ | \ | \ 0.0 +----------------------------->f ^ ^ ^ | | (1+a)·Fs/2 | | | Fs/2 | (1-a)·Fs/2
'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
For some reason (because the maths isn't too hard?), most implemenations use a "raised cosine" response. The rolloff section between (1-a)·Fs/2 and (1+a)·Fs/2 is actually a half cycle of a cosine wave.
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.
2. 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.
(This was inherent in the maths. If you want to know more, look at a textbook.) 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.
3. Combining 1 and 2 results in the following:
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.
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.
Must meet transmit spectral mask because:
Both of these place limits on the tx filter. This will entail:
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)
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?)
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.
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.)
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).