V.17 specifies a training sequence at the start of transmission, which makes the design of a V.17 receiver relatively straightforward. The first stage of the training sequence consists of 256 symbols, alternating between two constellation positions. The receiver monitors the signal power, to sense the possible presence of a valid carrier. When the alternating signal begins, the power rising above a minimum threshold (-43dBm0) causes the main receiver computation to begin. The initial measured power is used to quickly set the gain of the receiver. After this initial settling, the front end gain is locked, and the adaptive equalizer tracks any subsequent signal level variation. The signal is oversampled to 24000 samples/second (i.e. signal, zero, zero, signal, zero, zero, ...) and fed to a complex root raised cosine pulse shaping filter. This filter has been modified from the conventional root raised cosine filter, by shifting it up the band, to be centred at the nominal carrier frequency. This filter interpolates the samples, pulse shapes, and performs a fractional sample delay at the same time. 192 sets of filter coefficients are used to achieve a set of finely spaces fractional sample delays, between zero and one sample. By choosing every fifth sample, and the appropriate set of filter coefficients, the properly tuned symbol tracker can select data samples at 4800 samples/second from points within 0.28 degrees of the centre and mid-points of each symbol. The output of the filter is multiplied by a complex carrier, generated by a DDS. The result is a baseband signal, requiring no further filtering, apart from an adaptive equalizer. The baseband signal is fed to a T/2 adaptive equalizer. A band edge component maximisation algorithm is used to tune the sampling, so the samples fed to the equalizer are close to the mid point and edges of each symbol. Initially the algorithm is very lightly damped, to ensure the symbol alignment pulls in quickly. Because the sampling rate will not be precisely the same as the transmitter's (the spec. says the symbol timing should be within 0.01%), the receiver constantly evaluates and corrects this sampling throughout its operation. During the symbol timing maintainence phase, the algorithm uses a heavier damping.
The carrier is specified as 1800Hz +- 1Hz at the transmitter, and 1800 +-7Hz at the receiver. The receive carrier would only be this inaccurate if the link includes FDM sections. These are being phased out, but the design must still allow for the worst case. Using an initial 1800Hz signal for demodulation gives a worst case rotation rate for the constellation of about one degree per symbol. Once the symbol timing synchronisation algorithm has been given time to lock to the symbol timing of the initial alternating pattern, the phase of the demodulated signal is recorded on two successive symbols - once for each of the constellation positions. The receiver then tracks the symbol alternations, until a large phase jump occurs. This signifies the start of the next phase of the training sequence. At this point the total phase shift between the original recorded symbol phase, and the symbol phase just before the phase jump occurred is used to provide a coarse estimation of the rotation rate of the constellation, and it current absolute angle of rotation. These are used to update the current carrier phase and phase update rate in the carrier DDS. The working data already in the pulse shaping filter and equalizer buffers is given a similar step rotation to pull it all into line. From this point on, a heavily damped integrate and dump approach, based on the angular difference between each received constellation position and its expected position, is sufficient to track the carrier, and maintain phase alignment. A fast rough approximator for the arc-tangent function is adequate for the estimation of the angular error.
The next phase of the training sequence is a scrambled sequence of two particular symbols. We train the T/2 adaptive equalizer using this sequence. The scrambling makes the signal sufficiently diverse to ensure the equalizer converges to the proper generalised solution. At the end of this sequence, the equalizer should be sufficiently well adapted that is can correctly resolve the full QAM constellation. However, the equalizer continues to adapt throughout operation of the modem, fine tuning on the more complex data patterns of the full QAM constellation.
In the last phase of the training sequence, the modem enters normal data operation, with a short defined period of all ones as data. As in most high speed modems, data in a V.17 modem passes through a scrambler, to whiten the spectrum of the signal. The transmitter should initialise its data scrambler, and pass the ones through it. At the end of the ones, real data begins to pass through the scrambler, and the transmit modem is in normal operation. The receiver tests that ones are really received, in order to verify the modem trained correctly. If all is well, the data following the ones is fed to the application, and the receive modem is up and running. Unfortunately, some transmit side of some real V.17 modems fail to initialise their scrambler before sending the ones. This means the first 23 received bits (the length of the scrambler register) cannot be trusted for the test. The receive modem, therefore, only tests that bits starting at bit 24 are really ones.
The V.17 signal is trellis coded. Two bits of each symbol are convolutionally coded to form a 3 bit trellis code - the two original bits, plus an extra redundant bit. It is possible to ignore the trellis coding, and just decode the non-redundant bits. However, the noise performance of the receiver would suffer. Using a proper trellis decoder adds several dB to the noise tolerance to the receiving modem. Trellis coding seems quite complex at first sight, but is fairly straightforward once you get to grips with it.
Trellis decoding tracks the data in terms of the possible states of the convolutional coder at the transmitter. There are 8 possible states of the V.17 coder. The first step in trellis decoding is to find the best candidate constellation point for each of these 8 states. One of thse will be our final answer. The constellation has been designed so groups of 8 are spread fairly evenly across it. Locating them is achieved is a reasonably fast manner, by looking up the answers in a set of space map tables. The disadvantage is the tables are potentially large enough to affect cache performance. The trellis decoder works over 16 successive symbols. The result of decoding is not known until 16 symbols after the data enters the decoder. The minimum total accumulated mismatch between each received point and the actual constellation (termed the distance) is assessed for each of the 8 states. A little analysis of the coder shows that each of the 8 current states could be arrived at from 4 different previous states, through 4 different constellation bit patterns. For each new state, the running total distance is arrived at by inspecting a previous total plus a new distance for the appropriate 4 previous states. The minimum of the 4 values becomes the new distance for the state. Clearly, a mechanism is needed to stop this distance from growing indefinitely. A sliding window, and several other schemes are possible. However, a simple single pole IIR is very simple, and provides adequate results.
For each new state we store the constellation bit pattern, or path, to that state, and the number of the previous state. We find the minimum distance amongst the 8 new states for each new symbol. We then trace back through the states, until we reach the one 16 states ago which leads to the current minimum distance. The bit pattern stored there is the error corrected bit pattern for that symbol.
So, what does Trellis coding actually achieve? TCM is easier to understand by looking at the V.23bis modem spec. The V.32bis spec. is very similar to V.17, except that it is a full duplex modem and has non-TCM options, as well as the TCM ones in V.17.
V32bis defines two options for pumping 9600 bits per second down a phone line - one with and one without TCM. Both run at 2400 baud. The non-TCM one uses simple 16 point QAM on the raw data. The other takes two out of every four raw bits, and convolutionally encodes them to 3. Now we have 5 bits per symbol, and we need 32 point QAM to send the data.
The raw error rate from simple decoding of the 32 point QAM is horrible compared to decoding the 16 point QAM. If a point decoded from the 32 point QAM is wrong, the likely correct choice should be one of the adjacent ones. It is unlikely to have been one that is far away across the constellation, unless there was a huge noise spike, interference, or something equally nasty. Now, the 32 point symbols do not exist in isolation. There was a kind of temporal smearing in the convolutional coding. It created a well defined dependency between successive symbols. If we knew for sure what the last few symbols were, they would lead us to a limited group of possible values for the current symbol, constrained by the behaviour of the convolutional coder. If you look at how the symbols were mapped to constellation points, you will see the mapping tries to spread those possible symbols as far apart as possible. This will leave only one that is pretty close to the received point, which must be the correct choice. However, this assumes we know the last few symbols for sure. Since we don't, we have a bit more work to do to achieve reliable decoding.
Instead of decoding to the nearest point on the constellation, we decode to a group of likely constellation points in the neighbourhood of the received point. We record the mismatch for each - that is the distance across the constellation between the received point and the group of nearby points. To avoid square roots, recording x2 + y2 can be good enough. Symbol by symbol, we record this information. After a few symbols we can stand back and look at the recorded information.
For each symbol we have a set of possible symbol values and error metric pairs. The dependency between symbols, created by the convolutional coder, means some paths from symbol to symbol are possible and some are not. It we trace back through the possible symbol to symbol paths, and total up the error metric through those paths, we end up with a set of figures of merit (or more accurately figures of demerit, since larger == worse) for the likelihood of each path being the correct one. The path with the lowest total metric is the most likely, and gives us our final choice for what we think the current symbol really is.
That was hard work. It takes considerable computation to do this selection and traceback, symbol by symbol. We need to get quite a lot from this. It needs to drive the error rate down so far that is compensates for the much higher error rate due to the larger constellation, and then buys us some actual benefit. Well in the example we are looking at - V.32bis at 9600bps - it works out the error rate from the TCM option is like using the non-TCM option with several dB more signal to noise ratio. That's nice. The non-TCM option is pretty reasonable on most phone lines, but a better error rate is always a good thing. However, V32bis includes a 14,400bps option. That uses 2400 baud, and 6 bit symbols. Convolutional encoding increases that to 7 bits per symbol, by taking 2 bits and encoding them to 3. This give a 128 point QAM constellation. Again, the difference between using this, and using just an uncoded 64 point constellation is equivalent to maybe 5dB of extra signal to noise ratio. However, in this case it is the difference between the modem working only on the most optimal lines, and being widely usable across most phone lines. TCM absolutely transformed the phone line modem business.