Precoding for partial response signaling schemes

Introduction to precoding

Intersymbol interference (ISI) is a common problem in telecommunication systems, such as terrestrial television broadcasting, digital data communication systems, and cellular mobile communication systems. Dispersive effects in high-speed data transmission and multipath fading are the main reasons for ISI. To maximize the capacity, the transmission bandwidth must be extended to the entire usable bandwidth of the channel and that also leads to ISI.

To mitigate the effect of ISI, equalization techniques can be applied at the receiver side. Under the assumption of correct decisions, a zero-forcing decision feedback equalization (ZF-DFE) completely removes the ISI and leaves the white noise uncolored. It was also shown that ZF-DFE in combination with powerful coding techniques, allows transmission to approach the channel capacity [1]. DFE is adaptive and works well in the presence of spectral nulls and hence suitable for various PR channels that has spectral nulls. However, DFE suffers from error propagation and is not flexible enough to incorporate itself with powerful channel coding techniques such as trellis-coded modulation (TCM) and low-density parity codes (LDPC).

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These problems can be practically mitigated by employing precoding techniques at the transmitter side. Precoding eliminates error propagation effects at the source if the channel state information is known precisely at the transmitter. Additionally, precoding at transmitter allows coding techniques to be incorporated in the same way as for channels without ISI. In this text, a partial response (PR) signaling system is taken as an example to demonstrate the concept of precoding.

Precoding system using filters

In a PR signaling scheme, a filter is used at the transmitter to introduce a controlled amount of ISI into the signal. The introduced ISI can be compensated for, at the receiver by employing an inverse filter . In the case of PR1 signaling, the filters would be

Generally, the filter is chosen to be of FIR type and therefore its inverse at the receiver will be of IIR type. If the received signal is affected by noise, the usage of IIR filter at the receiver is prone to error propagation. Therefore, instead of compensating for the ISI at the receiver, a precoder can be implemented at the transmitter as shown in Figure 1.

Since the precoder is of IIR type, the output can become unbounded. For example, let’s filter a binary data sequence through the precoder used for PR1 signaling scheme .

% Matlab code snippet
>> d=[1,0,1,0,1,0,1,0,1,0]
>> filter(1,[1 1],d)
ans = 1  -1  2  -2  3  -3  4  -4  5  -5

The result indicates that the output becomes unbounded and some additional measure has to be taken to limit the output. Assuming M-ary signaling schemes like MPAM is used for transmission, the unbounded output of the precoder can be bounded by incorporating modulo-M operation.

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Reference

[1] R. Price, Nonlinear Feedback Equalized PAM versus Capacity for Noisy Filter Channels, in Proceedings of the Int. Conference on Comm. (ICC ’72), 1972, pp. 22.12-22.17

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Topics in this chapter

Pulse Shaping, Matched Filtering and Partial Response Signaling ● Introduction ● Nyquist Criterion for zero ISI ● Discrete-time model for a system with pulse shaping and matched filtering  □ Rectangular pulse shaping  □ Sinc pulse shaping  □ Raised-cosine pulse shaping  □ Square-root raised-cosine pulse shaping ● Eye Diagram ● Implementing a Matched Filter system with SRRC filtering  □ Plotting the eye diagram  □ Performance simulation ● Partial Response Signaling Models  □ Impulse response and frequency response of PR signaling schemes ● Precoding  □ Implementing a modulo-M precoder  □ Simulation and results