Tag: PR signaling

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).

This article is part of the book Wireless Communication Systems in Matlab, ISBN: 978-1720114352 available in ebook (PDF) format (click here) and Paperback (hardcopy) format (click here).

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.

A pre-equalization system incorporating a modulo-M precoder
Figure 1: A pre-equalization system incorporating a modulo-M precoder

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

Partial response schemes: impulse & frequency response

Impulse response and frequency response of PR signaling schemes

Consider a minimum bandwidth system in which the filter is represented as a cascaded combination of a partial response filter and a minimum bandwidth filter . Since is a brick-wall filter, the frequency response of the whole system is equivalent to frequency response of the FIR filter , whose transfer function, for various partial response schemes, was listed in Table 1 in the previous post (shown below).

Table 1: Partial response signaling schemes

The hand-crafted Matlab function (given in the book) generates the overall partial response signal for the given transfer function . The function records the impulse response of the filter by sending an impulse through it. These samples are computed at each symbol sampling instants. In order to visualize the pulse shaping functions and to compute the frequency response, the impulse response of are oversampled by a factor . This converts the samples from symbol rate domain to sampling rate domain. The oversampled impulse response of filter is convolved with a sinc filter that satisfies the Nyquist first criterion. This results in the overall response of the equivalent filter (refer Figure 2 in the previous post).

This article is part of the book Wireless Communication Systems in Matlab, ISBN: 978-1720114352 available in ebook (PDF) format (click here) and Paperback (hardcopy) format (click here).

The Matlab code to simulate both the impulse response and the frequency response of various PR signaling schemes, is given next (refer book for the Matlab code). The simulated results are plotted in the following Figure.

Figure: Impulse response and frequency response of various Partial response (PR) signaling schemes

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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

Partial response (PR) signaling Model

Consider the generic baseband communication system model and its equivalent representation, shown in Figure 1, where the various blocks in the system are represented as filters. To have no ISI at the symbol sampling instants, the equivalent filter should satisfy Nyquist’s first criterion.

Figure 1: A generic communication system model and its equivalent representation

If the system is ideal and noiseless, it can be characterized by samples of the desired impulse response . Let’s represent all the non-zero sample values of the desired impulse response, taken at symbol sampling spacing , as , for .

This article is part of the book
● Wireless Communication Systems in Matlab (second edition), ISBN: 979-8648350779 available in ebook (PDF) format and Paperback (hardcopy) format.

The partial response signaling model, illustrated in Figure 2, is expressed as a cascaded combination of a tapped delay line filter with tap coefficients set to and a filter with frequency response . The filter forces the desired sample values. On the other hand, the filter bandlimits the system response and at the same time it preserves the sample values from the filter . The choice of filter coefficients for the filter and the different choices for for satisfying Nyquist first criterion, result in different impulse response , but renders identical sample values in Figure 2 [1].

Figure 2: A generic partial response (PR) signaling model

To have a system with minimum possible bandwidth, the filter is chosen as

The inverse Fourier transform of results in a sinc pulse. The corresponding overall impulse response of the system is given by

If the bandwidth can be relaxed, other ISI free pulse-shapers like raised cosine can be considered for the filter.

Given the nature of real world channels, it is not always desirable to satisfy Nyquist’s first criterion. For example, the channel in magnetic recording, exhibits spectral null at certain frequencies and therefore it defines the channel’s upper frequency limit. In such cases, it is very difficult to satisfy Nyquist first criterion. An alternative viable solution is to allow a controlled amount of ISI between the adjacent samples at the output of the equivalent filter shown in Figure 2. This deliberate injection of controlled amount of ISI is called partial response (PR) signaling or correlative coding.

Partial Response Signaling Schemes

Several classes of PR signaling schemes and their corresponding transfer functions represented as (where is the delay operator) are shown in Table 1. The unit delay is equal to a delay of 1 symbol duration () in a continuous time system.

Table 1: Partial response signaling schemes

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References

[1] Peter Kabal and Subbarayan Pasupathy, Partial-response signaling, IEEE Transactions on Communications, Vol. 23, No. 9, pp. 921-934, September 1975.↗

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

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Python
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Matlab
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart
Hand-picked Best books on Communication Engineering
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