Performance Evaluation of a Novel Compacting Code System and Digital Subscriber Line System under Gaussian Noise and Intersymbol Interference

Mazen Alkhatib1, M. S. Alam2, and K. S. Tai2

1Department of Information Technology, Colorado Technical University, Malkhatib@faculty.ctuonline.edu 2 Department of Electrical and Computer Engineering, University of South Alabama, malam@southalabama.edu

Abstract

In this paper, we investigated the error level performance of fast digital subscriber line (VDSL2 Plus)

under Gaussian noise and intersymbol interference. We also investigated the error level performance of

a novel intersymbol interference (ISI) compacting code, and we compared the performance of this new

code to DSL systems. Simulation results show that the performance of the proposed novel ISI

compacting code system is significantly better than the performance of VDSL2 Plus system for signal

compression above 40%.

1. Introduction

Bit error rate plays the most critical role toward effective data transmission in any communication

system. Current communication systems require very high speed data transmission rates. This

requirement is challenged by the presence of intersymbol interference (ISI) and other sources of noise.

To achieve ultrahigh bit transmission rates an efficient pulse shaping and energy usage scheme can be

used as reported in [1] and [2]. It is also possible to tackle the ISI challenge by treating the hazards

resulting from the effect of ISI as the bit transmission rate is increased. Various studies have been

reported in the literature during the last few years [3-8] to calculate the probability of error under

Gaussian noise and ISI.

VDSL2 Plus is a standard that was approved in 2006 [10] for fast transmission of digital information at

speeds up to 200 Mbps. However, it is difficult to achieve such a high speed in the presence of Gaussian

noise and intersymbol interference (ISI). To overcome this problem, recently we designed a novel code

[9] that is capable of reducing the effect of ISI. This code depends on searching for code words that are

mapped to signals in a manner to minimize the effect of ISI.

In this paper, we developed detailed simulation models to measure the bit error rates (BER) of VDSL2

Plus systems and we compared it to the BER of our coding system in the presence of Gaussian noise and

ISI. We verified the effectiveness of the proposed model for different compression ratios and measured

the corresponding BERs.

The rest of the paper is organized as follows. Section 2 describes the simulation of the VDSL2

Plus system, Section 3 deals with the simulation for the novel code system. Section 4 shows the results

obtained, and section 5 includes the concluding remarks.

2. VDSL2 Plus System Simulation Model

The proposed simulation model for the VDSL2 Plus system consists of 4-bit symbols. Ten such symbols

are called a Discrete Multi-Tone (DMT) symbol. The simulated system consists of a transmitter that has

Trellis Coded Modulation (TCM) 3-bit generator, and a mapper to a 4-bit output that is connected to an

array of 4 parallel queues. The output of these queues is connected to four Binary Phase Shift Keying

(BPSK) modulator subsystems.

The receiver consists of BPSK demodulators, Viterbi decoder, and comparators. Figure 1 shows

the structure of the simulated transmitter model, and Figure 2 shows the TCM diagram where the Viterbi

algorithm was used at the receiver end. As shown in Figure 1, at the transmitter, three bits enter the

TCM sub-module which are then mapped into 4 bits (V1V0W1W0). These bits are then queued in four

parallel queues, and are BPSK modulated for transmission using four different carrier signals at different

frequencies.

Once the signals are transmitted, the simulation model adds Gaussian noise and ISI components to these

signals. The Gaussian noise is generated using the Box-Muller technique [7], which uses inverse

transformation to turn two uniformly distributed random variable U1 and U2 into two unit normal

random variables X and Y in the interval (0, 1). The random variables U1 and U2 are generated in such a

way that U1 ≠ 0. The unit normal random variables X and Y can be defined as

(1)

and

. (2)

Figure 1. Structure of the simulated transmitter model.

At the receiver end, signals are demodulated and then Viterbi decoded. The resulting digital pattern is

then compared to the transmitted pattern to determine the bit error rate. Figure 2 shows a Trellis

diagram for the Viterbi algorithm that was applied at the receiver end.

Figure 2. Trellis diagram where Viterbi algorithm was applied at the receiver end.

The simulation program was designed to provide the capability to compress the generated pulses before

transmission. This compression affects the ISI component added to the transmitted signal. Figure 3

shows a flow chart for the main simulation model used for implementing the VDSL2 Plus system.

Figure 3. Flow chart of the simulation program model for the VDSL2 Plus system.

3. Novel coding to reduce ISI:

Intersymbol interference occurs due to overlapping of pulses as signals are compressed. As the

result, the receiver will detect more than one pulse in a prescribed time period. The probability of

receiving an erroneous pulse is denoted as . In order to reduce , a new coding system is developed in

which it can achieve that is less than the value calculated by the series method developed by

Beaulieu in Reference 1. For the sake of simplicity, is measured in terms of BER in the simulation of

the new code for transmission.

The pulse code is developed based on trinary signaling levels i.e., the pulse code belongs to the

set {-1, 0, +1}. The series of bits (bit words) that contain values of {-1, +1} are mapped to series of

pulses (pulse series) with certain coefficients of {-1, 0, +1} where some of the pulses are omitted to

reduce the effect of intersymbol interference. Consider a word of bits which may lead to possible

bit words. As a result, in the new coding system, pulse codes are required to map to -bits

length words. Figure 4 shows the general structure of the novel coding system.

Figure 4: General structure of the novel coding system.

The following criteria were used in generating the novel code [9]:

Number of pulse coefficients within a series per code is .

Each coefficient within a pulse code series is selected from {-1, 0, +1}.

Percentage of the zero coefficients (PZ) should be as high as possible.

Percentage of the negative and positive coefficients (PN and PP) should be as equal as

possible for each type.

All hazard cases should be prevented as much as possible. These pattern are (+1, -1, 0) or (-1,

+1, 0) or (+1, -1, +1) or (-1, +1, -1) or (0, +1, -1) or (0, -1, +1) or (+1, 0, +1) or (-1, 0, -1).

Pulse code series starting with coefficients (0, 0) or (0, +1), (-1, 0) or (-1, +1) are not

allowed.

Pulse code series ending with coefficients (0, -1) or (+1, 0) or (+1, -1) or (-1, 0) or (-1, +1) or

(-1, -1) are not allowed.

Sequence Free Distance (FD) separation should be greater than or equal to minimum free

distance MFD. Values of MFD can be 2, 3, and 4. FD is calculated as the absolute

differences of pulse coefficients for any two pulses series in a code.

3.1. Novel code structure:

A detailed simulation software was designed and developed to simulate the new coding system. There

are two sub-programs in this simulation model. The first sub-program generates the N pulse series or

code words which will meet all the criteria as indicated previously. The second sub-program simulates

the performance of transmission under the presence of intersymbol interference and Gaussian noise

using the novel code words.

3.1.1 Code words generation:

Figure 5 shows the flow chart of the code words generation software model. This program model allows

the user to input the number of signals and the number of included code words to be generated. In

addition, the program model allows the user to set the number of allowed hazard cases within a code

word, the number of minimum zero coefficients, and the minimum free distance in between code words.

A value of 4 for the minimum free distance was used, a value of 0 for the minimum zero coefficients

within a code word was used, and no hazard cases were allowed in the code words.

Figure 5(a): Flow chart of code-generation program (Continued to Fig. 5(b))

Figure 5(b): Continuation of flow chart of the code generation program.

Numerous code search trials were run to find a code that yields the best performance under

Gaussian noise and ISI. A code of 64 words, where each code word is mapped to 12 Gaussian pulses

representing 6 bits each was found to have the best performance.

3.1.2 Transmission modeling:

Gaussian pulses are used to transmit data which is encoded using the optimized code obtained using the

procedure described in the previous section. Pulses are compressed at different levels to measure the

performance in terms of the level of received erroneous bits. Soft decision decoding using the maximum

likely-hood sequence detection principle is used to decode the received pulses into the original data bits.

Figure 6 shows the flow chart for the novel code transmission modeling simulation.

Figure 6: Flow chart of transmission simulation (Continued next page).

Figure 6: Continuation of flow chart of transmission simulation.

4. Results and Discussion

A detailed simulation software was developed for the VDSL system with 10 stages utilizing a

convolutional encoder at the transmitter and a trellis decoder (Viterbi decoder) at the receiver. Another

detailed simulation software was also developed for the novel coding system utilizing trinary signaling

levels with 64 code words with each code word consisting of 12 pulses representing 6 bits. Soft decision

decoding using the maximum likelihood sequence detection principle is used to decode the received

pulses at the receiver.

Gaussian pulses with various compression ratios ranging from 10% to 90% were used for

transmission between the transmitter and the receiver. Gaussian pulses with signal-to-noise ratio of 15

dB were used to be comparable to the results of the research performed in [3]. The resulting bit error

rates as a function of the compression ratio for the TCM and the novel code are summarized in Figure 7.

From Fig. 7, it is evident that when pulses are compressed above 40%, there is a considerable

degradation in the value of BER of the DSL/TCM system. For example, the BER rises sharply from

2.67x10-05 at a pulse compression ratio of 40% to 0.056283 for a compression ratio of 50%. Whereas for

the case of the novel code, BER remains less than 1x10-7 until the pulse compression ratio approaches

60%.BER. Thereafter, the BER rises to 2.69x10-05, which corresponds to the best performance obtained

under ISI and Gaussian noise. Table 1 shows the BER for DSL/TCM and new ISI compacting code

system versus pulse compression percentage.

Table 1: BER for DSL/TCM and new ISI compacting code system versus pulse compression percentage.Percentage of pulse compression% BER(DSL/TCM) BER Novel Code

90 2.27E-01 2.56E-0580 1.94E-01 2.18E-0570 1.60E-01 1.79E-0560 0.200943 2.69E-0550 0.056283 <1.00E-0740 2.67E-05 <1.00E-0730 1.55E-05 <1.00E-0720 1.33E-05 <1.00E-0710 1.11E-05 <1.00E-07

Figure 7: BER of DSL/TCM and the novel code systems as a function of the pulse compression ratio.

5. Conclusions

In this paper, we purposed and simulated a complete and comprehensive method for intersymbol

interference reduction under Gaussian noise and ISI, and compared the performance of the proposed

technique with VDSL systems utilizing TCM. The proposed method enables a new code to transmit 6

data bits per word by using 12 pulses. Simulation results verify that the performance of VDSL2 plus

systems degrades dramatically when the pulse compression ratio exceeds 40%, whereas for our

developed code, simulation results show that it is more reliable under ISI and Gaussian noise under all

values of pulse compression ratio. It outperforms the VDSL 2 plus system considerably for compression

ratios above 40%.

The proposed code uses trinary signaling levels and the maximum likelihood detection method at

the receiver. It would be interesting to further investigate the feasibility of expanding the proposed code

to deal with M-ary signaling level systems utilizing several bandwidth efficient pulse shaping methods

to achieve the most possible reliable faster than Nyquist signaling (FTN).

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