Wireless Communications and Mobile Computing

Volume 2017 (2017), Article ID 7071648, 10 pages

https://doi.org/10.1155/2017/7071648

## Quick Performance Assessment of Improved Nyquist Pulses

Department of Telecommunications, “Gheorghe Asachi” Technical University of Iaşi, Bd. Carol I, No. 11A, 700506 Iaşi, Romania

Correspondence should be addressed to Felix Diaconu; or.isaiut.itte@unocaidf

Received 7 July 2016; Accepted 20 September 2016; Published 11 January 2017

Academic Editor: Javier Del Ser

Copyright © 2017 Nicolae Dumitru Alexandru and Felix Diaconu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An explanation is proposed for the improved behavior of the improved Nyquist pulses with an asymptotic decay rate of when sampled with a timing offset. Three figures of merit that indicate the energy distribution into the sidelobes of the time response and allow a quick assessment of their performance in terms of error probability when the impulse response is sampled with a timing error have been proposed and verified on several improved Nyquist pulses reported in the literature. In order to check the validity of the proposed figures of merit a novel family of Nyquist pulses denoted as power sine was introduced. Using the proposed approach the design process was expedited as the volume of necessary calculations was significantly decreased. To explain the difference in close pulse performance a figure of merit based on limited ISI distortion was introduced.

#### 1. Introduction

In the design of a Nyquist filter the robustness to timing jitter is a prime factor. A solution is to redistribute the tail energy of impulse response by diminishing the size of the largest sidelobe and in turn increase the size of the subsequent sidelobes. The result is a decrease of the intersymbol interference (ISI), which manifests itself in a decrease of the error rate when the impulse response is sampled at the receiver’s site with a timing error. This approach was first proposed and used in [1].

Beaulieau et al. [1, 2] demonstrated that a Nyquist pulse that decays asymptotically as performs better than a pulse with an asymptotic decay rate (ADR) of , such as the standard raised cosine (RC or rcos) pulse, in terms of error probability when sampled with a timing offset. In practice the pulse is generated digitally using a truncated version of the impulse response of the Nyquist filter and is digitally filtered at the receiver, using an adapted filter to its truncated version, which introduces spectral re-growth.

The new pulses with a slower ADR ( versus ) require a bigger number of taps when implemented digitally in a finite impulse response (FIR) structure, as compared with the RC pulse. Stated in another way, for a given truncated length, the spectral regrowth of a slowly decaying pulse is worse than that of a fast decaying pulse. In order to obtain comparable performance in terms of spectral regrowth, one should increase the number of filter taps, resulting in bigger latency and implementation costs.

In some applications where the latency is not critical these disadvantages are counteracted by the fact that one obtains increased performance in terms of lower bit error rate (BER).

Several researchers embraced this idea known as improved Nyquist filter (INF) or pulse and produced on heuristic bases novel pulses that performed better than the ones previously reported [3–11]. However, the mechanism behind the improved performance of Nyquist pulses was not completely understood so far. In the sequel we investigated the behavior of several improved Nyquist pulses that decay asymptotically as and and proposed three figures of merit to quickly assess their performance in terms of error probability when the impulse response is sampled with a timing error.

Also, a novel family of Nyquist pulses, denoted as power sine was introduced in order to check the validity of the approach to expedite the design process.

#### 2. Fractional Energy

To make the tails of the impulse response less damped, that is, to increase its oscillatory feature it is necessary to transfer an amount of energy from the lower frequency part of the transition region into the higher one [7, 9]. The parameter is known as excess bandwidth or roll-off factor.

This is to say the frequency characteristic should be concave in the range and in view of odd-symmetry convex in the range. Otherwise stated, we wish to decrease the fractional energy contained in the lower frequency band of the time-unlimited Nyquist pulse by transferring some energy into the higher frequency band , that is, to increase the fractional energy contained in the frequency band up to some level.

As a figure of merit regarding the efficiency of this transfer we propose to take the ratio of the pulse energy contained either in the frequency interval or to the total energy contained in the frequency interval , by taking into account only the positive frequency components. We will denote it as fractional energy and , respectively,

Also,

For a Nyquist filter the energy carried in the frequency interval is obviously for . So,

This is justified by the fact that has unit energy, as this is a Nyquist filter.

A higher value of the fractional energy denotes that more energy is transferred within the transition band of the filter from to the frequency interval.

A similar approach was found in [12] where a similar metric was used. This is calculated by taking the ratio of the root-Nyquist pulse energy between and infinity, to its total energy.

Table 1 reports the analytic expressions of fractional energy for several Nyquist pulses that are mathematically tractable. For other pulses one must use numerical integration for given values of excess bandwidth .