Optimal Waveform of the Partial-Respond Signal Based on Minimum Out-of-Band Radiation Criterion

Abstract

This paper presents a novel method designing an optimized waveform with high spectral efficiency based on minimum out-of-band radiation criterion. Introducing the Pan-function model, Fourier series, and constraints, we achieve the numeric solution of the optimized baseband signals. The optimal waveform provides an excellent bandwidth efficiency, and its bit error rate (BER) performance is improved using the minimum cross-correlation function. To verify the proposed model, we present a simulation of the signal modulation and demodulation schemes using MATLAB software. The results are consistent with expected results, verifying the correctness of this method.

1. Introduction

Optimized signal waveform with features, higher utilization rate, lower peak-to-average power ratio (PAPR), and higher main lobe energy can be applied in many significant applications, such as mobile telecommunication, satellite communication, aircraft measurement, tracking, and controlling systems [1,2,3]. Previous studies have provided many effective optimization methods for different scenarios. In order to higher the utilization rate in band limited channel, modulations based on orthogonal frequency division multiplexing (OFDM) and quadrature amplitude modulation (QAM) are provided, both of which suffer from high PAPR [3,4,5]. Nonlinear characteristics of the transmitter amplifier lead to undesirable out-of-band radiation, which restrict the use of OFDM and QAM. Moreover, modulation with high PAPR is not suitable in nonlinear channel, where the signal spectrum could also be extended [1]. Efforts have been made in many studies to reduce the out-of-band radiation. In the work [6], minimum shift keying (MSK) is proposed, which is featured with constant envelope and has less out-of-band radiation compared with quadrature phase shift keying (QPSK). Furthermore, Gaussian filtered minimum shift keying (GMSK) shows better outband reduction compared with MSK, which takes the sacrifice of the constant envelope [7]. Pauli et al. [8] proposed the bandwidth efficient reduction of the crest factor (BERC) method.

In most canonical communication systems featured with constant or quasi-constant envelope, a rectangular base band signal is applied in QPSK, offset QPSK (OQPSK), and frequency-shift-keying (FSK)modulation systems; cosine signal is used in MSK and GMSK after filtering. These kinds of modulation systems show a lesser utilization rate compared with multiple carrier modulation. However, the contradiction between the utilization rate, main lode energy, and PAPR still needs further study. In recent years, there have been few studies giving mathematical optimization models based on certain criteria. In this paper, a novel method to design an optimized base band waveform is proposed based on minimum out-of-band radiation criterion and additional restrictions. In this method, we first give the target Pan-function based on the minimum out-of-band radiation criterion. Then, we construct the boundary function with the restrictions, symbol boundary condition, single-symbol signal energy condition, constant-envelope condition, and minimum cross correlation function. Hence, Lagrange’s equation is derived. After that, the base band waveform is discreted using Fourier series. The optimization problem is transmitted to finding the coefficient of the Fourier series, which could minimize the Lagrange’s equation. To verify the proposed method, a communication system simulation model is set up. The numerical results show that the optimized waveform presents sharper roll-off, small out-of-band radiation, quasi-constant envelope, and good bit error rate (BER) performance.

2. Optimized Mathematical Modeling of the Optimized Waveform

2.1. Minimum Out-of-Band Radiation Criterion

In order to optimize the baseband symbol signals with higher spectral roll-off rate, an optimized mathematical function is established which minimizes the out-of-band radiation criterion [9] as

$$J=\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}g(\omega ){\left|S(\omega )\right|}^{2}d\omega$$

(1)

where $a(t)$ is the symbol signal waveform. It is an even function in the symbol period $[-T/2,T/2]$. ${\left|S\left(\omega \right)\right|}^2$ is the power spectrum of $a(t)$, and $g(\omega )$ is an increasing function that determines the roll-off rate of the spectral density function $S(\omega )$. The value of n determines the degree of suppressed out-of-band radiation. To achieve a convergent and integrable J, the roll-off rate of ${\left|S\left(\omega \right)\right|}^2$ should be greater than the rate of increase in $g(\omega )$ to limit the roll-off rate of the power spectrum ${\left|S\left(\omega \right)\right|}^2$ of the signal $a(t)$. Figure 1 plots the increasing function, $g(\omega )={\omega }^{2n}(\omega >0)$ versus n.

Following a series of mathematical manipulations (including Taylor series expansion [10]), Equation (1) may be written as

$$J={(-1)}^n{\displaystyle \underset{-T/2}{\overset{T/2}{\int }}}a(t)a^{(2n)}(t)dt$$

(2)

where $a^{\left(2n\right)}(t)$ represents the $2n$-order derivative of the symbol signal. Equation (2) reveals that the criterion for minimum out-of-band radiation may be transformed into the problem of unraveling the symbol function $a(t)$ when minimizing the Pan-functional J.

2.2. Additional Constraints of the Pan-Function

As an example, we consider the partial-response quadrature modulation with a symbol length of 4T. The analytic solution of the optimized mathematical model is fairly complex under restriction of the symbol boundary condition, single-symbol signal energy condition, constant-envelope condition, and minimum cross-correlation function. However, we may obtain the optimized baseband signal waveforms using numerical calculation.

2.2.1. Restriction on the Symbol Boundary Condition

As is shown in [11], if $a(t)$, $a^{{}^{\prime }}(t)$, $a^{{}^{″}}(t)$, $\dots$, $a^{(n-1)}(t)$ are continuous without skipping, and $a^{(n)}(t)$ is finite, then there is no skipping in the end point of the interval (i.e., the power spectrum of $a(t)$ could decay rapidly with the frequency). Herein, assuming that function $a(t)$ is within the interval $[-2T,2T]$, it should satisfy the following boundary condition [1]:

$$a\left(\pm 2T\right)=a^{{}^{\prime }}\left(\pm 2T\right)=\dots =a^{\left(n-1\right)}\left(\pm 2T\right)=0$$

(3)

2.2.2. Restriction on the Single-Symbol Signal Energy Condition

Considering the signal $a(t)$ with the length of 4T, the energy per transmission can be expressed as

$$E={\displaystyle \underset{-2T}{\overset{2T}{\int }}}{a}^2\left(t\right)dt$$

(4)

2.2.3. Restriction on the Constant-Envelope Condition of the Quadrature-Modulated Signal

The constant-envelope modulation represents that the envelope of the modulated wave should be constant, the generated modulation signal is band-limited by the transmitter, and only a small spectrum spreading is produced after the signal passes through the nonlinear processing components [12]. Thus, the following conditions should be satisfied with t in the interval $[0,4T]$

$$I^2(t)+Q^2(t)=const$$

(5)

Since the signal is symmetric, we only need to satisfy Equation (5) for t in the interval $[0,T]$. Then, the condition will be satisfied in the rest of the initial interval; i.e., $[T,4T]$. In this case, the four symbols in the interval are defined as $d_{-1},d_0,d_1,d_2$. The baseband signal of $I(t)$ and $Q(t)$ in the interval $[0,T]$ can be written as

$$I(t)=d_{-1}a(t+T)+d_{1}a(t-T)$$

(6)

$$Q(t)=d_{0}a(t)+d_{2}a(t-2T)$$

(7)

where $d_{-1}$, $d_0$, $d_1$, $d_2$ are the uncorrelated binary data symbols, each taking one of the values $\pm 1$. The different conditions are as follows: For $d_{-1}{d}_1=1$, the adjacent baseband symbol signals take the same sign in the I channel.

If $d_0{d}_2=1$, the adjacent baseband symbol signals take the same sign in the Q channel, and to meet the constant-envelope condition, we should introduce the attenuation factor r in $[0,1]$, yielding

$$\begin{array}{c}\hfill I_{11}^2(t)=r^2{a}^2(t-T)+2ra(t+T)a(t-T)+a^2(t+T)\\ \hfill Q_{21}^2(t)=r^2{a}^2(t)+a^2(t-2T)+2ra(t)a(t-2T)\end{array}$$

(8)

If $d_0{d}_2=-1$, the adjacent baseband symbol signals take opposite signs in the Q channel, we have

$$\begin{array}{c}\hfill I_{12}^2(t)=a^2(t-T)+2a(t+T)a(t-T)+a^2(t+T)\\ \hfill Q_{22}^2(t)=r^2{a}^2(t)+a^2(t-2T)-2ra(t)a(t-2T)\end{array}$$

(9)

The constant value of the envelope is set to be $a(0)$, which is the value of $a(t)$ at time $t=0$, and is also the maximum value of the signal $a(t)$. To ensure that the modulated signal always has a constant envelope in the interval $[0,T]$, the constant-envelope condition is rewritten as

$${\displaystyle \underset{0}{\overset{T}{\int }}}{(I_{11}^2(t)+Q_{21}^2(t)-a^2(0))}^{2}dt=0$$

(10)

$${\displaystyle \underset{0}{\overset{T}{\int }}}{(I_{12}^2(t)+Q_{22}^2(t)-a^2(0))}^{2}dt=0$$

(11)

Similarly, if $d_{-1}{d}_1=1$, the adjacent baseband symbol signals take the opposite signs in the I channel. The constant envelope conditions may be obtained as

$${\displaystyle \underset{0}{\overset{T}{\int }}}{(I_{13}^2(t)+Q_{23}^2(t)-a^2(0))}^{2}dt=0$$

(12)

$${\displaystyle \underset{0}{\overset{T}{\int }}}{(I_{14}^2(t)+Q_{24}^2(t)-a^2(0))}^{2}dt=0$$

(13)

Thus, the constant-envelope conditions in the interval $[0,T]$ can be expressed as follows:

$$\begin{array}{c}\hfill {\int }_0^T\left\{\begin{array}{c}{\left[I_{11}^2(t)+Q_{21}^2(t)-a^2(0)\right]}^2+{\left[I_{12}^2(t)+Q_{22}^2(t)-a^2(0)\right]}^2\hfill \\ +{\left[I_{13}^2(t)+Q_{23}^2(t)-a^2(0)\right]}^2+{\left[I_{14}^2(t)+Q_{24}^2(t)-a^2(0)\right]}^2\hfill \end{array}\right\}dt=0\end{array}$$

(14)

2.2.4. Restriction on the Minimum Cross-Correlation Function

The intersymbol interference (ISI) lowers the correlation performance in a partial-response system, increasing the bit error rate (BER). In order to improve the receiving reliability, additional restrictions are introduced, including single-symbol energy, boundary condition, constant-envelope, and minimum cross-correlation function. Three random baseband symbols and eight types of waveforms exist in the interval $\left[-2T,2T\right]$, in orthogonal or in-phase channel. Eight different waveforms yield 36 combinations in Euclidean distances. We may express the minimum Euclidean distance between each two waveforms as

$$d_{min}^2=min\left\{d_{\left\{a_i,a_j\right\}}^2\right\}=\underset{i,j}{min}{\displaystyle \underset{-2T}{\overset{2T}{\int }}}{\left(a_i\left(t\right)-a_j\left(t\right)\right)}^{2}dt=min\left\{d_{i,j}^2\right\}$$

(15)

where $a_i\left(t\right)$ is the waveform in the interval $\left[-2T,2T\right]$, $i=1,2,\dots ,8$. Figure 2 shows all possible composed modes of three random baseband symbols in the interval $\left[-2T,2T\right]$. In contrast, Figure 3 plots the synthetic waveform of three random baseband symbols in the interval $\left[-2T,2T\right]$.

Table 1. Euclidean distance between any two waveforms.

${\mathit{d}}_{\mathit{ij}}$	$\mathit{j}=1$	$\mathit{j}=2$	$\mathit{j}=3$	$\mathit{j}=4$	$\mathit{j}=5$	$\mathit{j}=6$	$\mathit{j}=7$	$\mathit{j}=8$
$i=1$	0	$2E$	$4E$	$6E+8K$	$2E$	$4E$	$6E+8K$	$8E+8K$
$i=2$	-	0	$6E-8K$	$4E$	$4E$	$2E$	$8E$	$6E+8K$
$i=3$	-	-	0	$2E$	$6E-8K$	$8E-16K$	$2E$	$4E$
$i=4$	-	-	-	0	$8E$	$6E-8K$	$4E$	$2E$
$i=5$	-	-	-	-	0	$2E$	$4E$	$6E+8K$
$i=6$	-	-	-	-	-	0	$6E-8K$	$4E$
$i=7$	-	-	-	-	-	-	0	$2E$
$i=8$	-	-	-	-	-	-	-	0

lists the Euclidean distance between two waveforms.

$$\begin{array}{ccc}\hfill {d^2}_{i,j}& =& {\displaystyle \underset{-2T}{\overset{2T}{\int }}}{(a_i\left(t\right)-a_j\left(t\right))}^{2}dt\hfill \\ \hfill K& =& {\displaystyle \underset{-2T}{\overset{2T}{\int }}}a(t)·a(t-2T)dt\hfill \\ \hfill E& =& {\displaystyle \underset{-2T}{\overset{2T}{\int }}}{a}^2(t)dt\hfill \end{array}$$

(16)

We obviously see that decreasing K leads to increasing the minimum Euclidean distance, for a constant symbol energy E. Therefore, the minimum cross-correlation function K may restrict the improvement of the BER performance. The autocorrelation function in $\left[0,2T\right]$ can be defined as

$$K={\displaystyle \underset{0}{\overset{2T}{\int }}}a\left(t\right)·a\left(t-2T\right)dt$$

(17)

we have

$${\displaystyle \underset{0}{\overset{2T}{\int }}}\left(a\left(t\right)·a\left(t-2T\right)-K/2T\right)dt=0$$

(18)

Introducing the Fourier series and additional restrictions, single-symbol energy, constant envelope condition after modulation and minimum cross-correlation function, the Pan-function can be expressed as

$$\begin{array}{ccc}\hfill H& =& J+\lambda \left({\displaystyle \underset{-2T}{\overset{2T}{\int }}}{a}^2\left(t\right)dt-E\right)+\mu {\displaystyle \underset{0}{\overset{2T}{\int }}}[a(t)·a(t-2T)-K/2T]dt\hfill \\ & +& \gamma {\displaystyle \underset{0}{\overset{T}{\int }}}\{{[I_{11}^2(t)+Q_{21}^2(t)-a^2(0)]}^2+{[I_{12}^2(t)+Q_{22}^2(t)-a^2(0)]}^2\hfill \\ & +& {[I_{13}^2(t)+Q_{23}^2(t)-a^2(0)]}^2+{[I_{14}^2(t)+Q_{24}^2(t)-a^2(0)]}^2\}dt\hfill \end{array}$$

(19)

where $\lambda$ ,$\mu$, and $\gamma$ are Lagrange constants. H is the Lagrange function.

2.3. Signal Solving Process Using Fourier Series

A periodic function in the interval $[-2T,2T]$ may be expressed by Fourier series expansion [13]. Therefore, the efficient spectrum signal $a(t)$—as an even function with the width of $4T$—can be written as

$$a(t)=\frac{a_0}{2}+{\displaystyle \sum _{k=1}^m}(a_{k}cos(\frac{\pi }{2T}kt))$$

(20)

where

$$a_0=(1/2T){\displaystyle \underset{2T}{\overset{2T}{\int }}}a(t)dt$$

$$a_k=(1/2T){\displaystyle \underset{-2T}{\overset{2T}{\int }}}a(t)cos(\pi kt/2T)dt$$

Then, substituting Equation (20) into Equation (19), yielding

$$\begin{array}{ccc}\hfill H& =& 2T{\displaystyle \sum _{k=1}^m}{a}_k^2{(\frac{\pi }{2T}k)}^{2n}+\lambda \{\frac{E}{2T}-{[2{\displaystyle \sum _{k=1}^m}{(-1)}^{k+1}{a}_k]}^2-{\displaystyle \sum _{k=1}^m}{a}_k^2\}\hfill \\ & +& \mu {\displaystyle \underset{0}{\overset{2T}{\int }}}[a(t)·a(t-2T)-K/2T]dt+\gamma {\displaystyle \underset{0}{\overset{T}{\int }}}({(I_{11}^2(t)+Q_{21}^2(t)-a^2(0))}^2\hfill \\ & +& {(I_{12}^2(t)+Q_{22}^2(t)-a^2(0))}^2+{(I_{13}^2(t)+Q_{23}^2(t)-a^2(0))}^2\hfill \\ & +& {(I_{14}^2(t)+Q_{24}^2(t)-a^2(0))}^2)dt\hfill \end{array}$$

(21)

Based on the extreme value theory of the variational calculus [14], Equation (21) should satisfy the following partial differential equations

$$\begin{array}{ccc}\hfill \frac{\partial H}{\partial a_k}& =& 0;k=1,2,\dots ,m\hfill \\ \hfill \frac{\partial H}{\partial \lambda }& =& 0,\frac{\partial H}{\partial \mu }=0,\frac{\partial H}{\partial \gamma }=0\hfill \end{array}$$

(22)

The normalizations are defined as $E=1$ and $T=1$. Thus, the coefficients, $a_0$, $a_1$, …, $a_m$ of the Fourier series, and the Lagrange constants, $\lambda$ and $\mu$, are the unknown parameters. The solution of simultaneous equations is used to determine the values of the unknown coefficients, $a_0$, $a_1$, …, $a_m$ of the Fourier series. We introduce envelope fluctuation coefficient to determine the constant envelope properties. The envelope fluctuation coefficient is defined as

$$\epsilon =\frac{e_{max}-e_{\mathrm{ave}}}{e_{\mathrm{ave}}}$$

(23)

where $e_{max}$ represents the maximum value of the envelope and $e_{\mathrm{ave}}$ denotes the average value of the envelope.

Table 2. The unknown coefficient of the Fourier series in Equation (20).

n	m	${\mathit{a}}_0$	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{a}}_4$	${\mathit{a}}_5$	${\mathit{a}}_6$	${\mathit{a}}_7$
1	4	0.660	0.5000	0.1793	−0.001865	−0.01111	-	-	-
3	5	0.6364	0.4998	0.2152	0.01092	−0.03402	−0.01126	-	-
5	6	0.6280	0.4992	0.2266	0.02026	−0.03790	−0.02006	−0.003333	-
7	7	0.6310	0.4995	0.2224	0.01635	−0.03726	−0.01671	−0.001131	0.0004888

shows the coefficient of the Fourier series for different m and n. Figure 4 shows the corresponding time domain curves.

Figure 5 shows the normalized power spectra curves of optimized signal, where curve 1 corresponds to the MSK and curves 2–5 correspond to $[n=1,m=4]$, $[n=3,m=5]$, $[n=5,m=6]$, and $[n=7,m=7]$, respectively.

Table 3. Comparison of the characteristics of the signals of the energy spectrum and K. MSK: minimum shift keying.

Waveform	$\mathbf{\Delta }{\mathit{F}}_{-40\mathbf{dB}}$	$\mathbf{\Delta }{\mathit{F}}_{-60\mathbf{dB}}$	$\mathbf{\Delta }{\mathit{F}}_{90\%}$	$\mathbf{\Delta }{\mathit{F}}_{99\%}$	$\mathit{\epsilon }$$({10}^{-2})$	$\mathit{K}({10}^{-5})$
n=1, m=4	1.09	1.21	0.72	1.06	6.70	6.84
n=3, m=5	1.30	1.44	0.78	1.17	0.74	−7.52
n=5, m=6	1.43	1.61	0.79	1.20	0.45	−2.64
n=7, m=7	1.35	1.50	0.80	1.22	0.52	−11.52
MSK	2.5	8.0	0.78	1.2	0.0	0.0

considers different pairs of n and m, and presents the bandwidth, where the normalized power spectra of the optimized signals decreased as $-40$ dB ($\Delta F_{-40\mathrm{dB}}$) and $-60$ dB ($\Delta F_{-60\mathrm{dB}}$). Table 3 also lists the value of the bandwidth, where $90\%$ and $99\%$ of the signal energy is achieved (i.e., $\Delta F_{90\%}$ and $\Delta F_{99\%}$, respectively). Besides, the corresponding envelope fluctuation coefficient $\epsilon$ and the cross-correlation function K of the optimized signal are presented in the last two columns of Table 3.

Figure 5 and Table 3 reveal that for the minimum out-of-band energy radiation criterion with the restrictions, the optimized efficient spectrum signal presents the bandwidth of $1.09/T$ (for $n=1$ and $m=4$, calculated according to $-40$ dB) and $1.21/T$ (for $n=1$ and $m=4$, calculated according to $-60$ dB). Moreover, the bandwidths of $\Delta F_{-40\mathrm{dB}}$ and $\Delta F_{-60\mathrm{dB}}$ have been decreased to $43.6$% and $15.3$% of that of the MSK signal, respectively. This verifies that the signal’s out-of-band radiation is prevented. Applying the optimized efficient spectrum signal as the baseband modulation signal, the modulated signal after the quadrature modulation presents the envelope characteristics. Moreover, the cross-correlation function K of the optimized baseband signal is too small, which reduces the inter-symbol interference and increases the reliability of the receiving.

3. Simulation and Realization of the System

3.1. Scheme of the Modulation and Demodulation Systems

Figure 6 illustrates the block diagram of the adopted partial-respond quadrature modulation scheme. Three serial-to-parallel $(S/P)$ converters convert the group of symbols $d_i$, $i=0,1,\dots$, ∞, with a symbol duration of T into four blocks, each taking one of the values $\pm 1$, where the delay time is a different integer multiple of T while the symbol duration has increased to $4T$. The symbols enter cross-correlator, and according to the results of the cross-correlation operation, the symbol attenuation factor r was introduced. Then, the output signals of the pulse shaping filter provides the signals $I(t)$ and $Q(t)$ in two channels. The signals $I(t)$ and $Q(t)$ were multiplied to the quadrature carrier and added to achieve the modulated signal.

Figure 7 depicts the block diagram representation of the coherent demodulation of the signals. To remove the high-frequency components of the received modulated signal, it is coherently demodulated through the quadrature carriers and the low-pass filters. Since the signal in each channel introduces intersymbol interference, the baseband signal is filtered by a match filter. The signal on each channel is integrated, where the integration time is controlled by the integrator strobe pulse. Finally, the demodulation process follows the sampling, decision, and a $P/S$ converter.

3.2. Simulation Results

We simulated the quadrature modulation and coherent demodulation using the MATLAB software, where an optimized signal was used for the baseband symbol. Figure 8 presents the simulation results, where A represents the baseband symbol signal transmitted through the in-phase channel, B is the baseband symbol signal transmitted via the orthogonal channel, and C is the quadrature-modulated output signal. The envelope of the output signal after the modulation was found to be constant, and in agreement with the expected results.

Figure 9 plots the simulation results of the coherent demodulation, where A indicates the transmitted signal corrupted with additive white Gaussian noise (AWGN), which is the input of the receiver, B is the baseband signal demodulated in the in-phase channel, and C is the baseband signal demodulated in the orthogonal channel. The black solid and green dotted lines in D represent the baseband signal demodulated in the in-phase channel and the baseband signal through the matched filter, respectively, while the arrows show the best sampling time [1,15,16].

Figure 10 compares the BER performance of the optimized baseband signals achieved with and without minimum cross-correlation function [17]. We may notice from this figure that for small $E_b/N_0$s, the BER performances of the optimized baseband with and without minimum cross-correlation function restriction are close. However, when $BER={10}^{-3}$, the $E_b/N_0$s of the optimized baseband signal with and without minimum cross-correlation function restriction are $6.8$ dB and $7.0$ dB, reducing $0.2$ dB in the requirement of $E_b/N_0$. For, $BER={10}^{-4}$, the $E_b/N_0$s of the optimized baseband signal with and without minimum cross-correlation function restriction are $8.3$ dB and $8.6$ dB, respectively, which reduces $0.3$ dB in the requirement of $E_b/N_0$. The results show that the optimized baseband signals with minimum cross-correlation function have superior spectral efficiency and the BER performance can be improved by introducing the minimum cross-correlation function restriction.

4. Conclusions

A Pan-functional model was established to present optimized efficient spectrum signals based on the minimum out-band energy radiation criterion. The achieved signals provide higher out-band spectral roll-off rate with quasi-constant envelope. This is due to the introduction of the controlled ISI, which are greatly superior to that of the QPSK and MSK signals in frequency spectral. Limiting the minimum Euclidean distance, ISI may slightly affect the adjacent baseband symbol signal reception, and improve the performance of the system. We presented the simulation results of the signal modulation and demodulation schemes with controlled ISI, which have been carried out using the MATLAB software. The results reveal that by taking the restriction of the minimum cross-correlation function into account, for the BER of ${10}^{-4}$, the $E_b/N_0$ reduces by $0.3$ dB. As a future work, we will investigate how to optimize the multi-frequency signals (e.g., OFDM systems) to further improve the achievable spectral efficiency.