M-ary energy detection of a Gaussian FSK UWB system

Abstract

The energy detection M-ary Gaussian frequency-shift keying (FSK) system is proposed in this paper. The system performance is analyzed in additive white Gaussian noise channels, multipath channels, and in the presence of synchronization errors. The numerical results show that the M-ary modulation achieves the higher data rate than the binary modulation. However, it also results in performance degradation.

1 Introduction

Energy detection (ED) is a popular technology in ultra-wideband (UWB) communication systems. When compared to Rake receivers, ED has as many advantages, such as simple receiver structure [1, 2], robustness to synchronization errors, and no channel estimation requirement. The popular ED schemes include on-off keying (OOK) and pulse position modulation (PPM) [3]. In our previous publication [4], a new ED scheme based on using different-order derivatives of the Gaussian pulse was proposed. This new scheme is called Gaussian frequency-shift keying (GFSK) energy detection UWB. It was proved that this new scheme has better bit error rate (BER) performance than an energy detection PPM UWB system in multipath channels and in the presence of synchronization errors.

In [4], the energy detection Gaussian FSK system is based on binary modulation so we extend the research into M-ary modulation in this paper.

Although there exist some M-ary UWB methods, our method is different to these methods. In [5], the M-ary modulation is achieved by using the orthogonality of modified Hermite polynomials $h_n\left(t\right)={(-1)}^n{e}^{\frac{t^2}{4}}\frac{d^n}{d{t}^n}\left(e^{\frac{-t^2}{2}}\right)$. Where n is the order of the polynomials and n=0,1,|…, and −∞<t<∞. In [6], the waveform is obtained from the Battle-Lemarie wavelet $\psi \left(t\right)=\sqrt{2}\sum _l\sum _k{g}_l{c}_k{\beta }_n(2t-l-k)$, where g _l and c _k are coefficients, and β _n (t) is a B-spline function of order n. In [7], the author uses the pulsed sinusoidal waveform as carrier to realize the M-ary FSK modulation. In this paper, we will construct the M-ary FSK waveforms using different order derivatives of the Gaussian pulse. The spectra of these different-order derivatives of the Gaussian pulses are separated entirely in frequency domain so the orthogonality is achieved. Our M-ary FSK system still transmits baseband pulses without inducing carrier modulation. UWB is carrier-less system, so our method conforms to UWB regulation. At the receiver, the demodulation can be achieved by using filters with different passband frequencies to separate the pulses directly. This makes it very simple and easy to implement the demodulation procedure.

The structure of this paper is as follows: Section 2 introduces the system models. Section 3 evaluates system performance in additive white Gaussian noise (AWGN) channels. Section 4 evaluates system performance in multipath channels. The effect of synchronization errors on system performance is analyzed in Section 5. In Section 6, the numerical results are analyzed. Section 7 is the conclusion.

2 System models

Before we develop the M-ary Gaussian FSK system, we will briefly summarize the system model of binary modulation in [4]. The Fourier transform X _f and center frequency f _c of the k th-order derivative are given by

$$X_f\propto f^{k}exp(-\pi f^2{\alpha }^2/2)$$

(1)

$$f_c=\sqrt{k}/\left(\alpha \sqrt{\pi }\right)$$

(2)

where k is the order of the derivative and f is the frequency. The pulse shape factor is denoted by α. When the value of α is fixed and the value of k is changed, we obtain spectral curves for different-order derivatives of the Gaussian pulse. Those curves have similar shapes and bandwidths but their center frequencies are different. If the values of k and α are appropriately chosen, it is always possible to separate the spectra of the two pulses. Exploiting the spectral characteristics of the pulses, the transmitter of the Gaussian FSK system is as follows.

$$s{\left(t\right)}_{\text{GFSK}}=\sum _j\sqrt{E_p}\left(b_j{p}_1\right(t-j{T}_f)+(1-b_j\left)p_2\right(t-j{T}_f\left)\right)$$

(3)

where p₁(t) and p₂(t) denote the pulses of different-order derivatives with normalized energy and E _p is the signal energy. The j th transmitted bit is denoted by b _j . The frame period is denoted by T _f . The modulation is carried out as follows: when bit 1 is transmitted, the value of b _j and 1−b _j are 1 and 0, respectively, so p₁(t) is transmitted. Similarly, the transmitted pulse for bit 0 is p₂(t). The above transmitter considers a single user only, and a bit is transmitted once. The receiver includes two branches, and each branch is a conventional energy detection receiver. The only difference between the two branches is the passband frequency ranges of filters. The filter in the first branch is designed to pass the signal energy of p₁(t) and reject that of p₂(t), and the filter in the second branch passes the signal energy of p₂(t) and rejects that of p₁(t). Finally, the captured energies of two branches are subtracted to generate a decision variable and then this variable is compared with decision threshold to determine the transmitted bit. If the decision variable is greater than decision threshold, the transmitted bit is 1, otherwise it is 0. The abovementioned is the brief description of the transmitter and receiver of the energy detection GFSK UWB system in [4].

In this paper, we will develop the M-ary system as follows: The transmitter of the M-ary system is

$$s\left(t\right)=\sum _j\sqrt{E_p}{p}^{\left(M_j\right)}(t-j{T}_f)$$

(4)

where $p^{\left(M_j\right)}$ denotes the transmitted pulse for the j th symbol and $p^{\left(M_j\right)}$ is the derivative of the Gaussian pulse. The order of the derivative is given by the value of M _j . We will use an example to show how to assign a value for M _j according to the transmitted data. For example, we use a 4-ary system in the followed analysis and every two bits are encoded in to a symbol using Gray code. Figure 1 shows the spectral curves of the pulses for the different bit groups. The four curves have the same bandwidth and different center frequencies. They are orthogonal so they can be separated at the receiver. The bit groups of 00, 01, 11, and 10 are corresponding to the spectral curves from the left to right in Figure 1. The orders of the derivatives of these four curves are 52, 110, 190, and 290, respectively. To make sure that the spectra of the selected curves meet the Federal Communications Commission (FCC) emission mask, we add the emission mask. For convenience in observation, we convert the linear scale of Figure 1 into the logarithmic scale of Figure 2. In Figure 2, it is straightforward to see that all of the curves meet the emission requirement set by the FCC. The receiver structure is shown in Figure 3. The receiver has four branches. The only difference of these branches is the passbands of the filters. The filters in each branch can capture the energy of only one signal and suppress the energy of the other signals. In Figure 3, branches 1, 2, 3, and 4 capture the energy of the pulses with orders of 52, 110, 190, and 290, respectively. The shape factor is α=1×10⁻⁹, and the width of the pulses is 2.4×10⁻⁹. In other words, these four branches capture the signal energies corresponding to bit groups 00, 01, 11, and 10, respectively. Finally, the decision is based on comparing the captured energies of the four branches. The branch with the largest energy is chosen.

Figure 1

PSD of pulses versus FCC emission mask (logarithmic scale).

Figure 2

PSD of pulses versus FCC emission mask (linear scale).

Figure 3

Receiver of a GFSK UWB system.

3 SER performance in AWGN channels

3.1 SER performance of GFSK in AWGN channels

Assuming symbols 0, 1, 2, and 3 are transmitted with equal probability, the symbol error rate (SER) is calculated as

$$P_e=\frac{1}{4}{p}_{e_0}+\frac{1}{4}{p}_{e_1}+\frac{1}{4}{p}_{e_2}+\frac{1}{4}{p}_{e_3}$$

(5)

where $p_{e_0}$, $p_{e_1}$, $p_{e_2}$, and $p_{e_3}$ denote the SER when symbols 0, 1, 2, and 3 are transmitted, respectively. We will define how a detection error occurs as follows. When a symbol is transmitted, the captured signal energy in the branch corresponding to that symbol is not the largest, we can claim a detection error occurs. For example, when symbol 0 is transmitted, the error detection occurs if any of the captured signal energies from branch 1, 2, and 3 is greater than that of branch 0. When symbol 0 is transmitted, the SER can be expressed as

$${P_e}_0=1-{P_c}_0$$

(6)

where P _{c 0} denotes the probability of the successful detection when symbol 0 is transmitted. When the captured energy of branch 0 is greater than that of all the other branches, the signal is successfully detected. So we have

$$P_{c_0}=P_{c_{01}}{P}_{c_{02}}{P}_{c_{03}}=(1-P_{e_{01}})(1-P_{e_{02}})(1-P_{e_{03}})$$

(7)

where $P_{c_{01}}$, $P_{c_{02}}$, and $P_{c_{03}}$ denote the probability that the captured energy of branch 0 is greater than those of branches 1, 2, and 3, respectively. And $P_{e_{01}}$, $P_{e_{02}}$, and $P_{e_{03}}$ denote the probability that the captured energy of branch 0 is less than those of branches 1, 2, and 3, respectively. Substituting (7) into (6), we have

$$P_{e_0}=1-(1-P_{e_{01}})(1-P_{e_{02}})(1-P_{e_{03}})$$

(8)

The calculation of $p_{e_0}$ is converted to calculate $P_{e_{01}}$, $P_{e_{02}}$, and $P_{e_{03}}$. In Figure 3, Z_j,0, Z_j,1, Z_j,2, and Z_j,3 are chi-square variables with approximately a degree 2TW[8], where T denotes the integration time and W denotes the bandwidth of the filtered signals. Gaussian approximation has been a popular method to simplify the derivation of BER equation in energy detection UWB systems. When UWB signals pass through the channel, the long channel delay leads to a very large 2TW value and Gaussian approximation can achieve accurate result. The application of Gaussian approximation in energy detection UWB systems can be seen in [2–4, 9–14]. The chi-square variable can be approximated as Gaussian variables [4, 15].

$$\mu =N_0\mathit{\text{TW}}+E$$

(9)

$${\sigma }^2=N_0^2\mathit{\text{TW}}+2N_{0}E$$

(10)

where μ and σ² are the mean value and variance, respectively. The double-sided power spectral density of AWGN is N₀/2. And T denotes the integration time, W denotes the bandwidth of filter. The signal energy which passes through the filter is denoted by E. If the filter rejects all of the signal energy, then E=0. In Figure 3, when symbol 0 is transmitted, the signal energy passes through filter 0 and is rejected by the other filters. The probability density function (pdf) of Z_j,0 is $Z_{j,0}\sim N(N_0\mathit{\text{TW}}+E_s,N_0^2\mathit{\text{TW}}+2N_0{E}_s)$, where E _s denotes the symbol energy and j denotes the j th transmitted symbol. When symbol 0 is transmitted, Z_j,1, Z_j,2, and Z_j,3 have the same probability density function $Z_{j,1}=Z_{j,2}=Z_{j,3}\sim N(N_0\mathit{\text{TW}},N_0^2\mathit{\text{TW}})$. Firstly, we calculate $P_{e_{01}}$ as follow.

$$P_{e_{01}}=P(Z_{j,0}<Z_{j,1})$$

(11)

It can be converted to calculate $P_{e_{01}}=P(Z_{j,0}-Z_{j,1}<0)$. The probability density function of Z_j,0−Z_j,1 is $Z_{j,0}-Z_{j,1}\sim N(E_s,2N_0^2\mathit{\text{TW}}+2N_0{E}_s)$. We can achieve

$$\begin{array}{ll}{P}_{e_{01}}& =\underset{-\infty }{\overset{0}{\int }}\frac{1}{\sqrt{2\pi }\sqrt{2N_0^2\mathit{\text{TW}}+2N_0{E}_s}}{e}^{-\frac{{(x-E_s)}^2}{2(2N_0^2\mathit{\text{TW}}+2N_0{E}_s)}}\mathit{\text{dx}}\\ =Q\left(\frac{E_s}{\sqrt{2N_0^2\mathit{\text{TW}}+2N_0{E}_s}}\right)\end{array}$$

(12)

where Q(·) is the complementary error function. Using the same method, we achieve

$$P_{e_{02}}=P_{e_{03}}=Q\left(\frac{E_s}{\sqrt{2N_0^2\mathit{\text{TW}}+2N_0{E}_s}}\right)$$

(13)

Substituting (12) and (13) into (8), we achieve

$$P_{e_0}=1-{\left(1-Q\left(\frac{E_s}{\sqrt{2N_0^2\mathit{\text{TW}}+2N_0{E}_s}}\right)\right)}^3$$

(14)

Similarly, we can achieve

$$P_{e_1}=P_{e_2}=P_{e_3}=1-{\left(1-Q\left(\frac{E_s}{\sqrt{2N_0^2\mathit{\text{TW}}+2N_0{E}_s}}\right)\right)}^3$$

(15)

Substituting (14) and (15) into (5), we achieve the SER in AWGN channels

$$P_e=1-{\left(1-Q\left(\frac{E_s}{\sqrt{2N_0^2\mathit{\text{TW}}+2N_0{E}_s}}\right)\right)}^3$$

(16)

The above equation can be extended to the M-ary modulation as

$$P_e=1-{\left(1-Q\left(\frac{E_s}{\sqrt{2N_0^2\mathit{\text{TW}}+2N_0{E}_s}}\right)\right)}^{M-1}$$

(17)

If M=2, Equation 17 will be reduced to

$$P_e=Q\left(\frac{E_s}{\sqrt{2N_0^2\mathit{\text{TW}}+2N_0{E}_s}}\right)$$

(18)

Equation 18 is the SER of binary GFSK in AWGN channels, and it is the same as that in [4].

4 SER performance in multipath channels

In this section, the SER performances of PPM and GFSK in multipath channels are researched. The channel model of the IEEE 802.15.4a standard [16] is used in this paper. After the signal travels through a multipath channel, it is convolved with the channel impulse response. The received signal becomes

$$r\left(t\right)=s\left(t\right)\otimes h\left(t\right)+n\left(t\right)$$

(19)

where h(t) denotes the channel impulse response and n(t) is AWGN. The symbol ⊗ denotes the convolution operation. The IEEE 802.15.4a model is an extension of the Saleh-Valenzeula (S-V) model. The channel impulse response is

$$h\left(t\right)=\sum _{l=0}^L\sum _{k=0}^K{\alpha }_{k,l}exp\left(j{\varphi }_{k,l}\right)\delta (t-T_l-{\tau }_{k,l})$$

(20)

where δ(t) is the Dirac delta function, and α_k,l is the tap weight of the k th component in the l th cluster. The delay of the l th cluster is denoted by T _l and τ_k,l is the delay of the k th multipath component relative to T _l . The phase ϕ_k,l is uniformly distributed in the range [ 0,2π].

Figure 4 is the frame structure of M-ary GFSK in multipath channels. The integration interval T₀ is less or equal to the maximum channel spread of GFSK. The synchronization is assumed to be perfect in analysis. The guard interval is T _g , and the frame period is set to T _f =T₀+T _g =D. This will achieve the maximum data rate and prevent inter-frame interference (IFI) simultaneously. This frame structure is applied to all symbols. Assuming symbol 0 is transmitted, the pdfs of Z_j,0 is $Z_{j,0}\sim N(N_0\mathit{\text{TW}}+\lambda E_s,N_0^2\mathit{\text{TW}}+2N_0\lambda E_s)$. And Z_j,1, Z_j,2, and Z_j,3 have the same pdfs $Z_{j,1}=Z_{j,2}=Z_{j,3}\sim N(N_0\mathit{\text{TW}},N_0^2\mathit{\text{TW}})$. Following the same method in Section 3, we achieve the SER in multipath channels

$$P_e=1-{\left(1-Q\left(\frac{\lambda E_s}{\sqrt{2N_0^2\mathit{\text{TW}}+2N_0\lambda E_s}}\right)\right)}^3$$

(21)

Figure 4

A GFSK frame structure in multipath channels.

where λ denotes the ratio of the captured energy of a symbol. The value of λ is related to the integration interval. The longer integration interval will capture more energy. Due to the randomness of the channel parameters, there is no direct equation to calculate the value of λ[4]. We can follow the method in [4] to achieve the value of λ. The SER value should be averaged over many channel realizations. We will explain these more detailed in numerical analysis of Section 6. The above equation can be extended to the M-ary modulation as follows:

$$P_e=1-{\left(1-Q\left(\frac{\lambda E_s}{\sqrt{2N_0^2\mathit{\text{TW}}+2N_0\lambda E_s}}\right)\right)}^{M-1}$$

(22)

If binary modulation is used, M=2, and Equation 22 is reduced to

$$P_e=Q\left(\frac{\lambda E_s}{\sqrt{2N_0^2\mathit{\text{TW}}+2N_0\lambda E_s}}\right)$$

(23)

Equation 23 is SER of binary GFSK in multipath channels, and it is the same as in [4].

5 Performance analysis in the presence of synchronization errors

Figure 5 depicts the GFSK frame structures when synchronization errors ε occur. Assuming that coarse synchronization has been achieved, the SER performance of GFSK are analyzed in the range ε∈ [ 0,D/2]. To prevent IFI, the frame length is set to T _f =2D+T _g , where the guard interval T _g equals to D/2, the maximum synchronization error used in this paper. Assuming symbol 0 is transmitted, the pdfs of Z_j,0 is $Z_{j,0}\sim N(N_0\mathit{\text{TW}}+\rho E_s,N_0^2\mathit{\text{TW}}+2N_0\rho E_s)$. And Z_j,1, Z_j,2, and Z_j,3 have the same pdfs $Z_{j,1}=Z_{j,2}=Z_{j,3}\sim N(N_0\mathit{\text{TW}},N_0^2\mathit{\text{TW}})$. Following the same method in Section 3, we achieve the SER in the presence of synchronization errors.

$$P_e=1-{\left(1-Q\left(\frac{\rho E_s}{\sqrt{2N_0^2\mathit{\text{TW}}+2N_0\rho E_s}}\right)\right)}^3$$

(24)

Figure 5

A GFSK frame structure in the presence of synchronization errors.

And the equation for M-ary modulation is

$$P_e=1-{\left(1-Q\left(\frac{\rho E_s}{\sqrt{2N_0^2\mathit{\text{TW}}+2N_0\rho E_s}}\right)\right)}^{M-1}$$

(25)

where ρ denotes the ratio of the captured signal energy when synchronization error ε occurs. Due to the randomness of channel parameters, there is no direct equation to express ρ according to ε[4]. We can follow the method in [4] to achieve the value of ρ. The SER must be averaged over many channel realizations. In Section 6, the detailed method will be described. When M=2, Equation 25 is reduced to

$$P_e=Q\left(\frac{\rho E_s}{\sqrt{2N_0^2\mathit{\text{TW}}+2N_0\rho E_s}}\right)$$

(26)

It is the same as that in [4].

6 Numerical results, analysis and discussions

The SER equations for binary GFSK are proven validated in [4]. From the above analysis, we can know that the M-ary SER equation can be converted to the SER equation for binary modulation when the value of M=2. So, the SER equations of the M-ary modulation are general equations which can be applied to any M-ary modulation including the binary modulation. In the following, we will compare the performance of binary and M-ary modulation using both the simulated and analytical SER curves. We use the SER equations to generate analytical SER curves directly. In this paper, we use 4-ary as an example to compare with binary modulation. We consider the pulses in Figure 1. The filter bandwidth is 1.66 Ghz.

Figures 6 and 7 show the SER performance comparisons of binary and 4-ary GFSK in multipath channels. The CM4 model [16] is used in SER calculation. Synchronization is perfect, and the maximum channel spread D is truncated to 80 ns. The frame length is designed using the method mentioned in Section 4, so IFI is avoided in SER calculation. In analytical SER calculation, we need to know the value of λ in Equations 21 and 23. Since there is no direct equation to calculate the captured energy respective to the integration interval, we follow the method in [4] to achieve the value of λ as follows. We use the MATLAB code in [16] to generate the realization of channel impulse response h(t). And then we calculate the ratio of the captured energy in specific interval to the total energy of the channel realization to achieve the value of λ. Both the simulated and analytical SER is averaged over 100 channel realizations. We will compare the performance of these two modulations under different integration intervals. In Figure 6, when the integration interval T₀ is 80, 40, and 25 ns, the binary GFSK all achieves a 0.4-dB improvement at S E R=10⁻⁵. The simulated and analytical SER curves all match well.

Figure 6

Comparisons of SER performances of binary and 4-ary GFSK in multipath channels. CM4 model, D=80 ns, T₀=80 and 40 ns.

Figure 7

Comparisons of SER performances of binary and 4-ary GFSK in multipath channels. CM4 model, D=80 ns, T₀=25 ns.

In Figures 8 and 9, the performances of binary GFSK and 4-ary GFSK are compared in the presence of synchronization errors. In SER calculation, the frame length is set to the maximum channel spread D. And the SER is achieved by using Equations 21 and 23 directly. Similarly, the value ρ in Equations 21 and 23 are achieved by using the similar method above. When synchronization errors are ε=0,2,and 4 ns, the binary GFSK all achieve a 0.4-dB improvement at S E R=10⁻⁵. The simulated SER curves all match analytical SER curves well.

Figure 8

Comparisons of SER performances of binary and 4-ary GFSK in the presence of synchronization errors. CM4 model, T₀=D=80 ns, ε=0 and 2 ns.

Figure 9

Comparisons of SER performances of binary and 4-ary GFSK in the presence of synchronization errors. CM4 model, T₀=D=80 ns, ε=4 ns.

From above numerical analysis, we can know that binary GSFK achieves a consistent 0.4-dB improvement under different integration intervals T₀ or synchronization errors. However, the 4-ary can deliver a speed that is twice that of the binary modulation. It is significant that the 0.4-dB loss of performance can achieve a double speed. There is always a trade-off between binary and M-ary modulations. The M-ary can deliver higher data rate but it causes performance loss. The designers should consider the actual applications to decide what kind of modulation to choose. If the speed is the primary requirement, the M-ary is a better option. If the performance is the most important condition to consider, binary is more suitable.

But if we use BER to compare the performance, the conclusion is different. When we look at Figure 1, because a symbol can be converted to any other three symbols with the same probability, any symbol error will lead to the $\frac{1}{2}\ast \frac{1}{3}+1\ast \frac{1}{3}+\frac{1}{2}\ast \frac{1}{3}=\frac{2}{3}$ bit error. And the relationship between E _b /N₀ and E _s /N₀ is $E_s/N_0=\underset{2}{log}4\ast E_b/N_0=2\frac{E_b}{N_0}$. We can convert Equation 22 into

$$P_b=\frac{2}{3}\left(1-{\left(1-Q\left(\frac{\lambda 2E_b}{\sqrt{2N_0^2\mathit{\text{TW}}+2N_0\lambda 2E_b}}\right)\right)}^3\right)$$

(27)

Figure 10 shows the simulated and analytical BER curves. The analytical BER curves are generated using Equation 27. When T₀=80 ns, the 4-ary achieves about 5.6-dB improvement at BER =10⁻⁵. When T₀=40 ns, the 4-ary achieves about 5.8-dB improvement at BER =10⁻⁵. Because in binary, a symbol error leads to an entire bit error. However, the 4-ary still can achieve some correct bits. Also, every bit in 4-ary convey less energy. So when we use BER to compare performance, 4-ary actually achieves better BER performance.

Figure 10

Comparisons of BER performances of binary and 4-ary GFSK in multipath channels. CM4 model, D=80 ns, T₀=80 and 40 ns.

The implementation of high-order derivatives of the Gaussian pulse can be achieved by two approaches. The first one is the digital approach. The discreet points of waveforms are stored in the memory of application-specific integrated circuit (ASIC) or field programmable gate array (FPGA) chips. These points are sent to digital to analog converter (DAC), and then the DAC outputs the analog waveform of the pulse. This approach is straightforward and easy to achieve the waveform accurately. UWB pulses usually have large bandwidth, so it needs high sampling rate of the digital signal to recover the analog signal. The DAC needs to work in high clock rate. The burden of digital implementation is added on DAC. If we implement the pulse waveform by analog approach, it will not need high-speed DAC. The system will send the control signal to activate the corresponding pulse generator respective to the specific data. The design of analog pulse generator is not so straightforward as the digital approach. There are many analog research achievements on UWB pulse generators. In [17], a 7th-order pulse generator is proposed. In [18], the pulse generator arrives to the 13th order. The pulse generator in [19] can generate the pulse with a center frequency at 34 GHz. The higher the order of the pulse is, the higher the center frequency is. The pulse generator in [19] has the ability to generate pulse with higher center frequency than the range of 3.1 to 10.6 GHz set by FCC. These research achievements have already provided feasible solutions for analog implementation of high-order derivative of the Gaussian pulse. We do not provide any specific circuit design in this paper and it is left for the hardware engineers or researchers.

7 Conclusion

The M-ary energy detection GFSK UWB system is proposed. The system performance is analyzed in AWGN channels, multiple channels, and in the presence of synchronization errors. The M-ary system can achieve higher data rate than binary system. However, it causes the performance loss.