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International Journal of Bifurcation and Chaos, Vol. 13, No. 2 (2003) 411–425 c World Scientific Publishing Company
PARAMETER-INDUCED STOCHASTIC RESONANCE AND BASEBAND BINARY PAM SIGNALS TRANSMISSION OVER AN AWGN CHANNEL FABING DUAN and BOHOU XU∗ Department of Mechanics, Zhejiang University, Hangzhou 310027, P.R. China ∗ [email protected] Received June 7, 2001; Revised December 12, 2001 From the point of information theory, the baseband binary pulse amplitude modulated (PAM) signals transmission, via tuning the nonlinear receiver’s parameters, is studied over an additive white Gaussian noise (AWGN) channel. It is demonstrated that the channel capacity of baseband binary communication systems, for a given signal added noise, can be maximized by optimal designed receivers. This new form of stochastic resonance (SR) is referred to as parameter-induced stochastic resonance (PSR) in a broad sense. PSR effect does not require that the input signals are subthreshold. The characteristic behavior of channel capacity versus transmission bit rate is also studied. For reproducing the original binary signals more correctly, time scale transformation method and the approach of ensemble average probability of error bits are introduced. We observe a marked enhancement of the channel capacity for binary PAM signals transmission by PSR in numerical simulations. With this theory and method, the practical applications are considered. Keywords: Stochastic resonance; binary PAM signals; bit error rate; signal processing.
1. Introduction
ler, 1996], biological systems [Douglass et al., 1993; Collins et al., 1996b; Levin & Miller, 1996], signal transmission [Chapeau-Blondeau, 1997; Godivier & Chapeau-Blondeau, 1998], signal detection [Hu et al., 1992; Inchiosa & Bulsara, 1996, 1998; Galdi et al., 1998], and image processing [Misono et al., 1998] within the framework of information theory. Recently, ASR effect has been demonstrated experimentally in a real system by Barbay et al. [2000]. Most previous papers realized SR or ASR by tuning the input noise density. However, a more realistic situation is that the input signal corrupted by noise is a given quantity, which can be tuned by increasing but not decreasing the input noise. It will cause some difficulties in practice. Why do we not study and design the system that has adjustable
Stochastic resonance (SR) has been the subject of both theoretical and experimental research in the past two decades [Benzi et al., 1981; McNamara & Wiesenfeld, 1989; Wiesenfeld & Moss, 1995; Collins et al., 1995a; Bulsara & Gammaitoni, 1996; Gammaitoni et al., 1998; Alfonsi et al., 2000; Barbay et al., 2000]. Collins et al. [1995a] proposed aperiod stochastic resonance (ASR) as a theory of SR for excitable systems with a broadband aperiodic input signal. ASR has then attracted a particular interest in improving actual useful signal detection and transmission. This research subject has recently been the focus of intensive investigation in neurosciences [Collins et al., 1995a, 1995b, 1996a; Stemm∗
Author for correspondence. 411
412 F. Duan & B. Xu
parameters and yields the maximum amount of information? This is important for practical application in those systems where adding or reducing noise might not be an option [Anishchenko et al., 1993; Bulsara & Gammaitoni, 1996; Jung, 1995; Xu et al., 2002]. In contexts of signal processing and communication, the system under study can be viewed as an algorithm by which the sampled signals are processed or transformed in some way, resulting in other signals as outputs. In this paper, the characteristics of the baseband binary communication system, which includes a nonlinear receiver with two adjustable parameters, are studied for binary pulse amplitude modulated (PAM) signals transmission. Here, it is no longer a case of optimizing the noise level, but of designing this nonlinear receiver in such a way that the performance of the baseband binary communication system can be improved in the presence of noise. Even if the power of noise added is much stronger than that of the input signal, it is demonstrated that the optimal receivers with design-parameters can yield the maximum channel capacity. We refer to this new kind of SR as parameter-induced stochastic resonance (PSR) in a broad sense. It is worth noting that our resonance points or region are optimal nonlinear receivers. PSR effect occurs when the input signals are either subthreshold or suprathreshold [Stocks, 2000]. The behavior of channel capacity, as a function of transmission bit rate, is also investigated with optimal receivers. Furthermore, for recovering the information-bearing binary PAM signals more correctly, time scale transformation method and the approach of ensemble average probability of error bits are introduced. Especially worthy of note is a recent research on the information capacity of a bistable dynamic system [Godivier & ChapeauBlondeau, 1998], in which this system is used to transmit a broadband aperiodic informative signal. Comparing with their study, the present investigation designs the optimal systems via the tuning of the system’s parameters, rather than the noise intensity, and develops the theoretical formula of bit error rate. The organization of the paper is as follows. Section 2 briefly describes the characterizations of baseband binary PAM signals and communication systems. Channel capacity, deduced from the nonstationary probability density function, is used to quantify the performance of the baseband communication system. In Sec. 3, the optimal receivers
via tuning of its parameters are studied for binary PAM signals transmission over an AWGN channel. We observe a marked enhancement of the channel capacity in numerical simulations and attempt to interpret the basic mechanism of PSR effect. Moreover, the time scale transformation method and the approach of ensemble average probability of error bits are discussed in detail. Simulation results agree well with the theory developed in Sec. 2. In conclusion, the practical applications are considered with this theory and method.
2. The Measurements of Baseband Binary Communication System: Bit Error Rate and Channel Capacity In this section, the baseband binary communication system (see Fig. 1) for conveying informationbearing signals is described. Its performance is briefly analyzed by channel capacity, which is a function of bit error rate. Figure 1 shows the structure of this baseband binary communication system, wherein a nonlinear dynamic model acts as the receiver. As indicated in Fig. 1, the messages produced by source are a sequence of binary digits. The modulator simply maps the binary digit 0 into a waveform s1 (t) and the binary digit 1 into a waveform s2 (t). The signal waveforms here can be represented as [Schwartz, 1980; Proakis, 1995] for
s1 (t) = −A,
s2 (t) = A,
(n − 1)Ts ≤ t ≤ nTs ,
n = 1, 2, . . . .
(1)
Such a signal is called binary pulse amplitude modulated (PAM) signal. The time interval T s is termed
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Fig. 1. Basic elements of a baseband binary communication system. Dashed frames indicate being chosen.
Parameter-Induced Stochastic Resonance and Baseband Binary PAM Signals Transmission 413
bit duration and bit rate is r = 1/Ts [Schwartz, 1980; Proakis, 1995]. The channel displayed in Fig. 1 denotes a physical transmission medium, in which the transmission signals are corrupted by an additional noise ξ(t). ξ(t) is a Gaussian white noise with zero-mean and the correlation hξ(t)ξ(0)i = 2Dδ(t). Here, h·i and δ(·) represent the average operator and the Diracdelta function, respectively, and D represents noise intensity. To simplify, we assume that this physical channel attenuation factor is unity, that is, the channel output is the sum of input signals and noise. In the design of a communication system, it is convenient to construct a mathematical model for evaluating the performance of the communication system. A general communication channel model is an AWGN channel, which actually comprises all portions, such as the modulator, sampler, the receiver, the physical channel, etc. Let us consider this AWGN channel as a discrete memoryless channel, which has input binary signals S = {s 1 , s2 } and the detector outputs reading Y = {y 1 , y2 }. Assume that the input binary symbols occur with equiprobabilities P (0) = P (1) = 0.5 and are statistically independent, so that there is no dependence among successively received binary symbols. P (0) and P (1) are the source digit probabilities of zeros and ones, respectively. For this binary memoryless channel, the probability of error bits P (0|1) occurs at if the output signal is less than the decision level even if the input signal level is A, or if the output signal exceeds the decision level when the input signal level is −A, denoted by P (1|0). Therefore, the sum of probability of error bits (bit error rate) is calculated as Pe = P (1)P (0|1) + P (0)P (1|0) ,
(2)
which is a measurement of how well the communication system performs. The decision level will be discussed Sec. 2.2. Another measurement, channel capacity, is defined as the maximum rate of reliable information transmission through the channel [Proakis, 1995; Stremler, 1982]. Using Eq. (2), the channel capacity of the noise channel can be written as the function of Pe C = r[1 + Pe log 2 Pe + (1 − Pe ) log 2 (1 − Pe )] . (3) It is seen that the calculation of two measurements of bit error rate and channel capacity are related to the probabilities of error bits P (0|1) and P (1|0),
which will be solved theoretically in the succeeding subsections.
2.1. The stationary and nonstationary conditional probability density functions The sampler, depicted in Fig. 1, samples the channel output in the form of zero-order hold with sampling time interval ∆t. Thus, the sampled signal of channel output can be represented as si (n∆t) + ξ(n∆t) ,
(4)
with i = 1 or 2 and n = 1, 2, . . . . It is emphasized here that the important characteristic of the noise is its root-mean-square (rms) amplitude σ, which remains finite for any realizable noise. The noise density D is then given by σ 2 ∆t/2 [Chapeau-Blondeau & Godivier, 1997]. In practice, the correlation time of the noise is just the sampling time interval and the noise cutoff frequency f is equal to 1/∆t. For accurate simulations, one can simulate the effect of white noise by using a correlation time much smaller than the bit interval of input binary PAM signals [Godivier & Chapeau-Blondeau, 1998; Proakis & Salehi, 1997]. The dynamics of the nonlinear receiver system under study can be described in terms of an overdamped model, which obeys the stochastic differential equation dx = ax − µx3 + si (n∆t) + ξ(n∆t) , dt for (n − 1)∆t ≤ t < n∆t,
(5)
n = 1, 2, . . .
where a and µ are positive and given in terms of the receiver’s parameters. Here the quartic potential function can be written as 1 1 U (x) = − ax2 + µx4 + sx , 2 4
(6)
where s = si (n∆t) is the signal term. The critical threshold value is scr =
s
4a3 , 27µ
(7)
by which the input signals are classified as suprathreshold signals when the absolute values of signal amplitudes A > Scr and subthreshold signals otherwise [Moss et al., 1994; Gammaitoni et al., 1998].
414 F. Duan & B. Xu
Because the input binary symbols always take a constant level A or −A in a bit symbol interval Ts , Eq. (5) can be rewritten as dx = ax − µx3 ± A + ξ(n∆t) , dt
(8)
then, the corresponding Fokker–Planck (FPK) equation for the probability density ρ(x, t) reads ∂ρ(x, t) ∂ ∂ 2 ρ(x, t) = − [c(x)ρ(x, t)] + D , ∂t ∂x ∂x2
(9)
where c(x) = ax − µx3 ± A, and ρ(x, t) satisfies natural boundary conditions as x → ±∞ lim
∂ n ρ(x, t)
x→±∞
∂xn
= 0,
n = 0, 1, 2, . . . .
A A= √ , D
µ = µD ,
∂ρ(y, t) ∂ = − [c(y)ρ(y, t)] + ∂t ∂y
∂ 2 ρ(y, t) ∂2y
t→+∞
−∞
Eq. (10) then reduces to ∂ψ(y, t) ∂ 2 ψ(y, t) = − V (y)ψ(y, t) , ∂t ∂y 2 2
, (10)
(11)
∞
exp[g(y)]dy −∞
√ = C Df0 (a, µ, ±A) = 1 , √ C = [ Df0 (a, µ, ±A)]−1 ,
(12)
where the following integrals are defined for simplicity fi (a, µ, ±A) =
where V (y) = 1/2g (y) + 1/4g 0 (y). On the righthand side of Eq. (14), a self-conjugate differential operator acts on ψ(y, t), which can be expressed as
Z
+∞
(15)
y i exp[g(y)]dy ,
−∞
for i = 0, 1, 2, . . . . Equation (11), deduced from the conditional probability in steady-state solution of FPK equation, is suitable for the sequential string of the same
(16)
Multiply both sides of Eq. (16) by u(y) and integral it from minus infinity to infinity with respect to y, Eq. (16) becomes a variational form
λ = st.
u6=0
√ Z ρ(x)dx = C D
(14)
−λu(y) = u00 (y) − V (y)u(y) .
where g(y) = 1/2ay 2 − 1/4µy 4 ± Ay and C is a constant determined by the normalization condition of probability density ρ(x) ∞
(13)
Substituting Eq. (15) into Eq. (14) yields
where c(y) = ay − µy 3 ± A. Hence, the stationary solution of Eq. (9) is given by ρ(x) = lim ρ(x, t) = C exp[g(y)] ,
#
1 ρ(y, t) = ψ(y, t) exp g(y) , 2
ψ(y, t) = u(y) exp[−λt] ,
then Eq. (9) becomes
Z
"
00
Define (Another translation is developed in Appendix A in detail.) x y=√ , D
input signal level. However, the input signal waveform changes between two amplitudes ±A, or the input signal has no information. Thus, there exist error bit symbols induced by the nonstationary conditional probability of the output signal at nonlinear receiver. Furthermore, a transformation extensively used in dealing with the FPK Eq. (10) is
Z
+∞
[V (y)u2 (y) + u02 (y)]dy
−∞
Z
+∞
,
(17)
u2 (y)dy
−∞
satisfying the limit conditions of lim x→±∞ u(y) = 0 and limx→±∞ u0 (y) = 0, where st. means the stationary value in variational problems. The solution of eigenvalues defined in Eq. (17) is discussed in Appendix B in detail. The minimum eigenvalue λ0 is zero and ρ(y, t) = exp[g(y)] is the steady-state solution. We present that all other eigenvalues λi , for i = 1, 2, . . ., are positive (cf., Appendix B), in which λ1 is the minimum positive eigenvalue. Note that λ1 is of importance and we called it the receiver response speed. The inverse of the eigenvalue λ1 leads directly to the characteristic time of the system under study, which dominates the time of system tending to the steady state. It can be demonstrated that λ 1 is not related to the sign of signal amplitudes, that is, the receiver has the same characteristic time to stabilize in each state (see Fig. 2). Note that when a bit
Parameter-Induced Stochastic Resonance and Baseband Binary PAM Signals Transmission 415 60
of ρ˜(x), that is
~ ρ(y|s )
~ ρ ( y | s1 )
2
Z
Conditional probability density functions
50
+∞ −∞
√ Z ρ˜(x)dx = C˜ D
30
10
0 −0.1
−0.08
−0.06
−0.04
−0.02
0 y
0.02
0.04
0.06
0.08
0.1
Fig. 2. The nonstationary conditional probability density functions of binary PAM signals. The solid line means that the behavior of the nonstationary conditional probability density function ρ˜(y|s1 ) when input signal level is −1. While, the dashed line denotes ρ˜(y|s2 ), which is corresponding to signal amplitude +1. The receiver’s parameters are a = 5, µ = 3 × 1011 . The input binary signal is spoiled by a Gaussian white noise with rms amplitude σ = 4. Sampling time interval is ∆t is 10−6 s.
symbol takes waveform A and the next symbol also does so, the system requires no time to switch to another state. However, if the next bit symbol has signal level −A, we suggest that the system requires enough time to reach the signal level −A. Considering receiver response speed λ1 , the signal level at time t is evaluated under the form −A{1 − 2 exp[−λ1 t]} , thus, we assume that the output nonstationary signal level is αA at the end of bit duration time T s , where 1 α = {1 + 1 − 2 exp[−λ1 Ts ]} 2 = 1 − exp[−λ1 Ts ] .
When the receiver response speed λ1 is large enough, α is approximate to 1 and the nonstationary probabilities degenerate into the stationary ones. In particular, the conditional probability density function of the output when the input signal level is −A, can be written as ρ˜(y|s1 ) = C˜−A exp[˜ g−A (y)] , 1 1 g˜−A (y) = ay 2 − µy 4 − αAy , 2 4 C˜−A =
"
√ Z D
+∞ −∞
exp[˜ g−A (y)]dy
#−1
,
(21a)
whereas the conditional probability density function of the output when the input signal level is A, is given by ρ˜(y|s2 ) = C˜A exp[˜ gA (y)] , 1 1 g˜A (y) = ay 2 − µy 4 + αAy 2 4 √ Z ˜ D CA = "
+∞ −∞
exp[˜ gA (y)]dy
#−1
.
(21b)
Because exp[˜ g−A (−y)] = exp[˜ gA (y)], it is seen that ρ˜(y|s1 ) and ρ˜(y|s2 ) are symmetric and C˜−A = C˜A = ˜ An example of the conditional nonstationary C. probability density functions is illustrated in Fig. 2.
2.2. Decision level, bit error rate and channel capacity (18)
The nonstationary conditional probability density can be calculated as ρ˜(x) = C˜ exp[˜ g (y)] ,
exp[˜ g (y)]dy −∞
√ = C˜ Df0 (a, µ, ±αA) = 1 , √ (20) C˜ = [ Df0 (a, µ, ±αA)]−1 .
40
20
+∞
(19)
where g˜(y) = 1/2ay 2 −1/4µy 4 ±αAy and C˜ is also a constant determined by the normalization condition
In this subsection, we further develop the theoretical analysis of the measurements of bit error rate and channel capacity for this baseband binary communication system. We first determine an optimum decision level. According to conditional probability density functions of ρ˜(y|s 1 ) and ρ˜(y|s2 ), the optimum decision level for binary baseband PAM signals transmission is zero, which makes Z
+∞
ρ˜(y|s1 )dx = 0
Z
0
ρ˜(y|s2 )dx . −∞
416 F. Duan & B. Xu
In this case, the functional element of sample decision displayed in Fig. 1 compares the nonlinear receiver output signal y with the decision level zero. If y < 0, the decision is made in favor of s 1 , and the probability of error bits P (1|0) is represented as Z +∞ √ Z +∞ P (1|0) = ρ˜(y|s1 )dy ρ˜(y|s1 )dx = D 0
0
Z
+∞
= Z0+∞ −∞
exp[˜ g−A (y)]dy .
(22a)
exp[˜ g−A (y)]dy
Similarly, if y > 0, the decision is made that s 2 was transmitted, and the probability of error bits P (0|1) is given by Z 0 √ Z 0 ρ˜(y|s1 )dy ρ˜(y|s2 )dx = D P (0|1) = −∞
−∞
Z
0
= Z −∞ +∞ −∞
exp[˜ gA (y)]dy .
(22b)
exp[˜ gA (y)]dy
Clearly, P (0|1) = P (1|0), and it indicates that this channel is a symmetric channel. Then bit error rate defined in Eq. (2) becomes 1 Pe = {P (0|1) + P (1|0)} 2 = P (1|0) = P (0|1) . (23) Thus in the nonstationary condition, the channel capacity of the noise channel defined in Eq. (3) can be figured out. In fact, the expression of bit error rate defined in Eq. (23) indicates the probability of the region from minus infinity to zero when the input binary signal level is A, or the probability of the region from zero to infinity when the input binary signal has the amplitude −A. In the following section, we will concentrate on optimizing the design of the nonlinear receiver under the direction of the theory developed in this section, which leads us into observing the PSR effect.
3. Numerical Simulation, PSR Phenomenon and Performance of the Optimal Nonlinear Receiver In this section, the principles of our numerical simulation are clarified by which the following experiments are directed. The block scheme for simulating the baseband binary communication system
Fig. 3. The block scheme for simulating baseband binary communication systems. This system is simulated by a widely used software package SIMULINK, which is built on the language MATLAB [Proakis & Salehi, 1997].
A
∑
C
dx/dt Integrator
x
B
ax a
− µx 3 −µ
x3
Fig. 4. Functional block of the nonlinear receiver element in this baseband binary communication system (see Eq. (8)), which is solved by means of a fourth order Runge–Kutta integrator.
is presented in Fig. 3. The baseband information formed by the pseudorandom binary signal generator is passed to the digital modulator. Then, the transmitted binary PAM signals are corrupted by a band-limited Gaussian white noise. After the optimal nonlinear receiver “filtered” noise form the receiving signal, a sample detector can recover the information-bearing binary PAM signals. An error bit meter shown in Fig. 3 can automatically dis-
Parameter-Induced Stochastic Resonance and Baseband Binary PAM Signals Transmission 417
3.1. Parameter-induced stochastic resonance (PSR) and the optimal design receiver Parameter-induced stochastic resonance is a cooperative effect of the unchangeable signal added noise and the adjustable nonlinear receivers. Figure 5(a) represents an original input binary PAM signal. The sampled signal, which includes the transmitted signal and noise in the baseband communication AWGN channel, is shown in Fig. 5(b). The evolving time of input binary PAM signals is twenty seconds. To fix the idea, we observe six different nonlinear receivers’ outputs, shown in Fig. 6. It can be seen that three nonlinear receivers are in favor of detecting the input binary PAM signals, whether the input signals are suprathreshold for receivers (c) and (d) or subthreshold for the receiver (e). In
Input binary signal
1 0.5 0 −0.5 −1 0
2
4
6
8
10
12
14
16
18
20
12
14
16
18
20
(a)
20
Signal added noise
play three items: received bits number, error bits number and bit error rate. Figure 4 shows the functional block of the nonlinear receiver element in this communication system. The signal in A is the sampled signal sent from the sampler block. The signal in B is assumed to represent the derivative dx/dt of the variable of interest at time t. In C, after the integration block, the output signal is fed into two input terminals: the terminal of gain block a, to obtain the signal ax, and the terminal of multiplier block, which yields x3 . The lateral signal is then inverted and amplified through a cascaded gaining block. Moreover, the signal in C is also sent to the sample decision block to judge which bit digit is transmitted. Finally, the sampled signal, ax and −µx3 are added together through block Σ to give the starting signal. Nowadays, such block scheme can thus be translated into electronic analog devices or finally analyzed by various accurate digital simulations [Gammaitoni et al., 1998; Xu et al., 2002]. According to the decision level, the input binary signal will be recovered by means of values of the receiver output signals at sampling instants. In terms of Eqs. (3), (11) and (19), the channel capacity deduced on the assumption of the nonstationary probability is smaller than that deduced on the assumption of the stationary probability. Therefore, we design and expect that the outputs of the optimal receiver tend to stationary in a bit duration. It is apparent that the appropriate sampling times occur at the end of each bit interval, since it is the time that the receiver outputs have already reached their steady states.
10
0
−10
−20
0
2
4
6
8
10 time ( s )
(b) Fig. 5. (a) Input information-bearing binary PAM signals of time duration twenty seconds with two signal levels ±1 and Ts = 0.5 s. (b) Channel output signals, which are corrupted by an additional Gaussian white noise with zero-mean and variance σ 2 = 16. The sampling time interval is ∆t = 10−3 s.
terms of Eq. (23), bit error rate of these three receivers of (c)–(e) are 8.7389 × 10−6 , 5.7547 × 10−3 and 9.6391 × 10−3 , respectively. Clearly, the receiver of (c) with parameters a = 1 and µ = 500 reproduces the original input information with less error bits. Here we are concerned more with the information-transmission ability of this baseband binary communication system and less with the feather of the input signals. From the Fig. 6, it can be seen that as receiver’s parameter µ increases, the receiver response speed λ1 increases too [Xu et al., 2002]. For a given input binary signal corrupted by noise, the frequency domain between bit rate r and cutoff frequency f can be determined. If λ1 is not large enough to be compared with bit rate r, the nonlinear receiver cannot trace the varying of input signals. Indeed, as the system output of a bit symbol has not reached its steady state, the succeeding binary symbol is passed into the receiver, which causes the confusion. The final output state in a binary symbol is strongly influenced by the receiver response speed λ1 [see Figs. 6(a) and 6(b)]. While if λ 1 is so large that the receiver can be capable of following the random noise and the output signal is in disorder as presented in Fig. 6(f). Thus, it is seen that there exists an optimal nonlinear receiver yielding the
418 F. Duan & B. Xu 2
(a)
0 −2 0 1
2
4
6
8
10
12
14
16
18
20
(b)
Output signals of nonlinear receivers
0 −1 0 0.2
2
4
6
8
10
12
14
16
18
20
(c)
0 −0.2 0 0.1
2
4
6
8
10
12
14
16
18
20
(d)
0 −0.1 0 0.1
2
4
6
8
10
12
14
16
18
20
(e)
0 −0.1 0 0.05
2
4
6
8
10
12
14
16
18
20
(f)
0 −0.05
0
2
4
6
8
10 time ( s )
12
14
16
18
20
Fig. 6. Plots of output signals of six different receivers. The parameters of these receivers are (a) a = 1, µ = 1; (b) a = 1, µ = 10; (c) a = 1, µ = 500; (d) a = 10, µ = 12000; (e) a = 100, µ = 105 and (f) a = 100, µ = 106 . The receiver response speeds of these systems are about 8.7 Hz, 17.8 Hz, 56.5 Hz, 179.8 Hz, 204.4 Hz and 614.4 Hz respectively. Input binary PAM signal levels are ±1 and maybe greater or less than the critical thresholds Scr of potential function U (x) in these receivers.
minimum of bit error rate, that is, the maximum of channel capacity. This is a new kind of cooperative phenomenon, i.e. parameter-induced stochastic resonance (PSR). As contrast to the conventional SR or ASR, PSR effect is realized by means of tuning not the input noise intensity but the nonlinear receiver’s parameters. We argue that the basic mechanism of PSR is the competition among three factors: bit rate r, receiver response speed λ 1 and cutoff frequency f of noise, satisfying r < λ1 f .
(24)
The existence of the PSR effect can also be confirmed by calculation of bit error rate P e or channel capacity C. The behaviors of Pe and C against receiver response speed λ1 for the same input binary PAM signals are presented in Fig. 7. We notice here that channel capacity C first increases with increasing of receiver response speed λ1 , reaches and keeps a maximum value, and then decreases again. This is the typical signature of SR.
In this spirit, we begin considering the question how to tune the nonlinear receivers to achieve the AWGN channel’s optimal performance. We know that the measurements of channel capacity C and bit error rate Pe are functions of the input PAM signal levels, bit rate, noise intensity and nonlinear receiver’s parameters a and µ. For a noisy channel without input signals, noise measurement unit can estimate the variance of noise. According to the bit rate of binary symbols and noise cutoff frequency, one should choose an appropriate receiver response speed, which is assumed about twenty times as large as bit rate in the above numerical example. Consequently, with the same adopted receiver response speed, there are many pairs of receiver’s parameters a and µ, from which one can find a pair of parameters satisfying the desired bit error rate such as 10−5 . It is worth noting that the communication system with this pair of parameters might not have the minimum bit error rate, but it can be considered as the optimal design one. Thus, using Eqs. (17)
Parameter-Induced Stochastic Resonance and Baseband Binary PAM Signals Transmission 419
Input binary signal
1 0.5 0 −0.5 −1 0
(a)
0.01
0.02
0.03
0.04
0.06
0.07
0.08
0.09
0.1
0.06
0.07
0.08
0.09
0.1
(a)
20
Channel output signal
0.05
10
0
−10
−20
0
0.01
0.02
0.03
0.04
0.05 time (s)
(b) (b) Fig. 7. Plots of (a) bit error rate Pe and (b) channel capacity C against receiver response speed λ1 .
and (23), the optimal nonlinear receiver’s parameters a and µ can be obtained, regardless of whether the input binary signals are suprathreshold or subthreshold. As bit rate is low, we find that the channel transmits information with almost no loss by the optimal designed receivers, that is, the channel capacity is almost equal to bit rate. But for very large bit rate, bit error rate also increases, even though we tune the nonlinear receiver parameters very large. A numerical simulation example will be demonstrated in a detailed way in the Sec. 3.2, in which bit rate r is 1000 bits/s.
3.2. A numerical simulation example In this subsection, we show in detail that, by a numerical simulation illustration, how to design an optimal receiver for a given signal corrupted by noise. Suppose that we know the bit rate of input binary PAM signal r = 1000 bits/s and the noise rms level σ = 4. Considering Eq. (24), the sampling interval ∆t is taken as 10−6 s. The required bit error rate is 10−5 . The input binary PAM information-bearing signal is shown in Fig. 8(a) with two kinds of signal waveforms, which are generated by an equiprobable pseudorandom binary signal generator. An additive Gaussian noise with rms amplitude σ = 4 is added. It is carefully tuned that the receiver response speed
Fig. 8. (a) Input binary information PAM signal with signal levels ±1 transmitted over a time of 0.1 s. (b) Channel output signals, in which a Gaussian white noise with rms amplitude σ = 4 and cutoff frequency f = 1 MHz is added.
λ1 agrees with Eq. (24), which is adopted at about 55 KHz. And then, we can calculate the bit error rate of the corresponding receivers with the same receiver response speed. The receiver, being up to the required bit error rate 10−5 , is chosen as the optimal one. In this illustration, the nonlinear receiver with parameters a = 5 and µ = 3 × 10 11 is appropriate. The theoretical calculation of bit error rate Pe equals to 2.6191 × 10−6 , which is less than the desired bit error rate. Equation (5) can be concretely rewritten as dx = 5x − 3 × 1011 x3 + si (t) + ξ(t) , dt for 0 ≤ t < nTs , n = 1, 2, . . . .
(25)
Figure 9(a) presents the output signals at the end of the designed nonlinear receiver and Fig. 9(b) shows the recovered binary PAM signals. In the time duration 0.1 seconds, there are no error bit symbols, but error bits do occur in the processing over a long time. We compare our theory of optimal design-parameter receiver with the experiment’s results using the extensive Monte Carlo simulations over a long time of one thousand seconds. When we transmit one million bits of binary symbols by this communication system, only two or three error bits are observed. The error bit meter displays that bit error rate is in the range 2 × 10−6 to 3 × 10−6 , which greatly supports the theoretical analysis.
420 F. Duan & B. Xu −4
Output signal of receiver
x 10 2 1 0 −1 −2 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
(a)
X=
1
Recovered signal
for n = 1, 2, . . . , Ts f . Here we assume that Ts f is an integer. On assuming of Eq. (18) in Sec. 2.1, the amplitude of nonlinear receiver output turns into zero level when the factor α is null, that is, at time t = ln 2/λ1 . Thus, there are n0 = (ln 2/λ1 )f sample values to be taken out. In a bit duration, we then make use of n − n0 sample values. Suppose
0.5
1 (xn0 +1 + xn0 +2 + · · · + xn ) . n − n0
As a consequence, the ensemble average probability of error bits (cf. Appendix C) is bounded by
0 −0.5 −1 0
0.01
0.02
0.03
0.04
0.05 time (s)
0.06
0.07
0.08
0.09
0.1
(b) Fig. 9. (a) Output signal of the optimal nonlinear receiver. (b) The recovered binary PAM signal by the zero threshold decision level.
As indicated in Sec. 3.1, the optimal design receiver might not have the minimum bit error rate. We here present another nonlinear receiver with parameters a = 5000 and µ = 3 × 1011 , by which the baseband binary communication system has the theoretical bit error rate of 1.5072 × 10 −8 . However, if the error bit rate of 10−5 is all that is required, the receiver with parameters a = 5 and µ = 3 × 1011 can be employed and assumed as the optimal one. It should be stressed that the bit error rate of 1.5072 × 10−8 has not been demonstrated in numerical simulation because it needs a very long time to verify. For the illustrations of the above subsection, bit error rates proved to be coincident with our theoretical analysis in numerical experiments, as well as the example of the receiver with parameters a = 5 and µ = 3 × 1011 . In the next subsection, we further decrease, with the approach of ensemble average probability of error bits, bit error rate in the illustration of this subsection.
3.3. Ensemble average probability of error bits A widely used method, averaging, extracts signal buried in noise. Since bit rate is r = 1/T s and cutoff frequency is f = 1/∆t, there are n = T s f independent sample values xi in each bit duration Ts ,
1 Pea = Pr{|X − E[X]| ≥ (|E[X]| − 0|)} 2 ≤
f2 (a, µ, ±αA)f0 (a, µ, ±αA) − f12 (a, µ, ±αA) . 2(n − n0 )f12 (a, µ, ±αA) (26)
where fi (a, µ, ±αA) are defined in Sec. 2.1. With the above approach, we can further decrease bit error rate in the binary PAM signals transmission via tuning nonlinear receivers. Since the receiver response speed adopted in this numerical illustration is about 55 KHz, then we can obtain the number of ensemble n 0 = (ln 2/λ1 )f ≈ 0. In terms of Eq. (26), we can obtain Pea ≤ 1.2701 × 10−5 . Although 1.2701 × 10−5 is greater than Pe = 2.6191 × 10−6 , it is only an upper bound. In fact, it is really observed that this method can be used to decrease the bit error rate.
3.4. Time scale transformation method In Sec. 3.2, it is seen that when the bit rate is as high as 1000 Hz and the noise rms increases as large as 4, the receiver’s parameter µ is so large as 3 × 1011 . As the bit rate increases further, the receiver’s parameters a and µ are tuned larger and larger so that the receiver response speed can satisfy Eq. (24). Here we demonstrate that time scale transformation method can improve the fundamental limit of the nonlinear receiver for binary PAM signals transmission at very high bit rate or, equivalently, a discussion of limits of low frequency SR or ASR in [Hibbs et al., 1995; Kaufman et al., 1996; Gammaitoni et al., 1998]. The influence of the noise cutoff frequency on PSR effect in nonlinear receiver systems is practically significant. Previous studies on realizing SR or ASR via tuning input noise have
Parameter-Induced Stochastic Resonance and Baseband Binary PAM Signals Transmission 421
concluded that noise with a large cutoff frequency is favorable [Misono et al., 1998; Zhou & Moss, 1990; Gammaitoni et al., 1989]. In this letter, we show that such kinds of noise are also favorable for baseband binary PAM signals transmission via PSR. In the above subsection, it is known that there are n = Ts f independent sample values in each bit duration. Consequently, a discrete time sequence is formed. It is here worth noting that the time interval value of this sequence can be regarded as arbitrary time unit. If the time interval extends N times, that is, ∆τ = N ∆t(N > 1), then the sampled signals in Eq. (4) become sm (n∆τ ) + ξ(n∆τ ) ,
(27)
where hξ(τ )ξ(0)i = 2N Dδ(τ ) is the character of white Gaussian noise. Equation (27) can be explained as follows: the frequency of input signal reduces to 1/N times of that of the originally signal whereas the noise intensity is amplified N times, i.e. D 0 = N σ 2 ∆t/2. This method is time scale transformation. Consider that the cutoff frequency increases by N times, that is, f 0 = N/∆t. The sampling time interval is then ∆t/N . If the time interval of this discrete time sequence is taken as ∆t, it is interesting to note that the noise intensity D 0 = N σ 2 ∆t/(2N ) is equal to D = σ 2 ∆t/2, but the bit rate decreases to 1/N times. According to Eqs. (10) and (23), √ the probability of error bits is a function of A = A/ D and the receiver’s parameters a and µ. With the time scale transformation method, the optimal nonlinear receiver is only dependent on n = T s f , but not related to the bit rate. Thus, such an optimal receiver is not limited by the bit rate, high or low. For an example of a binary signal with levels ±1 and bit rate r = 10000 bits/s, the noise rms amplitude σ = 4 and cutoff frequency f = 10 MHz, and the nonlinear receiver’s parameters are a = 5, µ = 3 × 1011 . If time extends ten times, then bit error rate Pe is also 2.6191 × 10−6 as well as the illustration given in Sec. 3.2. So it is possible to transmit at the bit rate of 10000 bits/s, but at the expense of much larger cutoff frequency and more complex communication systems. However, it indicates that we can optimize the baseband binary communication systems not only by designing the nonlinear receiver but also appealing to time scale transformation approach.
4. Conclusion We now summarize the main results of this work. The AWGN channel, transmitting a binary information-bearing PAM signals, was studied by connecting with a nonlinear dynamic model as the receiver. We demonstrated that PSR phenomenon appears when the receiver systems are adjustable while the input signal added noise is a given quantity and its basic mechanism is explained. It was observed that a marked parameter-induced enhancement of the channel capacity, PSR effect, occurs in numerical simulations for the baseband binary PAM signals transmission. In the view of information theory, it is emphasized that the realization of PSR does not require the subthreshold signals. The consequence of these results is of potential importance for technical applications. Moreover, we suggest that an application of the nonlinear receiver with tunable parameters is useful for communication systems under the noisy channel. Undoubtedly, the channel that has yielded the biggest successes in information theory is Gaussian channel [Schwartz, 1980]. However, for the capacity of bandlimited channel of bandwidth W , studies of ASR or PSR in context of information theory will be more meaningful. We have already started to research the binary signals transmission in a bandwidth limited communication system, in which the signal’s bandwidth is equal to the noise’s. We raise here another important problem: for a linear receiver by which we desire to improve the baseband communication systems performance, is it possible to observe PSR phenomenon? It is guessed that the gain of the linear receiver with an inherent frequency is very dependent upon the input frequency, resulting in being unsuitable for a periodic signal processing. Thus, PSR or ASR effect might not occur. This question is open.
Acknowledgment The financial support by Zhejiang Province Natural Science Foundation No. 601089 for this work is greatly appreciated.
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422 F. Duan & B. Xu
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Parameter-Induced Stochastic Resonance and Baseband Binary PAM Signals Transmission 423
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above research [Xu et al., 2002], we have already discussed this question in detail.
Appendix B B.2. The solution of eigenvalues defined in Eq. (17) Assume [Xu et al., 2002] "
Appendix A A.1. Synthesis of the receiver’s parameters a and µ It would be useful to synthesize the influences of the two important receiver’s parameters a and µ on the performance of baseband communication systems. Redefine √ µD A x a √ y= , µ= 2 , A= √ , τ = at , a D aD
satisfying limx→±∞ u(y) = 0 and limx→±∞ u0 (y) = 0. Substituting Eq. (B.1) into Eq. (17) yields ([K] − λ[M ]){d} = 0 ,
{d} = [d0 for
(A.1)
mij =
Z
kij =
Z
∂ρ(y, τ ) ∂ ∂ 2 ρ(y, τ ) , (A.2) = − [c(y)ρ(y, τ )] + ∂τ ∂y ∂y 2
d n ]T ,
ρ(x) = lim ρ(x, t) = C exp[g(y)] ,
+∞
y i+j exp[g(y)]dy ,
−∞ +∞ −∞
("
1 + iy i−1 + g 0 (y)y i 2 1 + g 0 (y)y j 2
(A.3)
t→∞
#
1 1 00 g (y) + g 02 (y) y i+j 2 4
"
where c(y) = y − µy 3 ± A. Hence, the stationary solution of Eq. (A.2) is given by
#)
#"
jy j−1
exp[g(y)]dy .
It is easily found that − 1/4µy 4
where g(y) = ± Ay and C is a constant determined by the normalization condition of probability density ρ(x) √
ρ(x)dx = C D −∞
K
i, j = 0, 1, 2, K, n ,
where hΓ(τ )Γ(0)i = 2δ(τ ), then the corresponding FPK equation becomes
+∞
d1
[M ] = [mij ], [K] = [kij ] ,
dy = y − µy 3 ± A + Γ(n∆τ ) , dτ
Z
(B.2)
where
then Eq. (8) can be rewritten as
1/2y 2
#
1 u(y) = exp g(y) (d0 + d1 y + · · · + dn y n ) (B.1) 2
Z
+∞
√
exp[g(y)]dy/ a −∞
√ √ = C Df0 (µ, ±A)/ a = 1 , √ √ C = a[ Df0 (µ, ±A)]−1 .
It is seen that, for a fixed noise level and signal amplitude, there is only one system parameter µ. However, the input signal level A is variable. In the
mij = mji , kij = kji . Therefore, the matrices [M ] and [K] are symmetric. It can be demonstrated that the matrix [K] is semi-positive definite and the matrix [M ] is positive definite. Assuming u(y) = p(y) exp[1/2g(y)], it can be demonstrated Z
∞ −∞
("
#
)
1 1 00 g (y) + g 02 (y) u2 (y) + u(y)02 dy ≥ 0 . 2 4 (B.3)
424 F. Duan & B. Xu
for p(y) = const. Inequality (B.3) becomes the equality equation. Indeed, the integrated function is ("
#
1 00 1 g (y) + g 02 (y) p2 (y) 2 4 "
1 + p (y) + p(y)g 0 (y) 2 =
(
0
#2 )
exp[g(y)]
1 1 0 [g (y)p2 (y)]0 + g 02 (y)p2 (y) 2 2
+ p02 (y) exp[g(y)] ,
(B.4)
where g(y) was defined in Sec. 2. Since ∞
[g 0 (y)p2 (y)]0 exp[g(y)]dy
The second way is averaging sample values x i of the nonlinear receiver’s output, whose mean value is
= g 0 (y)p2 (y) exp[g(y)]k∞ −∞ ∞
− =−
Z
g 02 (y)p2 (y) exp[g(y)]dy
−∞ ∞
where the term 1/2 is introduced, since the probability of error bits is only the area of the conditional probability in the interval (−∞, 0), when the input signal level is A, thus σ2 1 . Pe = [P (1|0) + P (0|1)] ≤ 2 2nA2
−∞
Z
1 Pr{|X − E[X]| 2 D[X] ≥ (| ± A| − 0)} ≤ 2A2 σ2 D[x] = , = 2nA2 2nA2
P (1|0) = P (0|1) =
)
Z
with only two signal levels ±A and decision level zero, obtained from the Chebyshev inequality, the conditional probabilities of error bits are
E[X] = g 02 (y)p2 (y) exp[g(y)]dy ,
(B.5)
= E[x] ,
−∞
then
1 (E[x1 ] + E[x2 ] + · · · + E[xn ]) n
and its degree-two moment is Z
∞ −∞
("
1 00 1 g (y) + g 02 (y)]p2 (y) 2 4
"
1 + p0 (y) + p(y)g 0 (y) 2 =
Z
∞ −∞
#2 )
1 E[X ] = 2 n 2
exp[g(y)]dy
p02 (y) exp[g(y)]dy ≥ 0 .
(B.6)
When p0 (y) = 0, i.e. p(y) = const. inequality (B.3) becomes the equality equation. If λ 0 = 0 is the simple characteristic root, we can infer that λ1 > 0.
Appendix C C.3. Ensemble average probability of error bits Here, we present and compare two ways to calculate ensemble average probability of error bits. The first is the way without the nonlinear receiver processing. Consider the binary PAM signals transmitted
=
n X
E[x2i ]
i=1
+
n X
E[xi xj ]
i6=j
!
1 {nE[x2 ] + n(n − 1)E[x]2 } . n2
Hence, the variance D[X] can be written as D[X] = E[X 2 ] − E[X]2 =
1 (E[x2 ] − E[x]2 ) n
=
D[x] . n
For binary PAM signals, the mean value of receiver’s output is given by E[X] = E[x] = C˜
Z
+∞ −∞
x exp[˜ g±A (y)]dx
√ Df1 (a, µ, ±αA) , = f0 (a, µ, ±αA)
Parameter-Induced Stochastic Resonance and Baseband Binary PAM Signals Transmission 425
In Sec. 3.2, (n − n0 ) sample values are used for averaging. As a consequence, the ensemble average probability of error bits is bounded by
the degree-two moment can be expressed as E[x ] = C˜ 2
Z
+∞ −∞
x2 exp[˜ g±A (y)]dx
√ Z = C˜ D 3
1 Pe = Pr{|X − E[X]| ≥ (E[x] − 0)} 2
+∞ −∞
y 2 exp[˜ g±A (y)]dy
Df2 (a, µ, ±αA) , = f0 (a, µ, ±αA)
and the variance of receiver’s output signals is (
f2 (a, µ, ±αA) f12 (a, µ, ±αA) D[x] = D − f0 (a, µ, ±αA) f02 (a, µ, ±αA)
)
.
≤
D[x] 2(n − n0 )E[x]2
=
f2 (a, µ, ±αA)f0 (a, µ, ±αA) − f12 (a, µ, ±αA) . 2(n − n0 )f12 (a, µ, ±αA)
PARAMETER-INDUCED STOCHASTIC RESONANCE AND BASEBAND BINARY PAM SIGNALS TRANSMISSION OVER AN AWGN CHANNEL FABING DUAN and BOHOU XU∗ Department of Mechanics, Zhejiang University, Hangzhou 310027, P.R. China ∗ [email protected] Received June 7, 2001; Revised December 12, 2001 From the point of information theory, the baseband binary pulse amplitude modulated (PAM) signals transmission, via tuning the nonlinear receiver’s parameters, is studied over an additive white Gaussian noise (AWGN) channel. It is demonstrated that the channel capacity of baseband binary communication systems, for a given signal added noise, can be maximized by optimal designed receivers. This new form of stochastic resonance (SR) is referred to as parameter-induced stochastic resonance (PSR) in a broad sense. PSR effect does not require that the input signals are subthreshold. The characteristic behavior of channel capacity versus transmission bit rate is also studied. For reproducing the original binary signals more correctly, time scale transformation method and the approach of ensemble average probability of error bits are introduced. We observe a marked enhancement of the channel capacity for binary PAM signals transmission by PSR in numerical simulations. With this theory and method, the practical applications are considered. Keywords: Stochastic resonance; binary PAM signals; bit error rate; signal processing.
1. Introduction
ler, 1996], biological systems [Douglass et al., 1993; Collins et al., 1996b; Levin & Miller, 1996], signal transmission [Chapeau-Blondeau, 1997; Godivier & Chapeau-Blondeau, 1998], signal detection [Hu et al., 1992; Inchiosa & Bulsara, 1996, 1998; Galdi et al., 1998], and image processing [Misono et al., 1998] within the framework of information theory. Recently, ASR effect has been demonstrated experimentally in a real system by Barbay et al. [2000]. Most previous papers realized SR or ASR by tuning the input noise density. However, a more realistic situation is that the input signal corrupted by noise is a given quantity, which can be tuned by increasing but not decreasing the input noise. It will cause some difficulties in practice. Why do we not study and design the system that has adjustable
Stochastic resonance (SR) has been the subject of both theoretical and experimental research in the past two decades [Benzi et al., 1981; McNamara & Wiesenfeld, 1989; Wiesenfeld & Moss, 1995; Collins et al., 1995a; Bulsara & Gammaitoni, 1996; Gammaitoni et al., 1998; Alfonsi et al., 2000; Barbay et al., 2000]. Collins et al. [1995a] proposed aperiod stochastic resonance (ASR) as a theory of SR for excitable systems with a broadband aperiodic input signal. ASR has then attracted a particular interest in improving actual useful signal detection and transmission. This research subject has recently been the focus of intensive investigation in neurosciences [Collins et al., 1995a, 1995b, 1996a; Stemm∗
Author for correspondence. 411
412 F. Duan & B. Xu
parameters and yields the maximum amount of information? This is important for practical application in those systems where adding or reducing noise might not be an option [Anishchenko et al., 1993; Bulsara & Gammaitoni, 1996; Jung, 1995; Xu et al., 2002]. In contexts of signal processing and communication, the system under study can be viewed as an algorithm by which the sampled signals are processed or transformed in some way, resulting in other signals as outputs. In this paper, the characteristics of the baseband binary communication system, which includes a nonlinear receiver with two adjustable parameters, are studied for binary pulse amplitude modulated (PAM) signals transmission. Here, it is no longer a case of optimizing the noise level, but of designing this nonlinear receiver in such a way that the performance of the baseband binary communication system can be improved in the presence of noise. Even if the power of noise added is much stronger than that of the input signal, it is demonstrated that the optimal receivers with design-parameters can yield the maximum channel capacity. We refer to this new kind of SR as parameter-induced stochastic resonance (PSR) in a broad sense. It is worth noting that our resonance points or region are optimal nonlinear receivers. PSR effect occurs when the input signals are either subthreshold or suprathreshold [Stocks, 2000]. The behavior of channel capacity, as a function of transmission bit rate, is also investigated with optimal receivers. Furthermore, for recovering the information-bearing binary PAM signals more correctly, time scale transformation method and the approach of ensemble average probability of error bits are introduced. Especially worthy of note is a recent research on the information capacity of a bistable dynamic system [Godivier & ChapeauBlondeau, 1998], in which this system is used to transmit a broadband aperiodic informative signal. Comparing with their study, the present investigation designs the optimal systems via the tuning of the system’s parameters, rather than the noise intensity, and develops the theoretical formula of bit error rate. The organization of the paper is as follows. Section 2 briefly describes the characterizations of baseband binary PAM signals and communication systems. Channel capacity, deduced from the nonstationary probability density function, is used to quantify the performance of the baseband communication system. In Sec. 3, the optimal receivers
via tuning of its parameters are studied for binary PAM signals transmission over an AWGN channel. We observe a marked enhancement of the channel capacity in numerical simulations and attempt to interpret the basic mechanism of PSR effect. Moreover, the time scale transformation method and the approach of ensemble average probability of error bits are discussed in detail. Simulation results agree well with the theory developed in Sec. 2. In conclusion, the practical applications are considered with this theory and method.
2. The Measurements of Baseband Binary Communication System: Bit Error Rate and Channel Capacity In this section, the baseband binary communication system (see Fig. 1) for conveying informationbearing signals is described. Its performance is briefly analyzed by channel capacity, which is a function of bit error rate. Figure 1 shows the structure of this baseband binary communication system, wherein a nonlinear dynamic model acts as the receiver. As indicated in Fig. 1, the messages produced by source are a sequence of binary digits. The modulator simply maps the binary digit 0 into a waveform s1 (t) and the binary digit 1 into a waveform s2 (t). The signal waveforms here can be represented as [Schwartz, 1980; Proakis, 1995] for
s1 (t) = −A,
s2 (t) = A,
(n − 1)Ts ≤ t ≤ nTs ,
n = 1, 2, . . . .
(1)
Such a signal is called binary pulse amplitude modulated (PAM) signal. The time interval T s is termed
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Fig. 1. Basic elements of a baseband binary communication system. Dashed frames indicate being chosen.
Parameter-Induced Stochastic Resonance and Baseband Binary PAM Signals Transmission 413
bit duration and bit rate is r = 1/Ts [Schwartz, 1980; Proakis, 1995]. The channel displayed in Fig. 1 denotes a physical transmission medium, in which the transmission signals are corrupted by an additional noise ξ(t). ξ(t) is a Gaussian white noise with zero-mean and the correlation hξ(t)ξ(0)i = 2Dδ(t). Here, h·i and δ(·) represent the average operator and the Diracdelta function, respectively, and D represents noise intensity. To simplify, we assume that this physical channel attenuation factor is unity, that is, the channel output is the sum of input signals and noise. In the design of a communication system, it is convenient to construct a mathematical model for evaluating the performance of the communication system. A general communication channel model is an AWGN channel, which actually comprises all portions, such as the modulator, sampler, the receiver, the physical channel, etc. Let us consider this AWGN channel as a discrete memoryless channel, which has input binary signals S = {s 1 , s2 } and the detector outputs reading Y = {y 1 , y2 }. Assume that the input binary symbols occur with equiprobabilities P (0) = P (1) = 0.5 and are statistically independent, so that there is no dependence among successively received binary symbols. P (0) and P (1) are the source digit probabilities of zeros and ones, respectively. For this binary memoryless channel, the probability of error bits P (0|1) occurs at if the output signal is less than the decision level even if the input signal level is A, or if the output signal exceeds the decision level when the input signal level is −A, denoted by P (1|0). Therefore, the sum of probability of error bits (bit error rate) is calculated as Pe = P (1)P (0|1) + P (0)P (1|0) ,
(2)
which is a measurement of how well the communication system performs. The decision level will be discussed Sec. 2.2. Another measurement, channel capacity, is defined as the maximum rate of reliable information transmission through the channel [Proakis, 1995; Stremler, 1982]. Using Eq. (2), the channel capacity of the noise channel can be written as the function of Pe C = r[1 + Pe log 2 Pe + (1 − Pe ) log 2 (1 − Pe )] . (3) It is seen that the calculation of two measurements of bit error rate and channel capacity are related to the probabilities of error bits P (0|1) and P (1|0),
which will be solved theoretically in the succeeding subsections.
2.1. The stationary and nonstationary conditional probability density functions The sampler, depicted in Fig. 1, samples the channel output in the form of zero-order hold with sampling time interval ∆t. Thus, the sampled signal of channel output can be represented as si (n∆t) + ξ(n∆t) ,
(4)
with i = 1 or 2 and n = 1, 2, . . . . It is emphasized here that the important characteristic of the noise is its root-mean-square (rms) amplitude σ, which remains finite for any realizable noise. The noise density D is then given by σ 2 ∆t/2 [Chapeau-Blondeau & Godivier, 1997]. In practice, the correlation time of the noise is just the sampling time interval and the noise cutoff frequency f is equal to 1/∆t. For accurate simulations, one can simulate the effect of white noise by using a correlation time much smaller than the bit interval of input binary PAM signals [Godivier & Chapeau-Blondeau, 1998; Proakis & Salehi, 1997]. The dynamics of the nonlinear receiver system under study can be described in terms of an overdamped model, which obeys the stochastic differential equation dx = ax − µx3 + si (n∆t) + ξ(n∆t) , dt for (n − 1)∆t ≤ t < n∆t,
(5)
n = 1, 2, . . .
where a and µ are positive and given in terms of the receiver’s parameters. Here the quartic potential function can be written as 1 1 U (x) = − ax2 + µx4 + sx , 2 4
(6)
where s = si (n∆t) is the signal term. The critical threshold value is scr =
s
4a3 , 27µ
(7)
by which the input signals are classified as suprathreshold signals when the absolute values of signal amplitudes A > Scr and subthreshold signals otherwise [Moss et al., 1994; Gammaitoni et al., 1998].
414 F. Duan & B. Xu
Because the input binary symbols always take a constant level A or −A in a bit symbol interval Ts , Eq. (5) can be rewritten as dx = ax − µx3 ± A + ξ(n∆t) , dt
(8)
then, the corresponding Fokker–Planck (FPK) equation for the probability density ρ(x, t) reads ∂ρ(x, t) ∂ ∂ 2 ρ(x, t) = − [c(x)ρ(x, t)] + D , ∂t ∂x ∂x2
(9)
where c(x) = ax − µx3 ± A, and ρ(x, t) satisfies natural boundary conditions as x → ±∞ lim
∂ n ρ(x, t)
x→±∞
∂xn
= 0,
n = 0, 1, 2, . . . .
A A= √ , D
µ = µD ,
∂ρ(y, t) ∂ = − [c(y)ρ(y, t)] + ∂t ∂y
∂ 2 ρ(y, t) ∂2y
t→+∞
−∞
Eq. (10) then reduces to ∂ψ(y, t) ∂ 2 ψ(y, t) = − V (y)ψ(y, t) , ∂t ∂y 2 2
, (10)
(11)
∞
exp[g(y)]dy −∞
√ = C Df0 (a, µ, ±A) = 1 , √ C = [ Df0 (a, µ, ±A)]−1 ,
(12)
where the following integrals are defined for simplicity fi (a, µ, ±A) =
where V (y) = 1/2g (y) + 1/4g 0 (y). On the righthand side of Eq. (14), a self-conjugate differential operator acts on ψ(y, t), which can be expressed as
Z
+∞
(15)
y i exp[g(y)]dy ,
−∞
for i = 0, 1, 2, . . . . Equation (11), deduced from the conditional probability in steady-state solution of FPK equation, is suitable for the sequential string of the same
(16)
Multiply both sides of Eq. (16) by u(y) and integral it from minus infinity to infinity with respect to y, Eq. (16) becomes a variational form
λ = st.
u6=0
√ Z ρ(x)dx = C D
(14)
−λu(y) = u00 (y) − V (y)u(y) .
where g(y) = 1/2ay 2 − 1/4µy 4 ± Ay and C is a constant determined by the normalization condition of probability density ρ(x) ∞
(13)
Substituting Eq. (15) into Eq. (14) yields
where c(y) = ay − µy 3 ± A. Hence, the stationary solution of Eq. (9) is given by ρ(x) = lim ρ(x, t) = C exp[g(y)] ,
#
1 ρ(y, t) = ψ(y, t) exp g(y) , 2
ψ(y, t) = u(y) exp[−λt] ,
then Eq. (9) becomes
Z
"
00
Define (Another translation is developed in Appendix A in detail.) x y=√ , D
input signal level. However, the input signal waveform changes between two amplitudes ±A, or the input signal has no information. Thus, there exist error bit symbols induced by the nonstationary conditional probability of the output signal at nonlinear receiver. Furthermore, a transformation extensively used in dealing with the FPK Eq. (10) is
Z
+∞
[V (y)u2 (y) + u02 (y)]dy
−∞
Z
+∞
,
(17)
u2 (y)dy
−∞
satisfying the limit conditions of lim x→±∞ u(y) = 0 and limx→±∞ u0 (y) = 0, where st. means the stationary value in variational problems. The solution of eigenvalues defined in Eq. (17) is discussed in Appendix B in detail. The minimum eigenvalue λ0 is zero and ρ(y, t) = exp[g(y)] is the steady-state solution. We present that all other eigenvalues λi , for i = 1, 2, . . ., are positive (cf., Appendix B), in which λ1 is the minimum positive eigenvalue. Note that λ1 is of importance and we called it the receiver response speed. The inverse of the eigenvalue λ1 leads directly to the characteristic time of the system under study, which dominates the time of system tending to the steady state. It can be demonstrated that λ 1 is not related to the sign of signal amplitudes, that is, the receiver has the same characteristic time to stabilize in each state (see Fig. 2). Note that when a bit
Parameter-Induced Stochastic Resonance and Baseband Binary PAM Signals Transmission 415 60
of ρ˜(x), that is
~ ρ(y|s )
~ ρ ( y | s1 )
2
Z
Conditional probability density functions
50
+∞ −∞
√ Z ρ˜(x)dx = C˜ D
30
10
0 −0.1
−0.08
−0.06
−0.04
−0.02
0 y
0.02
0.04
0.06
0.08
0.1
Fig. 2. The nonstationary conditional probability density functions of binary PAM signals. The solid line means that the behavior of the nonstationary conditional probability density function ρ˜(y|s1 ) when input signal level is −1. While, the dashed line denotes ρ˜(y|s2 ), which is corresponding to signal amplitude +1. The receiver’s parameters are a = 5, µ = 3 × 1011 . The input binary signal is spoiled by a Gaussian white noise with rms amplitude σ = 4. Sampling time interval is ∆t is 10−6 s.
symbol takes waveform A and the next symbol also does so, the system requires no time to switch to another state. However, if the next bit symbol has signal level −A, we suggest that the system requires enough time to reach the signal level −A. Considering receiver response speed λ1 , the signal level at time t is evaluated under the form −A{1 − 2 exp[−λ1 t]} , thus, we assume that the output nonstationary signal level is αA at the end of bit duration time T s , where 1 α = {1 + 1 − 2 exp[−λ1 Ts ]} 2 = 1 − exp[−λ1 Ts ] .
When the receiver response speed λ1 is large enough, α is approximate to 1 and the nonstationary probabilities degenerate into the stationary ones. In particular, the conditional probability density function of the output when the input signal level is −A, can be written as ρ˜(y|s1 ) = C˜−A exp[˜ g−A (y)] , 1 1 g˜−A (y) = ay 2 − µy 4 − αAy , 2 4 C˜−A =
"
√ Z D
+∞ −∞
exp[˜ g−A (y)]dy
#−1
,
(21a)
whereas the conditional probability density function of the output when the input signal level is A, is given by ρ˜(y|s2 ) = C˜A exp[˜ gA (y)] , 1 1 g˜A (y) = ay 2 − µy 4 + αAy 2 4 √ Z ˜ D CA = "
+∞ −∞
exp[˜ gA (y)]dy
#−1
.
(21b)
Because exp[˜ g−A (−y)] = exp[˜ gA (y)], it is seen that ρ˜(y|s1 ) and ρ˜(y|s2 ) are symmetric and C˜−A = C˜A = ˜ An example of the conditional nonstationary C. probability density functions is illustrated in Fig. 2.
2.2. Decision level, bit error rate and channel capacity (18)
The nonstationary conditional probability density can be calculated as ρ˜(x) = C˜ exp[˜ g (y)] ,
exp[˜ g (y)]dy −∞
√ = C˜ Df0 (a, µ, ±αA) = 1 , √ (20) C˜ = [ Df0 (a, µ, ±αA)]−1 .
40
20
+∞
(19)
where g˜(y) = 1/2ay 2 −1/4µy 4 ±αAy and C˜ is also a constant determined by the normalization condition
In this subsection, we further develop the theoretical analysis of the measurements of bit error rate and channel capacity for this baseband binary communication system. We first determine an optimum decision level. According to conditional probability density functions of ρ˜(y|s 1 ) and ρ˜(y|s2 ), the optimum decision level for binary baseband PAM signals transmission is zero, which makes Z
+∞
ρ˜(y|s1 )dx = 0
Z
0
ρ˜(y|s2 )dx . −∞
416 F. Duan & B. Xu
In this case, the functional element of sample decision displayed in Fig. 1 compares the nonlinear receiver output signal y with the decision level zero. If y < 0, the decision is made in favor of s 1 , and the probability of error bits P (1|0) is represented as Z +∞ √ Z +∞ P (1|0) = ρ˜(y|s1 )dy ρ˜(y|s1 )dx = D 0
0
Z
+∞
= Z0+∞ −∞
exp[˜ g−A (y)]dy .
(22a)
exp[˜ g−A (y)]dy
Similarly, if y > 0, the decision is made that s 2 was transmitted, and the probability of error bits P (0|1) is given by Z 0 √ Z 0 ρ˜(y|s1 )dy ρ˜(y|s2 )dx = D P (0|1) = −∞
−∞
Z
0
= Z −∞ +∞ −∞
exp[˜ gA (y)]dy .
(22b)
exp[˜ gA (y)]dy
Clearly, P (0|1) = P (1|0), and it indicates that this channel is a symmetric channel. Then bit error rate defined in Eq. (2) becomes 1 Pe = {P (0|1) + P (1|0)} 2 = P (1|0) = P (0|1) . (23) Thus in the nonstationary condition, the channel capacity of the noise channel defined in Eq. (3) can be figured out. In fact, the expression of bit error rate defined in Eq. (23) indicates the probability of the region from minus infinity to zero when the input binary signal level is A, or the probability of the region from zero to infinity when the input binary signal has the amplitude −A. In the following section, we will concentrate on optimizing the design of the nonlinear receiver under the direction of the theory developed in this section, which leads us into observing the PSR effect.
3. Numerical Simulation, PSR Phenomenon and Performance of the Optimal Nonlinear Receiver In this section, the principles of our numerical simulation are clarified by which the following experiments are directed. The block scheme for simulating the baseband binary communication system
Fig. 3. The block scheme for simulating baseband binary communication systems. This system is simulated by a widely used software package SIMULINK, which is built on the language MATLAB [Proakis & Salehi, 1997].
A
∑
C
dx/dt Integrator
x
B
ax a
− µx 3 −µ
x3
Fig. 4. Functional block of the nonlinear receiver element in this baseband binary communication system (see Eq. (8)), which is solved by means of a fourth order Runge–Kutta integrator.
is presented in Fig. 3. The baseband information formed by the pseudorandom binary signal generator is passed to the digital modulator. Then, the transmitted binary PAM signals are corrupted by a band-limited Gaussian white noise. After the optimal nonlinear receiver “filtered” noise form the receiving signal, a sample detector can recover the information-bearing binary PAM signals. An error bit meter shown in Fig. 3 can automatically dis-
Parameter-Induced Stochastic Resonance and Baseband Binary PAM Signals Transmission 417
3.1. Parameter-induced stochastic resonance (PSR) and the optimal design receiver Parameter-induced stochastic resonance is a cooperative effect of the unchangeable signal added noise and the adjustable nonlinear receivers. Figure 5(a) represents an original input binary PAM signal. The sampled signal, which includes the transmitted signal and noise in the baseband communication AWGN channel, is shown in Fig. 5(b). The evolving time of input binary PAM signals is twenty seconds. To fix the idea, we observe six different nonlinear receivers’ outputs, shown in Fig. 6. It can be seen that three nonlinear receivers are in favor of detecting the input binary PAM signals, whether the input signals are suprathreshold for receivers (c) and (d) or subthreshold for the receiver (e). In
Input binary signal
1 0.5 0 −0.5 −1 0
2
4
6
8
10
12
14
16
18
20
12
14
16
18
20
(a)
20
Signal added noise
play three items: received bits number, error bits number and bit error rate. Figure 4 shows the functional block of the nonlinear receiver element in this communication system. The signal in A is the sampled signal sent from the sampler block. The signal in B is assumed to represent the derivative dx/dt of the variable of interest at time t. In C, after the integration block, the output signal is fed into two input terminals: the terminal of gain block a, to obtain the signal ax, and the terminal of multiplier block, which yields x3 . The lateral signal is then inverted and amplified through a cascaded gaining block. Moreover, the signal in C is also sent to the sample decision block to judge which bit digit is transmitted. Finally, the sampled signal, ax and −µx3 are added together through block Σ to give the starting signal. Nowadays, such block scheme can thus be translated into electronic analog devices or finally analyzed by various accurate digital simulations [Gammaitoni et al., 1998; Xu et al., 2002]. According to the decision level, the input binary signal will be recovered by means of values of the receiver output signals at sampling instants. In terms of Eqs. (3), (11) and (19), the channel capacity deduced on the assumption of the nonstationary probability is smaller than that deduced on the assumption of the stationary probability. Therefore, we design and expect that the outputs of the optimal receiver tend to stationary in a bit duration. It is apparent that the appropriate sampling times occur at the end of each bit interval, since it is the time that the receiver outputs have already reached their steady states.
10
0
−10
−20
0
2
4
6
8
10 time ( s )
(b) Fig. 5. (a) Input information-bearing binary PAM signals of time duration twenty seconds with two signal levels ±1 and Ts = 0.5 s. (b) Channel output signals, which are corrupted by an additional Gaussian white noise with zero-mean and variance σ 2 = 16. The sampling time interval is ∆t = 10−3 s.
terms of Eq. (23), bit error rate of these three receivers of (c)–(e) are 8.7389 × 10−6 , 5.7547 × 10−3 and 9.6391 × 10−3 , respectively. Clearly, the receiver of (c) with parameters a = 1 and µ = 500 reproduces the original input information with less error bits. Here we are concerned more with the information-transmission ability of this baseband binary communication system and less with the feather of the input signals. From the Fig. 6, it can be seen that as receiver’s parameter µ increases, the receiver response speed λ1 increases too [Xu et al., 2002]. For a given input binary signal corrupted by noise, the frequency domain between bit rate r and cutoff frequency f can be determined. If λ1 is not large enough to be compared with bit rate r, the nonlinear receiver cannot trace the varying of input signals. Indeed, as the system output of a bit symbol has not reached its steady state, the succeeding binary symbol is passed into the receiver, which causes the confusion. The final output state in a binary symbol is strongly influenced by the receiver response speed λ1 [see Figs. 6(a) and 6(b)]. While if λ 1 is so large that the receiver can be capable of following the random noise and the output signal is in disorder as presented in Fig. 6(f). Thus, it is seen that there exists an optimal nonlinear receiver yielding the
418 F. Duan & B. Xu 2
(a)
0 −2 0 1
2
4
6
8
10
12
14
16
18
20
(b)
Output signals of nonlinear receivers
0 −1 0 0.2
2
4
6
8
10
12
14
16
18
20
(c)
0 −0.2 0 0.1
2
4
6
8
10
12
14
16
18
20
(d)
0 −0.1 0 0.1
2
4
6
8
10
12
14
16
18
20
(e)
0 −0.1 0 0.05
2
4
6
8
10
12
14
16
18
20
(f)
0 −0.05
0
2
4
6
8
10 time ( s )
12
14
16
18
20
Fig. 6. Plots of output signals of six different receivers. The parameters of these receivers are (a) a = 1, µ = 1; (b) a = 1, µ = 10; (c) a = 1, µ = 500; (d) a = 10, µ = 12000; (e) a = 100, µ = 105 and (f) a = 100, µ = 106 . The receiver response speeds of these systems are about 8.7 Hz, 17.8 Hz, 56.5 Hz, 179.8 Hz, 204.4 Hz and 614.4 Hz respectively. Input binary PAM signal levels are ±1 and maybe greater or less than the critical thresholds Scr of potential function U (x) in these receivers.
minimum of bit error rate, that is, the maximum of channel capacity. This is a new kind of cooperative phenomenon, i.e. parameter-induced stochastic resonance (PSR). As contrast to the conventional SR or ASR, PSR effect is realized by means of tuning not the input noise intensity but the nonlinear receiver’s parameters. We argue that the basic mechanism of PSR is the competition among three factors: bit rate r, receiver response speed λ 1 and cutoff frequency f of noise, satisfying r < λ1 f .
(24)
The existence of the PSR effect can also be confirmed by calculation of bit error rate P e or channel capacity C. The behaviors of Pe and C against receiver response speed λ1 for the same input binary PAM signals are presented in Fig. 7. We notice here that channel capacity C first increases with increasing of receiver response speed λ1 , reaches and keeps a maximum value, and then decreases again. This is the typical signature of SR.
In this spirit, we begin considering the question how to tune the nonlinear receivers to achieve the AWGN channel’s optimal performance. We know that the measurements of channel capacity C and bit error rate Pe are functions of the input PAM signal levels, bit rate, noise intensity and nonlinear receiver’s parameters a and µ. For a noisy channel without input signals, noise measurement unit can estimate the variance of noise. According to the bit rate of binary symbols and noise cutoff frequency, one should choose an appropriate receiver response speed, which is assumed about twenty times as large as bit rate in the above numerical example. Consequently, with the same adopted receiver response speed, there are many pairs of receiver’s parameters a and µ, from which one can find a pair of parameters satisfying the desired bit error rate such as 10−5 . It is worth noting that the communication system with this pair of parameters might not have the minimum bit error rate, but it can be considered as the optimal design one. Thus, using Eqs. (17)
Parameter-Induced Stochastic Resonance and Baseband Binary PAM Signals Transmission 419
Input binary signal
1 0.5 0 −0.5 −1 0
(a)
0.01
0.02
0.03
0.04
0.06
0.07
0.08
0.09
0.1
0.06
0.07
0.08
0.09
0.1
(a)
20
Channel output signal
0.05
10
0
−10
−20
0
0.01
0.02
0.03
0.04
0.05 time (s)
(b) (b) Fig. 7. Plots of (a) bit error rate Pe and (b) channel capacity C against receiver response speed λ1 .
and (23), the optimal nonlinear receiver’s parameters a and µ can be obtained, regardless of whether the input binary signals are suprathreshold or subthreshold. As bit rate is low, we find that the channel transmits information with almost no loss by the optimal designed receivers, that is, the channel capacity is almost equal to bit rate. But for very large bit rate, bit error rate also increases, even though we tune the nonlinear receiver parameters very large. A numerical simulation example will be demonstrated in a detailed way in the Sec. 3.2, in which bit rate r is 1000 bits/s.
3.2. A numerical simulation example In this subsection, we show in detail that, by a numerical simulation illustration, how to design an optimal receiver for a given signal corrupted by noise. Suppose that we know the bit rate of input binary PAM signal r = 1000 bits/s and the noise rms level σ = 4. Considering Eq. (24), the sampling interval ∆t is taken as 10−6 s. The required bit error rate is 10−5 . The input binary PAM information-bearing signal is shown in Fig. 8(a) with two kinds of signal waveforms, which are generated by an equiprobable pseudorandom binary signal generator. An additive Gaussian noise with rms amplitude σ = 4 is added. It is carefully tuned that the receiver response speed
Fig. 8. (a) Input binary information PAM signal with signal levels ±1 transmitted over a time of 0.1 s. (b) Channel output signals, in which a Gaussian white noise with rms amplitude σ = 4 and cutoff frequency f = 1 MHz is added.
λ1 agrees with Eq. (24), which is adopted at about 55 KHz. And then, we can calculate the bit error rate of the corresponding receivers with the same receiver response speed. The receiver, being up to the required bit error rate 10−5 , is chosen as the optimal one. In this illustration, the nonlinear receiver with parameters a = 5 and µ = 3 × 10 11 is appropriate. The theoretical calculation of bit error rate Pe equals to 2.6191 × 10−6 , which is less than the desired bit error rate. Equation (5) can be concretely rewritten as dx = 5x − 3 × 1011 x3 + si (t) + ξ(t) , dt for 0 ≤ t < nTs , n = 1, 2, . . . .
(25)
Figure 9(a) presents the output signals at the end of the designed nonlinear receiver and Fig. 9(b) shows the recovered binary PAM signals. In the time duration 0.1 seconds, there are no error bit symbols, but error bits do occur in the processing over a long time. We compare our theory of optimal design-parameter receiver with the experiment’s results using the extensive Monte Carlo simulations over a long time of one thousand seconds. When we transmit one million bits of binary symbols by this communication system, only two or three error bits are observed. The error bit meter displays that bit error rate is in the range 2 × 10−6 to 3 × 10−6 , which greatly supports the theoretical analysis.
420 F. Duan & B. Xu −4
Output signal of receiver
x 10 2 1 0 −1 −2 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
(a)
X=
1
Recovered signal
for n = 1, 2, . . . , Ts f . Here we assume that Ts f is an integer. On assuming of Eq. (18) in Sec. 2.1, the amplitude of nonlinear receiver output turns into zero level when the factor α is null, that is, at time t = ln 2/λ1 . Thus, there are n0 = (ln 2/λ1 )f sample values to be taken out. In a bit duration, we then make use of n − n0 sample values. Suppose
0.5
1 (xn0 +1 + xn0 +2 + · · · + xn ) . n − n0
As a consequence, the ensemble average probability of error bits (cf. Appendix C) is bounded by
0 −0.5 −1 0
0.01
0.02
0.03
0.04
0.05 time (s)
0.06
0.07
0.08
0.09
0.1
(b) Fig. 9. (a) Output signal of the optimal nonlinear receiver. (b) The recovered binary PAM signal by the zero threshold decision level.
As indicated in Sec. 3.1, the optimal design receiver might not have the minimum bit error rate. We here present another nonlinear receiver with parameters a = 5000 and µ = 3 × 1011 , by which the baseband binary communication system has the theoretical bit error rate of 1.5072 × 10 −8 . However, if the error bit rate of 10−5 is all that is required, the receiver with parameters a = 5 and µ = 3 × 1011 can be employed and assumed as the optimal one. It should be stressed that the bit error rate of 1.5072 × 10−8 has not been demonstrated in numerical simulation because it needs a very long time to verify. For the illustrations of the above subsection, bit error rates proved to be coincident with our theoretical analysis in numerical experiments, as well as the example of the receiver with parameters a = 5 and µ = 3 × 1011 . In the next subsection, we further decrease, with the approach of ensemble average probability of error bits, bit error rate in the illustration of this subsection.
3.3. Ensemble average probability of error bits A widely used method, averaging, extracts signal buried in noise. Since bit rate is r = 1/T s and cutoff frequency is f = 1/∆t, there are n = T s f independent sample values xi in each bit duration Ts ,
1 Pea = Pr{|X − E[X]| ≥ (|E[X]| − 0|)} 2 ≤
f2 (a, µ, ±αA)f0 (a, µ, ±αA) − f12 (a, µ, ±αA) . 2(n − n0 )f12 (a, µ, ±αA) (26)
where fi (a, µ, ±αA) are defined in Sec. 2.1. With the above approach, we can further decrease bit error rate in the binary PAM signals transmission via tuning nonlinear receivers. Since the receiver response speed adopted in this numerical illustration is about 55 KHz, then we can obtain the number of ensemble n 0 = (ln 2/λ1 )f ≈ 0. In terms of Eq. (26), we can obtain Pea ≤ 1.2701 × 10−5 . Although 1.2701 × 10−5 is greater than Pe = 2.6191 × 10−6 , it is only an upper bound. In fact, it is really observed that this method can be used to decrease the bit error rate.
3.4. Time scale transformation method In Sec. 3.2, it is seen that when the bit rate is as high as 1000 Hz and the noise rms increases as large as 4, the receiver’s parameter µ is so large as 3 × 1011 . As the bit rate increases further, the receiver’s parameters a and µ are tuned larger and larger so that the receiver response speed can satisfy Eq. (24). Here we demonstrate that time scale transformation method can improve the fundamental limit of the nonlinear receiver for binary PAM signals transmission at very high bit rate or, equivalently, a discussion of limits of low frequency SR or ASR in [Hibbs et al., 1995; Kaufman et al., 1996; Gammaitoni et al., 1998]. The influence of the noise cutoff frequency on PSR effect in nonlinear receiver systems is practically significant. Previous studies on realizing SR or ASR via tuning input noise have
Parameter-Induced Stochastic Resonance and Baseband Binary PAM Signals Transmission 421
concluded that noise with a large cutoff frequency is favorable [Misono et al., 1998; Zhou & Moss, 1990; Gammaitoni et al., 1989]. In this letter, we show that such kinds of noise are also favorable for baseband binary PAM signals transmission via PSR. In the above subsection, it is known that there are n = Ts f independent sample values in each bit duration. Consequently, a discrete time sequence is formed. It is here worth noting that the time interval value of this sequence can be regarded as arbitrary time unit. If the time interval extends N times, that is, ∆τ = N ∆t(N > 1), then the sampled signals in Eq. (4) become sm (n∆τ ) + ξ(n∆τ ) ,
(27)
where hξ(τ )ξ(0)i = 2N Dδ(τ ) is the character of white Gaussian noise. Equation (27) can be explained as follows: the frequency of input signal reduces to 1/N times of that of the originally signal whereas the noise intensity is amplified N times, i.e. D 0 = N σ 2 ∆t/2. This method is time scale transformation. Consider that the cutoff frequency increases by N times, that is, f 0 = N/∆t. The sampling time interval is then ∆t/N . If the time interval of this discrete time sequence is taken as ∆t, it is interesting to note that the noise intensity D 0 = N σ 2 ∆t/(2N ) is equal to D = σ 2 ∆t/2, but the bit rate decreases to 1/N times. According to Eqs. (10) and (23), √ the probability of error bits is a function of A = A/ D and the receiver’s parameters a and µ. With the time scale transformation method, the optimal nonlinear receiver is only dependent on n = T s f , but not related to the bit rate. Thus, such an optimal receiver is not limited by the bit rate, high or low. For an example of a binary signal with levels ±1 and bit rate r = 10000 bits/s, the noise rms amplitude σ = 4 and cutoff frequency f = 10 MHz, and the nonlinear receiver’s parameters are a = 5, µ = 3 × 1011 . If time extends ten times, then bit error rate Pe is also 2.6191 × 10−6 as well as the illustration given in Sec. 3.2. So it is possible to transmit at the bit rate of 10000 bits/s, but at the expense of much larger cutoff frequency and more complex communication systems. However, it indicates that we can optimize the baseband binary communication systems not only by designing the nonlinear receiver but also appealing to time scale transformation approach.
4. Conclusion We now summarize the main results of this work. The AWGN channel, transmitting a binary information-bearing PAM signals, was studied by connecting with a nonlinear dynamic model as the receiver. We demonstrated that PSR phenomenon appears when the receiver systems are adjustable while the input signal added noise is a given quantity and its basic mechanism is explained. It was observed that a marked parameter-induced enhancement of the channel capacity, PSR effect, occurs in numerical simulations for the baseband binary PAM signals transmission. In the view of information theory, it is emphasized that the realization of PSR does not require the subthreshold signals. The consequence of these results is of potential importance for technical applications. Moreover, we suggest that an application of the nonlinear receiver with tunable parameters is useful for communication systems under the noisy channel. Undoubtedly, the channel that has yielded the biggest successes in information theory is Gaussian channel [Schwartz, 1980]. However, for the capacity of bandlimited channel of bandwidth W , studies of ASR or PSR in context of information theory will be more meaningful. We have already started to research the binary signals transmission in a bandwidth limited communication system, in which the signal’s bandwidth is equal to the noise’s. We raise here another important problem: for a linear receiver by which we desire to improve the baseband communication systems performance, is it possible to observe PSR phenomenon? It is guessed that the gain of the linear receiver with an inherent frequency is very dependent upon the input frequency, resulting in being unsuitable for a periodic signal processing. Thus, PSR or ASR effect might not occur. This question is open.
Acknowledgment The financial support by Zhejiang Province Natural Science Foundation No. 601089 for this work is greatly appreciated.
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422 F. Duan & B. Xu
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Parameter-Induced Stochastic Resonance and Baseband Binary PAM Signals Transmission 423
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above research [Xu et al., 2002], we have already discussed this question in detail.
Appendix B B.2. The solution of eigenvalues defined in Eq. (17) Assume [Xu et al., 2002] "
Appendix A A.1. Synthesis of the receiver’s parameters a and µ It would be useful to synthesize the influences of the two important receiver’s parameters a and µ on the performance of baseband communication systems. Redefine √ µD A x a √ y= , µ= 2 , A= √ , τ = at , a D aD
satisfying limx→±∞ u(y) = 0 and limx→±∞ u0 (y) = 0. Substituting Eq. (B.1) into Eq. (17) yields ([K] − λ[M ]){d} = 0 ,
{d} = [d0 for
(A.1)
mij =
Z
kij =
Z
∂ρ(y, τ ) ∂ ∂ 2 ρ(y, τ ) , (A.2) = − [c(y)ρ(y, τ )] + ∂τ ∂y ∂y 2
d n ]T ,
ρ(x) = lim ρ(x, t) = C exp[g(y)] ,
+∞
y i+j exp[g(y)]dy ,
−∞ +∞ −∞
("
1 + iy i−1 + g 0 (y)y i 2 1 + g 0 (y)y j 2
(A.3)
t→∞
#
1 1 00 g (y) + g 02 (y) y i+j 2 4
"
where c(y) = y − µy 3 ± A. Hence, the stationary solution of Eq. (A.2) is given by
#)
#"
jy j−1
exp[g(y)]dy .
It is easily found that − 1/4µy 4
where g(y) = ± Ay and C is a constant determined by the normalization condition of probability density ρ(x) √
ρ(x)dx = C D −∞
K
i, j = 0, 1, 2, K, n ,
where hΓ(τ )Γ(0)i = 2δ(τ ), then the corresponding FPK equation becomes
+∞
d1
[M ] = [mij ], [K] = [kij ] ,
dy = y − µy 3 ± A + Γ(n∆τ ) , dτ
Z
(B.2)
where
then Eq. (8) can be rewritten as
1/2y 2
#
1 u(y) = exp g(y) (d0 + d1 y + · · · + dn y n ) (B.1) 2
Z
+∞
√
exp[g(y)]dy/ a −∞
√ √ = C Df0 (µ, ±A)/ a = 1 , √ √ C = a[ Df0 (µ, ±A)]−1 .
It is seen that, for a fixed noise level and signal amplitude, there is only one system parameter µ. However, the input signal level A is variable. In the
mij = mji , kij = kji . Therefore, the matrices [M ] and [K] are symmetric. It can be demonstrated that the matrix [K] is semi-positive definite and the matrix [M ] is positive definite. Assuming u(y) = p(y) exp[1/2g(y)], it can be demonstrated Z
∞ −∞
("
#
)
1 1 00 g (y) + g 02 (y) u2 (y) + u(y)02 dy ≥ 0 . 2 4 (B.3)
424 F. Duan & B. Xu
for p(y) = const. Inequality (B.3) becomes the equality equation. Indeed, the integrated function is ("
#
1 00 1 g (y) + g 02 (y) p2 (y) 2 4 "
1 + p (y) + p(y)g 0 (y) 2 =
(
0
#2 )
exp[g(y)]
1 1 0 [g (y)p2 (y)]0 + g 02 (y)p2 (y) 2 2
+ p02 (y) exp[g(y)] ,
(B.4)
where g(y) was defined in Sec. 2. Since ∞
[g 0 (y)p2 (y)]0 exp[g(y)]dy
The second way is averaging sample values x i of the nonlinear receiver’s output, whose mean value is
= g 0 (y)p2 (y) exp[g(y)]k∞ −∞ ∞
− =−
Z
g 02 (y)p2 (y) exp[g(y)]dy
−∞ ∞
where the term 1/2 is introduced, since the probability of error bits is only the area of the conditional probability in the interval (−∞, 0), when the input signal level is A, thus σ2 1 . Pe = [P (1|0) + P (0|1)] ≤ 2 2nA2
−∞
Z
1 Pr{|X − E[X]| 2 D[X] ≥ (| ± A| − 0)} ≤ 2A2 σ2 D[x] = , = 2nA2 2nA2
P (1|0) = P (0|1) =
)
Z
with only two signal levels ±A and decision level zero, obtained from the Chebyshev inequality, the conditional probabilities of error bits are
E[X] = g 02 (y)p2 (y) exp[g(y)]dy ,
(B.5)
= E[x] ,
−∞
then
1 (E[x1 ] + E[x2 ] + · · · + E[xn ]) n
and its degree-two moment is Z
∞ −∞
("
1 00 1 g (y) + g 02 (y)]p2 (y) 2 4
"
1 + p0 (y) + p(y)g 0 (y) 2 =
Z
∞ −∞
#2 )
1 E[X ] = 2 n 2
exp[g(y)]dy
p02 (y) exp[g(y)]dy ≥ 0 .
(B.6)
When p0 (y) = 0, i.e. p(y) = const. inequality (B.3) becomes the equality equation. If λ 0 = 0 is the simple characteristic root, we can infer that λ1 > 0.
Appendix C C.3. Ensemble average probability of error bits Here, we present and compare two ways to calculate ensemble average probability of error bits. The first is the way without the nonlinear receiver processing. Consider the binary PAM signals transmitted
=
n X
E[x2i ]
i=1
+
n X
E[xi xj ]
i6=j
!
1 {nE[x2 ] + n(n − 1)E[x]2 } . n2
Hence, the variance D[X] can be written as D[X] = E[X 2 ] − E[X]2 =
1 (E[x2 ] − E[x]2 ) n
=
D[x] . n
For binary PAM signals, the mean value of receiver’s output is given by E[X] = E[x] = C˜
Z
+∞ −∞
x exp[˜ g±A (y)]dx
√ Df1 (a, µ, ±αA) , = f0 (a, µ, ±αA)
Parameter-Induced Stochastic Resonance and Baseband Binary PAM Signals Transmission 425
In Sec. 3.2, (n − n0 ) sample values are used for averaging. As a consequence, the ensemble average probability of error bits is bounded by
the degree-two moment can be expressed as E[x ] = C˜ 2
Z
+∞ −∞
x2 exp[˜ g±A (y)]dx
√ Z = C˜ D 3
1 Pe = Pr{|X − E[X]| ≥ (E[x] − 0)} 2
+∞ −∞
y 2 exp[˜ g±A (y)]dy
Df2 (a, µ, ±αA) , = f0 (a, µ, ±αA)
and the variance of receiver’s output signals is (
f2 (a, µ, ±αA) f12 (a, µ, ±αA) D[x] = D − f0 (a, µ, ±αA) f02 (a, µ, ±αA)
)
.
≤
D[x] 2(n − n0 )E[x]2
=
f2 (a, µ, ±αA)f0 (a, µ, ±αA) − f12 (a, µ, ±αA) . 2(n − n0 )f12 (a, µ, ±αA)
