Receiver sensitivity of type-II return-to-zero signals having finite extinction ratios

Jihoon Lee; Hoon Kim

doi:10.1364/OE.475326

1. Introduction

Comparison of the return-to-zero (RZ) and the nonreturn-to-zero (NRZ) on-off keying (OOK) modulation formats is a recurring topic in optical communications. The receiver sensitivity, which is the required average optical power to achieve a target bit-error ratio (BER), is one aspect of such comparison. Ignoring chromatic dispersion and fiber nonlinearities, it is known that the RZ format prevails over the NRZ format with respect to receiver sensitivity [1–9]. Even utilizing a receiver having bandwidths optimized to yield the highest sensitivity for NRZ signals (about 0.7 times the data rate), a sensitivity gain of 1-2 dB can be obtained by using the RZ format rather than the NRZ format [5,8]. Contrary to the NRZ format, the RZ format is unsusceptible to performance degradations from inter-symbol interference (ISI), giving it the upper hand. If we were to optimize the receiver for the RZ signal by expending a larger bandwidth, greater sensitivity improvements can be achieved [4]. The sensitivity improvement along with the bandwidth requirement is amplified by lowering the duty cycle of the RZ signal.

Despite the potential sensitivity gains, RZ signals having low duty cycles (e.g., < 0.1) were of little interest in conventional fiber-optic transmission systems. Along with the increased demands on the receiver, low duty cycle signals are prone to harsh sensitivity degradations induced by both dispersion and nonlinear effects in the fiber due to having large spectral widths and high peak powers. But, in the absence of such sensitivity degrading effects as in free-space optic (FSO) systems, exhausting the full potential of the RZ format by utilizing low-duty-cycle signals becomes more plausible.

The RZ format was frequently discussed as a candidate for optical intersatellite links [7–9], as the RZ format offers good receiver sensitivities together with requiring a simple intensity modulation (IM) and direct detection (DD) receiver. As is the case with pulse position modulation, modulation formats providing excellent receiver sensitivities in cost-effective IM/DD systems are mostly based on optical pulses. Recent trends show an increase in low earth orbit (LEO) satellites and CubeSat microsatellites which have tight power constraints owing to their small size [10]. The size and power constraints of the small satellites would make the employment of the direct detection of RZ format in space laser communication links even more compelling than the costly coherent detection. An interesting and emerging application for the RZ format is in joint sensing-communication systems using pulsed lidars [11]. Based on the mature concept of radar in radio frequency (RF) [12], a joint sensing-communication system can provide accurate ranging and communication by using optics. When we embed information on the short optical pulses via intensity modulation, we have an RZ signal having very low duty cycles.

To the best of our knowledge, the latest work applicable to low duty cycles is the one done by Winzer and Kalmar [4]. They identified the dependence of the receiver sensitivity on the duty cycle in the presence of different receiver noise. In [4], however, they assumed that the extinction ratio (ER) of the RZ signal is infinite. It was implied in [9] that signals having lower duty cycles suffer larger sensitivity penalties from a poor ER. This suggests that the ER is a critical factor that needs to be considered with the duty cycle when employing the RZ format. Overlooking the effect of finite ER, the theory developed in [4] would yield overly optimistic sensitivity estimates for practical RZ signals, especially for those having low duty cycles.

This paper presents a theory on the receiver sensitivity of RZ signals having finite ERs and arbitrary duty cycles. We derive a closed-form lower bound for the sensitivity of DD receivers assuming that the signal-independent noise at the receiver is white (i.e., having a flat power spectral density). Based on this lower bound, we obtain an approximation of the receiver sensitivity in the presence of non-white (or colored) signal-independent noise. We validate this approximation through an experiment carried out by using two DD receivers: an optically pre-amplified receiver and a PIN receiver. We show that the receiver sensitivity degrades as the duty cycle decreases when the signal-dependent noise limits the system performance. In signal-independent noise-limited receivers, an optimum duty cycle that minimizes the receiver sensitivity exists. We also confirm that the sensitivity penalty due to a finite ER is severe for RZ signals having low duty cycles. Regardless of the receiver used, we predict that the sensitivity is degraded, and the dominant noise at the receiver becomes signal-dependent noise, as the duty cycle approaches 0.

There are two known ways to model an RZ signal having a finite ER [5]. RZ signals have a mark representing a logical 1 bit. For the representation of a logical 0 bit, the RZ signal can have either an attenuated version of the mark or a constant optical power (i.e., a space). This difference in how the 0 bit is represented determines the way the RZ signal is modeled. Termed Type I and II in [6], this paper is focused on the latter, Type-II RZ signals.

2. Signal and receiver model

We formulate an expression of the receiver sensitivity using the receiver structure shown in Fig. 1. The optical power of the Type-II RZ signal incident on this receiver is given by

(1)$$s(t )= \sum\limits_{k = 0}^{M - 1} {{b_k}p({t - k{T_b}} )} + {P_0}$$

where M is the length of the bit sequence, b_k ∈ {0,1} is the k^th input bit, T_b is the bit duration, p(t) ≥ 0 is the RZ pulse, and P₀ is the power corresponding to a space. Assuming that bits are equiprobable, we can express the average signal power at the receiver input as

(2)$${P_{\textrm{avg}}} = \frac{{D({\varepsilon - 1} )+ 2}}{{2({\varepsilon - 1} )}}\mathop {\max }\limits_t \{{p(t )} \}.$$

0 < D ≤ 1 is the duty cycle, which is defined as

(3)$$D = {{{T_p}} / {{T_b}}}$$

where T_p is the effective pulse width of p(t),

(4)$${T_p} = \frac{{\int_{ - \infty }^\infty {p(t )} dt}}{{{{\max }_t}\{{p(t )} \}}}.$$

Also, ɛ > 1 is the ER, defined as

(5)$$\varepsilon = {{{P_1}} / {{P_0}}}$$

where P₁= max_t{p(t)} + P₀. At the receiver, the optical signal can be optically pre-amplified and bandpass filtered before photo-detection. When the signal impinges on the photo-detector (PD), the optical signal is converted to photo-current, electrically amplified, and low-pass filtered. The impulse response of the detection chain is denoted by h(t). For the sake of simplicity, we set $\int_{ - \infty }^\infty {h(t )dt} = 1$, making the electrical gain in the receiver unity. At the output of the receiver, the photo-current is given by [13]

(6)$$i(t )= K({s \ast h} )(t )$$

where K is the overall conversion factor (A/W) of the receiver. When we have a gain G, e.g., an optical gain from an optical amplifier, K = GR, where R is the detector responsivity. For receivers employing an avalanche detector (APD), K = MR, where M is the multiplication factor of the APD. In the absence of such a gain or multiplication process, i.e., when utilizing a simple PIN detector, K = R. The symbol ⁎ denotes a convolution,

(7)$$({f \ast g} )(t )= f(t )\ast g(t )= \int_{ - \infty }^\infty {f(\tau )g({t - \tau } )d\tau } .$$

We assume the receiver noise to be additive and Gaussian-distributed. The variance of the noise at the receiver can be expressed as

(8)$${\sigma ^2}(t )= \sigma _{\textrm{dep}}^2(t )+ \sigma _{\textrm{indep}}^2(t )$$

where $\sigma _{\textrm{dep}}^2(t )$ and $\sigma _{\textrm{indep}}^2(t )$ are the contributions of the signal-dependent and signal-independent noise, respectively. A general form of $\sigma _{\textrm{dep}}^2(t )$ is given by [14,15]

(9)$$\sigma _{\textrm{dep}}^2(t )= C({s \ast {h^2}} )(t )$$

where C is an appropriate constant. For the signal-dependent, shot noise in a PIN detector, C = Rq, where q is the electric charge. When an APD is utilized, C = RqM²F_APD for shot noise, where F_APD is the excess noise factor of the APD. For the beat noise produced between the signal and the amplified spontaneous (ASE) noise when utilizing an optical amplifier, C = 2GR²N_ASE, where N_ASE is the power spectral density of the ASE noise per polarization mode [15]. In the presence of multiple sources of signal-dependent noise, C is the sum of constants. For example, we write C = 2GR²N_ASE+ Rq when both the signal-ASE beat noise and the shot noise contribute to the signal-dependent noise. $\sigma _{\textrm{indep}}^2(t )$ is the variance of the signal-independent noise, which can be expressed as

(10)$$\sigma _{\textrm{indep}}^2(t )= \int_0^\infty {N(f ){{|{H(f )} |}^2}df}$$

where H(f) is the Fourier transform of h(t) and N(f) is the (single-sided) power spectral density of the signal-independent noise. In many cases, N(f) is the sum of the thermal noise from the receiver and the dark current of the detector. When an optical amplifier is utilized at the receiver, the ASE-ASE beat noise is included in N(f). The variance of the ASE-ASE beat noise is expressed as $2{R^2}N_{\textrm{ASE}}^2({{B_o} - f} ),$ where B_o is the optical filter bandwidth [15]. The Q-factor is defined as

(11)$$Q = \mathop {\max }\limits_t \left\{ {\frac{{{i_1}(t )- {i_0}(t )}}{{{\sigma_1}(t )+ {\sigma_0}(t )}}} \right\}$$

where the subscripts 1 and 0 denote the photo-current and noise standard deviation generated by an (isolated, ISI-free) mark and space, respectively.

Fig. 1. Model of a direct-detection receiver. A Type-II RZ signal having an optical power of s(t) is incident on the receiver. The receiver consists of an optional optical amplifier and optical BPF, along with a PD, an electrical amplifier, and an electrical LPF.

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3. Expression of receiver sensitivity

For notational convenience, we assume that the ratio in Eq. (11) is maximized by setting t = T_s. We identify the optical power of a mark as s₁(t) = p(t) + P₀ and a space as s₀(t) = P₀. Substituting Eq. (6), Eq. (8), Eq. (9), and Eq. (10) in Eq. (11), we obtain

(12)$$\begin{aligned} Q &= K({p \ast h} )({{T_s}} )\\ &\quad \times \left( {{{\left( {C\left[ {({p \ast {h^2}} )({{T_s}} )+ {P_0}\int_{ - \infty }^\infty {{h^2}(t )dt} } \right] + \int_0^\infty {N(f ){{|{H(f )} |}^2}df} } \right)}^{1/2}}} \right.\\ &{\left. {\quad + {{\left( {C{P_0}\int_{ - \infty }^\infty {{h^2}(t )dt} + \int_0^\infty {N(f ){{|{H(f )} |}^2}df} } \right)}^{1/2}}} \right)^{ - 1}}. \end{aligned}$$

Using Eq. (2) and Eq. (5), we express the Q-factor in terms of P_avg as

(13)$$\begin{aligned} Q &= \frac{{2({\varepsilon - 1} ){P_{\textrm{avg}}}}}{{D({\varepsilon - 1} )+ 2}}K({\bar{p} \ast h} )({{T_s}} )\\ &\quad \times \left( {{{\left( {\frac{{2{P_{\textrm{avg}}}}}{{D({\varepsilon - 1} )+ 2}}C\left[ {({\varepsilon - 1} )({\bar{p} \ast {h^2}} )({{T_s}} )+ \int_{ - \infty }^\infty {{h^2}(t )dt} } \right] + \int_0^\infty {N(f ){{|{H(f )} |}^2}df} } \right)}^{1/2}}} \right.\\ &{\left. {\quad + {{\left( {\frac{{2{P_{\textrm{avg}}}}}{{D({\varepsilon - 1} )+ 2}}C\int_{ - \infty }^\infty {{h^2}(t )dt} + \int_0^\infty {N(f ){{|{H(f )} |}^2}df} } \right)}^{1/2}}} \right)^{ - 1}} \end{aligned}$$

where $\bar{p}(t )= {{p(t )} / {{{\max }_t}\{{p(t )} \}}}$. The receiver sensitivity for a given Q-factor can be written as

(14)$$\begin{aligned} {P_{\textrm{rec}}} &= \frac{{Q[{D({\varepsilon - 1} )+ 2} ]}}{{K({\varepsilon - 1} )}}\left( {\frac{{CQ\left[ {({\varepsilon - 1} )({\bar{p} \ast {h^2}} )({{T_s}} )+ 2\int_{ - \infty }^\infty {{h^2}(t )dt} } \right]}}{{2K({\varepsilon - 1} ){{[{({\bar{p} \ast h} )({{T_s}} )} ]}^2}}}} \right.\\ &\left. {\quad + {{\left( {\frac{{{{({CQ} )}^2}\left[ {({\varepsilon - 1} )({\bar{p} \ast {h^2}} )({{T_s}} )+ \int_{ - \infty }^\infty {{h^2}(t )dt} } \right]\int_{ - \infty }^\infty {{h^2}(t )dt} }}{{{{({K({\varepsilon - 1} ){{[{({\bar{p} \ast h} )({{T_s}} )} ]}^2}} )}^2}}} + \frac{{\int_0^\infty {N(f ){{|{H(f )} |}^2}df} }}{{{{[{({\bar{p} \ast h} )({{T_s}} )} ]}^2}}}} \right)}^{{1 / 2}}}} \right). \end{aligned}$$

We wish to minimize P_rec in Eq. (14) by optimizing the pulse shape for a given duty cycle and the receiver’s impulse response. However, this is extremely difficult due to the non-white nature of N(f). Thus, we derive a lower bound for P_rec assuming N(f) = N, i.e., the signal-independent noise is white, then approximate the minimum receiver sensitivity for an arbitrary N(f) using this lower bound.

3.1 Lower bound for white signal-independent noise

Suppose N(f) = N. We can then use the conjugate symmetry of H(f) [i.e., H(f) = H*(−f)] and Parseval’s relation, $\int_{ - \infty }^\infty {{{|{H(f ) } |}^2}df} = \int_{ - \infty }^\infty {{h^2}(t )dt}$, and express P_rec as

(15)$$\begin{aligned} {P_{\textrm{rec}}} &= \frac{{Q[{D({\varepsilon - 1} )+ 2} ]}}{{K({\varepsilon - 1} )}}\left( {\frac{{CQ\left[ {({\varepsilon - 1} )({\bar{p} \ast {h^2}} )({{T_s}} )+ 2\int_{ - \infty }^\infty {{h^2}(t )dt} } \right]}}{{2K({\varepsilon - 1} ){{[{({\bar{p} \ast h} )({{T_s}} )} ]}^2}}}} \right.\\ &\left. {+ {{\left( {\frac{{{{({CQ} )}^2}\left[ {({\varepsilon - 1} )({\bar{p} \ast {h^2}} )({{T_s}} )+ \int_{ - \infty }^\infty {{h^2}(t )dt} } \right]\int_{ - \infty }^\infty {{h^2}(t )dt} }}{{{{({K({\varepsilon - 1} ){{[{({\bar{p} \ast h} )({{T_s}} )} ]}^2}} )}^2}}} + \frac{{N\int_{ - \infty }^\infty {{h^2}(t )dt} }}{{2{{[{({\bar{p} \ast h} )({{T_s}} )} ]}^2}}}} \right)}^{{1 / 2}}}} \right). \end{aligned}$$

We find

(16)$$\int_{ - \infty }^\infty {{h^2}(t )dt} \ge ({\bar{p} \ast {h^2}} )({{T_s}} ).$$

From Eq. (15) and Eq. (16) we have

(17)$$\begin{aligned} {P_{\textrm{rec}}} &\ge \frac{{Q[{D({\varepsilon - 1} )+ 2} ]}}{{K({\varepsilon - 1} )}}\left( {\frac{{CQ({\varepsilon + 1} )}}{{2K({\varepsilon - 1} )}}\frac{{({\bar{p} \ast {h^2}} )({{T_s}} )}}{{{{[{({\bar{p} \ast h} )({{T_s}} )} ]}^2}}}} \right.\\ &\left. {+ {{\left( {{{\left( {\frac{{CQ\sqrt \varepsilon }}{{K({\varepsilon - 1} )}}\frac{{({\bar{p} \ast {h^2}} )({{T_s}} )}}{{{{[{({\bar{p} \ast h} )({{T_s}} )} ]}^2}}}} \right)}^2} + \frac{N}{2}\frac{{({\bar{p} \ast {h^2}} )({{T_s}} )}}{{{{[{({\bar{p} \ast h} )({{T_s}} )} ]}^2}}}} \right)}^{{1 / 2}}}} \right). \end{aligned}$$

Using the Cauchy-Schwarz inequality along with Eq. (3) and Eq. (4), we find

(18)$$\frac{{({\bar{p} \ast {h^2}} )({{T_s}} )}}{{{{[{({\bar{p} \ast h} )({{T_s}} )} ]}^2}}} \ge \frac{1}{{\int_{ - \infty }^\infty {\bar{p}(t )dt} }} = \frac{1}{{D{T_b}}}.$$

From Eq. (17) and Eq. (18), we obtain a closed-form lower bound for P_rec as

(19)$${P_{\textrm{dep + indep}}} = \frac{{Q[{D({\varepsilon - 1} )+ 2} ]}}{{K({\varepsilon - 1} )}}\left( {\frac{{CQ({\varepsilon + 1} )}}{{2DK{T_b}({\varepsilon - 1} )}} + {{\left( {{{\left( {\frac{{CQ\sqrt \varepsilon }}{{DK{T_b}({\varepsilon - 1} )}}} \right)}^2} + \frac{N}{{2D{T_b}}}} \right)}^{{1 / 2}}}} \right).$$

Through a proper choice of $\bar{p}(t )$ and h(t), P_rec can be made equal to P_{dep + indep}. In deriving Eq. (19), we use two inequalities, Eq. (16) and Eq. (18). Equality in both holds if

(20)$$\bar{p}(t )= \left\{ \begin{array}{l} 1,\quad 0 \le t \le D{T_b}\\ 0,\quad \textrm{otherwise} \end{array} \right.$$

and h(t) is matched to $\bar{p}(t )$. This implies that the use of an elementary rectangular pulse and a matched filter at the receiver gives the best receiver sensitivity in the presence of white signal-independent noise [16]. With matched filtering in the electrical domain, the signal-to-noise ratio (SNR) depends only on the electrical power of the received signal [17]. Note that the photo-detector translates optical power to electrical current, not electrical power. So even if the received signal has a fixed average optical power, it may have varying electrical powers, resulting in varying SNRs, depending on the shape of the optical signal.

3.2 Approximation for colored signal-independent noise

By modifying Eq. (19), we now generalize the receiver sensitivity in the presence of colored signal-independent noise. If we define the power equivalent bandwidth of the detection chain as

(21)$${B_h} = \int_0^\infty {{{|{H(f )} |}^2}df} ,$$

the bandwidth of the receiver matched to the signal is found as

(22)$${B_h} = \frac{1}{{2D{T_b}}}.$$

Given a single-sided noise power spectrum N(f), we substitute N in Eq. (19) with

(23)$$N \approx \frac{1}{{{B_h}}}\int_0^{{B_h}} {N(f )df} = 2D{T_b}\int_0^{{1 / {(2D{T_b})}}} {N(f )df}$$

where Eq. (22) is used in the second step. Finally, we write

(24)$${P_{\textrm{dep + indep}}} = \frac{{Q[{D({\varepsilon - 1} )+ 2} ]}}{{K({\varepsilon - 1} )}}\left( {\frac{{CQ({\varepsilon + 1} )}}{{2DK{T_b}({\varepsilon - 1} )}} + {{\left( {{{\left( {\frac{{CQ\sqrt \varepsilon }}{{DK{T_b}({\varepsilon - 1} )}}} \right)}^2} + \int_0^{{1 / {(2D{T_b})}}} {N(f )df} } \right)}^{{1 / 2}}}} \right).$$

3.3 Simplified forms

If either signal-dependent or signal-independent noise dominates the receiver, P_{dep + indep} in Eq. (24) can be written in a simpler form. By setting N(f) = 0, i.e., considering only the signal-dependent noise at the receiver, we can reduce P_{dep + indep} to

(25)$${P_{\textrm{dep}}} = \frac{{C{Q^2}[{D({\varepsilon - 1} )+ 2} ]}}{{2D{K^2}{T_b}{{\left( {\sqrt \varepsilon - 1} \right)}^2}}}.$$

If the ER is infinite, Eq. (25) becomes P_dep = CQ²/(2K²T_b). Thus, the receiver sensitivity becomes independent of the duty cycle [3,4]. Rearranging this equation gives Q² = K(2P_depT_b)/(C/K). This expression is identical to Eq. (20) in [4]. In the case of finite ER, as ∂P_dep/∂D < 0, we find that the sensitivity of a receiver dominated by signal-dependent noise degrades as the duty cycle decreases. We now consider the case when the receiver performance is limited by signal-independent noise. By setting C = 0 in Eq. (24), we have

(26)$${P_{\textrm{indep}}} = \frac{{Q[{D({\varepsilon - 1} )+ 2} ]}}{{K({\varepsilon - 1} )}}{\left( {\int_0^{{1 / {(2D{T_b})}}} {N(f )df} } \right)^{{1 / 2}}}.$$

It was shown in [4] that when signal-independent noise dominates the receiver, lowering the duty cycle improves the sensitivity. This improvement is obtained under the assumption that the ER is infinite and the signal-independent noise limiting the system performance is white. Plugging ɛ = ∞ and N(f) = N in Eq. (26), we obtain P_indep = Q/K(DN/2T_b)^1/2. From this equation, we see that P_indep decreases monotonically with the decrease of the duty cycle, which supports the analysis reported in [4]. But if the ER is finite, assuming N(f) is white, we find that P_indep is minimized at

(27)$$D = \frac{2}{{\varepsilon - 1}}.$$

In other words, if the signal-independent noise limits the receiver performance, an optimum duty cycle that minimizes the receiver sensitivity exists. These findings indicate that Eq. (25) and Eq. (26) are generalizations of the theory given in [4], valid even for finite ERs.

4. Experiment and discussion

We verify our theory through an experiment. Figure 2 shows the experimental setup. Continuous-wave light from a laser diode (LD) is modulated by an electro-absorption modulator (EAM). Driving the EAM is an electrical Gaussian-shaped RZ signal that is generated by an arbitrary waveform generator (AWG). Since the maximum sampling rate of the AWG is 64 Gsample/s, the narrowest pulse width we can achieve in our experiment is limited to 40 ps. Thus, we chose a data rate of 62.5 Mb/s to validate our theory for a wide range of duty cycles. For example, by varying the pulse width from 6.4 ns to 40 ps in the AWG, we adjust the duty cycle from 0.4 to 0.0025 in the electrical domain. The optical RZ signals are sent to either an optically pre-amplified receiver or a PIN receiver.

Fig. 2. The experimental setup. The setup consists of (a) a transmitter with a variable optical attenuator connected to either (b) an optically pre-amplified receiver or (c) a PIN receiver.

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4.1 Optically pre-amplified receiver

Figure 2(b) shows the schematic diagram of the optically pre-amplified receiver used in the experiment. The received optical signal is first amplified by a two-stage erbium-doped fiber amplifier (EDFA) having an optical gain of 52 dB and a noise figure of 3.5 dB. An optical bandpass filter (OBPF) is placed after each stage to reduce the ASE noise from the preceding amplifier. The OBPF1 sandwiched between the EDFAs has a bandwidth of 3.2 nm. The filter bandwidth following the second stage is set to 3/(T_bD) (= 0.0015/D nm at 1550 nm). This is to maintain a constant ratio between the optical filter bandwidth and the RZ signal’s spectral width of roughly, 1/(T_bD). However, the narrowest filter bandwidth available in our experiment is 0.06 nm. Thus, the bandwidth of the OBPF2 is fixed to 0.06 nm when D ≥ 0.025. The amplified optical signal is then detected by a PIN detector, electrically amplified, and sampled by using a real-time oscilloscope. Offline processing at the scope involves low-pass filtering using a 5^th-order Bessel filter, resampling, and decision. The BER is obtained through direct error counting. The low-pass filter (LPF) bandwidth is chosen such that it minimizes the BER for each duty cycle. The relevant parameters are summarized in Table 1.

Table 1. Summary of receiver parameters

View Table

As stated in Section 2, we set K = GR for optically pre-amplified signals. In this receiver, the signal-ASE beat noise is the dominant source of signal-dependent noise. Thus, by neglecting the shot noise, we write C = 2GR²N_ASE. We also assume that the ASE-ASE beat noise dominates the thermal noise. Thus, we have $N(f) = 2{R^2}N_{\textrm{ASE}}^2({{B_o} - f} )$.

Figure 3(a) shows the measured receiver sensitivities (at a BER of 10⁻³) as a function of the inverse of the duty cycle. Also plotted in this figure are the theoretical results. We observe that our theory agrees very well with the experimental data. The sensitivity differences between the theory and the experiment are measured to be less than 0.5 dB. The results show an optimum duty cycle minimizing the receiver sensitivity. This optimum duty cycle depends on the ER, e.g., when the ER is 13 dB, the minimum receiver sensitivity is obtained at D = 0.25, and as the ER increases to 17 dB, the optimum duty cycle decreases to D = 0.1.

Fig. 3. (a) Measured receiver sensitivities of the optically pre-amplified receiver as a function of the inverse of the duty cycle. (b) Theoretical receiver sensitivities, showing the contribution of signal-dependent and signal-independent noise for ɛ = 13 dB. (c) The receiver sensitivity versus duty cycle for various ERs when signal-dependent noise limits the system performance.

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Figure 3(b) shows the breakdown of the receiver sensitivity for ɛ = 13 dB. When D < 0.025, P_{dep + indep} closely follows P_dep. This implies that the signal-independent, ASE-ASE beat noise constitutes an insignificant portion of the total receiver noise, and thus negligible. For D ≥ 0.025, however, P_{dep + indep} deviates from P_dep (and follows P_indep for large duty cycles). While the bandwidth of the OBPF2 is fixed for D ≥ 0.025, the spectral width of the RZ signal decreases with the increase of the duty cycle. As a result, the contribution of the ASE-ASE beat noise grows larger as D increases in excess of 0.025.

Figure 3(c) shows P_dep versus the duty cycle for various ERs when the system performance is limited by signal-dependent noise. As discussed in Section 3.3, we see that the receiver sensitivity degrades as the duty cycle decreases if the ER is finite, but becomes independent of the duty cycle in the case of infinite ER. We also confirm that the sensitivity penalty induced by finite ERs increases as the duty cycle decreases [9]. For example, a finite ER of 13 dB incurs a sensitivity penalty of 3 dB for D = 1, but the penalty exceeds 20 dB for D < 10⁻³.

4.2 PIN receiver

The schematic diagram of the PIN receiver is depicted in Fig. 2(c). The received optical signal is detected by a PIN-TIA detector, electrically amplified, and captured by using a real-time oscilloscope for offline processing. Identical to the optically pre-amplified receiver, the signal processing consists of low-pass filtering with a 5^th-order Bessel filter, resampling, decision, and direct error counting. The bandwidth of the LPF is optimized for the BER per duty cycle.

We set K = R with a PIN receiver. The signal-dependent noise in this receiver is the shot noise. Thus, we have C = Rq. To obtain N(f), we measured the power spectrum of the noise after the electrical amplifier, and the result is shown in Fig. 4. The measured frequency range is from 400 kHz to 30 GHz. We modeled this noise spectrum as the dashed line (the expression is given in Table 1). We assume that the noise is white beyond 30 GHz.

Fig. 4. Measured and modeled noise spectrum after the electrical amplifier.

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Figure 5(a) shows the measured sensitivities (at a BER of 10⁻³) of the PIN receiver as a function of the inverse of the duty cycle. Also plotted for comparison are the theoretical results. Our theory agrees within 1.6 dB with the experimental data for different duty cycles and ERs. Similar to the sensitivity curves of Fig. 3(a), the sensitivity curves of Fig. 5(a) show that there exists an optimum duty cycle that minimizes the receiver sensitivity. For the RZ signal having an ER of 13 dB, the best receiver sensitivity is estimated to be obtained when D = 0.06. For an ER of 17 dB, the optimum duty cycle is estimated to be D = 0.025.

Fig. 5. (a) Measured receiver sensitivities as a function of the inverse of the duty cycle when the PIN receiver is employed. (b) Theoretical receiver sensitivities for various ERs.

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Figure 5(b) shows the breakdown of the PIN receiver sensitivity for ɛ = 13 and 53 dB and also for ɛ = ∞. The performance of the system utilizing a PIN receiver is typically limited by the signal-independent thermal noise. When D > 10⁻⁶, it is indeed the signal-independent noise that limits the receiver performance. In this range of duty cycles, we see that the receiver sensitivity has an optimum duty cycle when the ER is finite, as mentioned in Section 3.3. Non-white characteristics of N(f) make the optimum duty cycle deviate from the duty cycle given in Eq. (27) (which is 0.1 for ɛ = 13 dB and 10⁻⁵ for ɛ = 53 dB). Nevertheless, we can see that the inverse relationship between the ER and the optimum duty cycle holds. In the case of an infinite ER, the sensitivity improves monotonically, also brought up in Section 3.3.

Surprisingly, however, as the duty cycle becomes extremely low, e.g., D < 10⁻⁸, the signal-dependent shot noise dominates the thermal noise and eventually limits the receiver sensitivity, regardless of the ER. It is worth noting that this is not specific to the receiver type. As shown in Eq. (24), P_{dep + indep} = P_dep as D approaches 0, regardless of the values of C and N(f), granted, N(f) is bounded below some value and does not diverge. We see from Eq. (22) that detecting an RZ signal having a low duty cycle demands a large receiver bandwidth. Since the total noise power increases with the bandwidth, a low-duty-cycle signal requires a large eye opening (i.e., P₁ − P₀) to achieve the target BER. This implies that the contribution of shot noise increases and eventually dominates the thermal noise as the RZ pulse approaches an impulse. In the case of infinite ER, the sensitivity improves as the duty cycle decreases when the system performance is limited by signal-independent noise [4], but eventually saturates as the signal-dependent noise becomes dominant. Again, by comparing the curves for when the ER is finite ER and ɛ = ∞, we observe that the sensitivity penalty increases as the duty cycle decreases.

Figure 6 shows the optimum bandwidths for receiver sensitivity when the optically pre-amplified or PIN receiver is employed. Our theoretical analysis given in Eq. (22) agrees very well with the experimental data.

Fig. 6. The optimum receiver bandwidth for receiver sensitivity versus the inverse of the duty cycle.

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5. Conclusions

We have developed a theory on the receiver sensitivity of Type-II RZ signals having finite extinction ratios and verified it through an experiment. We present a single expression of the receiver sensitivity affected by both signal-dependent and signal-independent noise. When signal-dependent noise limits the system performance, the sensitivity degrades monotonically as the duty cycle decreases. On the other hand, there exists an optimum duty cycle for receiver sensitivity in signal-independent noise-limited systems. Our analysis also predicts that as the RZ pulse approaches an impulse, the signal-dependent noise eventually limits the system performance, regardless of the receiver. Finally, we quantitatively show that the sensitivity penalties due to a finite extinction ratio increase as the duty cycle decreases.

Funding

Institute for Information and Communications Technology Promotion (2022-0-00239).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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Optically pre-amplified receiver	PIN receiver
Data rate = 62.5 Mb/s	Data rate = 62.5 Mb/s
Responsivity, R = 0.74 A/W	Responsivity, R = 0.67 A/W
EDFA gain, G = 52 dB
EDFA noise figure = 3.5 dB
Optical filter bandwidth (before the PD), $B_{o} = {\begin{cases} 0.06 nm D \geq 0.025 \\ 0.0015 / D nm D < 0.025 \end{cases}$
T_b = 1.6 × 10⁻⁸ s	T_b = 1.6 × 10⁻⁸ s
K = 1.17 × 10⁵ A/W	K = 0.67 A/W
C = 3.95 × 10⁻⁹ A²s/W	C = 1.07 × 10⁻¹⁹ A²s/W
N(f) = 5.67 × 10⁻²⁸ (B_o − f) A²/Hz	N(f) = 1.7 × 10⁻²² + 2 × 10⁻¹⁸ f ^−1/2 A²/Hz

Receiver sensitivity of type-II return-to-zero signals having finite extinction ratios

Abstract

1. Introduction

2. Signal and receiver model

3. Expression of receiver sensitivity

3.1 Lower bound for white signal-independent noise

3.2 Approximation for colored signal-independent noise

3.3 Simplified forms

4. Experiment and discussion

4.1 Optically pre-amplified receiver

4.2 PIN receiver

5. Conclusions

Funding

Disclosures

Data availability

References

Data availability

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Figures (6)

Tables (1)

Equations (27)

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