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

Jihoon Lee; Hoon Kim

doi:10.1364/OE.484253

1. Introduction

Free-space optical (FSO) communication systems suffer significant reductions in received optical power due to atmospheric attenuation, turbulence, and beam divergence. In such systems, it is therefore necessary to minimize the required power at the receiver. It is well known that coherent detection offers excellent receiver sensitivities. However, atmospheric turbulence severely distorts the wavefront, thereby harshly lowering the fiber-coupling efficiency [1]. Thus, the sensitivity benefits of this technique could be greatly diminished in the presence of atmospheric turbulence.

The return-to-zero (RZ) on-off keying (OOK) modulation format offers a simple and cost-effective way to implement intensity-modulation (IM)/ direct-detection (DD) systems having decent receiver sensitivity. In the absence of fiber dispersion and nonlinear effects, one can exploit short, highly-peaked RZ pulses and achieve the required receiver performance, even at very low average powers [2]. Utilizing short-pulsed (i.e., low-duty-cycle) RZ signals in airborne FSO communication systems is particularly interesting, because achieving high receiver sensitivities without sacrificing size, weight, and power is one of the prime design goals of such systems.

When utilizing low-duty-cycle RZ signals, high extinction ratios (ERs) are required. For a fixed average power, the power leakage into the “off” state ultimately limits the achievable peak power and, thus, the receiver performance [3]. However, the impact of ER on the receiver sensitivity of RZ signals has not yet been properly analyzed. To the best of our knowledge, prior works quantifying the impact of ER cover only a limited range of duty cycles (e.g., ≥ 0.05) [4–6] or assume a suboptimum receiver [7,8].

In this work, we report on the receiver sensitivity of RZ signals having finite ERs. We optimize the signal pulse shape and the receiver’s impulse response and evaluate the minimum (or best-case) receiver sensitivity for arbitrary duty cycles. RZ signals can be classified as Type I or II based on how they are generated, which we illustrate in Fig. 1 [8]. We can see that the waveform for each type is distinct, with Type-I RZ signals requiring two ERs to describe their shape, whereas Type-II RZ signals require only one. As a result, the impact of finite ER on receiver sensitivity varies significantly between the two types. Thus, we perform a two-part analysis, one for each type of RZ signal.

Fig. 1. Generation of two types of RZ signals, Type I and II. (a) Continuous-wave light is intensity-modulated by an NRZ data modulator (which initially creates the dashed waveform in gray), then is passed through a pulse carver. (b) A train of optical pulses generated via a pulsed source (dashed waveform in gray) is intensity-modulated by an NRZ data modulator. (c) Continuous-wave light is phase-modulated by differentially encoded NRZ data. The subsequent delay-line interferometer is adjusted to create destructive interference in the absence of phase changes, thereby creating pulses having widths equal to the optical delay. (d) An LD is directly modulated by RZ data.

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In part 1, we presented the receiver sensitivity of Type-II RZ signals having finite ERs [9]. If signal-dependent noise dominates the receiver, the sensitivity increases (or degrades) with the decrease of the duty cycle. If signal-independent noise dominates, there exists an optimum duty cycle for sensitivity.

This paper serves as part 2, where we present the receiver sensitivity of Type-I RZ signals having finite ERs. We begin in Section 2 by modeling the Type-I RZ signal and the DD receiver. Then in Section 3, we derive the minimum sensitivity of a DD receiver, whose performance is limited by signal-dependent or signal-independent noise. When signal-dependent noise limits the receiver performance, the receiver sensitivity is independent of the duty cycle. For receivers limited by signal-independent noise, there exists an optimum duty cycle for sensitivity. We prove our findings to be generalizations of the theory in [10], even applicable to RZ signals having finite ERs. As a by-product of our derivation, we obtain the optimum receiver conditions that minimize the sensitivity. An integrate-and-dump receiver is optimum for dominating signal-dependent noise, regardless of the duty cycle. For dominating signal-independent noise, the optimum is a matched filter. In the remainder of the section, we approximate the receiver sensitivity affected by both signal-dependent and signal-independent noise. We verify this approximation in Section 4, through an experiment using a PIN receiver and an optically pre-amplified receiver. Finally, the concluding remarks and a summary of Type I and II are drawn in Section 5.

2. Signal and receiver model

Two ERs need to be defined in the modeling of Type-I RZ signals. To specify the origin of each ER, we assume that the signal is generated via the method shown in Fig. 1(b).

First, we define the output power of the pulsed source as

(1)$${p_{\textrm{train}}}(t )= \sum\limits_{k = 0}^{M - 1} {p({t - k{T_b}} )} + {P_{\textrm{pedestal}}}$$

where M is the number of pulses (which becomes the length of the bit sequence after intensity modulation), T_b is the pulse repetition time (i.e., bit duration), p(t) ≥ 0 is the unit pulse, and P_pedestal is the pedestal power of the pulse train.

The duty cycle, 0 < D ≤ 1, is defined as

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

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

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

We define the first ER, ε₁ > 1, as

(4)$${\varepsilon _1} = 1 + \frac{{{{\max }_t}\{{p(t )} \}}}{{{P_{\textrm{pedestal}}}}}.$$

In other words, ε₁ is the ER of the pulse train. If the pulses are generated via a pulsed source such as a mode-locked laser, ε₁ may range widely from 10 to 30 dB [11–13].

By intensity-modulating the pulse train with nonreturn-to-zero (NRZ) data, we obtain the Type-I RZ signal

(5)$$s(t )= \sum\limits_{k = 0}^{M - 1} {[{{P_1}({{b_k} \oplus 1} )+ {P_0}({{b_k} \oplus 0} )} ]{{\bar{p}}_b}({t - k{T_b}} )}$$

where ${\bar{p}_b}(t )$ is the pulse train during one bit slot, normalized to unit amplitude

(6)$${\bar{p}_b}(t )= \left\{ \begin{array}{l} \frac{{{p_{\textrm{train}}}(t )}}{{{{\max }_t}\{{{p_{\textrm{train}}}(t )} \}}},\quad 0 \le t \le {T_b}\\ 0,\quad \quad \quad \quad \quad \quad \;\;\textrm{otherwise} \end{array} \right.$$

b_k ∈ {0,1} is the k^th input bit, ${\oplus}$ is the bitwise XOR operator, and P₁ and P₀ are the peak powers corresponding to “1” and “0” bits, respectively. We define the second ER, ε₂ > 1, as

(7)$${\varepsilon _2} = {{{P_1}} / {{P_0}}}.$$

ε₂ is the ER determined by the NRZ data modulator. Commercially available data modulators such as Mach-Zehnder modulators (MZMs) and electro-absorption modulators (EAMs) have ERs limited to < 25 dB.

If the method shown in Fig. 1(a) is used to generate the Type-I RZ signal, ε₂ remains the ER determined by the NRZ data modulator, whereas ε₁ becomes the ER determined by the pulse carver.

Figure 2 shows the model of the DD receiver. The receiver consists of a photo-detector (PD), an electrical amplifier, and a low-pass filter (LPF). The impulse response of the receiver is denoted by h(t). To make the electrical gain unity, we normalize h(t) to unit area, $\int_{ - \infty }^\infty {h(t )dt} = 1$. We may add an optional optical amplifier and optical bandpass filter (OBPF) before the photo-detector, assuming that the OBPF passes the optical signal undistorted.

Fig. 2. Model of a Type-I RZ signal and a DD receiver. For the sake of simplicity, we illustrate the Type-I RZ signal assuming that the NRZ data modulator has zero insertion loss. Thus, P₁ = max_t{p(t)} + P_pedestal. The DD receiver consists of an optional optical amplifier and optical BPF (both in dashed lines), and a detection chain having an impulse response of h(t). The detection chain comprises a PD, an electrical amplifier, and an electrical LPF.

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The photo-current generated at the receiver output is given by [14]

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

where K is the overall conversion factor (A/W) of the receiver and the symbol ⁎ denotes a convolution. For a PIN detector, K = R, where R is the detector responsivity. For an APD having a multiplication factor of M, K = MR. If we utilize an optical amplifier, we have K = GR, where G is the amplifier gain.

If we assume that the receiver noise is additive and Gaussian-distributed, the variance of the receiver noise can be written as

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

Here, $\sigma _{\textrm{dep}}^2(t )$ is the variance of the signal-dependent noise, typically expressed as [15,16]

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

where C is an appropriate constant. For the shot noise in a PIN detector, C = Rq, where q is the electric charge. For the shot noise in an APD, C = RqM²F_APD, where F_APD is the APD’s excess noise factor [14]. If an optical amplifier is utilized (with a PIN detector), the beat noise produced between the signal and the amplified spontaneous emission (ASE) noise is added to the existing shot noise. Thus, we write C = 2GR²N_ASE+ Rq, where N_ASE is the power spectral density of the ASE noise per polarization mode.

$\sigma _{\textrm{indep}}^2(t )$ is the variance of the signal-independent noise that can be expressed as

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

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. The thermal noise from the receiver and the dark current of the detector constitutes N(f). If an optical amplifier is utilized, the ASE-ASE beat noise is added to N(f). The variance of ASE-ASE noise can be expressed as $2{R^2}N_{\textrm{ASE}}^2({{B_o} - f} ),$ where B_o is the OBPF bandwidth [16].

As a measure of system performance, we define the Q-factor as

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

The subscripts 1 and 0 denote the photo-current and the standard deviation of the receiver noise generated by a single 1 and 0 bit, respectively.

3. Expression of receiver sensitivity

From Eq. (5), the received optical power for a 1 bit can be obtained as ${s_1}(t )= {P_1}{\bar{p}_b}(t )$, and for a 0 bit, it can be obtained as ${s_0}(t )= {P_0}{\bar{p}_b}(t )$. Thus, we write

(13)$$\begin{aligned} Q &= K({{P_1} - {P_0}} )({{{\bar{p}}_b} \ast h} )({{T_s}} )\\ &\quad \times {\left\{ {\left[ {C{P_1}({{{\bar{p}}_b} \ast {h^2}} )({{T_s}} )+ \int_0^\infty {N(f ){{|{H(f )} |}^2}df} } \right]} \right.^{1/2}}\\ &\quad{\left. {\quad \quad + {{\left[ {C{P_0}({{{\bar{p}}_b} \ast {h^2}} )({{T_s}} )+ \int_0^\infty {N(f ){{|{H(f )} |}^2}df} } \right]}^{1/2}}} \right\}^{ - 1}} \end{aligned}$$

where T_s is the optimum (sample) time that maximizes the Q-factor.

Assuming bits are equiprobable, we express P₁ in terms of the average optical power at the receiver input, P_avg, as

(14)$${P_1} = \frac{{2{\varepsilon _1}{\varepsilon _2}{P_{\textrm{avg}}}}}{{[{D({{\varepsilon_1} - 1} )+ 1} ]({{\varepsilon_2} + 1} )}}.$$

Using Eq. (7) and Eq. (14), we express the Q-factor in terms of P_avg as

(15)$$\begin{aligned} Q &= \frac{{2K{\varepsilon _1}({{\varepsilon_2} - 1} ){P_{\textrm{avg}}}({{{\bar{p}}_b} \ast h} )({{T_s}} )}}{{[{D({{\varepsilon_1} - 1} )+ 1} ]({{\varepsilon_2} + 1} )}}\\ & \times \left( {{{\left\{ {\frac{{2C{\varepsilon_1}{\varepsilon_2}{P_{\textrm{avg}}}({{{\bar{p}}_b} \ast {h^2}} )({{T_s}} )}}{{[{D({{\varepsilon_1} - 1} )+ 1} ]({{\varepsilon_2} + 1} )}} + \int_0^\infty {N(f ){{|{H(f )} |}^2}df} } \right\}}^{1/2}}} \right.\\ &{\left. {\quad + {{\left\{ {\frac{{2C{\varepsilon_1}{P_{\textrm{avg}}}({{{\bar{p}}_b} \ast {h^2}} )({{T_s}} )}}{{[{D({{\varepsilon_1} - 1} )+ 1} ]({{\varepsilon_2} + 1} )}} + \int_0^\infty {N(f ){{|{H(f )} |}^2}df} } \right\}}^{1/2}}} \right)^{ - 1}}. \end{aligned}$$

The receiver sensitivity for a given Q-factor is found as

(16)$$\begin{aligned} {P_{\textrm{rec}}} &= \frac{{Q[{D({{\varepsilon_1} - 1} )+ 1} ]({{\varepsilon_2} + 1} )}}{{K{\varepsilon _1}({{\varepsilon_2} - 1} )}}\left[ {\frac{{CQ({{\varepsilon_2} + 1} )({{{\bar{p}}_b} \ast {h^2}} )({{T_s}} )}}{{2K({{\varepsilon_2} - 1} ){{[{({{{\bar{p}}_b} \ast h} )({{T_s}} )} ]}^2}}}} \right.\\ &\left. {\quad + {{\left( {{{\left\{ {\frac{{CQ\sqrt {{\varepsilon_2}} ({{{\bar{p}}_b} \ast {h^2}} )({{T_s}} )}}{{K({{\varepsilon_2} - 1} ){{[{({{{\bar{p}}_b} \ast h} )({{T_s}} )} ]}^2}}}} \right\}}^2} + \frac{{\int_0^\infty {N(f ){{|{H(f )} |}^2}df} }}{{{{[{({{{\bar{p}}_b} \ast h} )({{T_s}} )} ]}^2}}}} \right)}^{{1 / 2}}}} \right]. \end{aligned}$$

3.1 Signal-dependent noise limited receiver

If signal-dependent noise dominates in the receiver, we can set N(f) = 0 in Eq. (16). By using the Cauchy-Schwarz inequality, we find

(17)$$\begin{aligned} {P_{\textrm{rec}}} &= \frac{{C{Q^2}[{D({{\varepsilon_1} - 1} )+ 1} ]({{\varepsilon_2} + 1} )}}{{2{K^2}{\varepsilon _1}{{\left( {\sqrt {{\varepsilon_2}} - 1} \right)}^2}}}\frac{{({{{\bar{p}}_b} \ast {h^2}} )({{T_s}} )}}{{{{[{({{{\bar{p}}_b} \ast h} )({{T_s}} )} ]}^2}}}\\ &\ge \frac{{C{Q^2}[{D({{\varepsilon_1} - 1} )+ 1} ]({{\varepsilon_2} + 1} )}}{{2{K^2}{\varepsilon _1}{{\left( {\sqrt {{\varepsilon_2}} - 1} \right)}^2}}}\frac{1}{{\int_{ - \infty }^\infty {{{\bar{p}}_b}(t )} dt}} = \frac{{C{Q^2}({{\varepsilon_2} + 1} )}}{{2{K^2}{T_b}{{\left( {\sqrt {{\varepsilon_2}} - 1} \right)}^2}}}. \end{aligned}$$

Equality holds if

(18)$$[{{h^2}(t )- a} ]{\bar{p}_b}(t )= 0$$

where a is an arbitrary constant. When utilizing an integrate-and-dump receiver, i.e.,

(19)$$h(t )= \left\{ \begin{array}{l} {1 / {{T_b}}},\quad 0 \le t \le {T_b}\\ 0,\quad \quad \textrm{otherwise} \end{array} \right.$$

Equation (18) can be satisfied regardless of ${\bar{p}_b}(t )$.

The minimum receiver sensitivity when signal-dependent noise dominates,

(20)$${P_{\textrm{dep}}} = \frac{{C{Q^2}({{\varepsilon_2} + 1} )}}{{2{K^2}{T_b}{{\left( {\sqrt {{\varepsilon_2}} - 1} \right)}^2}}},$$

is independent of D and ε₁. If ε₂ is infinite, P_dep = CQ²/(2K²T_b). Type-I and -II RZ signals become indistinguishable when the ER (both ε₁ and ε₂ for Type I and ε for Type II) is infinite. This equation is thus identical to Eq. (24) in [9] when ε = ∞. Rearranging this equation gives Q² = K(2P_depT_b)/(C/K), which is identical to the form of Eq. (20) in [10].

3.2 Signal-independent noise limited receiver

If signal-independent noise dominates, we can set C = 0 in Eq. (16). We first assume N(f) = N, i.e., the noise is white. Using $\int_0^\infty {{{|{H(f )} |}^2}df} = {{\int_{ - \infty }^\infty {{h^2}(t )dt} } / 2}$ and the Cauchy-Schwarz inequality, we find

(21)$$\begin{aligned} {P_{\textrm{rec}}} &= \frac{{\sqrt N Q[{D({{\varepsilon_1} - 1} )+ 1} ]({{\varepsilon_2} + 1} )}}{{\sqrt 2 K{\varepsilon _1}({{\varepsilon_2} - 1} )}}{\left\{ {\frac{{\int_{ - \infty }^\infty {{h^2}(t )dt} }}{{{{[{({{{\bar{p}}_b} \ast h} )({{T_s}} )} ]}^2}}}} \right\}^{{1 / 2}}}\\ & \ge \frac{{\sqrt N Q[{D({{\varepsilon_1} - 1} )+ 1} ]({{\varepsilon_2} + 1} )}}{{\sqrt 2 K{\varepsilon _1}({{\varepsilon_2} - 1} )}}{\left\{ {\frac{1}{{\int_{ - \infty }^\infty {\bar{p}_b^2(t )} dt}}} \right\}^{{1 / 2}}} \ge \frac{{\sqrt N Q[{D({{\varepsilon_1} - 1} )+ 1} ]({{\varepsilon_2} + 1} )}}{{K{{\{{2{T_b}[{D({\varepsilon_1^2 - 1} )+ 1} ]} \}}^{{1 / 2}}}({{\varepsilon_2} - 1} )}}. \end{aligned}$$

P_rec is minimized if the pulse shape, given D and ε₁, is

(22)$${\bar{p}_b}(t )= \left\{ \begin{array}{l} 1,\quad \quad \;0 \le t \le D{T_b}\\ {1 / {{\varepsilon_1}}},\quad D{T_b} < t \le {T_b}\\ 0,\quad \quad \textrm{otherwise} \end{array} \right.$$

i.e., when using a train of rectangular pulses, and the receiver response is matched to Eq. (22).

The minimum receiver sensitivity when signal-independent noise dominates,

(23)$${P_{\textrm{indep}}} = \frac{{\sqrt N Q[{D({{\varepsilon_1} - 1} )+ 1} ]({{\varepsilon_2} + 1} )}}{{K{{\{{2{T_b}[{D({\varepsilon_1^2 - 1} )+ 1} ]} \}}^{{1 / 2}}}({{\varepsilon_2} - 1} )}},$$

is maximized at D = 1, minimized at

(24)$$D = \frac{1}{{{\varepsilon _1} + 1}},$$

and converges to P_indep (D = 1) as the duty cycle approaches 0.

If ε₁ = ε₂ = ∞, P_indep = Q/K[DN/(2T_b)]^1/2. This is identical to Eq. (26) in [9] when ε = ∞ and N(f) = N. The sensitivity of a receiver limited by white signal-dependent noise decreases (or improves) monotonically with the decrease of the duty cycle, which supports the analysis reported in [10].

We now generalize P_indep for an arbitrary N(f). The frequency response of the receiver matched to ${\bar{p}_b}(t )$ can be approximated as

(25)$$H(f )\approx \left\{ \begin{array}{l} 1,\quad \quad \quad \quad \quad \;\;0 \le f \le {1 / {(2{T_b})}}\\ \frac{{D({{\varepsilon_1} - 1} )}}{{D({{\varepsilon_1} - 1} )+ 1}},\quad {1 / {(2{T_b})}} < f \le {1 / {(2D{T_b})}}\\ 0,\quad \quad \quad \quad \quad \;\;\textrm{otherwise} \end{array} \right.$$

Thus, the noise variance for an arbitrary N(f) can be approximated as

(26)$$\int_0^\infty {N(f ){{|{H(f )} |}^2}df} \approx \int_0^{{1 / {(2{T_b})}}} {N(f )df} + {\left[ {\frac{{D({{\varepsilon_1} - 1} )}}{{D({{\varepsilon_1} - 1} )+ 1}}} \right]^2}\int_{{1 / {(2{T_b})}}}^{{1 / {(2D{T_b})}}} {N(f )df} .$$

Using Eq. (25), Eq. (26), and $N = {{\int_0^\infty {N(f ){{|{H(f )} |}^2}df} } / {\int_0^\infty {{{|{H(f )} |}^2}df} }}$, we rewrite Eq. (23) as

(27)$$\begin{aligned} {P_{\textrm{indep}}} &= \frac{{Q{{[{D({{\varepsilon_1} - 1} )+ 1} ]}^2}({{\varepsilon_2} + 1} )}}{{K[{D({\varepsilon_1^2 - 1} )+ 1} ]({{\varepsilon_2} - 1} )}}\\ &\quad\times {\left\{ {\int_0^{{1 / {(2{T_b})}}} {N(f )df} + {{\left[ {\frac{{D({{\varepsilon_1} - 1} )}}{{D({{\varepsilon_1} - 1} )+ 1}}} \right]}^2}\int_{{1 / {(2{T_b})}}}^{{1 / {(2D{T_b})}}} {N(f )df} } \right\}^{{1 / 2}}}. \end{aligned}$$

3.3 Generalized form

When we consider both the signal-dependent and signal-independent noise in the receiver, it is difficult to minimize P_rec. The conditions for minimization depend upon the parameters C and N(f), making them hard to generalize. For example, if C >> N(f) (signal-dependent noise dominates), an integrate-and-dump receiver minimizes the sensitivity. However, it is Eq. (22) and a matched receiver that minimize the sensitivity if C << N(f) (signal-independent noise dominates). Thus, we resort to an approximation.

First, by substituting Eq. (19) and Eq. (22) into Eq. (16), we obtain

(28)$${P_{\textrm{ID}}} = \frac{{Q({{\varepsilon_2} + 1} )}}{{K({{\varepsilon_2} - 1} )}}\left( {\frac{{CQ({{\varepsilon_2} + 1} )}}{{2K{T_b}({{\varepsilon_2} - 1} )}} + {{\left\{ {{{\left[ {\frac{{CQ\sqrt {{\varepsilon_2}} }}{{K{T_b}({{\varepsilon_2} - 1} )}}} \right]}^2} + \int_0^{{1 / {(2{T_b})}}} {N(f )df} } \right\}}^{{1 / 2}}}} \right)$$

where we also utilize Eq. (26), evaluated at D = 1. Since the output of the integrate-and-dump receiver depends upon the signal energy rather than its peak power, P_ID is independent of the duty cycle.

Next, by substituting Eq. (22), an h(t) matched to Eq. (22), and Eq. (26) into Eq. (16), we obtain

(29)$$\begin{aligned} {P_{\textrm{matched}}} &= \frac{{Q[{D({{\varepsilon_1} - 1} )+ 1} ][{D({\varepsilon_1^3 - 1} )+ 1} ]({{\varepsilon_2} + 1} )}}{{K{{[{D({\varepsilon_1^2 - 1} )+ 1} ]}^2}({{\varepsilon_2} - 1} )}}\\ &\quad \times \left[ {\frac{{CQ({{\varepsilon_2} + 1} )}}{{2K{T_b}({{\varepsilon_2} - 1} )}} + \left( {{{\left[ {\frac{{CQ\sqrt {{\varepsilon_2}} }}{{K{T_b}({{\varepsilon_2} - 1} )}}} \right]}^2} + \frac{{{{[{D({{\varepsilon_1} - 1} )+ 1} ]}^2}{{[{D({\varepsilon_1^2 - 1} )+ 1} ]}^2}}}{{{{[{D({\varepsilon_1^3 - 1} )+ 1} ]}^2}}}} \right.} \right.\\ &\left. {{{\left. {\quad \times \left\{ {\int_0^{{1 / {(2{T_b})}}} {N(f )df} + {{\left[ {\frac{{D({{\varepsilon_1} - 1} )}}{{D({{\varepsilon_1} - 1} )+ 1}}} \right]}^2}\int_{{1 / {(2{T_b})}}}^{{1 / {(2D{T_b})}}} {N(f )df} } \right\}} \right)}^{{1 / 2}}}} \right]. \end{aligned}$$

Finally, we approximate the minimum receiver sensitivity by

(30)$${P_{\textrm{dep + indep}}} = \min ({{P_{\textrm{ID}}},{P_{\textrm{matched}}}} ).$$

If C >> N(f), P_{dep + indep} = P_dep, whereas if C << N(f), P_{dep + indep} = P_indep. It is worth noting that for D = 1 (which corresponds to an NRZ signal), an integrate-and-dump receiver is a receiver matched to the signal [P_ID = P_matched (D = 1)]. Thus, P_{dep + indep} ≤ P_{dep + indep} (D = 1), i.e., the receiver sensitivity Type-I RZ signals are upper bounded.

4. Experiment and discussion

We verify our theory through an experiment. The experimental setup is shown in Fig. 3. The EAM1 driven by the arbitrary waveform generator 1 (AWG1) carves out a train of Gaussian pulses from the continuous-wave light emitted by the laser diode (LD). Since the maximum sampling rate of the AWG1 is 64 Gsample/s, the narrowest pulse width we can generate in our experiment is limited to 40 ps. To experiment using a wide range of duty cycles, we set the pulse repetition rate to 62.5 MHz, achieving duty cycles as low as 0.0025. The EAM2 driven by the AWG2 imprints NRZ data on the optical pulse train at a rate of 62.5 Mb/s via intensity modulation. The generated Type-I RZ signals are sent to either a PIN receiver or an optically pre-amplified receiver. The relevant receiver parameters are summarized in Table 1.

Fig. 3. Experimental setup. The setup consists of (a) a transmitter with a variable optical attenuator connected to either (b) a PIN receiver or (c) an optically pre-amplified receiver.

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Table 1. Summary of receiver parameters

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4.1 PIN receiver

The schematic diagram of the PIN receiver used in the experiment is shown in Fig. 3(b). The received optical signal is detected by a PIN-TIA detector. Before sampling the detected signal via a real-time oscilloscope, we amplify the signal using an electrical amplifier to reduce the adverse effects of quantization noise from the scope. After the signal is sampled, it is processed offline; the signal processing consists of low-pass filtering using a 5^th-order Bessel filter, resampling, and decision. Finally, the bit-error ratio (BER) is obtained through direct error counting. The LPF bandwidth is chosen such that it minimizes the BER per duty cycle.

When utilizing a PIN detector, we set K = R and C = Rq. To find N(f), we measured the noise spectrum after the electrical amplifier. The measured result is shown in Fig. 4 in [9]. The measured frequency range is from 400 kHz to 30 GHz. Beyond 30 GHz, we assume that the noise is white. The modeled N(f) is given in Table 1.

Fig. 4. Sensitivity of the PIN receiver versus the inverse of the duty cycle for three pairs of (ε₁, ε₂). Insets show the optimum LPF bandwidth, B_h, normalized by the data rate, R_b, for each (ε₁, ε₂).

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The measured and theoretical sensitivities (for a BER of 10⁻³) of the PIN receiver versus D⁻¹ are shown in Fig. 4. Also plotted in Fig. 4 are the simulation results obtained by numerically solving Eq. (16) using a Gaussian pulse shape for p(t), i.e.,

(31)$${\bar{p}_b}(t )= \left\{ \begin{array}{l} \frac{1}{{{\varepsilon_1} + 1}}\left[ {{\varepsilon_1}\exp \left( { - \frac{{{{({t - {{{T_b}} / 2}} )}^2}}}{{2{\sigma^2}}}} \right) + 1} \right],\quad 0 \le t \le {T_b}\\ 0,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\textrm{otherwise} \end{array} \right.$$

where σ = DT_b /(2π)^1/2, and a 5^th-order Bessel filter impulse response for h(t). The bandwidth of h(t) is optimized for receiver sensitivity and the appropriate parameters are substituted from Table 1. Note that the numerical simulation does not support 0.5 ≤ D ≤ 1. This is because the pulses having duty cycles larger than 0.5 do not “return to zero” during a single bit duration.

We can see that for different duty cycles and ERs, the theoretical values are consistent with both the experimental and simulation values within a margin of 1.5 dB. As mentioned in Section 3.2, the receiver sensitivity is optimized at a certain duty cycle when signal-independent noise limits the system performance. We can see that the optimum duty cycle depends on ε₁ but not on ε₂; when (ε₁, ε₂) = (13 dB, 6 dB) and (ε₁, ε₂) = (13 dB, 10 dB) the optimum is at D = 0.045, whereas when (ε₁, ε₂) = (17 dB, 10 dB) the optimum decreases to D = 0.026. Increasing ε₂ from 6 to 10 dB simply brings a sensitivity improvement of 1.4 dB for all duty cycles.

The insets of Fig. 4 show the optimum LPF bandwidths (with respect to receiver sensitivity) versus D⁻¹ for different (ε₁, ε₂). The bandwidths are normalized by the data rate. We see from the insets that the bandwidth shifts abruptly from > 10²R_b to 0.6R_b at a certain duty cycle. This causes a corresponding discontinuity in the slope of the receiver sensitivity curves. Defining the point of discontinuity as D*, D* = 10⁻³ when ε₁= 13 dB and D* = 1.7 × 10⁻⁴ when ε₁= 17 dB.

As the duty cycle is decreased, we see from see Eq. (14) that the peak power increases monotonically. However, if ε₁ is finite, the peak power converges to a finite value. By contrast, the spectral width of the signal, and hence the required bandwidth to properly receive it increases without bound. Thus, reducing the duty cycle below a certain value, i.e., an optimum duty cycle, leads to worse receiver sensitivities, since the increase in noise variance brought by the increase in receiver bandwidth eventually exceeds the increase in peak power.

To explain the discontinuity in slope, we illustrate in Fig. 5, the photo-current waveform for Type-I and -II RZ signals having a duty cycle of 0.025. When these signals pass through a receiver having a bandwidth of 0.6R_b, which is much narrower than the signal’s spectral width, the pulses disappear. Despite the absence of these pulses, we can still distinguish between “on” and off states for Type-I RZ signals. This is because the pedestal also carries information; the data are imprinted on the pulse train via NRZ coding. In other words, one may give up the high signal peak power in favor of reduced noise variance, and always receive the Type-I RZ signal as an NRZ signal by using a narrow bandwidth of say, 0.6R_b. It is worth noting that when doing so, the sensitivity becomes independent of D and ε₁, which can be inferred from Fig. 4. Type-II RZ signals, on the other hand, do not share this feature. The absence of the pulse is simply the absence of data. Thus, a large receiver bandwidth is necessary to decode data for low-duty-cycle Type-II RZ signals. This is why the sensitivity of Type-I RZ signals is upper-bounded and experiences an abrupt change in slope, whereas the sensitivity of Type-II RZ signals diverges as the RZ pulse approaches an impulse.

Fig. 5. Generated photo-current for a received (a) Type-I and (b) -II RZ signal, both having a duty cycle of 0.025. The gray line illustrates when the receiver bandwidth is sufficiently large, and the black line illustrates when the receiver bandwidth is 0.6 times the data rate (narrow).

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Figure 6 shows the theoretical sensitivity of the PIN receiver as a function of ε₁ and ε₂ for different duty cycles. We can see that a higher ER leads to better receiver performance. But the improvement in receiver sensitivity is insignificant when the ER is already quite high; if D = 0.25, the sensitivity can be regarded as being independent of ε₁ (ε₂) for ε₁ > 30 dB (ε₂ > 20 dB). By comparing Fig. 6(a) and (b), three changes in receiver sensitivity are apparent upon reducing the duty cycle. Firstly, a higher ε₁ is required to regard the receiver sensitivity as being independent of ε₁; it is ε₁ > 30 dB when D = 0.25, but for D = 0.025, it is ε₁ > 40 dB. Secondly, a lower receiver sensitivity can be achieved; for D = 0.25, the receiver sensitivity cannot be made lower than −36 dBm, when −41 dBm is achievable for D = 0.025. Finally, the receiver sensitivity becomes more susceptible to variations in ε₁; when ε₂ = 20 dB, increasing ε₁ from 10 to 20 dB brings a sensitivity gain of 2 dB for D = 0.25, whereas for D = 0.025, the same increase in ε₁ brings a near 6-dB improvement.

Fig. 6. Theoretical sensitivity of the PIN receiver as a function of ε₁ and ε₂ when (a) D = 0.25 and (b) D = 0.025. The contour lines are separated by 2 dB.

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4.2 Optically pre-amplified receiver

Next, the schematic diagram of the optically pre-amplified receiver is shown in Fig. 3(c). 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 4.1 dB. An OBPF is placed after each stage to reduce the ASE noise from the preceding amplifier. To avoid truncating the power of the RZ signal, we set the bandwidth of the OBPF1 and OBPF2 to 3.2 and 0.5 nm, respectively. The amplified optical signal is then detected by a PIN detector, electrically amplified, and sampled by using a real-time oscilloscope. The offline signal processing is identical to the one done for the PIN receiver.

For the optically pre-amplified receiver, we set K = GR. Since the optical gain from the two-stage EDFA is sufficiently large, we assume that the signal-ASE beat noise dominates the shot noise. Thus, we write C = 2GR²N_ASE. Likewise, we assume that the ASE-ASE beat noise dominates the thermal noise and write, $N(f) = 2{R^2}N_{\textrm{ASE}}^2({{B_o} - f} )$.

Figure 7 shows the receiver sensitivities (for a BER of 10⁻³) versus D⁻¹. The simulation results shown in Fig. 7 are obtained by numerically solving Eq. (16) using Eq. (31) and a 5^th-order Bessel filter impulse response for h(t). Again, the bandwidth of h(t) is optimized for receiver sensitivity and the parameters are taken from Table 1.

Fig. 7. Sensitivity of the optically pre-amplified receiver versus the inverse of the duty cycle for three pairs of (ε₁, ε₂). Insets show the LPF bandwidth, B_h, normalized by the data rate, R_b, for each (ε₁, ε₂).

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The theoretical values are consistent with both the experimental and simulation values within a margin of 1 dB. We find that the receiver sensitivity curves follow a similar trend as that of the PIN receiver. The receiver sensitivity is optimized at a certain duty cycle, and there exists a discontinuity in the slope of the curves. An abrupt shift in the optimum LPF bandwidth is responsible for the discontinuity, as evidenced by the insets of Fig. 7.

Due to the increased contribution of signal-dependent noise, however, the sensitivity curves do exhibit some notable differences when compared to that of the PIN receiver. Namely, both the optimum duty cycle and D* are now functions of ε₂; when (ε₁, ε₂) = (13 dB, 6 dB) the optimum is at D = 0.11 and D* = 0.03, when (ε₁, ε₂) = (13 dB, 10 dB), the optimum is at D = 0.08 and D^*= 0.018, and lastly, when (ε₁, ε₂) = (17 dB, 10 dB) the optimum is at D = 0.05 and D* = 0.005. This is because increasing ε₂ reduces the required average optical power at the receiver, which in turn reduces the contribution of signal-dependent noise.

Additionally, for the same (ε₁, ε₂), the optimum duty cycle and D* are larger in comparison. Since an increase in signal peak power and receiver bandwidth increases the variance of signal-dependent noise, reducing the duty cycle begins to degrade the sensitivity at much larger duty cycles compared to when the receiver is dominated by signal-independent noise. Thus, the larger the contribution of signal-dependent noise, the larger the optimum duty cycle and D*. If signal-dependent noise dominates the receiver, both the optimum duty cycle and D* become 1, i.e., the receiver sensitivity becomes independent of the duty cycle.

Figure 8(a) and (b) show the theoretical sensitivity of the optically pre-amplified receiver as a function of ε₁ and ε₂ for two duty cycles. The findings are identical to those of the PIN receiver. If ε₁ (ε₂) is sufficiently high, the receiver sensitivity can be regarded as being independent of ε₁ (ε₂). Three changes in the receiver sensitivity are notable when decreasing the duty cycle. Firstly, a higher ε₁ is required to ignore its impact. Secondly, a lower sensitivity can be achieved. And finally, the sensitivity becomes more susceptible to changes in ε₁. In Fig. 8(c), we show the sensitivity of the optically pre-amplified receiver when N(f) = 0, i.e., the signal-independent noise is negligible. We can see that the receiver sensitivity is independent of ε₁ (and also D), as mentioned in Section 3.1.

Fig. 8. Theoretical sensitivity of the optically pre-amplified receiver as a function of ε₁ and ε₂ when (a) D = 0.25, (b) D = 0.025, and (c) N(f) = 0. The contour lines are separated by 2 dBm.

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5. Concluding remarks

We have developed a theory on the receiver sensitivity of Type-I RZ signals having finite extinction ratios and verified it through an experiment. We have presented an approximation for the minimum (or best-case) receiver sensitivity affected by both signal-dependent and signal-independent noise. The receiver sensitivity is independent of the duty cycle if signal-dependent noise limits the system performance. Otherwise, there exists an optimum duty cycle for receiver sensitivity. The receiver sensitivity becomes more susceptible to changes in the extinction ratio as the duty cycle is decreased.

We have shown that low-duty-cycle RZ signals typically require receiver bandwidths much larger than the data rate. However, in practical scenarios where the data rate is high, it might be challenging to obtain sufficiently large receiver bandwidths. In the Appendix, we show that when utilizing a receiver bandwidth larger than 30% of the optimum value, the sensitivity penalty is less than 2 dB, regardless of the receiver being used.

Finally, the receiver sensitivity and the optimum receiver conditions [assuming N(f) = N] of Type-I and -II RZ signals are summarized in Tables 2 and 3.

Table 2. Summary of receiver sensitivity

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Table 3. Summary of optimum receiver conditions

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Appendix

In this section, we analyze the sensitivity penalty when the receiver bandwidth is narrower than the optimum value. The penalty is obtained through numerical simulation. For this purpose, we solve Eq. (16) using Eq. (31) and a 5^th-order Bessel filter impulse response for h(t). The receiver parameters are listed in Table 1. We ignore the effects of inter-symbol interference from narrow filtering by assuming that the bandwidth of h(t) is larger than the data rate, R_b [7].

The sensitivity penalty as a function of filter bandwidth, B_h, normalized by the optimum filter bandwidth, B_opt, is plotted in Fig. 9. Since the optimum bandwidth is larger for lower duty cycles, the bandwidth can be reduced further before it reaches R_b. This opens up the potential for larger penalties.

Fig. 9. Sensitivity penalty versus B_h/B_opt obtained via simulation for (a) the PIN and (b) the optically pre-amplified receivers. B_opt is the optimum filter bandwidth that minimizes the receiver sensitivity. The curves are cut below the point where B_h becomes smaller than R_b.

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For the PIN receiver in Fig. 9(a), we can see that the penalty curves for varying duty cycles exhibit a similar trend. Slight deviations are observed between the curves, where larger duty cycles experience larger penalties for the same B_h/B_opt. These deviations should be attributed to the fact that N(f) is not white. If we assume the noise is white [i.e., N(f) = N], the decrease in noise variance from a decrease in receiver bandwidth would be independent of the duty cycle. Additionally, B_opt = k/D, where k is a constant. Thus, when decreasing the receiver bandwidth from B_opt, the photo-current after the filter would decrease by a same factor for all duty cycles. This counter-example confirms that it is the non-white nature of noise that makes the penalty depend upon the duty cycle.

We can see that the penalty increases with ε₁. This is because signals having small ε₁ have poor performance even at the optimum bandwidth, thus decreasing the bandwidth does not lead to a large degradation in performance. If ε₁ is finite, we can see a decrease in slope as B_h/B_opt is reduced. As mentioned in Section 4.1, one may receive the Type-I RZ signal as an NRZ signal when the receiver bandwidth is narrow. Thus, if ε₁ is poor and D is low, utilizing narrow bandwidths becomes beneficial in terms of performance. The values of ε₂ are not specified in the plot, since the penalty curves are independent of ε₂.

For the optically pre-amplified receiver in Fig. 9(b), the findings are similar to those of the PIN receiver, except that the penalty curves are also a function of ε₂. The sensitivity penalty is smaller than the PIN receiver for the same B_h/B_opt. since the variance of signal-dependent noise decreases faster than the variance of signal-independent noise as the bandwidth is decreased.

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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6. M. Pfennigbauer, M. M. Strasser, M. Pauer, and P. J. Winzer, “Dependence of optically preamplified receiver sensitivity on optical and electrical filter bandwidths—Measurement and simulation,” IEEE Photon. Technol. Lett. 14(6), 831–833 (2002). [CrossRef]

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PIN receiver	Optically pre-amplified receiver
Data rate = 62.5 Mb/s Responsivity, R = 0.67 A/W	Data rate = 62.5 Mb/s
	Responsivity, R = 0.62 A/W
	EDFA gain, G = 52 dB
	EDFA noise figure = 4.1 dB
	OBPF2 bandwidth, B_o = 0.5 nm
T_b = 1.6 × 10⁻⁸ s	T_b = 1.6 × 10⁻⁸ s
K = 0.67 A/W	K = 9.83 × 10⁴ A/W
C = 1.07 × 10⁻¹⁹ A²s/W	C = 2.77 × 10⁻⁹ A²s/W
N(f) = 1.7 × 10⁻²² + 1.5 × 10⁻¹⁸ f ^−1/2 A²/Hz	N(f) = 2.49 × 10^−17–3.98 × 10⁻²⁸ f A²/Hz
Q = 3.09 (corresponds to a BER of 10⁻³)	Q = 3.09 (corresponds to a BER of 10⁻³)

	Type I	Type II
Signal-dependent noise limited receiver	Independent of D	Increases monotonically with the decrease of D,
Signal-dependent noise limited receiver	Independent of D	diverges as D → 0
Signal-independent noise limited receiver	Minimized at D = 1 / (ε₁ + 1), converges as D → 0	Minimized at D = 2 / (ε − 1), diverges as D → 0

PIN receiver	Optically pre-amplified receiver
Data rate = 62.5 Mb/s Responsivity, R = 0.67 A/W	Data rate = 62.5 Mb/s
	Responsivity, R = 0.62 A/W
	EDFA gain, G = 52 dB
	EDFA noise figure = 4.1 dB
	OBPF2 bandwidth, B_o = 0.5 nm
T_b = 1.6 × 10⁻⁸ s	T_b = 1.6 × 10⁻⁸ s
K = 0.67 A/W	K = 9.83 × 10⁴ A/W
C = 1.07 × 10⁻¹⁹ A²s/W	C = 2.77 × 10⁻⁹ A²s/W
N(f) = 1.7 × 10⁻²² + 1.5 × 10⁻¹⁸ f ^−1/2 A²/Hz	N(f) = 2.49 × 10^−17–3.98 × 10⁻²⁸ f A²/Hz
Q = 3.09 (corresponds to a BER of 10⁻³)	Q = 3.09 (corresponds to a BER of 10⁻³)

	Type I	Type II
Signal-dependent noise limited receiver	Independent of D	Increases monotonically with the decrease of D,
Signal-dependent noise limited receiver	Independent of D	diverges as D → 0
Signal-independent noise limited receiver	Minimized at D = 1 / (ε₁ + 1), converges as D → 0	Minimized at D = 2 / (ε − 1), diverges as D → 0

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

Abstract

1. Introduction

2. Signal and receiver model

3. Expression of receiver sensitivity

3.1 Signal-dependent noise limited receiver

3.2 Signal-independent noise limited receiver

3.3 Generalized form

4. Experiment and discussion

4.1 PIN receiver

4.2 Optically pre-amplified receiver

5. Concluding remarks

Appendix

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Tables (3)

Equations (31)

Optics Express