Question 5 of 6: Receiver Front-End Choice and the Erbium-Doped Fiber Amplifier

Question 5: Receiver Front-End Choice and the Erbium-Doped Fiber Amplifier (equal value)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Find. (a) The unequalized 3 dB bandwidth of each front-end option, which to choose, and what an equalizer buys option B. (b) EDFA operating principle, characteristics, and its advantages/disadvantages in a WDM system.

Approach. Each front end is, without equalization, a single-pole RC low-pass whose resistance is the total resistance the detector's current source sees: $R_b\parallel R_{in}$ for the plain high-impedance amplifier, and the much smaller feedback-reduced input resistance $R_f/(1+A_{ol})$ for the transimpedance amplifier.

Part (a)

1. Option B — high-input-impedance front end. The detector's bias resistor and the amplifier's input resistance appear in parallel to the signal current: $$R_{eq,B}=\frac{R_bR_{in,B}}{R_b+R_{in,B}}=\frac{(680\times10^3)(1\times10^6)}{680\times10^3+1\times10^6}=405\ \text{k}\Omega.$$ $$f_{3dB,B}=\frac{1}{2\pi R_{eq,B}C_T}=\frac{1}{2\pi(4.05\times10^5)(5\times10^{-12})}=\boxed{78.6\ \text{kHz}}.$$
2. Option A — transimpedance amplifier. Shunt-shunt feedback around the open-loop gain $A_{ol}$ reduces the resistance the detector sees to approximately $$R_{eq,A}=\frac{R_f}{1+A_{ol}}=\frac{220\times10^3}{1+500}=439\ \Omega,$$ $$f_{3dB,A}=\frac{1}{2\pi R_{eq,A}C_T}=\frac{1}{2\pi(439)(5\times10^{-12})}=\boxed{72.5\ \text{MHz}}.$$
3. Choice. Without any equalization, the TIA (option A) delivers roughly 900× more bandwidth (72.5 MHz vs. 78.6 kHz) because feedback divides the effective input resistance by $(1+A_{ol})$. On noise, feedback lowers the input impedance without adding the noise of a physical 439 Ω resistor. The TIA's thermal-noise current is set by the feedback resistor in parallel with the bias resistor, $R_f\parallel R_b=166$ kΩ. Option B's is set by $R_b\parallel R_{in}=405$ kΩ. The TIA's mean-square noise current $4kTB/R$ is therefore about $405/166=2.4$ times larger (about 3.9 dB), a modest penalty that is far outweighed by not needing post-detection equalization at all — option A is the correct choice for any link needing more than a few hundred kHz of bandwidth. Option B's huge $R_b\parallel R_{in}$ gives excellent noise performance (small $4kT/R_{eq}$) but only inside its very narrow unequalized bandwidth.
4. Benefit of adding an equalizer after amplifier B. An equalizer with a rising ($+20$ dB/decade) response beyond $f_{3dB,B}$ can flatten the receiver's overall frequency response well past 78.6 kHz, recovering usable bandwidth from the high-$R$, low-noise front end without giving up its excellent low-frequency SNR. The cost is that the equalizer boosts noise at high frequencies right along with the signal, so the net SNR improvement is smaller than the raw bandwidth gain suggests — a classic high-impedance-front-end/ integrate-and-differentiate trade-off, whereas the TIA achieves its bandwidth without paying this noise-boost penalty at all.

Part (b) — erbium-doped fiber amplifier (EDFA)

An EDFA is a length of silica fiber whose core is doped with trivalent erbium ions (Er$^{3+}$), optically pumped (typically at 980 nm or 1480 nm, via a wavelength-division coupler) into an excited metastable state. Signal photons near 1550 nm — matching the Er$^{3+}$ transition from the metastable $^4I_{13/2}$ level down to the ground $^4I_{15/2}$ level — stimulate emission from the population-inverted ions as they pass through, providing gain directly in the optical domain with no optical-electrical-optical conversion. Key characteristics: gain typically 20–30 dB over a usable band of roughly 30–40 nm (conventional C-band, extendable with L-band doping profiles), noise figure typically 4–6 dB (set by amplified spontaneous emission, ASE), and gain that is largely polarization- and bit-rate-independent (a slow, population-based gain medium, unlike a semiconductor optical amplifier). In a WDM system, the chief advantages are that a single EDFA amplifies all wavelength channels simultaneously (no per-channel electronics), is transparent to modulation format and bit rate, and introduces comparatively low noise and no electronic bandwidth bottleneck. The main disadvantages are gain non-uniformity across the amplification band (requiring gain-flattening filters in long cascades), gain saturation and cross-gain effects when channels are added/dropped (transient power excursions on surviving channels), accumulated ASE noise limiting cascadability, and the fact that EDFAs only operate in the 1530–1610 nm window (no equivalent standard rare-earth amplifier for the 850 nm or 1310 nm bands).