Multiplexing: FDM, TDM, and WDM

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1. Overview of Multiplexing

Multiplexing is the technique of combining multiple independent data signals from multiple sources into a single composite signal transmitted over a single shared high-capacity physical link.

  Input Line 1 ──\                                            /── Output Line 1
  Input Line 2 ───\  +-------------+        +---------------+  /─── Output Line 2
  Input Line 3 ────+ | MULTIPLEXER | =====> | DEMULTIPLEXER | +──── Output Line 3
  Input Line n ───/  |   (MUX)     |  Link  |    (DEMUX)    |  \─── Output Line n
                   +-------------+        +---------------+

Multiplexing strategies are classified based on the physical dimension utilized:

                              MULTIPLEXING
                                   |
         +-------------------------+-------------------------+
         |                                                   |
      ANALOG                                              DIGITAL
         |                                                   |
  +------+------+                                            |
  |             |                                            |
 FDM           WDM                                          TDM
(Frequency)  (Optical Wavelength)                            |
                                              +--------------+--------------+
                                              |                             |
                                        Synchronous TDM               Statistical TDM

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2. Frequency Division Multiplexing (FDM)

FDM is an analog multiplexing technique applied when the physical transmission link bandwidth exceeds the combined bandwidth of the signals to be transmitted.

 Baseband Channel 1 (0-4 kHz) ──> [ Modulator (fc1) ] ──\
 Baseband Channel 2 (0-4 kHz) ──> [ Modulator (fc2) ] ────+---> [ Composite FDM Signal ]
 Baseband Channel 3 (0-4 kHz) ──> [ Modulator (fc3) ] ──/

• Operating Principle: Each baseband signal modulates a distinct high-frequency carrier frequency ($f_{c1}, f_{c2}, \dots$).
• Guard Bands: Unused frequency intervals inserted between adjacent channels to eliminate spectral overlap and prevent crosstalk.
• Bandwidth Calculation:

• Applications: AM/FM radio, Cable TV (CATV), analog telephone FDM hierarchies.

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3. Time Division Multiplexing (TDM)

TDM is a digital multiplexing technique where multiple digital streams share the same physical link by taking turns in time. The entire bandwidth of the link is allocated to each channel for a fraction of time called a Time Slot.

A. Synchronous TDM

• Fixed Round-Robin Slots: Every input line has a dedicated time slot in every frame, whether it has data to transmit or is idle.
• Empty Slot Wastage: If an input device is idle, its assigned slot is transmitted blank, reducing channel efficiency.
• Framing Bit: 1 synchronization bit is added per frame to synchronize receiver demultiplexers.
• Classic Example (T-1 Carrier):
  • 24 digitized voice channels (PCM 8-bit, 8000 samples/sec).
  • Frame length: $(24 \times 8) + 1 = 193\text{ bits}$.
  • Frame rate: $8{,}000\text{ frames/sec}$.
  • Output Bit Rate: $193 \times 8{,}000 = 1.544\text{ Mbps}$.

B. Statistical (Asynchronous) TDM

• Dynamic Slot Allocation: Slots are allocated dynamically on-demand only to active input channels.
• Address Overhead: Because slots are not tied to fixed positions, each slot must carry an explicit channel address header.
• High Efficiency: Overcomes idle slot wastage; aggregate input rates can exceed physical link bandwidth ($\sum R_{\text{input}} > R_{\text{link}}$).

  Synchronous TDM:
  Frame: [ Ch 3 |  EMPTY  | Ch 1 ]   (Wastes slot when Ch 2 is idle)

  Statistical TDM:
  Frame: [ Addr 3 | Data 3 ][ Addr 1 | Data 1 ]  (No empty slots; includes address)

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4. Wavelength Division Multiplexing (WDM)

WDM is an optical multiplexing technique designed for fiber optic communications. It combines multiple light beams of different wavelengths ($\lambda$) (optical frequencies) onto a single optical fiber.

 Laser 1 (λ1) ──\   +--------+                                  +--------+   /─── λ1 ──> Sensor 1
 Laser 2 (λ2) ───+->| PRISM  | ===== [ Single Fiber Strand ] ===>| PRISM  |──+──── λ2 ──> Sensor 2
 Laser 3 (λ3) ──/   |  MUX   |                                  | DEMUX  |   \─── λ3 ──> Sensor 3
                    +--------+                                  +--------+

• Fundamental Wave Law: $\lambda = \frac{c}{f}$ (speed of light $c = 3 \times 10^8\text{ m/s}$).
• Optical Combining & Splitting: Prisms or diffraction gratings combine light beams at the transmitter and refract them into separate wavelengths at the receiver based on Snell's Law.
• CWDM vs. DWDM:
  • Coarse WDM (CWDM): Wide channel spacing ($20\text{ nm}$), up to 18 channels, uncooled lasers, metro networks ($\leq 50\text{ km}$).
  • Dense WDM (DWDM): Ultra-narrow spacing ($\leq 0.8\text{ nm}$ / $100\text{ GHz}$ grid), up to $160+$ channels, cooled lasers, EDFA optical amplifiers, trans-oceanic multi-terabit backbones.

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5. Summary Comparison of FDM, TDM, and WDM

Parameter	FDM	TDM	WDM
Domain	Frequency ($f$ in Hz)	Time ($t$ in seconds)	Optical Wavelength ($\lambda$ in nm)
Signal Nature	Analog	Digital	Optical (Light pulses)
Medium	Copper (Twisted Pair, Coax), Air	Copper, Microwave, Fiber	Strictly Optical Fiber
Interference Prevention	Guard Bands (Hz)	Guard Times / Framing Bits	Channel Spacing ($\Delta\lambda$)
Slot / Band Allocation	Continuous simultaneous frequencies	Dedicated / Dynamic time slices	Continuous simultaneous optical wavelengths
Typical Data Rate	Kilobits to Megabits/sec	Megabits to Gigabits/sec	Multi-Terabits per second (Tbps)

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6. Key Exam Questions & Numericals

1. Calculate the bandwidth required for an FDM system multiplexing 6 voice channels (4 kHz each) with 500 Hz guard bands.

2. Why does Statistical TDM outperform Synchronous TDM in data networking?
   • Computer traffic is bursty, with long periods of silence. Synchronous TDM wastes bandwidth transmitting blank slots for idle devices, whereas Statistical TDM allocates slots only to nodes with active data.
3. What is the optical counterpart of FDM?
   • WDM (Wavelength Division Multiplexing) is the optical analog of FDM, operating at optical light frequencies ($> 190\text{ THz}$) using wavelengths in nanometers.