Multiplexing: FDM, TDM & Their Successors

Multiplexing makes one physical medium behave like many logical channels. The trick is always the same: map each flow into a region of some resource space, combine the signals, then give the receiver enough structure to separate them again. The resource may be time, frequency, optical wavelength, code, polarization, or spatial mode.

Sharing is never free. Fixed partitions strand unused capacity; tight partitions require sharper filters or better synchronization; flexible scheduling needs queues, control signalling, and fairness. A multiplex is designed by choosing which inefficiency and failure mode the system can tolerate.

Separation is a receiver property #

Write a composite signal as

$$x(t)=\sum_{i=0}^{K-1}a_i\phi_i(t).$$

If the basis waveforms are orthogonal over the receiver's observation interval,

$$\int\phi_i(t)\phi_j^*(t)\,dt=0\qquad(i\ne j),$$

a correlator or filter bank can recover each coefficient without energy from the others. FDM approximates this with non-overlapping filters. TDM makes users non-overlapping in time. OFDM permits overlapping spectra but chooses subcarriers that are exactly orthogonal over the useful symbol interval.

Real systems preserve that separation with guards: guard bands, guard time, cyclic prefixes, pulse shaping, code correlation, or spatial nulls. Drift, filter leakage, multipath, and nonlinear products break the ideal orthogonality and appear as cross-channel interference.

flowchart LR
  A[source A] --> MAP[map to resource slots]
  B[source B] --> MAP
  C[source C] --> MAP
  MAP --> COMB[combine]
  COMB --> CH[shared medium]
  CH --> SYNC[synchronize and estimate channel]
  SYNC --> SEP[filter, correlate, FFT, or beamform]
  SEP --> OA[output A]
  SEP --> OB[output B]
  SEP --> OC[output C]

The synchronization block is load-bearing. A mathematically orthogonal resource grid is not separable at the receiver until time, frequency, and channel phase are known well enough for the chosen demultiplexer.

Frequency-division multiplexing #

FDM translates each baseband signal to a different carrier band and sums the modulated channels. A bank of band-pass filters separates them at the receiver. If channel $i$ occupies bandwidth $B_i$ and neighbouring filters need guard $G_i$, the composite allocation is approximately

$$B_{total}\approx\sum_iB_i+\sum_iG_i.$$

Guard bands pay for finite filter transition width, oscillator error, Doppler, and transmitter spectral regrowth. Making them narrower increases spectral efficiency but demands more selective filters, cleaner oscillators, tighter masks, or coordination.

FDM fits continuous sources and simple receivers: broadcast stations occupy persistent channels, cable systems stack carriers, and analogue telephony historically assembled voice channels into groups. Its central weakness is rigid reservation—a silent source still owns its band.

Time-division multiplexing #

Synchronous TDM assigns each source a recurring time slot whether or not it has data. If a frame contains $N$ slots of $b$ bits and repeats at $f_F$, its aggregate raw rate is

$$R=Nbf_F.$$

ITU-T G.704's 2048 kbit/s primary frame is the canonical calculation: 32 time slots × 8 bits × 8000 frames/s gives

$$32\times8\times8000=2.048\ \text{Mbit/s}.$$

The slots do not all carry user payload; framing, alarms, signalling, and synchronization consume defined positions. Fixed TDM therefore buys deterministic delay and simple demultiplexing at the cost of idle-slot waste and permanent framing overhead.

Time slots need guard time when transmitters take turns on a shared radio or when propagation delays differ. If a slot of useful duration $T_u$ requires guard $T_g$, its time efficiency is

$$\eta_t=\frac{T_u}{T_u+T_g}.$$

A 100 μs payload slot with 5 μs guard is only 95.24% time-efficient before pilots, headers, and coding.

Statistical multiplexing shares queues, not fixed slots #

Packet networks assign capacity only to active flows. This statistical multiplexing exploits burstiness: if many sources rarely peak together, a link can be smaller than the sum of their peak rates while carrying far more than the sum of fixed reserved averages.

That gain replaces deterministic ownership with a queue. Let aggregate arrival rate be $A(t)$ and service rate $C$. Whenever $A(t)>C$, backlog grows at $A(t)-C$; if the long-term offered load reaches capacity, mean delay becomes unstable. Average utilization alone is insufficient because burst duration, correlation among sources, buffer size, and the acceptable loss/delay tail determine whether the multiplex works.

Schedulers turn the queue into policy. Strict priority protects one class but can starve another. Weighted fair queueing approximates reserved shares while lending unused capacity. Deadline-aware scheduling may minimize late packets rather than equalize throughput. These are multiplexing decisions because they determine who owns the shared transmission opportunity.

Wavelength-division multiplexing is optical FDM #

WDM puts independently modulated optical carriers onto one fibre. A multiplexer combines wavelengths; wavelength-selective filters or coherent receivers separate them. “Wavelength” is convenient language, but the standards grid is defined in optical frequency, where equal spacing is linear.

ITU-T G.694.1's in-force DWDM grid is anchored at 193.1 THz. Fixed grids use channel spacings from 12.5 GHz through 100 GHz and wider. The flexible grid defines nominal central-frequency positions in 6.25 GHz steps and slot widths in 12.5 GHz multiples. A route can therefore allocate a wider spectral slot to a higher-baud or less spectrally compact signal without abandoning a common reference grid.

WDM adds optical constraints absent from low-frequency FDM: laser frequency stability, filter passband narrowing across many reconfigurable nodes, chromatic dispersion, nonlinear fibre effects, amplifier gain flatness, and optical signal-to-noise ratio. A slot present on a planning screen is not proof that a modulation format can traverse the selected route.

OFDM overlaps spectra without losing orthogonality #

For useful symbol duration $T_u$, OFDM chooses subcarrier spacing

$$\Delta f=\frac{1}{T_u}.$$

Over exactly $T_u$, the inner product of any two subcarriers separated by an integer multiple of $\Delta f$ is zero. Their sinc-shaped spectra overlap, yet an FFT separates them at the correct timing and frequency. This is denser than placing independent filtered carriers behind wide guard bands.

The transmitter maps data to subcarrier bins, takes an inverse FFT, and prepends a cyclic prefix (CP). If the channel impulse response fits inside the CP, linear convolution appears circular over the useful block and each subcarrier sees approximately one complex gain. A frequency-selective equalizer becomes one multiplication per bin.

The CP is discarded, so its direct efficiency is

$$\eta_{CP}=\frac{T_u}{T_u+T_{CP}}.$$

With $T_u=66.67\ \mu s$ and $T_{CP}=4.76\ \mu s$, $\eta_{CP}\approx93.3\%$. The prefix is worthwhile only because it turns a long time-domain equalizer into many simple scalar equalizers and protects orthogonality against delayed copies within its span.

OFDMA schedules a two-dimensional grid #

OFDMA gives different users different subcarriers and OFDM symbols. 3GPP TS 38.211 defines the 5G NR physical resource grid, reference signals, numerologies, and channel mappings that make those assignments reproducible. The scheduler works in time-frequency units, but a usable allocation also needs pilots, control channels, power limits, coding rate, and a channel estimate.

Resource allocation can follow frequency selectivity: give each user subcarriers where its channel is strong. That gain is bounded by channel-state age and feedback cost. A schedule optimized for yesterday's fade pattern can be worse than a robust uniform allocation.

Code and space are additional dimensions #

Code-division multiplexing lets users overlap in time and frequency while assigning distinguishable spreading sequences. Separation depends on correlation at the actual delay and fails under near–far power imbalance or multipath that destroys alignment; spread spectrum develops those limits.

Spatial multiplexing uses multiple antenna modes to transmit independent streams at the same time and frequency. The receiver separates them only when the channel matrix has enough independent rank and usable singular values. Diversity sends redundant information to improve reliability; spatial multiplexing sends different information to increase rate. Both use multiple antennas, but they spend the spatial degrees of freedom differently.

Account for every overhead in the same denominator #

Raw modulation rate is not payload rate. For a regular framed link, a useful decomposition is

$$R_{payload}=R_{raw}\,\eta_{guard}\eta_{pilot}\eta_{frame}\eta_{FEC}\eta_{retransmit},$$

where each factor is defined against the output of the previous stage. Do not add percentage overheads drawn from different denominators. For statistical multiplexing, also report offered load, loss, and a delay percentile; a single average throughput hides queueing failure.

Multiplexing and multiple access are the same physical operation viewed from different ownership. A multiplexer combines tributaries under one administrator; a multiple-access protocol coordinates independent transmitters. Central scheduling can enforce timing and power tightly. Random access spends collisions and retries to avoid constant grants. The receiver still has to separate the resulting signals in one of the dimensions above.

Verification workflow #

1. Draw the resource grid including guards, pilots, framing, control, and unavailable regions—not only payload rectangles.
2. Recompute raw-to-payload efficiency from symbol, slot, frame, code, and retransmission counters observed at one boundary.
3. Inject worst-case clock/frequency error, propagation delay, filter drift, and amplifier nonlinearity; measure leakage into every neighbour.
4. For statistical multiplexing, replay correlated bursts and report queue occupancy, loss, and tail delay, not only mean utilization.
5. For OFDM, sweep fractional-bin frequency offset and channel delay beyond the CP; inspect inter-carrier and inter-symbol interference separately.
6. For WDM, verify the end-to-end passband and OSNR after every route element, not only transmitter centre frequency.
7. Confirm that scheduler grants and receiver demapping use the same numerology, slot boundary, and resource indexing.

The unifying question is not “FDM or TDM?” It is: which resource dimension makes users distinguishable at the receiver, what guard and state preserve that distinction, and what happens when demand or synchronization violates the partition?