Channelization: Structured Channel Sharing

When radio was first invented, communication was simple: one transmitter, one receiver, one frequency. But as wireless communication exploded—from military radios to commercial broadcasting to cellular networks—a fundamental challenge emerged: the electromagnetic spectrum is finite, yet demand for wireless communication is virtually unlimited.

Consider the scale of the problem: a single cellular tower might need to serve thousands of simultaneous phone calls, each requiring a dedicated communication channel. How can we carve out thousands of distinct channels from a limited frequency allocation? This is the Multiple Access Problem, and its solutions form the foundation of all modern wireless communication systems.

Frequency Division Multiple Access (FDMA) represents the most intuitive solution to this challenge: divide the available spectrum into non-overlapping frequency bands and assign each user their own band. It's conceptually elegant, historically significant, and still forms the backbone of many modern communication systems.

To understand FDMA, we must first grasp how electromagnetic signals occupy spectrum. When you transmit a signal at a particular frequency, you don't use just that single frequency point—you use a band of frequencies centered around your carrier frequency. This band, called the signal bandwidth, is determined by the information content of your signal.

The Bandwidth-Information Relationship:

Fourier analysis tells us that any time-varying signal can be decomposed into a sum of sinusoidal components at different frequencies. A signal that changes rapidly over time contains high-frequency components; a slowly varying signal contains mostly low-frequency components. The bandwidth of a signal is the range of frequencies it occupies:

$$B = f_{\text{max}} - f_{\text{min}}$$

For a voice signal, typical speech contains frequencies from about 300 Hz to 3400 Hz, giving a bandwidth of approximately 3.1 kHz. For high-quality music, we need 20 Hz to 20,000 Hz—a bandwidth of about 20 kHz. For HD video, the bandwidth extends to several MHz.

The Orthogonality Principle:

FDMA works because signals at different frequencies are orthogonal—they don't interfere with each other. Mathematically, two sinusoidal signals at different frequencies, when multiplied together and integrated over time, produce zero:

$$\int_{0}^{T} \sin(2\pi f_1 t) \cdot \sin(2\pi f_2 t) , dt = 0 \quad \text{for } f_1 eq f_2$$

This orthogonality means that a receiver tuned to frequency $f_1$ will completely reject signals at frequency $f_2$. In practice, this separation is achieved through bandpass filters that pass only the desired frequency range while blocking all others.

This is why you can tune your radio to exactly one station at a time—the filter in your receiver passes only the narrow band assigned to that station and rejects all adjacent channels.

An FDMA system divides a given frequency band into N non-overlapping sub-bands, each serving as an independent communication channel. The architecture requires careful planning of frequency assignments and sophisticated filtering at both transmitters and receivers.

System Components:

Frequency Band Organization:

The total available bandwidth $W_{\text{total}}$ is divided among $N$ channels, each with bandwidth $B_{\text{ch}}$. However, channels cannot be placed immediately adjacent to each other due to practical filter limitations. Guard bands of width $B_{\text{guard}}$ must separate adjacent channels:

$$W_{\text{total}} = N \cdot B_{\text{ch}} + (N-1) \cdot B_{\text{guard}} + 2 \cdot B_{\text{edge}}$$

where $B_{\text{edge}}$ represents guard bands at the edges of the allocated spectrum to prevent interference with other systems.

Guard bands are perhaps the most important practical consideration in FDMA system design. They exist to solve three critical problems that arise from real-world hardware limitations.

Calculating Guard Band Requirements:

The minimum guard band width depends on several factors:

$$B_{\text{guard}} \geq 2 \cdot \Delta f_{\text{drift}} + 2 \cdot f_{\text{Doppler}} + B_{\text{transition}}$$

where:

• $\Delta f_{\text{drift}}$ is the maximum expected carrier frequency drift
• $f_{\text{Doppler}}$ is the maximum Doppler shift
• $B_{\text{transition}}$ is the filter transition bandwidth

Example Calculation:

Consider an FDMA cellular system with:

• Carrier frequency: 900 MHz
• Oscillator stability: ±0.5 ppm → $\Delta f_{\text{drift}} = 900 \times 10^6 \times 0.5 \times 10^{-6} = \pm 450$ Hz
• Maximum vehicle speed: 120 km/h → $f_{\text{Doppler}} = \frac{v \cdot f_c}{c} = \frac{120/3.6 \times 900 \times 10^6}{3 \times 10^8} \approx \pm 100$ Hz
• Filter transition bandwidth: 3 kHz

Minimum guard band: $2 \times 450 + 2 \times 100 + 3000 = 4100$ Hz ≈ 4.1 kHz

In practice, guard bands are typically set to 5-10 kHz to provide additional margin.

The capacity of an FDMA system—measured as the number of simultaneous users it can support—is a function of the total available bandwidth, the per-channel bandwidth requirement, and the guard band overhead.

Number of Channels:

For a system with total bandwidth $W$, channel bandwidth $B_c$, and guard band $B_g$:

$$N = \left\lfloor \frac{W - B_{\text{edge}}}{B_c + B_g} \right\rfloor$$

where $\lfloor \cdot \rfloor$ denotes the floor function (we can't have fractional channels).

Spectral Efficiency:

Spectral efficiency measures how well we utilize the available bandwidth:

$$\eta_{\text{spectral}} = \frac{N \cdot B_c}{W} = \frac{\text{Useful bandwidth}}{\text{Total bandwidth}}$$

Due to guard bands, this is always less than 1.0. Typical FDMA systems achieve $\eta_{\text{spectral}}$ between 0.7 and 0.9.

Capacity Comparison with Theoretical Maximum:

Shannon's theorem tells us the maximum data rate achievable over a noisy channel:

$$C = B \log_2(1 + \text{SNR})$$

For an FDMA system, each user gets their full channel bandwidth but only during their assigned frequency. If we compare the aggregate capacity:

$$C_{\text{FDMA-total}} = N \cdot B_c \cdot \log_2(1 + \text{SNR}) = (W - \text{overhead}) \cdot \log_2(1 + \text{SNR})$$

The guard band overhead directly reduces total system capacity. This is why FDMA is often considered less spectrally efficient than TDMA or CDMA in pure theoretical terms, though practical considerations often favor FDMA for specific applications.

Real communication systems require two-way (duplex) communication. FDMA systems typically achieve this through Frequency Division Duplexing (FDD), which allocates separate frequency bands for uplink (mobile to base) and downlink (base to mobile) transmission.

FDD System Organization:

The total spectrum allocation is split into two equal-sized bands separated by a duplex gap:

• Uplink band: Frequencies used by mobile devices to transmit to the base station
• Downlink band: Frequencies used by the base station to transmit to mobile devices
• Duplex gap: A guard band between uplink and downlink to prevent self-interference

For each call, a user is assigned a pair of frequencies—one in the uplink band and a corresponding one in the downlink band.

Frequency Pair Assignment:

In FDD systems, uplink and downlink channels are paired with a fixed frequency offset. For AMPS:

$$f_{\text{downlink}} = f_{\text{uplink}} + 45 \text{ MHz}$$

Channel numbers are assigned based on the uplink frequency:

$$f_{\text{uplink}} = 825.030 + 0.030 \times (N - 1) \text{ MHz} \quad \text{for } N = 1 \text{ to } 799$$

This systematic relationship simplifies hardware design—the duplexer only needs to handle one fixed frequency offset, and channel selection requires only one synthesizer with offset mixing.

FDMA's dominance in early wireless systems and continued use in specific applications reflects its distinct advantages. However, its limitations have driven the development of alternative multiple access techniques.

FDMA has been the foundation of numerous communication systems. Understanding these applications reveals both the technique's strengths and the contexts where it excels.

We've explored FDMA from fundamental physics through practical system design. Let's consolidate the essential concepts:

Critical Formulas:

$$N = \left\lfloor \frac{W_{\text{total}} - B_{\text{edge}}}{B_{\text{channel}} + B_{\text{guard}}} \right\rfloor$$

$$\eta_{\text{spectral}} = \frac{N \cdot B_{\text{channel}}}{W_{\text{total}}}$$

$$B_{\text{guard}} \geq 2 \cdot \Delta f_{\text{drift}} + 2 \cdot f_{\text{Doppler}} + B_{\text{transition}}$$

What's Next:

In the next page, we explore TDMA (Time Division Multiple Access), which takes a fundamentally different approach: instead of dividing frequency, it divides time. Each user gets the full bandwidth but only during their assigned time slots. This approach eliminates guard bands in the frequency domain but introduces new challenges in synchronization and buffering.