Code Division Multiplexing (CDM)

Code Division Multiplexing: Sharing by Secret Codes

Code Division Multiplexing is a remarkable technique where multiple users transmit simultaneously on the same frequency at the same time — yet their signals can be perfectly separated at the receiver. This seemingly impossible feat is achieved through the mathematical magic of orthogonal spreading codes. Each user's data is multiplied by a unique code, spreading it across a wide bandwidth, and only a receiver knowing that specific code can extract the signal.

The Cocktail Party Analogy

Think of it this way: imagine a cocktail party where everyone speaks simultaneously. Normally, you could not understand anyone. But what if each person spoke a different language? If you only understand English, you can focus on the English speaker and all other languages sound like background noise. CDM works similarly — each user "speaks" in a unique mathematical "language" (spreading code), and the receiver "listens" for only one specific code.

How CDM Works

At the transmitter:

1. User data: a stream of bits at rate Rb (e.g., 10 kbps)
2. Spreading code: a much faster sequence at rate Rc (chip rate, e.g., 1.23 Mcps)
3. Multiply data by spreading code → signal bandwidth expands by factor Rc/Rb
4. All users' spread signals are summed and transmitted on the same frequency

At the receiver:

1. Received signal = sum of all users' spread signals + noise
2. Multiply received signal by desired user's spreading code
3. Integrate over one data bit period (correlation/despreading)
4. Desired signal: code × code = 1 → data recovered
5. Interfering signals: their code × desired code ≈ 0 → rejected!

The key property: when you multiply a signal by its own code, you recover the original data. When you multiply by a different user's code, the result averages to zero (orthogonality).

Spreading Factor and Processing Gain

Spreading factor (SF): N = Rc/Rb = chip rate / data rate

This determines how many chips represent each data bit.

Processing gain (PG): PG = 10 × log₁₀(N) dB

Example: UMTS (3G) with chip rate 3.84 Mcps and data rate 12.2 kbps:

• SF = 3,840,000 / 12,200 = 315
• PG = 10 × log₁₀(315) = 25 dB

This 25 dB processing gain means interference is suppressed by 25 dB relative to the desired signal — a massive advantage that enables multiple users to coexist.

Types of Spreading Codes

Walsh Codes (Orthogonal):

• Generated using Hadamard matrices
• Perfectly orthogonal (zero cross-correlation when synchronized)
• Used for downlink (base station to mobiles) in CDMA systems
• Example (length 4): W0=[++++], W1=[+-+-], W2=[++--], W3=[+--+]

PN Sequences (Pseudo-Noise):

• Generated by Linear Feedback Shift Registers (LFSR)
• Near-zero cross-correlation on average (not perfectly orthogonal)
• Used for uplink (mobiles to base station) where synchronization is imperfect
• Maximum-length sequences have period 2^n - 1

Gold Codes:

• Combination of two m-sequences
• Good cross-correlation properties
• Large family size (2^n + 1 codes available)
• Used in GPS (each satellite has a unique Gold code)

The Near-Far Problem

CDM's biggest challenge in mobile communication: if a nearby user transmits at high power and a distant user at low power, the strong signal overwhelms the weak one despite code orthogonality:

Problem: Received power from near user might be 100× stronger than from far user. Even with processing gain, the near user's residual interference after despreading can bury the far user's signal.

Solution — Power Control:

• Open-loop: Mobile estimates path loss from base station signal
• Closed-loop: Base station measures received power, sends power-up/down commands
• In CDMA2000: power control at 800 Hz (every 1.25 ms)
• Target: all signals arrive at base station within 1 dB of each other

Power control is so critical that CDMA systems cannot function without it — it is the single most important implementation requirement.

CDMA System Capacity

The capacity of a CDMA system (maximum number of simultaneous users):

N ≈ (PG) / (Eb/N₀ required) × (1/(1+f)) × voice activity factor

Where f = ratio of other-cell interference to same-cell interference (~0.6)

Typical capacity gain over TDMA: 3-5× more users per cell for voice service, because:

• Voice activity factor (~0.4): Users only speak ~40% of the time; silence reduces interference
• Soft handoff: No hard cell boundaries; gradual transitions
• No frequency planning needed: All cells use same frequency

Applications

Advantages and Disadvantages

Advantages:

• Graceful degradation (more users = slightly more noise, no hard limit)
• No frequency planning (universal frequency reuse)
• Resistant to narrowband interference (spread spectrum)
• Inherent path diversity (RAKE receiver combines multipath)
• Security (spread signal appears as noise without code knowledge)

Disadvantages:

• Near-far problem requires precise power control
• Complexity of RAKE receiver and interference cancellation
• Limited by total interference (capacity is soft, not hard)
• Self-interference from multipath degrades code orthogonality

Key Takeaways

1. CDM allows multiple users to share the same frequency and time simultaneously by assigning unique spreading codes that enable separation through correlation.
2. Processing gain PG = 10×log₁₀(SF) dB determines how effectively interference is suppressed — higher spreading means better rejection.
3. Walsh codes provide perfect orthogonality for synchronized downlinks; PN/Gold codes handle asynchronous uplinks.
4. The near-far problem is CDM's critical challenge — strict power control ensures all signals arrive at similar levels.
5. CDMA capacity benefits from voice activity (silence reduces interference) and soft capacity limits (graceful degradation).
6. CDM provides inherent anti-jamming, security, and multipath resistance — key advantages for GPS, military, and mobile communication.