Channelization: Structured Channel Sharing

FDMA separates users by frequency. TDMA separates users by time. Code Division Multiple Access (CDMA) takes a radically different approach: it allows all users to transmit simultaneously on the same frequency and distinguishes them through unique mathematical codes.

This concept initially seems impossible—if everyone transmits at once on the same frequency, shouldn't the result be chaos, an unintelligible cacophony of overlapping signals? The answer lies in spread spectrum technology and the remarkable properties of orthogonal codes.

CDMA was originally developed for military communications due to its inherent resistance to jamming and low probability of intercept. The same properties that provide security also enable efficient multiple access—making CDMA the foundation of 3G cellular systems (CDMA2000, WCDMA) and GPS navigation.

Traditional communication systems transmit signals in the minimum bandwidth required—a principle called bandwidth efficiency. CDMA deliberately violates this principle: it spreads each user's signal across a bandwidth much wider than necessary.

Why Spread the Signal?

Spreading provides several profound benefits:

1. Multiple Access Capability — Spread signals can overlap in frequency and time while remaining distinguishable through their unique spreading codes.
2. Interference Resistance — Spread signals are robust against narrowband interference and jamming.
3. Security — Without knowing the spreading code, the signal appears as low-level noise, difficult to detect or intercept.
4. Multipath Resistance — Spread spectrum enables separating and combining multipath components constructively.

The Spreading Process:

Each data bit is multiplied by a high-rate spreading code consisting of many "chips." If the data rate is $R_b$ bits per second and the chip rate is $R_c$ chips per second, the spreading factor $SF$ is:

$$SF = \frac{R_c}{R_b}$$

A spreading factor of 64 means each data bit is represented by 64 chips—and the signal occupies approximately 64× the bandwidth of the original data.

Direct Sequence Spread Spectrum (DSSS):

The most common CDMA implementation uses Direct Sequence Spread Spectrum. The user's data signal $d(t)$ is multiplied by a spreading code $c(t)$:

$$s(t) = d(t) \cdot c(t) \cdot \cos(2\pi f_c t)$$

where:

• $d(t) \in {-1, +1}$ is the data signal (BPSK: Binary Phase Shift Keying representation)
• $c(t) \in {-1, +1}$ is the spreading code (running at chip rate)
• $f_c$ is the carrier frequency

The spread signal $s(t)$ occupies bandwidth approximately equal to the chip rate $R_c$, regardless of the original data rate $R_b$.

Spreading alone doesn't enable multiple access—what matters is the receiver's ability to despread the desired signal while rejecting all others. This is where the mathematical properties of spreading codes become crucial.

The Despreading Operation:

The receiver multiplies the incoming signal by the same spreading code used by the intended transmitter:

$$r(t) \cdot c(t) = [d(t) \cdot c(t)] \cdot c(t) = d(t) \cdot c^2(t) = d(t) \cdot 1 = d(t)$$

Since a spreading code multiplied by itself equals +1 (because $(+1)^2 = (-1)^2 = 1$), the despreading operation recovers the original data.

What Happens to Other Users' Signals?

If User A's receiver multiplies by User A's code $c_A(t)$, but receives User B's signal $d_B(t) \cdot c_B(t)$:

$$[d_B(t) \cdot c_B(t)] \cdot c_A(t) = d_B(t) \cdot [c_B(t) \cdot c_A(t)]$$

If codes $c_A$ and $c_B$ are orthogonal, their product averages to zero over each bit period:

$$\frac{1}{T_b} \int_0^{T_b} c_A(t) \cdot c_B(t) , dt = 0$$

This means User B's signal, after despreading with User A's code, becomes zero-mean noise that can be filtered out—User B effectively disappears!

The Correlation Receiver:

In CDMA, the receiver performs correlation—multiplying by the code and integrating over the bit period. The integration averages out:

1. Other users' signals (if codes are orthogonal)
2. Narrowband interference (spread by multiplication, then reduced by integration)
3. Some noise components (statistical averaging reduces noise power)

Mathematical Formulation:

The decision statistic for bit $i$ of User A is:

$$z_i = \int_{iT_b}^{(i+1)T_b} r(t) \cdot c_A(t) , dt$$

where $r(t)$ is the total received signal including all users and noise. If orthogonality holds and codes are synchronized:

$$z_i = A_A \cdot d_i^A + \sum_{k \neq A} A_k \cdot d_i^k \cdot \rho_{Ak} + n_i$$

where $\rho_{Ak}$ is the cross-correlation between codes $c_A$ and $c_k$. For orthogonal codes, $\rho_{Ak} = 0$ and the interference term vanishes.

One of CDMA's most powerful properties is processing gain—the improvement in signal-to-interference ratio achieved through the spreading and despreading process.

Definition of Processing Gain:

Processing gain $G_p$ is the ratio of the spread bandwidth to the data bandwidth, equivalently the spreading factor:

$$G_p = \frac{W}{R_b} = \frac{\text{Chip Rate}}{\text{Bit Rate}} = SF$$

where $W$ is the spread bandwidth (approximately equal to chip rate) and $R_b$ is the data rate.

In decibels: $$G_p \text{ (dB)} = 10 \log_{10}(SF)$$

Physical Interpretation:

Processing gain quantifies the suppression of interference through despreading:

1. Interference from other CDMA users is reduced by factor $G_p$ (for orthogonal codes)
2. Narrowband jamming is spread across the entire bandwidth, reducing its power spectral density by factor $G_p$
3. Thermal noise power in the data bandwidth is the same, but the signal is coherently combined (integration gain)

Processing Gain and System Capacity:

Processing gain directly impacts CDMA system capacity. The number of simultaneous users $K$ a CDMA system can support is approximately:

$$K \approx 1 + \frac{G_p}{(E_b/N_0)_{\text{req}}} \cdot \frac{1}{v}$$

where:

• $G_p$ is processing gain
• $(E_b/N_0)_{\text{req}}$ is the required signal-to-noise ratio per bit for acceptable error rate
• $v$ is the voice activity factor (typically 0.4–0.5, accounting for silence periods)

Higher processing gain → more users. This is why higher chip rates and lower data rates enable more simultaneous users.

The choice of spreading codes is critical to CDMA system performance. Different code families offer different properties suited to different aspects of system operation.

Walsh Codes (Hadamard Sequences):

Walsh codes are generated from the Hadamard matrix, constructed recursively:

$$H_1 = [1]$$

$$H_{2N} = \begin{bmatrix} H_N & H_N \ H_N & -H_N \end{bmatrix}$$

For example, $H_4$:

$$H_4 = \begin{bmatrix} 1 & 1 & 1 & 1 \ 1 & -1 & 1 & -1 \ 1 & 1 & -1 & -1 \ 1 & -1 & -1 & 1 \end{bmatrix}$$

Each row is a Walsh code. The inner product of any two different rows is zero—they are perfectly orthogonal.

Critical Limitation: Walsh codes are orthogonal only when perfectly time-aligned. A time shift as small as one chip destroys orthogonality. This makes Walsh codes suitable for synchronous channels (like downlink) but problematic for asynchronous channels.

PN Sequences (Maximum-Length Sequences):

PN sequences are generated by Linear Feedback Shift Registers (LFSRs). An $n$-stage LFSR produces a sequence of period $2^n - 1$.

Key properties:

• Autocorrelation: Sharp peak at zero shift, approximately $-1/N$ elsewhere
• Balance: Nearly equal numbers of +1 and -1 (differs by exactly 1)
• Run property: Runs of consecutive same values follow specific pattern
• Spectral flatness: Approximately white noise spectrum

PN sequences are used for:

1. Cell identification — Each cell uses a different PN offset
2. Base scrambling — Provides security and separates cells
3. Uplink spreading — Users distinguished by PN sequence offsets or different PN codes

While CDMA's theoretical elegance is compelling, practical implementation faces a severe challenge: the near-far problem. This problem nearly killed CDMA as a viable technology and required brilliant engineering solutions to overcome.

The Problem:

Consider a base station receiving signals from two mobiles:

• Mobile A is 100 meters away (strong signal)
• Mobile B is 1 km away (weak signal)

Radio signal power decreases with distance according to the path loss law: $$P_r = P_t \cdot \left(\frac{\lambda}{4\pi d}\right)^n$$

where $n$ is the path loss exponent (typically 3-4 in urban environments). If $n = 4$:

$$\frac{P_A}{P_B} = \left(\frac{d_B}{d_A}\right)^4 = \left(\frac{1000}{100}\right)^4 = 10,000 = 40 \text{ dB}$$

Mobile A's signal is 10,000 times stronger than Mobile B's signal!

Why This Destroys CDMA:

In theory, orthogonal codes completely separate users. In practice:

1. Codes aren't perfectly orthogonal under real-world propagation conditions (multipath, timing errors)
2. Cross-correlation isn't exactly zero but some small value $\rho$
3. Strong signals leak into weak signal detection

If the cross-correlation between codes is even $\rho = 0.01$ (1%), Mobile A's leakage into Mobile B's channel is:

$$I_A = P_A \cdot \rho^2 = 10,000 \times 0.0001 = 1$$

The interference from Mobile A equals Mobile B's entire signal power! Mobile B is completely jammed by Mobile A, even though they're using "orthogonal" codes.

With 20 near users and one far user, the situation is hopeless—the far user is buried under interference.

The solution to the near-far problem is elegant: if near users are too strong, make them weaker. Power control adjusts each mobile's transmit power so that all signals arrive at the base station with equal power.

The Goal:

All mobile signals should arrive at the base station with the same received power:

$$P_{r,A} = P_{r,B} = P_{r,C} = ... = P_{\text{target}}$$

To achieve this, near mobiles transmit at low power while far mobiles transmit at high power:

$$P_{t,i} = P_{\text{target}} \cdot L_i$$

where $L_i$ is the path loss from mobile $i$ to the base station.

CDMA Power Control Mechanisms:

Power Control Speed Requirements:

The channel changes due to fading, mobility, and environment. Power control must track these changes to maintain equal received power:

• Slow fading (shadowing): Changes over seconds—handled by open-loop
• Fast fading (multipath): Changes over milliseconds—requires closed-loop

For a mobile moving at 100 km/h at 900 MHz, the Doppler frequency is approximately 83 Hz, meaning the channel can change completely in 6 ms. IS-95's 800 Hz update rate (1.25 ms period) is fast enough to track this fading with some margin.

CDMA offers several significant advantages over FDMA and TDMA that made it the technology choice for 3G cellular systems.

We've explored CDMA from spread spectrum fundamentals through the practical challenges and solutions that made it viable. Let's consolidate the essential concepts:

Critical Formulas:

$$SF = \frac{R_c}{R_b} = G_p$$

$$G_p \text{ (dB)} = 10 \log_{10}(SF)$$

$$K \approx 1 + \frac{G_p}{(E_b/N_0)_{\text{req}} \cdot v}$$

$$P_{t,i} = P_{\text{target}} \cdot L_i$$

What's Next:

In the next page, we dive deeper into code orthogonality—the mathematical foundation that makes CDMA possible. We'll explore Hadamard matrices, cross-correlation properties, and why perfect orthogonality requires perfect synchronization. Understanding orthogonality illuminates both CDMA's power and its practical limitations.