Channelization: Structured Channel Sharing

CDMA's ability to separate simultaneous transmissions on the same frequency rests on a profound mathematical concept: orthogonality. When we say two codes are orthogonal, we mean that when we multiply them together and sum the result, we get zero. This "zero cross-correlation" property is what allows a receiver to completely ignore all signals except the one intended for it.

Orthogonality is familiar from geometry—perpendicular vectors have zero dot product. But in CDMA, we extend this concept to sequences of numbers (codes), and the consequences are far-reaching. Understanding orthogonality deeply reveals why CDMA works, when it fails, and what design choices system architects must make.

This page explores orthogonality from first principles through advanced implications, giving you the mathematical intuition to understand CDMA's power and limitations.

Let's begin with the rigorous mathematical definition, then build intuition from there.

Discrete-Time Definition:

Two code sequences $c_1$ and $c_2$, each of length $N$, are orthogonal if:

$$\sum_{i=0}^{N-1} c_1[i] \cdot c_2[i] = 0$$

This sum is the inner product (or dot product) of the sequences. For CDMA codes using ${-1, +1}$ values, orthogonality means the codes have equal numbers of agreements and disagreements.

Continuous-Time Definition:

For continuous-time signals over period $T$:

$$\int_0^T c_1(t) \cdot c_2(t) , dt = 0$$

Normalized Cross-Correlation:

The cross-correlation $\rho_{12}$ is often normalized to the range $[-1, +1]$:

$$\rho_{12} = \frac{1}{N} \sum_{i=0}^{N-1} c_1[i] \cdot c_2[i]$$

For orthogonal codes: $\rho_{12} = 0$

For identical codes: $\rho_{11} = 1$ (autocorrelation at zero lag)

Understanding correlation functions is essential for analyzing CDMA code performance. Two types of correlation matter: cross-correlation (between different codes) and auto-correlation (a code with itself at different time shifts).

Cross-Correlation Function:

The cross-correlation of codes $c_1$ and $c_2$ as a function of time shift $\tau$ is:

$$R_{12}(\tau) = \sum_{i=0}^{N-1} c_1[i] \cdot c_2[(i + \tau) \mod N]$$

For CDMA, we want $R_{12}(0) = 0$ (orthogonality at zero lag), but what about other shifts?

Perfect vs. Practical Orthogonality:

• Walsh codes have $R_{12}(0) = 0$ but $R_{12}(\tau) \neq 0$ for $\tau \neq 0$
• PN sequences have $R_{12}(\tau) \approx 0$ for all $\tau$, but never exactly zero
• Gold codes have bounded $|R_{12}(\tau)| \leq t(n)$ where $t(n)$ depends on sequence length

Auto-Correlation Function:

The auto-correlation of a code with a time-shifted version of itself:

$$R_{11}(\tau) = \sum_{i=0}^{N-1} c_1[i] \cdot c_1[(i + \tau) \mod N]$$

Ideal auto-correlation has a sharp peak at $\tau = 0$ and zero elsewhere:

$$R_{11}(\tau) = \begin{cases} N & \tau = 0 \ 0 & \tau \neq 0 \end{cases}$$

This is crucial for synchronization—the receiver correlates with the code, and the peak tells it exactly when the code starts.

Walsh codes (also called Hadamard codes) provide perfect orthogonality when users are synchronized. They form the backbone of channel separation in CDMA systems.

Hadamard Matrix Construction:

Walsh codes are the rows of a Hadamard matrix, constructed recursively:

$$H_1 = [1]$$

$$H_2 = \begin{bmatrix} 1 & 1 \ 1 & -1 \end{bmatrix}$$

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

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

Key Properties:

1. $H_N$ is an $N \times N$ matrix with elements ${-1, +1}$
2. $H_N \cdot H_N^T = N \cdot I_N$ (each row is orthogonal to every other row)
3. All rows are mutually orthogonal
4. Only $N$ orthogonal codes of length $N$ exist

Walsh codes' perfect orthogonality holds only at zero time shift. In practice, signals arrive at different times due to propagation delays, multipath, and timing imperfections. Understanding how timing errors destroy orthogonality is crucial for CDMA system design.

Shifted Cross-Correlation:

Consider Walsh codes W1 and W2 (length 4):

• W1 = [+1, -1, +1, -1]
• W2 = [+1, +1, -1, -1]

At zero shift: $R_{12}(0) = (+1)(+1) + (-1)(+1) + (+1)(-1) + (-1)(-1) = 0$ ✓

At shift $\tau = 1$ (W2 shifted by one position):

• W2 shifted = [-1, +1, +1, -1] (cyclic shift)
• $R_{12}(1) = (+1)(-1) + (-1)(+1) + (+1)(+1) + (-1)(-1) = -1 - 1 + 1 + 1 = 0$

Actually, W1 and W2 happen to remain orthogonal at shift 1. Let's try W1 and W3:

• W1 = [+1, -1, +1, -1]
• W3 = [+1, -1, -1, +1]

At shift $\tau = 1$ (W3 shifted):

• W3 shifted = [+1, +1, -1, -1]
• $R_{13}(1) = (+1)(+1) + (-1)(+1) + (+1)(-1) + (-1)(-1) = 1 - 1 - 1 + 1 = 0$

Let me try W1 and W4:

• W1 = [+1, -1, +1, -1]
• W4 = [+1, +1, +1, +1]

At shift $\tau = 0$: $R_{14}(0) = 1 - 1 + 1 - 1 = 0$ ✓ At shift $\tau = 1$: W4 shifted = [+1, +1, +1, +1] (unchanged, all same) $R_{14}(1) = 1 - 1 + 1 - 1 = 0$ ✓ (W4 is constant)

Let's examine a more compelling example with length-8 codes:

W2 and W3 (length 8):

• W2 = [+1, +1, -1, -1, +1, +1, -1, -1]
• W3 = [+1, -1, -1, +1, +1, -1, -1, +1]

At $\tau = 0$: Inner product = 0 ✓

At $\tau = 1$ (W3 cyclic shifted right by 1):

• W3' = [+1, +1, -1, -1, +1, +1, -1, -1]

$R_{23}(1) = (+1)(+1) + (+1)(+1) + (-1)(-1) + (-1)(-1) + (+1)(+1) + (+1)(+1) + (-1)(-1) + (-1)(-1)$ $= 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 8$ ✗

Complete correlation! A one-chip shift made W3 identical to W2!

Multipath Impact:

Multipath propagation causes copies of the signal to arrive at different times. Consider a two-ray channel where:

• Direct path: arrives at time $t_0$
• Reflected path: arrives at time $t_0 + \tau$ with attenuation $\alpha$

Received signal: $r(t) = s(t) + \alpha \cdot s(t - \tau)$

When the receiver despreads:

• Direct path: fully correlated (orthogonality maintained)
• Reflected path: shifted, potentially non-orthogonal

The reflected path creates self-interference from other Walsh channels. In severe multipath (like indoor environments with delay spreads > 1 chip), significant orthogonality loss occurs.

Since Walsh codes lose orthogonality when shifted, CDMA systems need codes with better shifted correlation properties for asynchronous operation. Pseudo-Noise (PN) sequences fill this role.

Maximum-Length Sequences (m-sequences):

PN sequences generated by Linear Feedback Shift Registers (LFSRs) exhibit remarkable properties:

1. Period: $N = 2^n - 1$ for an $n$-stage LFSR
2. Balance: Contains $2^{n-1}$ ones and $2^{n-1} - 1$ zeros (nearly balanced)
3. Run property: Runs of consecutive same values follow specific pattern
4. Autocorrelation: Sharp peak at zero, approximately $-1/N$ elsewhere

Autocorrelation of m-sequence:

$$R(\tau) = \begin{cases} N & \tau = 0 \ -1 & \tau \neq 0 \end{cases}$$

This two-valued autocorrelation is ideal for synchronization—a sharp peak indicates alignment.

Gold Code Construction:

While m-sequences have excellent autocorrelation, their cross-correlation can take on many values, some quite large. Gold codes combine pairs of m-sequences to create larger sets with bounded cross-correlation.

Construction: For two m-sequences $a$ and $b$ of period $N = 2^n - 1$:

$$G^{(k)} = a \oplus T^k b \quad \text{for } k = 0, 1, ..., N-1$$

where $T^k b$ is sequence $b$ shifted by $k$ positions and $\oplus$ is XOR.

Gold code set size: $N + 2$ codes (the two original m-sequences plus $N$ XOR combinations)

Cross-correlation bound: For properly chosen preferred pairs: $$|R_{ij}(\tau)| \leq t(n) = \begin{cases} 2^{(n+1)/2} + 1 & n \text{ odd} \ 2^{(n+2)/2} + 1 & n \text{ even} \end{cases}$$

Modern CDMA systems need to support multiple data rates—voice at 12 kbps, video at 384 kbps, data at 2 Mbps—while maintaining orthogonality. Orthogonal Variable Spreading Factor (OVSF) codes solve this problem elegantly.

The Challenge:

Different data rates require different spreading factors:

• Low rate (voice): High SF (e.g., 256) → long codes
• High rate (data): Low SF (e.g., 4) → short codes

Short and long Walsh codes are not automatically orthogonal—a length-4 code might correlate with part of a length-64 code.

OVSF Tree Structure:

OVSF codes are organized as a tree where:

• Root: SF = 1, single code
• Each node spawns two children with doubled SF
• Codes from a parent-child line are not orthogonal
• Codes from different branches are orthogonal

OVSF Allocation Rules:

1. Ancestor blocking: When a code is allocated, all its ancestors and descendants become unavailable
2. Sibling orthogonality: Codes on different branches (not ancestor-descendant) remain orthogonal regardless of SF
3. Capacity trade-off: One SF=4 user blocks four potential SF=16 users

Code Allocation Example:

If we allocate C4_1 = (1,1,1,1), then:

• ✗ C2_1 = (1,1) is blocked (parent)
• ✗ C1 = (1) is blocked (grandparent)
• ✗ All descendants of C4_1 are blocked (if tree extended)
• ✓ C4_2, C4_3, C4_4 are available (siblings, cousins)
• ✓ C2_2 is available (uncle)

The mathematical properties of code orthogonality have profound practical implications for CDMA system design, capacity, and performance.

We've explored orthogonality from mathematical definition through practical system implications. Let's consolidate the essential concepts:

Critical Formulas:

$$\rho_{12} = \frac{1}{N} \sum_{i=0}^{N-1} c_1[i] \cdot c_2[i] = 0 \text{ (for orthogonal codes)}$$

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

$$C_{\text{actual}} = C_{\text{ideal}} \times \alpha \text{ (orthogonality factor)}$$

What's Next:

In the final page, we explore cellular applications of channelization—how FDMA, TDMA, and CDMA have been deployed in real cellular networks from 1G through 4G/LTE. We'll examine how hybrid systems combine these techniques and how spectral efficiency has evolved across mobile generations.