Ch. 12 — Multicarrier Modulation

Chapter 12 of Wireless Communications (2nd ed. Draft) — Andrea Goldsmith

Abstract

This chapter presents the theoretical foundations and system models for spread spectrum communication systems, specifically focusing on Direct-Sequence Spread Spectrum (DSSS) and Frequency-Hopping Spread Spectrum (FHSS). It establishes how spreading codes enable interference rejection, multipath resolution via RAKE receivers, and multiuser access through Code-Division Multiple Access (CDMA). The chapter quantitatively analyzes the performance differences between downlink and uplink channels, highlighting the near-far effect and discussing advanced multiuser detection techniques to mitigate interference while managing computational complexity.

Key Concepts

• Direct-Sequence Spread Spectrum (DSSS) System Model: The DSSS transmitter multiplies a linearly modulated data signal $x(t)$ by a wideband spreading code $s_c(t)$ with chip time $T_c$, followed by upconversion. The receiver performs downconversion, despreading using a synchronized copy of $s_c(t)$, and baseband demodulation. This process spreads the signal bandwidth to $1/T_c$, allowing narrowband interference and non-aligned multipath components to be suppressed by the matched filter integration bandwidth of approximately $1/T_s$.
• Spreading Code Autocorrelation and ISI Rejection: The ability of a DSSS system to reject inter-symbol interference (ISI) depends on the autocorrelation function ${\rho }_c(\tau )$ of the spreading code. For an m-sequence of length $N$, the autocorrelation is approximately a delta function $\delta (\tau )$, attenuating delayed multipath components by $-1/N$. Short codes (period $T=N{T}_c=Ts$) suffer residual ISI at integer multiples of the symbol time, while long codes ($N\gg Ts/T_c$) effectively remove interference from typical multipath delays.
• Synchronization and Acquisition: DSSS receivers require precise timing alignment between the local spreading code generator and the incoming signal. Acquisition involves a coarse search (e.g., feedback loop or parallel search) to align within a chip time $T_c$, followed by fine tracking to maximize the autocorrelation peak. Synchronization is complicated by multipath channels, where the receiver typically locks to the first component above a threshold, necessitating complex circuitry to handle severe ISI or interference.
• RAKE Receiver Diversity: A RAKE receiver utilizes multiple correlator branches to resolve and coherently combine multipath components separated by more than a chip time. Each branch is synchronized to a distinct delay ${\tau }_i$, effectively converting frequency-selective fading into diversity branches. The outputs are combined using selection combining, equal gain combining, or maximal ratio combining (MRC), improving performance relative to a single-branch receiver locked to only one path.
• Multiuser DSSS and Orthogonal Codes: In multiuser DSSS (CDMA), users are separated by unique spreading codes $s_{c_i}(t)$. Synchronous downlinks can utilize orthogonal codes like Walsh-Hadamard sequences, where cross-correlation ${\rho }_{ij}(0)=0$ eliminates mutual interference. However, orthogonality is destroyed by channel dispersion and asynchronous uplink transmission, introducing multiple access interference (MAI) dependent on the code cross-correlation properties and relative delays.
• The Near-Far Effect: In asynchronous uplink channels, users at different distances experience unequal path losses, leading to significant Received Signal Strength (RSS) disparities. If interference power ${\alpha }_j^2\gg {\alpha }_k^2$ for desired user $k$, the SIR degrades drastically. Power control strategies, such as channel inversion, are employed to equalize received power $P$ at the base station, mitigating this effect but potentially increasing interference to neighboring cells.
• Multiuser Detection (MUD): MUD techniques exploit knowledge of interfering users’ spreading codes to subtract interference rather than treating it as noise. Optimal detection uses a Viterbi algorithm with complexity $O(2^K)$, while suboptimal linear detectors like the Decorrelating detector (inverting cross-correlation matrix $R$) or MMSE detector offer lower complexity ($O(K)$) at the cost of noise enhancement or residual interference. Successive Interference Cancellation (SIC) approaches capacity by decoding strong users first.
• Multicarrier CDMA (MC-CDMA): This technique combines OFDM and CDMA by spreading data symbols across multiple independently fading orthogonal subcarriers in the frequency domain. Each subchannel is modulated with a chip of the spreading sequence. MC-CDMA leverages frequency diversity to combat ISI and allows multiple users to share the channel via code orthogonality across subcarriers, provided synchronization is maintained.

Key Equations and Algorithms

• DSSS Received Signal Model: The despread signal output is modeled as ${\hat{s}}_l={\alpha }_0{s}_l+\text{ISI terms}+\text{Interference terms}+n_l$. The desired signal $s_l$ is attenuated by the LOS gain ${\alpha }_0$, while interference and ISI are attenuated by the spreading code’s correlation properties, specifically ${\rho }_c(\tau )$. This model quantifies how processing gain suppresses out-of-band energy.
• Maximal Linear Code Autocorrelation: For a maximal-length sequence (m-sequence) with period $N$, the autocorrelation over a period is given by ${\rho }_c(\tau )=\{\begin{array}{ll}1-\frac{2|\tau |}{N{T}_c},& |\tau |<(N-1)T_c\\ -\frac{1}{N},& \text{elsewhere}\end{array}$ This equation shows a triangular peak at $\tau =0$ and a low sidelobe level of $-1/N$, which determines the system’s ability to distinguish the desired path from delayed multipath reflections.
• Downlink Signal-to-Interference Ratio (SIR): For a synchronous $K$-user system with $N$ chips per symbol and random spreading sequences, the SIR for user $k$ is approximated by $\text{SIR}\approx \frac{E_s}{I_{total}}\approx \frac{N}{K-1}$ where $N$ represents the processing gain. This indicates that capacity is linearly proportional to the processing gain and inversely proportional to the number of interferers.
• Uplink Signal-to-Interference Ratio with Near-Far: In an asynchronous uplink with channel gain disparity, the SIR for user $k$ is modified to ${\text{SIR}}_k\approx \frac{{\alpha }_k^2{E}_s}{{\sum }_{j\ne k}{\alpha }_j^2{E}_s\xi +\text{noise}}$ where $\xi$ characterizes the code cross-correlation statistics (typically $\xi =1$ for random codes). This highlights that a strong interferer $j$ with ${\alpha }_j\gg {\alpha }_k$ can overwhelm the desired signal regardless of processing gain.
• Decorrelating Multiuser Detector: The linear MUD estimates the bit vector $\mathbf{b}$ by computing $\hat{\mathbf{b}}=\text{sgn}({\mathbf{R}}^{-1}\mathbf{y})$ where $\mathbf{R}$ is the cross-correlation matrix of spreading codes and $\mathbf{y}$ is the matched filter bank output. This operation completely removes MAI in the noise-limited case but amplifies noise components orthogonal to the signal space.
• Minimum Mean-Square Error (MMSE) Detector: To balance interference suppression and noise enhancement, the MMSE detector computes $\mathbf{D}=(\mathbf{ARA}+{\sigma }^2\mathbf{I}{)}^{-1}\mathbf{A}$ where $\mathbf{A}$ contains channel gains and ${\sigma }^2$ is noise variance. This detector minimizes the expected MSE between the estimated and transmitted bits, providing better performance at low SNR compared to the decorrelator.
• RAKE Combining Output SNR: With $J$ branches in a RAKE receiver using maximal ratio combining (MRC), the total output SNR is the sum of individual branch SNRs: ${\gamma }_{total}={\sum }_{j=1}^J{\gamma }_j$ assuming the multipath components are separated by at least one chip time $T_c$ and are uncorrelated. This summation provides the diversity gain necessary to combat severe fading.

Key Claims and Findings

• Spread spectrum systems exhibit robust resistance to narrowband interference because the despreading process spreads the interference power over a bandwidth $1/T_c$, while the integration filter only passes bandwidth $1/T_s$, effectively attenuating the interference power by the processing gain $G\approx N$.
• Orthogonal Walsh-Hadamard codes effectively eliminate multiuser interference in synchronous downlink channels but fail in asynchronous uplink channels due to loss of orthogonality caused by timing offsets and channel multipath.
• The near-far effect in uplink CDMA systems significantly limits capacity unless power control is implemented; without power control, a user with 6 dB less received power supports substantially fewer total users compared to a balanced power scenario.
• Maximal linear (m-sequences) offer excellent autocorrelation properties for ISI rejection but limited cross-correlation properties for multiuser separation; Gold and Kasami codes are preferred for multiuser systems despite slightly inferior autocorrelation, due to their larger set sizes and better cross-correlations.
• Multiuser detection (MUD) improves system capacity by mitigating the near-far effect and MAI, though the optimal Viterbi-based detector scales exponentially with the number of users $2^K$, necessitating suboptimal linear or successive interference cancellation algorithms for practical implementation.
• RAKE receivers convert the multipath channel from a source of ISI into a source of diversity, provided that multipath delays exceed the chip duration $T_c$; combining techniques like MRC optimize performance by weighting branches according to their SNR.
• Multicarrier CDMA (MC-CDMA) combines the frequency diversity gains of OFDM with the multiuser access capabilities of CDMA, allowing data spreading in the frequency domain across independently fading subcarriers to mitigate spectral selectivity.

Terminology

• Chip Time ($T_c$): The duration of a single element in the spreading code sequence, defining the bandwidth of the spread signal as approx $1/T_c$. It determines the resolution of the multipath channel and the spacing between RAKE receiver branches.
• Processing Gain ($G$): The ratio of spread bandwidth to information bandwidth ($G\approx T_s/T_c$), quantifying the theoretical interference rejection capability of a spread spectrum system. A higher gain implies better suppression of narrowband interference and MAI.
• Despreading: The receiver operation of multiplying the incoming spread signal by a synchronized local copy of the spreading code $s_c(t)$. This operation collapses the desired signal bandwidth back to its original width while spreading interferers further, enhancing the SNR of the desired signal.
• Autocorrelation (${\rho }_c(\tau )$): A measure of the similarity between a spreading code and its time-shifted version. An ideal autocorrelation function is a delta function, indicating that the code distinguishes the correct timing delay from all others, crucial for synchronization and multipath rejection.
• Cross-Correlation (${\rho }_{ij}(\tau )$): A measure of the similarity between two different users’ spreading codes $s_{c_i}(t)$ and $s_{c_j}(t)$. Low cross-correlation values minimize multiple access interference (MAI) in multiuser CDMA systems.
• Orthogonal Code Set: A set of spreading codes where the cross-correlation ${\rho }_{ij}(0)$ is zero for all $i\ne j$ when synchronized. Walsh-Hadamard sequences are the primary example, used in synchronous downlinks where timing alignment is guaranteed.
• Near-Far Effect: A phenomenon in uplink CDMA where strong signals from proximate users drown out weak signals from distant users. It occurs because interference is proportional to received power, and without power control, the SIR of the distant user degrades significantly.
• Multiuser Detector (MUD): A receiver structure that jointly detects symbols from multiple users to cancel interference. Unlike single-user matched filters, MUDs utilize knowledge of other users’ codes to improve capacity, often classified as linear (Decorrelator, MMSE) or nonlinear (SIC, Parallel Cancellation).
• M-Sequence: A maximal-length linear feedback shift register sequence with period $N=2^n-1$. They are pseudorandom sequences with good statistical properties (balanced, run-length) and specific autocorrelation characteristics used for spreading code generation.
• RAKE Receiver: A specialized multi-branch receiver designed for spread spectrum systems that aligns its branches with the delays of significant multipath components. It performs coherent combining to exploit the energy contained in scattered signal paths, providing frequency diversity.