Ch. 11 — Equalization

Chapter 11 of Wireless Communications (2nd ed. Draft) — Andrea Goldsmith

Abstract

Chapter 12 introduces multicarrier modulation (MCM) as a method to combat frequency-selective fading by dividing a wideband channel into multiple narrowband subchannels. The central technical contribution is the discrete implementation of this technique via Orthogonal Frequency-Division Multiplexing (OFDM), which utilizes the Discrete Fourier Transform (DFT) and a cyclic prefix to convert linear convolution into circular convolution. This chapter establishes the trade-offs between spectral efficiency, implementation complexity, and robustness against intersymbol interference (ISI) and peak-to-average power ratio (PAR), positioning MCM as the preferred solution for high-data-rate wireless systems with moderate to large delay spreads compared to time-domain equalization.

Key Concepts

• Multicarrier Modulation (MCM) Principle: MCM splits a data stream into $N$ parallel substreams transmitted over orthogonal subchannels. The subchannel bandwidth $B_N$ is designed to be significantly smaller than the channel coherence bandwidth $B_c$, ensuring that each subchannel experiences relatively flat fading rather than frequency-selective fading, thereby minimizing ISI.
• Overlapping Subcarrier Orthogonality: To improve spectral efficiency, subchannels are allowed to overlap provided they remain orthogonal. The minimum frequency separation required for orthogonality on the interval $[0,T_N]$ is $\Delta f=1/T_N$, where $T_N$ is the symbol time per subchannel, enabling the use of sinc-shaped spectra without guard bands between carriers.
• Mitigation of Subcarrier Fading: While MCM eliminates ISI, individual subchannels suffer from flat fading. Mitigation techniques include coding with time/frequency interleaving to exploit frequency diversity, frequency equalization (which enhances noise), precoding (inverted at the transmitter), and adaptive loading based on channel state information.
• Cyclic Prefix and Circular Convolution: A cyclic prefix of length $\mu$ appended to each OFDM symbol transforms the channel’s linear convolution into circular convolution with respect to the useful samples. This allows recovery of the data via simple frequency-domain equalization ($Y[i]/H[i]$) without requiring complex time-domain equalizers, provided $\mu$ exceeds the channel delay spread.
• OFDM Discrete Implementation: OFDM replaces bank-of-filters with the Inverse Fast Fourier Transform (IFFT) at the transmitter and Fast Fourier Transform (FFT) at the receiver. This discrete implementation reduces complexity from $O(N^2)$ to $O(N\log N)$ and eliminates the need for complex analog filter banks for each subcarrier.
• Vector Coding (VC) vs. OFDM: Vector coding decomposes the channel matrix $H$ using Singular Value Decomposition (SVD) to create orthogonal subchannels without a cyclic prefix. While VC is theoretically optimal and more energy-efficient, it requires channel knowledge at the transmitter and high computational complexity for SVD, making OFDM preferable for wireless applications.
• Peak-to-Average Power Ratio (PAR): OFDM signals exhibit high PAR due to the coherent addition of subcarriers. For large $N$, the signal envelope approximates a Rayleigh distribution, leading to a maximum PAR of $N$. High PAR necessitates linear power amplifiers with large backoff, reducing energy efficiency.
• Timing and Frequency Offset Sensitivity: Orthogonality is compromised by carrier frequency offset $\delta$ and timing offset $\tau$. Frequency offset introduces Intercarrier Interference (ICI) that grows quadratically with $\delta$, while timing offset introduces phase rotation but minimal ICI if the cyclic prefix is not violated.

Key Equations and Algorithms

• Multicarrier Signal Model: The transmitted signal is the sum of $N$ modulated subcarriers: $s(t)={\sum }_{i=0}^{N-1}{s}_i(t)e^{j2\pi f_{i}t}$. Here $s_i(t)$ is the baseband symbol waveform for the $i$-th subcarrier, and $f_i$ is the carrier frequency, forming the composite passband signal.
• Overlapping System Bandwidth: For $N$ overlapping subchannels with pulse shape rolloff $\beta$ and time-limiting excess $ϵ$, the total bandwidth is approximately $B\approx N/T_N+(1+\beta +ϵ)/T_N$. This contrasts with non-overlapping schemes where bandwidth scales linearly with $N(1+\beta +ϵ)/T_N$.
• Water-Filling Power Allocation: To maximize capacity over $N$ subchannels with gain ${\alpha }_i$, power $P_i$ is allocated as $P_i=[1/\lambda -N_0{B}_N/{\alpha }_i^2{]}^{+}$. This “water-filling” over frequency assigns more power to stronger subchannels, subject to a total power constraint $P=\sum P_i$.
• Cyclic Prefix Construction: For an input block $x[n],0\le n\le N-1$, the prefix copies the last $\mu$ samples: $\tilde{x}[n]=x[N+n]$ for $-\mu \le n<0$. This redundancy converts linear convolution $y=h\ast x$ into circular convolution $y=h\odot x$ over the interval $0\le n<N$.
• PAR Calculation: The PAR of a discrete-time signal is defined as $\text{PAR}=10{\log }_{10}\left(\frac{{\max }_n|x[n]{|}^2}{\text{En}[|x[n]{|}^2]}\right)$ dB. For $N$ independent Gaussian subcarriers, the probability of exceeding a threshold $P_0$ is approximated by $1-(1-e^{-P_0}{)}^N$.
• Intercarrier Interference (ICI) Power: The ICI power on subcarrier $i$ due to normalized frequency offset $\delta$ is approximated as $P_{\text{ICI}}\approx C_0{\sum }_{m\ne 0}{\left|\frac{\sin (\pi (\delta -m))}{\pi (\delta -m)}\right|}^2$. This indicates that ICI increases with the magnitude of the frequency offset $\delta$.

Key Claims and Findings

• Flat Fading Condition: Multicarrier systems ensure flat fading on subchannels by setting the subchannel bandwidth $B_N\ll B_c$, which corresponds to a symbol time $T_N\gg T_m$ (channel delay spread).
• Spectral Efficiency of Overlap: Overlapping subcarriers reduce the required system bandwidth by a factor of roughly 2 compared to non-overlapping schemes, significantly improving spectral efficiency at the cost of requiring tight synchronization.
• Cyclic Prefix Overhead: While the cyclic prefix eliminates ISI between OFDM symbols, it introduces an overhead of $\mu /N$, reducing the effective data rate by a factor of $N/(N+\mu )$; this overhead is minimized when $N\gg \mu$.
• Vector Coding Complexity: Although Vector Coding achieves capacity via SVD without cyclic prefix overhead, the need for transmitter channel knowledge and SVD computation makes it prohibitive for rapidly time-varying wireless channels where OFDM is preferred.
• PAR Scaling: The Peak-to-Average Power Ratio increases approximately linearly with the number of subcarriers $N$, necessitating power amplifier backoff and high-resolution A/D converters to maintain linearity.
• ICI Sensitivity: Frequency offset sensitivity increases with $N$ because larger $N$ typically requires narrower subcarrier spacing ($\Delta f=1/T_N$), making the system more susceptible to Doppler spread and oscillator inaccuracies.

Terminology

• Coherence Bandwidth ($B_c$): The statistical range of frequencies over which the channel transfer function is essentially constant, approximated inversely proportional to the delay spread $T_m$ (i.e., $B_c\approx 1/T_m$).
• Cyclic Prefix: A redundant copy of the tail end of an OFDM symbol appended to the beginning, used to absorb channel delay spread and prevent inter-block interference.
• Circular Convolution: A discrete operation where the sequence is treated as periodic, enabling frequency-domain multiplication ($DFT(y)=DFT(h)\cdot DFT(x)$) via the DFT properties.
• Intercarrier Interference (ICI): Interference caused by the loss of orthogonality between subcarriers, typically resulting from frequency offset, timing error, or Doppler spread.
• Processing Gain: (In context of MCM) The ratio of total bandwidth to data rate, which in OFDM is determined by the number of subcarriers $N$ and the cyclic prefix overhead.
• Singular Value Decomposition (SVD): A matrix factorization $H=U\Sigma V^H$ used in Vector Coding to diagonalize the channel matrix into independent subchannels.
• Water-filling: A power allocation strategy that distributes transmit power across frequency subchannels inversely proportional to the channel noise-to-signal ratio to maximize capacity.
• Guard Band: Unused spectral space introduced between non-overlapping subchannels to prevent overlap, which reduces spectral efficiency compared to the overlap-permitting OFDM approach.