Prerequisites & Notation

Before You Begin

This chapter sits at the intersection of three strands: (i) massive MIMO beamforming and linear precoding, (ii) estimation/detection theory, and (iii) waveform design for delay-Doppler channels. The reader is expected to be comfortable moving between these viewpoints without friction.

Massive MIMO uplink/downlink precoding and SINR analysis(Review ch04)
Self-check: Can you write the SINR at user $k$ under an arbitrary linear precoder $\mathbf{W}$ and identify the per-user power constraint?
Cell-free architecture: coherent combining across APs, macro-diversity, fronthaul models(Review ch11)
Self-check: Can you explain why a user at the edge between two cells benefits from coherent combining of signals received at both APs?
Binary hypothesis testing, Neyman–Pearson lemma, ROC curves, non-central chi-square statistics(Review ch10)
Self-check: Can you derive the detection probability of a target embedded in Gaussian noise as a function of SNR at a fixed false-alarm probability?
Cramer–Rao bound and Fisher information matrix for parameter estimation(Review ch09)
Self-check: Can you write the Fisher information matrix $\mathbf{J}$ for a Gaussian observation model and derive the CRB on a parameter of interest?
OFDM and OTFS waveform basics: time–frequency vs delay–Doppler representation(Review ch02)
Self-check: Can you describe the discrete ambiguity function of a waveform and relate its mainlobe width to the achievable delay/Doppler resolution?
Semidefinite programming: relaxation of quadratic constraints $\mathbf{v}^H \mathbf{A}_i \mathbf{v}$ via lifting to $\mathbf{V} = \mathbf{v}\mathbf{v}^H \succeq 0$ (Review ch16)
Self-check: Can you recognize when an SDP relaxation is tight (e.g., when the optimal $\mathbf{V}$ has rank one)?

Notation for This Chapter

Symbols introduced or specialized in this chapter. Customizable symbols use $\ntn{}$ tokens; the values shown are the defaults from the notation registry. See the NGlobal Notation Table master table.

Symbol	Meaning	Introduced
$N_t$	Number of ISAC-BS transmit antennas (joint comm+sensing array)	s01
$K$	Number of communication users sharing the ISAC waveform	s01
$\mathbf{H}_{k}$	Downlink communication channel vector to user $k$ , $N_t \times 1$	s01
$\mathbf{a}(\theta)$	Transmit array steering vector toward angle $\theta$	s01
$\mathbf{R}_x$	Transmit signal covariance $\mathbb{E}[\mathbf{x}\mathbf{x}^H]$ , $N_t \times N_t$	s03
$P(\theta; \mathbf{R}_x)$	Transmit beampattern: $P(\theta; \mathbf{R}_x) = \mathbf{a}^H(\theta) \mathbf{R}_x \mathbf{a}(\theta)$	s03
$\mathbf{w}$	AWGN at the communication receiver / sensing receiver, $\sim \mathcal{CN}(\mathbf{0}, \sigma^2\mathbf{I})$	s01
$\text{SNR}$	Receive SNR (linear scale)	s01
$C$	Communication capacity (bits/s/Hz)	s02
$D$	Sensing distortion (MSE on the target parameter $\boldsymbol{\eta}$ )	s02
$\mathbf{J}$	Fisher information matrix of the sensing parameter $\boldsymbol{\eta}$	s02
$\mathcal{C}(D)$	Capacity–distortion function: max achievable rate for sensing distortion $\leq D$	s02
$P_d, P_{\text{fa}}$	Target detection probability and false-alarm probability	s04
$L$	Number of distributed APs in the cell-free ISAC network	s04
$\boldsymbol{\eta}$	Sensing parameter vector (e.g., target range, angle, Doppler, RCS)	s01
$\chi(\tau, \nu)$	Ambiguity function of the transmit waveform in delay $\tau$ and Doppler $\nu$	s05

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