Prerequisites

Before You Begin

• Linear algebra: complex matrices, eigendecompositions, Kronecker products(Review ch01)

  Self-check: Can you compute the eigendecomposition of a Hermitian matrix $\mathbf{R} = \mathbf{U}\boldsymbol{\Lambda}\mathbf{U}^H$ and apply the Kronecker product identity $\mathrm{vec}(\mathbf{A}\mathbf{X}\mathbf{B}) = (\mathbf{B}^T \otimes \mathbf{A})\mathrm{vec}(\mathbf{X})$?

• Signals and Fourier analysis: Fourier transform, matched filtering, spectral estimation(Review ch04)

  Self-check: Can you derive the matched filter output for a known waveform in additive white Gaussian noise and explain why the matched filter maximises the output SNR? Can you compute the time-frequency resolution trade-off from the uncertainty principle $\Delta t \cdot \Delta f \geq 1/(4\pi)$?

• Antennas and arrays: array response vectors, beamforming, spatial filtering(Review ch07)

  Self-check: Can you write the array steering vector $\mathbf{a}(\theta) = [1, e^{j2\pi d\sin\theta/\lambda}, \ldots, e^{j2\pi(N-1)d\sin\theta/\lambda}]^T$ and form a beam in a desired direction by applying appropriate phase weights?

• OFDM: subcarrier modulation, cyclic prefix, frequency-domain equalisation(Review ch14)

  Self-check: Can you describe the OFDM transmit signal as $s(t) = \sum_{k=0}^{K-1} X_k e^{j2\pi k \Delta f t}$ and explain how the cyclic prefix converts a linear convolution with the channel into a circular convolution, enabling per-subcarrier equalisation?

• MIMO systems: spatial multiplexing, channel capacity, precoding(Review ch15)

  Self-check: Can you state the MIMO capacity formula $C = \log_2\det\!\bigl(\mathbf{I} + \frac{P}{M}\mathbf{H}\mathbf{H}^{H}/\sigma^2\bigr)$ and explain how singular value decomposition decomposes the MIMO channel into parallel spatial streams?