Sources and References

Primary references — C. A. Balanis, Antenna Theory: Analysis and Design, 4th ed., Wiley, 2016; D. M. Pozar, Microwave Engineering, 4th ed., Wiley, 2012. Supplementary texts — C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed., Wiley, 2012; J. S. Seybold, Introduction to RF Propagation, Wiley, 2005. Online resources — MIT OpenCourseWare 6.013 Electromagnetics and Applications (D. Staelin); IEEE Antennas and Propagation Society educational resources.

---

Chapter 1: Maxwell’s Equations and Wave Propagation

1.1 Maxwell’s Equations in Differential and Integral Form

Maxwell’s equations form the complete mathematical description of classical electromagnetism. In a linear, isotropic medium with permittivity \(\epsilon\), permeability \(\mu\), and conductivity \(\sigma\), the differential (point) forms are:

\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]\[ \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} \]\[ \nabla \cdot \mathbf{D} = \rho_v \]\[ \nabla \cdot \mathbf{B} = 0 \]

where \(\mathbf{D} = \epsilon \mathbf{E}\), \(\mathbf{B} = \mu \mathbf{H}\), and \(\mathbf{J} = \sigma \mathbf{E}\) for a conducting medium. The corresponding integral forms, obtained via Stokes’ and divergence theorems, are:

\[ \oint_C \mathbf{E} \cdot d\mathbf{l} = -\frac{d}{dt} \iint_S \mathbf{B} \cdot d\mathbf{S} \]\[ \oint_C \mathbf{H} \cdot d\mathbf{l} = \iint_S \left(\mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}\right) \cdot d\mathbf{S} \]\[ \oiint_S \mathbf{D} \cdot d\mathbf{S} = \iiint_V \rho_v \, dV \]\[ \oiint_S \mathbf{B} \cdot d\mathbf{S} = 0 \]

For time-harmonic fields with \(e^{j\omega t}\) convention (phasor form), time derivatives \(\partial/\partial t\) are replaced by \(j\omega\):

\[ \nabla \times \mathbf{E} = -j\omega\mu\mathbf{H} \]\[ \nabla \times \mathbf{H} = \mathbf{J} + j\omega\epsilon\mathbf{E} = (\sigma + j\omega\epsilon)\mathbf{E} \]

1.2 Wave Equation Derivation

Starting from the phasor Maxwell equations in a source-free (\(\mathbf{J} = 0\), \(\rho_v = 0\)) lossless medium, take the curl of Faraday’s law:

\[ \nabla \times (\nabla \times \mathbf{E}) = -j\omega\mu (\nabla \times \mathbf{H}) \]

Using the vector identity \(\nabla \times (\nabla \times \mathbf{A}) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}\) and substituting Ampere’s law with \(\mathbf{J} = 0\):

\[ \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E} = -j\omega\mu (j\omega\epsilon \mathbf{E}) \]

Since \(\nabla \cdot \mathbf{E} = 0\) in a source-free region, this simplifies to the vector Helmholtz equation:

\[ \nabla^2 \mathbf{E} + k^2 \mathbf{E} = 0 \]

where the wavenumber is:

\[ k = \omega\sqrt{\mu\epsilon} = \frac{2\pi}{\lambda} \]

For free space: \(k_0 = \omega\sqrt{\mu_0\epsilon_0} = \omega/c\), where \(c = 1/\sqrt{\mu_0\epsilon_0} \approx 3 \times 10^8\) m/s. An analogous equation holds for \(\mathbf{H}\).

1.3 Uniform Plane Wave Solution

For a plane wave traveling in the \(+z\) direction with \(\hat{x}\)-polarization, the solution is:

\[ \mathbf{E}(z) = E_0 e^{-j\beta z} \hat{x} \]

where \(\beta = k = \omega\sqrt{\mu\epsilon}\) is the phase constant. The associated magnetic field from Faraday’s law is:

\[ \mathbf{H}(z) = \frac{E_0}{\eta} e^{-j\beta z} \hat{y} \]

where \(\eta = \sqrt{\mu/\epsilon}\) is the intrinsic impedance of the medium. For free space, \(\eta_0 = \sqrt{\mu_0/\epsilon_0} \approx 377\) \(\Omega\).

In time domain:

\[ \mathbf{E}(z,t) = E_0 \cos(\omega t - \beta z) \hat{x} \]

The phase velocity (velocity at which a constant-phase surface travels) is:

\[ v_p = \frac{\omega}{\beta} = \frac{1}{\sqrt{\mu\epsilon}} \]

1.3.1 Wave Polarization

Polarization describes the orientation of \(\mathbf{E}\). Consider two orthogonal components:

\[ \mathbf{E} = (E_x \hat{x} + E_y \hat{y}) e^{-j\beta z} \]

• Linear polarization: \(E_x\) and \(E_y\) are in phase (or 180° out of phase). The \(\mathbf{E}\) vector oscillates along a fixed line.
• Circular polarization: \(\lvert E_x \rvert = \lvert E_y \rvert\) with a 90° phase difference: \(\mathbf{E} = E_0(\hat{x} \mp j\hat{y})e^{-j\beta z}\). Right-hand circular polarization (RHCP) uses the minus sign.
• Elliptical polarization: The general case where amplitudes differ or phase difference is not exactly 90°. The tip of \(\mathbf{E}\) traces an ellipse in the transverse plane.

1.3.2 Poynting Vector and Power Density

The instantaneous power density carried by an electromagnetic wave is the Poynting vector:

\[ \mathbf{S} = \mathbf{E} \times \mathbf{H} \quad [\text{W/m}^2] \]

For time-harmonic fields, the time-averaged Poynting vector (average power density) is:

\[ \mathbf{S}_{\text{av}} = \frac{1}{2} \text{Re}(\mathbf{E} \times \mathbf{H}^*) \]

For a uniform plane wave in a lossless medium:

\[ \mathbf{S}_{\text{av}} = \frac{|E_0|^2}{2\eta} \hat{z} \quad [\text{W/m}^2] \]

1.3.3 Wave Propagation in Lossy Media and Skin Depth

In a lossy medium with conductivity \(\sigma \neq 0\), the effective complex permittivity is:

\[ \epsilon_c = \epsilon' - j\epsilon'' = \epsilon\left(1 - j\frac{\sigma}{\omega\epsilon}\right) \]

The complex propagation constant is:

\[ \gamma = j\omega\sqrt{\mu\epsilon_c} = \alpha + j\beta \]

The attenuation constant \(\alpha\) [Np/m] and phase constant \(\beta\) [rad/m] are:

\[ \alpha = \omega\sqrt{\frac{\mu\epsilon}{2}}\left[\sqrt{1 + \left(\frac{\sigma}{\omega\epsilon}\right)^2} - 1\right]^{1/2} \]\[ \beta = \omega\sqrt{\frac{\mu\epsilon}{2}}\left[\sqrt{1 + \left(\frac{\sigma}{\omega\epsilon}\right)^2} + 1\right]^{1/2} \]

The wave decays as \(e^{-\alpha z}\). The skin depth \(\delta_s\) is the depth at which the field amplitude falls to \(1/e\) of its surface value:

\[ \delta_s = \frac{1}{\alpha} \]

For a good conductor (\(\sigma \gg \omega\epsilon\)):

\[ \delta_s \approx \sqrt{\frac{2}{\omega\mu\sigma}} = \frac{1}{\sqrt{\pi f \mu \sigma}} \]

---

Chapter 2: Reflection and Transmission at Planar Interfaces

2.1 Normal Incidence

Consider a plane wave in medium 1 (\(\eta_1\)) incident normally on a planar boundary at \(z = 0\) separating medium 2 (\(\eta_2\)). Writing incident, reflected, and transmitted fields:

\[ \mathbf{E}^i = E_0^+ e^{-j\beta_1 z} \hat{x}, \quad \mathbf{E}^r = E_0^- e^{+j\beta_1 z} \hat{x}, \quad \mathbf{E}^t = E_0^t e^{-j\beta_2 z} \hat{x} \]

Applying boundary conditions (tangential \(\mathbf{E}\) and \(\mathbf{H}\) continuous at \(z = 0\)):

\[ E_0^+ + E_0^- = E_0^t \]\[ \frac{E_0^+}{\eta_1} - \frac{E_0^-}{\eta_1} = \frac{E_0^t}{\eta_2} \]

Solving, the reflection coefficient \(\Gamma\) and transmission coefficient \(\tau\) are:

\[ \Gamma = \frac{E_0^-}{E_0^+} = \frac{\eta_2 - \eta_1}{\eta_2 + \eta_1} \]\[ \tau = \frac{E_0^t}{E_0^+} = \frac{2\eta_2}{\eta_2 + \eta_1} \]

Note that \(\tau = 1 + \Gamma\). Power conservation requires \(|\Gamma|^2 + \frac{\eta_1}{\eta_2}|\tau|^2 = 1\).

2.2 Oblique Incidence

For a wave incident at angle \(\theta_i\) from the normal, we decompose the field into two fundamental polarizations:

2.2.1 Snell’s Law

Phase matching at the boundary requires the tangential wavenumber to be continuous:

\[ \beta_1 \sin\theta_i = \beta_2 \sin\theta_t \]\[ n_1 \sin\theta_i = n_2 \sin\theta_t \quad \text{(Snell's law)} \]

where \(n = c/v_p = \sqrt{\mu_r \epsilon_r}\) is the refractive index. The angle of reflection equals the angle of incidence: \(\theta_r = \theta_i\).

2.2.2 Perpendicular (TE) Polarization — Fresnel Equations

With \(\mathbf{E}\) perpendicular to the plane of incidence:

\[ \Gamma_\perp = \frac{\eta_2 \cos\theta_i - \eta_1 \cos\theta_t}{\eta_2 \cos\theta_i + \eta_1 \cos\theta_t} \]\[ \tau_\perp = \frac{2\eta_2 \cos\theta_i}{\eta_2 \cos\theta_i + \eta_1 \cos\theta_t} \]

2.2.3 Parallel (TM) Polarization — Fresnel Equations

With \(\mathbf{E}\) in the plane of incidence:

\[ \Gamma_\parallel = \frac{\eta_2 \cos\theta_t - \eta_1 \cos\theta_i}{\eta_2 \cos\theta_t + \eta_1 \cos\theta_i} \]\[ \tau_\parallel = \frac{2\eta_2 \cos\theta_i}{\eta_2 \cos\theta_t + \eta_1 \cos\theta_i} \]

2.2.4 Brewster Angle

The Brewster angle \(\theta_B\) is the incidence angle at which \(\Gamma_\parallel = 0\) — the parallel-polarized reflected wave vanishes. Setting the numerator to zero and using Snell’s law:

\[ \tan\theta_B = \frac{n_2}{n_1} = \sqrt{\frac{\epsilon_2}{\epsilon_1}} \quad \text{(for non-magnetic media)} \]

For the perpendicular polarization, a Brewster angle exists only if \(\mu_1 \neq \mu_2\), which is uncommon at radio frequencies.

2.2.5 Total Internal Reflection

When \(n_1 > n_2\) (denser-to-rarer), Snell’s law predicts \(\sin\theta_t = (n_1/n_2)\sin\theta_i > 1\) beyond the critical angle:

\[ \theta_c = \arcsin\left(\frac{n_2}{n_1}\right) \]

For \(\theta_i > \theta_c\), \(\lvert \Gamma \rvert = 1\) for both polarizations — total internal reflection. The transmitted wave becomes evanescent (exponentially decaying) in medium 2 with no real power transfer.

---

Chapter 3: Multilayer Structures and Transfer Matrix Method

3.1 Transfer Matrix (ABCD) Method

For a stack of \(N\) planar layers, the transfer matrix method provides a systematic way to compute the overall reflection and transmission. For layer \(i\) with thickness \(d_i\), propagation constant \(\beta_i\), and intrinsic impedance \(\eta_i\), the characteristic matrix relating tangential \(\mathbf{E}\) and \(\mathbf{H}\) across the layer is:

\[ M_i = \begin{pmatrix} \cos(\beta_i d_i) & -j\eta_i \sin(\beta_i d_i) \\ -j\frac{1}{\eta_i}\sin(\beta_i d_i) & \cos(\beta_i d_i) \end{pmatrix} \]

The total system matrix is the product:

\[ M = M_1 M_2 \cdots M_N = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \]

The reflection coefficient from medium 0 (incidence) into medium \(s\) (substrate) is:

\[ \Gamma = \frac{A\eta_s + B - (C\eta_0\eta_s + D\eta_0)}{A\eta_s + B + C\eta_0\eta_s + D\eta_0} \]

3.2 Anti-Reflection Coatings

A single-layer anti-reflection (AR) coating on a substrate of index \(n_s\) in air (\(n_0 = 1\)) achieves zero reflection when:

1. Thickness condition: \(d = \lambda_1/4\) (quarter-wave optical thickness), so \(\beta_1 d = \pi/2\).
2. Impedance condition: \(\eta_1 = \sqrt{\eta_0 \eta_s}\), i.e., \(n_1 = \sqrt{n_0 n_s} = \sqrt{n_s}\).

Under these conditions, the matrix for the single layer becomes:

\[ M_1 = \begin{pmatrix} 0 & -j\eta_1 \\ -j/\eta_1 & 0 \end{pmatrix} \]

Substituting into the reflection formula yields \(\Gamma = 0\) when \(\eta_1^2 = \eta_0\eta_s\). Multi-layer coatings extend the bandwidth over which low reflection is achieved, exploiting constructive and destructive interference between multiple reflected waves.

---

Chapter 4: Metallic Waveguides

4.1 Rectangular Waveguide Theory

A rectangular metallic waveguide is a hollow tube with inner dimensions \(a \times b\) (typically \(a > b\)) and PEC walls. The field inside satisfies the Helmholtz equation subject to boundary conditions: tangential \(\mathbf{E} = 0\) at all PEC walls.

Assuming propagation in the \(+z\) direction as \(e^{-j\beta z}\), the transverse fields are derived from either a \(z\)-component of \(\mathbf{E}\) (TM modes, \(H_z = 0\)) or \(\mathbf{H}\) (TE modes, \(E_z = 0\)).

4.2 TE\(_{mn}\) Modes

For TE modes (\(E_z = 0\)), \(H_z\) satisfies the 2D Helmholtz equation with Neumann BCs:

\[ H_z(x,y) = H_0 \cos\!\left(\frac{m\pi x}{a}\right)\cos\!\left(\frac{n\pi y}{b}\right) \]

where \(m = 0, 1, 2, \ldots\) and \(n = 0, 1, 2, \ldots\) (not both zero). The transverse fields are found from:

\[ \mathbf{E}_t = \frac{-j\omega\mu}{k_c^2}(\hat{z} \times \nabla_t H_z), \quad \mathbf{H}_t = \frac{-j\beta}{k_c^2}\nabla_t H_z \]

where the cutoff wavenumber is:

\[ k_c = \sqrt{\left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2} \]

4.3 Cutoff Frequency

The cutoff frequency of the TE\(_{mn}\) (or TM\(_{mn}\)) mode is:

\[ f_{c,mn} = \frac{k_c}{2\pi\sqrt{\mu\epsilon}} = \frac{c}{2}\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2} \]

For free-space filling. The propagation constant is:

\[ \beta_{mn} = \sqrt{k^2 - k_c^2} = \frac{2\pi}{c}\sqrt{f^2 - f_{c,mn}^2} \]

Propagation occurs only when \(f > f_{c,mn}\). Below cutoff, \(\beta\) becomes imaginary and the mode is evanescent.

4.4 Dominant TE\(_{10}\) Mode

The lowest-order mode (longest cutoff wavelength) is TE\(_{10}\) with \(m=1, n=0\):

\[ f_{c,10} = \frac{c}{2a} \]

The field components are:

\[ H_z = H_0 \cos\!\left(\frac{\pi x}{a}\right) e^{-j\beta z} \]\[ E_y = \frac{j\omega\mu a}{\pi} H_0 \sin\!\left(\frac{\pi x}{a}\right) e^{-j\beta z} \]\[ H_x = \frac{j\beta a}{\pi} H_0 \sin\!\left(\frac{\pi x}{a}\right) e^{-j\beta z} \]

with \(E_x = E_z = H_y = 0\). The electric field has only a \(y\)-component, sinusoidally distributed across the \(x\)-direction and uniform in \(y\). This single-mode operation is exploited in practice by choosing \(a < \lambda < 2a\).

4.5 Phase and Group Velocity

In a waveguide, the phase velocity exceeds the speed of light in the medium:

\[ v_p = \frac{\omega}{\beta} = \frac{c}{\sqrt{1 - (f_c/f)^2}} > c \]

The group velocity (velocity of energy and information transport) is:

\[ v_g = \frac{d\omega}{d\beta} = c\sqrt{1 - (f_c/f)^2} < c \]

Note that \(v_p v_g = c^2\). Near cutoff, \(v_g \to 0\) and energy propagation stalls. Far above cutoff, both approach \(c\).

4.6 TM Modes

For TM modes (\(H_z = 0\)), \(E_z\) satisfies Dirichlet BCs:

\[ E_z(x,y) = E_0 \sin\!\left(\frac{m\pi x}{a}\right)\sin\!\left(\frac{n\pi y}{b}\right) \]

Both \(m\) and \(n\) must be nonzero, so TM\(_{11}\) is the lowest TM mode. The cutoff frequency formula is identical to the TE case.

4.7 Power Transmission

The time-averaged power carried by the TE\(_{10}\) mode is:

\[ P = \frac{|E_0|^2 ab}{4\eta} \sqrt{1 - \left(\frac{f_c}{f}\right)^2} \]

where \(E_0\) is the peak electric field amplitude. Power is zero at cutoff and increases toward the TEM value \(|E_0|^2 ab / (4\eta)\) as \(f \gg f_c\).

---

Chapter 5: Radiation Theory

5.1 Retarded Potentials

Radiation arises from time-varying currents. The vector potential \(\mathbf{A}\) satisfies the inhomogeneous Helmholtz equation:

\[ \nabla^2 \mathbf{A} + k^2 \mathbf{A} = -\mu \mathbf{J} \]

The solution using the free-space Green’s function is:

\[ \mathbf{A}(\mathbf{r}) = \frac{\mu}{4\pi} \iiint_V \mathbf{J}(\mathbf{r}') \frac{e^{-jkR}}{R} \, dV' \]

where \(R = \lvert \mathbf{r} - \mathbf{r}' \rvert\) is the distance from source point \(\mathbf{r}'\) to observation point \(\mathbf{r}\). The fields are recovered as \(\mathbf{H} = \nabla \times \mathbf{A}/\mu\) and \(\mathbf{E} = -j\omega\mathbf{A} - j\nabla\Phi/\omega\mu\epsilon\).

5.2 Hertzian Dipole

The Hertzian dipole is an infinitesimally short current element \(\mathbf{J} = I_0 \Delta l \, \delta^3(\mathbf{r}) \hat{z}\). The vector potential is:

\[ \mathbf{A} = \frac{\mu I_0 \Delta l}{4\pi} \frac{e^{-jkr}}{r} \hat{z} \]

In spherical coordinates \((r, \theta, \phi)\), the fields are:

\[ E_\theta = j\eta \frac{k I_0 \Delta l}{4\pi} \left[1 + \frac{1}{jkr} - \frac{1}{(kr)^2}\right] \frac{e^{-jkr}}{r}\sin\theta \]\[ H_\phi = j \frac{k I_0 \Delta l}{4\pi} \left[1 + \frac{1}{jkr}\right] \frac{e^{-jkr}}{r}\sin\theta \]

5.2.1 Near-Field and Far-Field Regions

• Near field (reactive): \(r \ll \lambda/(2\pi)\), i.e., \(kr \ll 1\). The \(1/(kr)^2\) and \(1/(kr)^3\) terms dominate; fields are predominantly reactive (stored energy), decaying rapidly with distance.
• Intermediate (radiating near field / Fresnel): \(\lambda/(2\pi) \lesssim r \lesssim 2D^2/\lambda\).
• Far field (Fraunhofer): \(r \gg \lambda/(2\pi)\) and \(r \gg D^2/\lambda\). Only the \(1/r\) terms survive:

\[ E_\theta^{\text{far}} = j\eta \frac{k I_0 \Delta l}{4\pi} \frac{e^{-jkr}}{r} \sin\theta \]\[ H_\phi^{\text{far}} = \frac{E_\theta^{\text{far}}}{\eta} \]

The \(\sin\theta\) dependence produces a toroidal radiation pattern — maximum broadside to the dipole axis and zero along the axis.

5.3 Radiation Intensity, Directivity, and Gain

The radiation intensity \(U(\theta,\phi)\) [W/sr] is defined as:

\[ U(\theta,\phi) = r^2 \mathbf{S}_{\text{av}} \cdot \hat{r} = \frac{r^2 |E_\theta|^2}{2\eta} \]

The total radiated power is:

\[ P_{\text{rad}} = \oint U \, d\Omega = \int_0^{2\pi}\int_0^\pi U\sin\theta \, d\theta \, d\phi \]

Directivity is the ratio of radiation intensity in a given direction to the isotropic average:

\[ D(\theta,\phi) = \frac{4\pi U(\theta,\phi)}{P_{\text{rad}}} \]

For the Hertzian dipole, integrating the \(\sin^2\theta\) pattern:

\[ P_{\text{rad}} = \frac{\pi\eta}{3}\left(\frac{I_0 \Delta l}{\lambda}\right)^2 \]\[ D_{\text{max}} = \frac{4\pi U_{\text{max}}}{P_{\text{rad}}} = 1.5 \quad (1.76 \text{ dBi}) \]

Gain \(G = \eta_{\text{ant}} D\) accounts for ohmic losses through the radiation efficiency \(\eta_{\text{ant}}\).

---

Chapter 6: Transmitting and Receiving Antennas

6.1 Half-Wave Dipole

The half-wave dipole (\(\ell = \lambda/2\)) has a sinusoidal current distribution \(I(z) = I_0 \cos(kz)\) for \(-\lambda/4 \leq z \leq \lambda/4\). The far-field pattern is:

\[ E_\theta = j\eta \frac{I_0 e^{-jkr}}{2\pi r} \frac{\cos\!\left(\frac{\pi}{2}\cos\theta\right)}{\sin\theta} \]

Key parameters:

• Directivity: \(D = 1.64\) (2.15 dBi)
• Input impedance: \(Z_{\text{in}} = 73 + j42.5\) \(\Omega\) (slightly inductive; resonance achieved by shortening to \(\ell \approx 0.48\lambda\) to remove the reactive part)
• Radiation resistance: \(R_{\text{rad}} = 73\) \(\Omega\)

6.2 Effective Aperture and Receiving Antenna

A receiving antenna intercepts power from an incident wave. The effective aperture (or effective area) \(A_e\) relates received power \(P_r\) to incident power density \(S_i\):

\[ P_r = A_e \cdot S_i \]

The effective aperture is related to directivity by:

\[ A_e = \frac{\lambda^2 D}{4\pi} \]

For the isotropic radiator (\(D=1\)): \(A_{e,\text{iso}} = \lambda^2/(4\pi)\). For the half-wave dipole (\(D=1.64\)): \(A_e = 1.64\lambda^2/(4\pi)\).

6.3 Friis Transmission Equation

Consider a transmitter with power \(P_t\) and gain \(G_t\) and a receiver with gain \(G_r\) separated by distance \(r\) in free space. The power density at the receiver location is:

\[ S = \frac{P_t G_t}{4\pi r^2} \]

The received power is:

\[ P_r = S \cdot A_{e,r} = \frac{P_t G_t}{4\pi r^2} \cdot \frac{\lambda^2 G_r}{4\pi} \]

This is the Friis transmission equation:

\[ \boxed{\frac{P_r}{P_t} = G_t G_r \left(\frac{\lambda}{4\pi r}\right)^2} \]

The factor \((4\pi r/\lambda)^2\) is the free-space path loss \(L_{\text{fs}}\). In decibels:

\[ \frac{P_r}{P_t}\bigg|_{\text{dB}} = G_t|_{\text{dBi}} + G_r|_{\text{dBi}} - L_{\text{fs}}|_{\text{dB}} \]\[ L_{\text{fs}}|_{\text{dB}} = 20\log_{10}(r) + 20\log_{10}(f) + 20\log_{10}\!\left(\frac{4\pi}{c}\right) \approx 20\log_{10}(r) + 20\log_{10}(f) - 147.55 \]

with \(r\) in meters and \(f\) in Hz.

6.4 Radar Equation

A radar transmits power \(P_t\), which illuminates a target with radar cross section (RCS) \(\sigma_{\text{RCS}}\) [m²]. The re-radiated power returns to the receiver (co-located in monostatic radar):

\[ P_r = \frac{P_t G^2 \lambda^2 \sigma_{\text{RCS}}}{(4\pi)^3 r^4} \]

The fourth-power range dependence (\(r^{-4}\)) makes radar systems power-hungry — doubling detection range requires a 16× increase in transmit power.

---

Chapter 7: Antenna Arrays

7.1 Array Factor and Pattern Multiplication

An antenna array consists of multiple antenna elements whose combined radiation pattern is the product of the element pattern and the array factor (AF). The array factor depends only on element positions and excitation amplitudes and phases, not the individual element type.

For a uniform linear array (ULA) of \(N\) isotropic elements along the \(z\)-axis with element spacing \(d\) and progressive phase shift \(\alpha\) between adjacent elements:

\[ AF = \sum_{n=0}^{N-1} a_n e^{jn\psi} \]

where the array phase argument is:

\[ \psi = kd\cos\theta + \alpha = \frac{2\pi d}{\lambda}\cos\theta + \alpha \]

For uniform excitation (\(a_n = 1\)), the AF is a geometric series:

\[ AF = \frac{\sin(N\psi/2)}{\sin(\psi/2)} e^{j(N-1)\psi/2} \]

The normalized array factor magnitude is:

\[ |AF_n| = \frac{1}{N}\left\lvert\frac{\sin(N\psi/2)}{\sin(\psi/2)}\right\rvert \]

This has its main beam at \(\psi = 0\) (i.e., \(\cos\theta_0 = -\alpha/(kd)\)) and sidelobes.

7.2 Broadside and End-Fire Arrays

• Broadside array: Main beam perpendicular to the array axis (\(\theta_0 = 90°\)). Requires \(\alpha = 0\). Main beam at \(\psi = 0\) when \(\theta = 90°\).
• End-fire array: Main beam along the array axis (\(\theta_0 = 0°\) or \(180°\)). Requires \(\alpha = -kd\) (for \(\theta_0 = 0\)). Ordinary end-fire achieves \(D \approx 2N\cdot(d/\lambda)\); Hansen–Woodyard condition (\(\alpha = -(kd + \pi/N)\)) improves gain slightly.

7.3 Half-Power Beamwidth (HPBW)

The HPBW of a broadside ULA (found by solving \(|AF_n|^2 = 1/2\)):

\[ \text{HPBW} \approx \frac{0.886\lambda}{Nd} \text{ (radians)} = \frac{50.8\lambda}{Nd} \text{ (degrees)} \]

for large \(N\). Larger arrays produce narrower beams; doubling \(N\) halves the HPBW.

7.4 Grating Lobes

Grating lobes appear when \(\psi = \pm 2\pi\), creating secondary main beams of equal amplitude to the desired main lobe. The condition for no grating lobes in visible space is:

\[ d < \frac{\lambda}{1 + |\cos\theta_{\text{scan}}|} \]

For broadside (\(\theta_{\text{scan}} = 90°\)), the criterion simplifies to \(d < \lambda\). For end-fire (\(\theta_{\text{scan}} = 0°\)), it requires \(d < \lambda/2\). In phased arrays designed for scanning, \(d = \lambda/2\) is the standard spacing to prevent grating lobes across the full scan range.

7.5 Phased Arrays and Adaptive Beamforming

A phased array steers the main beam electronically by controlling the phase \(\alpha\) at each element. To steer to angle \(\theta_0\):

\[ \alpha = -kd\cos\theta_0 \]

This produces \(\psi = 0\) (peak AF) at \(\theta = \theta_0\). Modern phased arrays use digital phase shifters and can steer beams in microseconds.

Adaptive beamforming (e.g., LCMV, MVDR beamformers) extends this by adjusting complex weights \(w_n\) to maximize the signal-to-interference-plus-noise ratio (SINR). The weight vector:

\[ \mathbf{w} = \frac{R_{\text{in}}^{-1}\mathbf{a}(\theta_0)}{\mathbf{a}^H(\theta_0)R_{\text{in}}^{-1}\mathbf{a}(\theta_0)} \]

where \(R_{\text{in}}\) is the interference-plus-noise covariance matrix and \(\mathbf{a}(\theta_0)\) is the steering vector. Nulls are automatically placed in the directions of interferers.

---

Chapter 8: Radio-Wave Propagation Models

8.1 Free-Space Path Loss

In free space with no obstacles, the received-to-transmitted power ratio follows directly from the Friis equation. Defining free-space path loss as the loss experienced by a link with isotropic antennas (\(G_t = G_r = 1\)):

\[ L_{\text{fs}} = \left(\frac{4\pi r}{\lambda}\right)^2 = \left(\frac{4\pi r f}{c}\right)^2 \]

In dB:

\[ L_{\text{fs}}(\text{dB}) = 20\log_{10}(d) + 20\log_{10}(f) - 147.55 \]

Path loss increases by 6 dB per octave in distance (doubling of distance) or frequency.

8.2 Two-Ray Ground Reflection Model

The two-ray model accounts for a direct path and a ground-reflected path between a transmitter at height \(h_t\) and a receiver at height \(h_r\) separated by horizontal distance \(d\). For large \(d\) (far field), the received power simplifies to:

\[ P_r = P_t G_t G_r \left(\frac{h_t h_r}{d^2}\right)^2 \]

The path loss in the two-ray model scales as \(d^4\) (40 dB/decade) rather than \(d^2\) (20 dB/decade) in free space. This arises because the direct and reflected waves undergo destructive interference at large distances for grazing geometry, producing a \(d^{-4}\) power law.

The breakpoint distance at which the model transitions from \(d^{-2}\) to \(d^{-4}\) behavior is:

\[ d_{\text{break}} = \frac{4 h_t h_r}{\lambda} \]

For \(d < d_{\text{break}}\), the free-space model applies; for \(d > d_{\text{break}}\), the two-ray model governs.

8.3 Log-Distance Path Loss Model

Empirical measurements show that average path loss over a large ensemble of environments can be described by:

\[ \overline{PL}(d) = \overline{PL}(d_0) + 10n\log_{10}\!\left(\frac{d}{d_0}\right) \quad [\text{dB}] \]

where:

• \(d_0\) is a reference distance (typically 1 m for indoor, 100 m or 1 km for outdoor)
• \(n\) is the path loss exponent: \(n = 2\) for free space, \(n = 2.7\)–3.5 for urban macro, \(n = 1.6\)–1.8 for guided propagation in hallways, \(n = 4\)–6 for obstructed indoor environments
• \(\overline{PL}(d_0)\) is the mean path loss at reference distance (often taken as the free-space value)

To account for shadowing — random variation due to obstacles — a zero-mean log-normal random variable \(X_\sigma\) [dB] is added:

\[ PL(d) = \overline{PL}(d) + X_\sigma, \quad X_\sigma \sim \mathcal{N}(0, \sigma_s^2) \]

with \(\sigma_s\) typically 4–12 dB. This combined model is the log-normal shadowing model.

8.4 Okumura-Hata Model

The Okumura model is based on extensive measurements in Tokyo and is presented as a set of curves. Hata fit empirical equations to these curves for three environments:

Urban Macro (\(f\) in MHz, distances in km):

\[ L_{50}(\text{urban}) = 69.55 + 26.16\log f - 13.82\log h_b - a(h_m) + (44.9 - 6.55\log h_b)\log d \]

where \(h_b\) is the base station antenna height (m), \(h_m\) is the mobile antenna height (m), and \(a(h_m)\) is a correction factor:

• For small-medium cities: \(a(h_m) = (1.1\log f - 0.7)h_m - (1.56\log f - 0.8)\)
• For large cities (\(f > 300\) MHz): \(a(h_m) = 3.2(\log(11.75 h_m))^2 - 4.97\)

Validity: \(150 \leq f \leq 1500\) MHz, \(30 \leq h_b \leq 200\) m, \(1 \leq h_m \leq 10\) m, \(1 \leq d \leq 20\) km.

8.5 COST-231 Hata Model

The COST-231 extension covers 1500–2000 MHz (PCS/3G bands), updating the Hata formula with an additional correction term \(C_M\) (0 dB for medium cities and suburbs, 3 dB for metropolitan centers):

\[ L_{50} = 46.3 + 33.9\log f - 13.82\log h_b - a(h_m) + (44.9 - 6.55\log h_b)\log d + C_M \]

The COST-231 Walfish-Ikegami model further refines the prediction for urban environments by explicitly accounting for building height, street width, and street orientation.

---

Chapter 9: Radio Transmission Systems

9.1 Link Budget

A link budget is an accounting of all gains and losses in a communication link. The received signal power (in dBm or dBW) is:

\[ P_r = P_t + G_t + G_r - L_{\text{path}} - L_{\text{misc}} \]

where \(L_{\text{misc}}\) includes feeder/cable losses, atmospheric absorption, polarization mismatch, and implementation losses. The received power must exceed the receiver sensitivity \(P_{\text{min}}\):

\[ P_{\text{min}} = N_0 + NF + \text{SNR}_{\text{min}} + 10\log_{10}(B) \]

where \(N_0 = kT_0 = -174\) dBm/Hz (at room temperature \(T_0 = 290\) K), \(NF\) is the noise figure, and \(B\) is the noise bandwidth.

The link margin is the excess received power above sensitivity:

\[ \text{Margin} = P_r - P_{\text{min}} \quad [\text{dB}] \]

A positive margin provides robustness against fading, additional losses, and interference.

9.2 System Noise Figure

For a cascade of \(N\) stages each with gain \(G_i\) and noise figure \(F_i\), the Friis formula for noise gives the overall noise figure:

\[ F_{\text{total}} = F_1 + \frac{F_2 - 1}{G_1} + \frac{F_3 - 1}{G_1 G_2} + \cdots + \frac{F_N - 1}{\prod_{i=1}^{N-1} G_i} \]

The first stage dominates if its gain is large — hence the low-noise amplifier (LNA) is placed first in receiver chains to minimize total noise figure.

The noise figure \(NF\) in dB is related to the linear noise factor \(F\) by:

\[ NF = 10\log_{10}(F) \quad [\text{dB}] \]

The noise factor is defined as the degradation of SNR:

\[ F = \frac{SNR_{\text{in}}}{SNR_{\text{out}}} = \frac{S_i/N_i}{S_o/N_o} \]

9.3 Thermal Noise and SNR

Thermal noise power in bandwidth \(B\):

\[ N = kTB \quad [W] \]

At room temperature (\(T = 290\) K), \(N = -174 + 10\log_{10}(B)\) dBm. In a 200 kHz GSM channel: \(N = -174 + 53 = -121\) dBm. In a 20 MHz LTE channel: \(N = -174 + 73 = -101\) dBm.

9.4 Bit Error Rate and SNR Requirements

The relationship between BER and \(E_b/N_0\) (energy per bit to noise spectral density) depends on the modulation scheme. For BPSK in AWGN:

\[ BER = Q\!\left(\sqrt{\frac{2E_b}{N_0}}\right) = \frac{1}{2}\text{erfc}\!\left(\sqrt{\frac{E_b}{N_0}}\right) \]

where \(Q(x) = \frac{1}{2}\text{erfc}(x/\sqrt{2})\) is the Q-function. A BER of \(10^{-3}\) requires \(E_b/N_0 \approx 6.8\) dB for BPSK; higher-order modulations (16-QAM, 64-QAM) require progressively more \(E_b/N_0\) for the same BER.

The relationship between SNR (in bandwidth \(B\)) and \(E_b/N_0\) is:

\[ \text{SNR} = \frac{E_b}{N_0} \cdot \frac{R_b}{B} \]

where \(R_b\) is the bit rate and \(R_b/B\) is the spectral efficiency.

---

Chapter 10: Channel Capacity and Wireless Networks

10.1 Shannon Channel Capacity

The Shannon-Hartley theorem gives the maximum error-free data rate (channel capacity) over a channel with bandwidth \(B\) and SNR:

\[ C = B\log_2(1 + \text{SNR}) \quad [\text{bps}] \]

This is the Shannon limit — no coding scheme, however complex, can achieve rates above \(C\) with arbitrarily small error probability. The spectral efficiency \(\eta = C/B\) in b/s/Hz grows logarithmically with SNR.

10.2 Capacity in Fading Channels

In a fading channel where the instantaneous SNR \(\gamma\) varies, the ergodic capacity is:

\[ \bar{C} = B\,\mathbb{E}\!\left[\log_2(1 + \gamma)\right] = B\int_0^\infty \log_2(1+\gamma) f(\gamma)\,d\gamma \]

where \(f(\gamma)\) is the PDF of the instantaneous SNR. For Rayleigh fading, this is lower than the AWGN capacity at the same mean SNR. Outage capacity \(C_{\text{out}}\) is defined such that \(P(C < C_{\text{out}}) = p\), where \(p\) is the outage probability.

10.3 Multiple Access Schemes

Several orthogonalization strategies allow multiple users to share a radio channel:

10.4 OFDM and Multipath Robustness

Orthogonal Frequency-Division Multiplexing (OFDM) divides a wideband channel into \(N\) narrowband subcarriers, each of bandwidth \(B/N\), with subcarrier spacing \(\Delta f = 1/T_s\) chosen to ensure orthogonality. A cyclic prefix (CP) of length \(T_{\text{CP}} \geq \tau_{\text{max}}\) (maximum delay spread) converts the linear convolution of the channel into circular convolution, enabling single-tap frequency-domain equalization per subcarrier:

\[ Y_k = H_k X_k + W_k \]

where \(H_k\) is the complex channel gain on subcarrier \(k\), \(X_k\) is the transmitted symbol, and \(W_k\) is noise. Equalization requires only a division \(\hat{X}_k = Y_k/H_k\), with no inter-symbol interference (ISI) as long as \(T_{\text{CP}} \geq \tau_{\text{max}}\).

10.5 MIMO and Spatial Multiplexing

Multiple-Input Multiple-Output (MIMO) systems with \(N_t\) transmit and \(N_r\) receive antennas exploit the spatial dimension. For a rich-scattering channel with channel matrix \(\mathbf{H} \in \mathbb{C}^{N_r \times N_t}\), the capacity is:

\[ C = B\log_2\det\left(\mathbf{I} + \frac{\text{SNR}}{N_t}\mathbf{H}\mathbf{H}^H\right) \quad [\text{bps}] \]

The singular value decomposition \(\mathbf{H} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^H\) reveals \(\min(N_t, N_r)\) parallel sub-channels (spatial streams). In i.i.d. Rayleigh fading, capacity scales approximately linearly with \(\min(N_t, N_r)\) at high SNR:

\[ C \approx \min(N_t, N_r) \cdot B\log_2\!\left(\frac{\text{SNR}}{\min(N_t,N_r)}\right) \]

This spatial multiplexing gain is the foundation of 4G LTE-A and 5G NR massive MIMO.

---

Chapter 11: System-Level Design Considerations

11.1 Noise and Interference in Wireless Systems

Real receivers contend with both thermal noise and co-channel interference (CCI). The signal-to-interference-plus-noise ratio is:

\[ \text{SINR} = \frac{P_s}{P_I + N} \]

where \(P_I = \sum_i P_{I,i}\) is the total interference power and \(N = kTBF\) is the noise power including receiver noise figure \(F\). In interference-limited systems (\(P_I \gg N\)):

\[ \text{SINR} \approx \frac{P_s}{P_I} \]

Frequency reuse in cellular networks creates CCI. The carrier-to-interference ratio for a hexagonal cellular network with reuse factor \(K\) is:

\[ \frac{C}{I} = \frac{(\sqrt{3K})^n}{6} \]

where \(n\) is the path loss exponent and 6 is the number of co-channel interferers in the first tier.

11.2 Fade Margin and Coverage Planning

Coverage planning requires ensuring the received SNR exceeds the minimum threshold with a specified probability (e.g., 95% area coverage). For log-normal shadowing with standard deviation \(\sigma_s\):

\[ P(\text{coverage}) = Q\!\left(\frac{P_{\text{min}} - \bar{P}_r}{\sigma_s}\right) \]

The required fade margin \(F_m\) to achieve \(p\%\) coverage:

\[ F_m = Q^{-1}(1-p/100) \cdot \sigma_s \]

For 95% coverage with \(\sigma_s = 8\) dB: \(F_m = 1.645 \times 8 \approx 13.2\) dB.

11.3 Complete Link Budget Example

11.4 Review of Key Formulas

The following table summarizes the fundamental equations used throughout this course:

Quantity	Formula	Notes
Wavenumber	\(k = \omega\sqrt{\mu\epsilon}\)	Free space: \(k_0 = \omega/c\)
Intrinsic impedance	\(\eta = \sqrt{\mu/\epsilon}\)	Free space: \(\eta_0 \approx 377\,\Omega\)
Skin depth	\(\delta_s = 1/\sqrt{\pi f \mu \sigma}\)	Good conductors
Normal incidence \(\Gamma\)	\(\Gamma = (\eta_2-\eta_1)/(\eta_2+\eta_1)\)	PEC: \(\Gamma = -1\)
TE\(_{mn}\) cutoff	\(f_{c,mn} = \frac{c}{2}\sqrt{(m/a)^2+(n/b)^2}\)	Rectangular waveguide
Hertzian dipole \(D\)	\(D = 1.5\)	(1.76 dBi)
Half-wave dipole \(D\)	\(D = 1.64\)	(2.15 dBi)
Friis equation	\(P_r = P_t G_t G_r (\lambda/4\pi r)^2\)	Free space
ULA array factor	\(AF = \sum_{n=0}^{N-1} e^{jn\psi}\)	\(\psi = kd\cos\theta + \alpha\)
Shannon capacity	\(C = B\log_2(1+\text{SNR})\)	AWGN channel
Free-space PL (dB)	\(L = 20\log d + 20\log f - 147.55\)	\(d\) in m, \(f\) in Hz
Two-ray model	\(P_r \propto (h_t h_r)^2/d^4\)	\(d > d_{\text{break}}\)
Log-distance PL	\(PL = PL(d_0) + 10n\log(d/d_0)\)	Empirical model

---