Ferkans — Interactive Telecom Tutor

ex01-helmholtz-derivation

Easy

Starting from the vector wave equation $\nabla \times \nabla \times \mathbf{E} - \kappa^{2}(\mathbf{r})\mathbf{E} = 0$ :

(a) Show that for TM polarization ( $\mathbf{E} = E_z\hat{z}$ ) in a 2D geometry, the vector equation reduces to the scalar Helmholtz equation $\nabla^2 E_z + \kappa^{2}(\mathbf{r})E_z = 0$ .

(b) For TE polarization ( $\mathbf{H} = H_z\hat{z}$ ), derive the corresponding scalar equation and show it involves $\nabla \cdot (1/\varepsilon \nabla H_z)$ .

(c) Explain why TM is preferred for scalar imaging formulations.

Show Hint

For TM, $\nabla \times (E_z\hat{z}) = \hat{x}\partial_y E_z - \hat{y}\partial_x E_z$ .

In TE, the boundary conditions involve $1/\varepsilon$ , complicating the formulation.

Solution

TM reduction

For $\mathbf{E} = E_z\hat{z}$ and $\partial/\partial z = 0$ : $\nabla \times \nabla \times (E_z\hat{z}) = -\nabla^2 E_z\,\hat{z}$ (since $\nabla \cdot \mathbf{E} = 0$ for TM in 2D). The equation becomes $\nabla^2 E_z + \kappa^{2} E_z = 0$ .

TE derivation

From $\nabla \times \mathbf{E} = j\omega\mu_0\mathbf{H}$ and $\nabla \times \mathbf{H} = -j\omega\varepsilon\mathbf{E}$ , eliminating $\mathbf{E}$ : $\nabla \times (1/\varepsilon \nabla \times (H_z\hat{z})) + \omega^2\mu_0 H_z\hat{z} = 0$ , which gives $\nabla \cdot (1/\varepsilon \nabla H_z) + \omega^2\mu_0 H_z = 0$ .

TM preference

TM has continuous $E_z$ and $\partial E_z/\partial n$ at interfaces, yielding a standard Helmholtz equation. TE requires handling the $1/\varepsilon$ coefficient, which complicates numerical discretization.

ex02-greens-function-properties

Easy

For the 2D free-space Green's function $G(\mathbf{r}, \mathbf{r}') = \frac{j}{4}H_0^{(2)}(\kappa_{0}|\mathbf{r}-\mathbf{r}'|)$ :

(a) Verify that $G$ satisfies $\nabla^2 G + \kappa_{0}^{2} G = -\delta(\mathbf{r} - \mathbf{r}')$ for $\mathbf{r} \neq \mathbf{r}'$ .

(b) Show that $G$ satisfies the Sommerfeld radiation condition.

(c) Derive the far-field asymptotic form using the large-argument expansion of $H_0^{(2)}$ .

Show Hint

The large-argument expansion is $H_0^{(2)}(z) \sim \sqrt{2/(\pi z)}\,e^{-j(z - \pi/4)}$ .

Solution

Helmholtz verification

$H_0^{(2)}(\kappa_{0} r)$ satisfies $r^{-1}\partial_r(r\partial_r H_0^{(2)}) + \kappa_{0}^{2} H_0^{(2)} = 0$ for $r > 0$ (Bessel's equation of order 0).

Radiation condition

For $H_0^{(2)}(z) \sim \sqrt{2/(\pi z)}\,e^{-j(z-\pi/4)}$ : $\partial_r G \sim -j\kappa_{0} G$ as $r \to \infty$ , so $\sqrt{r}(\partial_r G - (-j\kappa_{0}) G) \to 0$ . Satisfied.

Far-field form

$G \sim \frac{j}{4}\sqrt{\frac{2}{\pi \kappa_{0} r}}\, e^{-j(\kappa_{0} r - \pi/4)} = \frac{e^{-j\pi/4}}{4}\sqrt{\frac{2}{\pi \kappa_{0} r}}\,e^{-j\kappa_{0} r}.$ $

ex03-lippmann-schwinger-1d

Easy

Consider 1D scattering: $u'' + \kappa_{0}^{2}(1+\chi(x))u = 0$ on $\mathbb{R}$ with $\chi(x) = \chi_0$ for $|x| < a$ and $\chi = 0$ otherwise. Incident field: $u^{\text{inc}} = e^{-j\kappa_{0} x}$ .

(a) Write the 1D Green's function: $G(x, x') = \frac{j}{2\kappa_{0}}e^{-j\kappa_{0}|x-x'|}$ .

(b) Write the Lippmann-Schwinger equation for this problem.

(c) For $\chi_0 = 0.1$ , $\kappa_{0} a = 0.5$ , compute $u^{\text{sca}}_{\text{Born}}(x)$ for $x > a$ .

Show Hint

The Born scattered field for $x > a$ involves integrating $e^{-j\kappa_{0} x'} e^{-j\kappa_{0}(x-x')}$ over $[-a, a]$ .

Solution

Lippmann-Schwinger equation

$u(x) = e^{-j\kappa_{0} x} + \kappa_{0}^{2}\chi_0 \int_{-a}^{a} \frac{j}{2\kappa_{0}}e^{-j\kappa_{0}|x-x'|}\,u(x')\,dx'.$ $

Born approximation for transmitted field

Replace $u(x') \approx e^{-j\kappa_{0} x'}$ inside:

$u^{\text{sca}}_{\text{Born}}(x) = \frac{j\kappa_{0}\chi_0}{2} \int_{-a}^{a} e^{-j\kappa_{0}(x-x')} e^{-j\kappa_{0} x'}\,dx' = \frac{j\kappa_{0}\chi_0}{2} e^{-j\kappa_{0} x} \cdot 2a = j\kappa_{0}\chi_0 a\,e^{-j\kappa_{0} x}.$

For $\chi_0 = 0.1$ , $\kappa_{0} a = 0.5$ : $|u^{\text{sca}}_{\text{Born}}| = 0.05$ , about 5% of the incident field.

ex04-born-cylinder

Medium

A circular cylinder of radius $a$ and uniform contrast $\chi_0$ is illuminated by a plane wave $u^{\text{inc}} = e^{-j\kappa_{0} x}$ .

(a) Using the Born approximation, show that the far-field scattering amplitude is proportional to $J_1(\kappa_{0} a\,|\hat{k}^{\text{inc}} - \hat{r}|)$ .

(b) Plot $|f(\theta)|^2$ for $\kappa_{0} a = 1, 3, 5$ .

(c) Compare with the exact Mie series for $\chi_0 = 0.1$ and $\chi_0 = 0.5$ .

Show Hint

The Fourier transform of a disk is an Airy pattern involving $J_1$ .

The scattering vector magnitude is $|\mathbf{K}| = 2\kappa_{0}\sin(\theta/2)$ .

Solution

Born scattering amplitude

$f(\theta) \propto -\kappa_{0}^{2}\chi_0 \int_{\text{disk}} e^{j\mathbf{K}\cdot\mathbf{r}'}\,d\mathbf{r}' = -\kappa_{0}^{2}\chi_0 \pi a^2 \frac{2J_1(Ka)}{Ka},$ $where$ K = 2\kappa_{0}\sin(\theta/2)$.

Lobe interpretation

The number of lobes increases with $\kappa_{0} a$ . The first null occurs at $Ka \approx 3.83$ , i.e., $\theta_{\text{null}} = 2\arcsin(3.83/(2\kappa_{0} a))$ .

ex05-fourier-coverage

Medium

Consider a 2D scattering experiment with $P$ equally spaced illumination angles and $Q$ equally spaced receiver angles on a circle of radius $R$ .

(a) Show that the sampled spatial frequencies $\mathbf{K}_{pq}$ lie on arcs of circles in Fourier space.

(b) Sketch the complete coverage for $P = Q = 36$ (10-degree spacing).

(c) Identify the "missing cone" for limited-angle configurations ( $P = 18$ , angles $[0, \pi]$ ).

(d) How does wideband illumination ( $\kappa_{0}$ to $2\kappa_{0}$ ) change the coverage?

Show Hint

For fixed $|\mathbf{k}_p| = |\mathbf{k}_q| = \kappa_{0}$ , the tips of $\mathbf{K}$ trace a circle of radius $\kappa_{0}$ .

Solution

Ewald sphere arcs

$\mathbf{K}_{pq} = \mathbf{k}_p - \mathbf{k}_q$ with $|\mathbf{k}_p| = |\mathbf{k}_q| = \kappa_{0}$ . For fixed $p$ , varying $q$ traces an arc of radius $\kappa_{0}$ centered at $\mathbf{k}_p$ . The union over all $p$ fills a disk of radius $2\kappa_{0}$ .

Missing cone

Limited-angle configurations leave a "missing cone" in Fourier space corresponding to the directions not illuminated. This causes directional resolution degradation in the reconstruction.

ex06-rytov-slab

Medium

A dielectric slab of thickness $L$ and contrast $\chi_0$ is illuminated at normal incidence.

(a) Compute the exact transmission coefficient $T$ .

(b) Compute the Born approximation at $x = L$ .

(c) Compute the Rytov phase $\phi_1 = u^{\text{sca}}_{\text{Born}}/u^{\text{inc}}$ .

(d) Compare $|T_{\text{exact}}|$ , $|1 + u^{\text{sca}}_{\text{Born}}/u^{\text{inc}}|$ , and $|e^{\phi_1}|$ for $\chi_0 = 0.1$ and $\kappa_{0} L \in [0.1, 20]$ .

Show Hint

The exact $T = 4kk_0/((k+k_0)^2 e^{-jkL} - (k-k_0)^2 e^{jkL})$ where $k = \kappa_{0}\sqrt{1+\chi_0}$ .

Solution

Born and Rytov comparison

For small $\kappa_{0} L$ , both Born and Rytov agree with exact. As $\kappa_{0} L$ increases with $\chi_0 = 0.1$ : Born deviates (accumulated amplitude error), while Rytov remains accurate (phase error stays bounded because $|\chi_0|$ is small). This demonstrates Rytov's advantage for large, weakly scattering objects.

ex07-dbim-steps

Hard

Implement DBIM for a 2D circular inhomogeneity with $\chi_0 = 0.5$ and $\kappa_{0} a = 2$ , using 16 transmitters and 16 receivers.

(a) Generate synthetic data using MoM.

(b) Initialize with the Born approximation.

(c) Run DBIM for 10 iterations with Tikhonov regularization.

(d) Plot reconstructions at iterations 0, 1, 3, 5, 10 and track the relative error.

(e) Compare with BIM (free-space Green's function throughout).

Show Hint

For 2D MoM, use pulse basis functions and point matching.

The system matrix is $\mathbf{I} - \mathbf{G}\text{diag}(\chi)$ .

Solution

Expected convergence

Born (iteration 0): ~30% error for $\kappa_{0} a|\chi_0| = 1$ . DBIM: error drops below 10% by iteration 5--8. BIM: slower convergence, reaching 10% error around iteration 15--20.

ex08-optical-theorem

Medium

For a 2D PEC cylinder of radius $a$ with $\kappa_{0} a = 3$ (TM incidence):

(a) Compute the forward scattering amplitude $f(0)$ using the Mie series.

(b) Compute $\sigma_{\text{ext}}$ via the optical theorem.

(c) Compute $\sigma_{\text{sca}}$ by integrating $|f(\theta)|^2$ .

(d) Verify agreement (PEC: no absorption, so $\sigma_{\text{ext}} = \sigma_{\text{sca}}$ ).

Show Hint

Mie coefficients for TM PEC cylinder: $a_n = -J_n(\kappa_{0} a)/H_n^{(2)}(\kappa_{0} a)$ .

Solution

Verification

Both methods should agree to numerical precision. The optical theorem $\sigma_{\text{ext}} = \sqrt{8\pi/\kappa_{0}}\,\text{Im}\{e^{j\pi/4}f(0)\}$ provides a single-measurement check on the total scattering power.

ex09-fresnel-number

Easy

A radar antenna of aperture $D = 1$ m operates at $f \in \{1, 3, 10, 30, 77\}$ GHz.

(a) Compute $\lambda$ and the far-field distance $R_{\text{ff}} = 2D^2/\lambda$ .

(b) Compute the Fresnel number $\mathcal{F} = D^2/(\lambda R)$ at $R \in \{10, 100, 1000\}$ m.

(c) Classify each case as Fraunhofer, Fresnel, or geometric optics.

(d) For automotive radar ( $f = 77$ GHz, $D = 5$ cm), find the Fraunhofer distance.

Show Hint

$R_{\text{ff}} = 2D^2/\lambda$ corresponds to $\mathcal{F} = 0.5$ .

Solution

Computation

$f$ (GHz)	$\lambda$ (cm)	$R_{\text{ff}}$ (m)
1	30	6.67
3	10	20
10	3	66.7
30	1	200
77	0.39	513

Automotive radar: $R_{\text{ff}} = 2(0.05)^2/0.0039 \approx 1.3$ m.

ex10-rcs-composite

Medium

Two circular PEC cylinders of radii $a_1 = 0.1\lambda$ and $a_2 = 0.2\lambda$ , separated by distance $d$ .

(a) For $d = 5\lambda$ , compute monostatic RCS using independent scattering.

(b) For $d = 0.5\lambda$ , explain why independent scattering fails.

(c) Identify angular directions of constructive/destructive interference.

Show Hint

For independent scattering, the total field is the sum with appropriate phase factors.

Solution

Independent vs coupled scattering

At $d = 5\lambda$ , the cylinders are well-separated and their interactions are weak (Rayleigh regime, small RCS). At $d = 0.5\lambda$ , mutual coupling is significant — the field scattered by one cylinder illuminates the other, producing additional scattering not captured by the independent model.

ex11-sensing-matrix-svd

Hard

For a 2D Born imaging setup with $P = 8$ transmitters and $Q = 16$ receivers on a circle, imaging a $20 \times 20$ grid:

(a) Construct $\mathbf{A} \in \mathbb{C}^{128 \times 400}$ .

(b) Compute the SVD and plot singular values.

(c) Relate the number of significant singular values to Ewald sphere coverage.

(d) Compute the condition number vs frequency.

(e) Reconstruct a point scatterer with Tikhonov regularization.

Show Hint

The number of significant singular values $\approx$ area of Fourier coverage / Fourier cell size.

Solution

SVD analysis

The singular value spectrum decays gradually. The number of significant values is approximately the number of independent Fourier samples within the Ewald sphere, bounded by $2PQ$ (number of measurements) and the grid resolution.

ex12-born-caire-derivation

Medium

Following Caire's research note, derive the discrete sensing model starting from the Born integral.

(a) Starting from $x(\mathbf{s}, \mathbf{r}; f) \propto \int_\Omega \frac{c(\mathbf{p})}{d(\mathbf{s}, \mathbf{p})d(\mathbf{p}, \mathbf{r})} e^{-j\kappa(d(\mathbf{s},\mathbf{p})+d(\mathbf{p},\mathbf{r}))}d\mathbf{p}$ , apply the first-order Taylor expansion of the distances around $\mathbf{p}_{0}$ .

(b) Define the transmit and receive wavenumber vectors $\boldsymbol{\kappa}_s$ , $\boldsymbol{\kappa}_r$ and show the integral becomes a Fourier transform of $c$ .

(c) Discretize on a grid of $Q$ voxels to obtain $\ntn{img} = \mathbf{A}\ntn{refl_vec} + \mathbf{w}$ .

Show Hint

The Taylor expansion is $d(\mathbf{s}, \mathbf{p}) \approx d(\mathbf{s}, \mathbf{p}_{0}) + \nabla_{\mathbf{p}} d|_{\mathbf{p}_{0}} \cdot (\mathbf{p} - \mathbf{p}_{0})$ .

Solution

Wavenumber decomposition

After Taylor expansion, the phase becomes $-j(\boldsymbol{\kappa}_s + \boldsymbol{\kappa}_r)^\mathsf{T}(\mathbf{p} - \mathbf{p}_{0})$ where $\boldsymbol{\kappa}_s = \kappa\frac{\mathbf{p}_{0} - \mathbf{s}}{d(\mathbf{s}, \mathbf{p}_{0})}$ . The integral is the Fourier transform of $c$ evaluated at $\ntn{comb_wavenum} = \boldsymbol{\kappa}_s + \boldsymbol{\kappa}_r$ .

ex13-contrast-source-inversion

Challenge

Implement CSI for a 2D phantom with two concentric cylinders (inner $\chi = 0.8$ , outer $\chi = 0.3$ ) using 24 Tx, 24 Rx at 1 GHz.

(a) Generate data via MoM.

(b) Implement CSI with alternating updates for $w$ and $\chi$ .

(c) Compare with Born inversion and DBIM in reconstruction quality and computation time.

(d) Add 10% noise and compare robustness.

Show Hint

CSI update: $w^{(k+1)} = w^{(k)} + \alpha \mathcal{G}_S^*(\mathbf{y} - \mathcal{G}_S w^{(k)}) + \beta \chi^{(k)}(u^{\text{inc}} + \mathcal{G}w^{(k)} - w^{(k)}/\chi^{(k)})$ .

Solution

Expected results

CSI converges faster per wall-clock time than DBIM (no forward solves), but may require more iterations. Born fails badly for $\chi = 0.8$ . DBIM provides the best reconstruction quality but is slowest. CSI offers the best speed-accuracy tradeoff.

ex14-greens-function-3d

Easy

A point source at the origin radiates at $f = 5$ GHz.

(a) Compute $|G_{\text{3D}}|$ at $r = 1$ cm, $10$ cm, and $1$ m.

(b) At what distance does the field magnitude drop to 1% of its value at $r = 1$ cm?

(c) How many wavelengths fit between $r = 0$ and $r = 1$ m?

Show Hint

$|G_{\text{3D}}| = 1/(4\pi r)$ . At 5 GHz, $\lambda = 6$ cm.

Solution

Magnitude and distance

$|G(1\text{cm})| = 7.96$ m $^{-1}$ , $|G(10\text{cm})| = 0.796$ m $^{-1}$ , $|G(1\text{m})| = 0.0796$ m $^{-1}$ .

1% level: $r = 100 \times 1\text{cm} = 1$ m. Wavelengths in 1 m: $1/0.06 \approx 16.7$ .

ex15-reciprocity-measurements

Medium

A monostatic radar system collects backscattering measurements from $N$ angles. A bistatic system uses $M$ transmitters and $N$ receivers.

(a) Without reciprocity, how many independent measurements does each system provide?

(b) With reciprocity, how many independent measurements remain?

(c) For $M = N = 16$ , compute the reduction factor.

(d) How does this affect the condition number of $\mathbf{A}$ ?

Show Hint

Reciprocity: $f(\hat{r}; \hat{k}) = f(-\hat{k}; -\hat{r})$ .

Solution

Measurement counts

(a) Monostatic: $N$ measurements. Bistatic: $MN$ measurements. (b) Monostatic: $N$ (already self-reciprocal). Bistatic: $M(N+1)/2$ unique (upper triangle + diagonal) when $M = N$ . (c) For $M = N = 16$ : from 256 to 136 unique measurements (47% reduction). (d) Reciprocity does not change the condition number — the redundant measurements carry no new information.

Exercises

ex01-helmholtz-derivation

TM reduction

TE derivation

TM preference

ex02-greens-function-properties

Helmholtz verification

Radiation condition

Far-field form

ex03-lippmann-schwinger-1d

Lippmann-Schwinger equation

Born approximation for transmitted field

ex04-born-cylinder

Born scattering amplitude

Lobe interpretation

ex05-fourier-coverage

Ewald sphere arcs

Missing cone

ex06-rytov-slab

Born and Rytov comparison

ex07-dbim-steps

Expected convergence

ex08-optical-theorem

Verification

ex09-fresnel-number

Computation

ex10-rcs-composite

Independent vs coupled scattering

ex11-sensing-matrix-svd

SVD analysis

ex12-born-caire-derivation

Wavenumber decomposition

ex13-contrast-source-inversion

Expected results

ex14-greens-function-3d

Magnitude and distance

ex15-reciprocity-measurements

Measurement counts