Wireless Channel Lab

The scene is a plan view of vertical walls standing on a flat ground. For that geometry the 3D problem separates: rays are found in 2D with the image method [24], [25], and the antenna heights only change each ray's unfolded length and add one ground-reflected twin. The transmitter's image tree is pruned by beam tracing [26], [27], which keeps only the images whose beam can actually reach a wall. This cuts the third-order tree of the office scene from 45,080 to 1,340 nodes with no change in the result.

Each wall is a single dielectric slab. Its permittivity and conductivity come from ITU-R P.2040 [21], and its reflection and transmission coefficients include the multiple internal bounces inside the slab:

Wall ends and convex corners diffract. The engine treats each one as a knife edge [22]. Behind the edge it adds a diffracted ray scaled by \(F(\nu)\). In front of the edge, where the obstruction is close to the direct path, it multiplies the direct ray by \(F(\nu)\), so the field stays continuous across the shadow boundary.

The map shows either the coherent sum, with small-scale fading, or the incoherent sum \(\sum_i|a_i|^2\), which approximates the local mean that tools like Wireless InSite [29] and Sionna RT [28] also report. RMS delay spread is \(\sigma_\tau=\sqrt{\overline{\tau^2}-\bar\tau^2}\) over the power-weighted rays [1].

Clarke's model [5] treats the received field as many plane waves arriving from uniformly random directions. The lab builds it as a sum of 32 sinusoids with random arrival angles and phases [7], which gives a complex Gaussian process whose envelope is Rayleigh distributed and whose autocorrelation approaches \(J_0(2\pi f_D\tau)\) [6]. A line-of-sight term turns this into Rician fading [8].

The stat tiles compare the generator against theory over a 10,000-Doppler-period run. For Rayleigh fading the level-crossing rate and average fade duration at normalized level \(\rho\) are [1]:

The coherence time uses the common rule \(T_c\approx 0.423/f_D\) [1].

In your link mode the taps are the rays from question 1, each with its own delay and complex amplitude, so the response is deterministic for a fixed laptop position. In statistical mode the taps follow an exponential power-delay profile, scaled so that its RMS delay spread equals the slider value exactly. Each tap is an independent complex Gaussian (a wide-sense stationary, uncorrelated scattering channel [18]), and the frequency response is \(H(f)=\sum_\ell h_\ell e^{-j2\pi f\tau_\ell}\).

The coherence bandwidth uses the 50%-correlation rule of thumb [1]. For this truncated exponential profile the exact 50% bandwidth is \(0.20\)–\(0.22/\sigma_\tau\), very close to the rule.

OFDM [19], [20] uses an inverse FFT to place data on orthogonal subcarriers. A cyclic prefix longer than the maximum excess delay keeps the echoes from leaking into the next symbol, so each subcarrier only needs a one-tap equalizer. Subcarriers that land in a deep fade are protected by coding and bit loading.

Symbols are drawn uniformly from a unit-energy constellation, complex Gaussian noise of variance \(N_0\) is added, and the receiver picks the nearest point, which is the maximum-likelihood decision for equally likely symbols in AWGN [4]. The measured SER accumulates across frames until a parameter changes.

For square M-QAM the symbol error rate in the key equation is exact [4]; QPSK is the \(M=4\) case and BPSK is \(Q(\sqrt{2E_s/N_0})\). With Gray labelling, one symbol error usually flips a single bit, so \(P_b\approx P_s/\log_2 M\) at useful SNRs.

With \(L\) independent Rayleigh branches combined by maximal-ratio combining [9], the BPSK bit error rate at average branch SNR \(\bar\gamma\) has the closed form [4], [11]:

The Alamouti scheme [10] sends two symbols over two antennas in two slots. Its decision statistic matches two-branch MRC at half the per-antenna SNR, so its curve is \(P_b^{\mathrm{MRC}}(\bar\gamma/2,2)\). The paper checks every curve here against Monte Carlo simulation; all points fall within 2.6 binomial standard deviations.

A uniform linear array with weights \(w_n\) and progressive phase \(\beta=2\pi d\sin\theta_0/\lambda\) has the array factor in the key equation [12], [13]. With uniform weights at broadside the half-power beamwidth is about \(0.886\,\lambda/(Nd)\) radians and the first sidelobe sits near \(-13.3\) dB. Grating lobes stay out of visible space only if

Delay versus phase. A true time delay \(\tau_n\) multiplies element \(n\) by \(e^{-j2\pi f\tau_n}\), which scales with frequency, so the beam points at \(\theta_0\) across the whole band [33], [35]. A phase shifter applies a fixed \(\phi_n = 2\pi f_0\tau_n\), so at frequency \(f\) the beam moves to

an effect called beam squint. A \(B\)-bit phase shifter also rounds each phase to a multiple of \(2\pi/2^B\), which costs about \(-20\log_{10}\!\big[\mathrm{sinc}(\pi/2^B)\big]\) dB of gain [36]. The paper checks both against the engine. Rotman's lens [34] is a classic way to build many true-time-delay beams at once.

Hamming and Blackman windows [14] lower the sidelobes at the cost of a wider beam and a small taper loss \(N\sum w_n^2/(\sum w_n)^2\). The Dolph-Chebyshev taper [15] gives the narrowest beam for a chosen sidelobe level, here \(-30\) dB, with all sidelobes equal.

The channel is \(\mathbf H=\sqrt{\tfrac{K}{K+1}}\,\mathbf a_r\mathbf a_t^H+\sqrt{\tfrac{1}{K+1}}\,\mathbf H_w\), where \(\mathbf H_w\) has i.i.d. \(\mathcal{CN}(0,1)\) entries and \(\mathbf a\) are half-wavelength array steering vectors. With no channel knowledge at the transmitter, power is split equally and capacity is the log-det formula of Telatar [16] and Foschini and Gans [17], computed here from the eigenvalues of \(\mathbf H\mathbf H^H\):

In the paper the Monte Carlo ergodic capacity matches Telatar's closed-form integral within 1.6% for 1×1 through 4×4. Turning scattering down shows the other side: at 20 dB a 4×4 link drops from 22.1 to 10.7 b/s/Hz as the K-factor goes from 0 to 100, because the line-of-sight term has rank one [3].