Diversity & Multiplexing Demo

Diversity vs spatial multiplexing demo

Two labs on one page. Receive diversity compares one Rayleigh branch with selection combining (SC) and maximum-ratio combining (MRC) for coherent BPSK/QPSK, as a BER curve or an outage CDF. Multiplexing vs diversity sketches the Zheng–Tse trade-off and ergodic MIMO capacity for an Nt×Nr i.i.d. Rayleigh channel. Controls that do not apply to the current lab are hidden.

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Picks which experiment (and which controls) are active.
BER shows reliability averaged over fading; the CDF shows how often the combined SNR falls below a threshold.
The frontier is the best diversity any scheme can reach at each multiplexing gain; capacity shows what the extra streams buy you.
Number of independent receive branches combined by SC and MRC. The BER slope at high SNR approaches L.
Eb/N0 of each branch (mean over the Rayleigh fading). Every branch has the same average SNR.
Versus Eb/N0 the two have the same BER, so the curves do not move; only the Es/N0 readout changes (QPSK: +3.01 dB).
The link is in outage when the post-combining SNR falls below this level (marked by the orange dashed line).
Light blue curves for every L, so you can see the diminishing returns of each extra branch.
Sets the diversity ceiling NtNr and the number of streams you can multiplex.
Streams are limited by min(Nt, Nr); extra receive antennas add diversity and array gain.

Try it (Lab A): L = 2 BER shows most of the gain from one extra branch; L = 8 approaches the AWGN curve. In outage view, watch how SC and MRC both fall like (γth/γ̄)L but MRC sits lower by a factor L!.

Try it (Lab B): 2×2 frontier versus 4×4, then switch to capacity to see the slope grow with min(Nt, Nr).

Background: Diversity techniques and spatial multiplexing · SIMO models · Selection combining · MRC · MIMO overview.

Controls explained

Lab
A compares combining schemes on a single-input, L-branch receiver. B compares what a multi-antenna link can spend its antennas on: streams (multiplexing) or reliability (diversity).
Plot (Lab A)
BER plots the average bit-error rate versus Eb/N0 per branch, with the unfaded AWGN curve as the limit. Outage CDF plots P(γout < γth) versus threshold at the average SNR you choose, on a log axis.
Branches L
How many independent Rayleigh branches the receiver combines. SC keeps the best branch; MRC co-phases and weights all of them by their SNR. Both reach diversity order L; MRC adds array gain.
Average SNR
Lab A: the mean Eb/N0 on each branch (total transmit power is fixed, so there is no penalty for adding branches except hardware). Marker lines and readouts are evaluated here. Lab B: average receive SNR ρ per receive antenna, with transmit power split equally over the Nt antennas.
Modulation
Coherent BPSK or Gray-coded QPSK. For i.i.d. Rayleigh and coherent detection the BER versus Eb/N0 is identical; the switch only changes the Es/N0 readout.
Outage threshold γth
Only in the outage plot. The probability that the post-combining SNR is below this level. The 1%-outage rows answer the reverse question: which SNR level is exceeded 99% of the time.
Show MRC family
Draws MRC for every L from 2 to 8 as light curves, behind the thick curves for your chosen L.
Nt, Nr (Lab B)
Antenna counts. The trade-off frontier connects the points (k, (Nt−k)(Nr−k)) for k = 0 … min(Nt, Nr). Capacity uses 1000 seeded Monte-Carlo channel draws with equal power per transmit antenna and no transmit CSI, so results are stable and shareable.
Formulas used

Per-branch average SNR γ̄ = Eb/N0, μ = sqrt(γ̄ / (1 + γ̄)).

1 branch: Pb = (1 − μ) / 2.

MRC, L branches: Pb = [(1 − μ)/2]^L · sum_{k=0..L-1} C(L−1+k, k) [(1 + μ)/2]^k.

SC, L branches: Pb = (1/2) · sum_{k=0..L} (−1)^k C(L,k) / sqrt(1 + k/γ̄). The page evaluates the equivalent integral (1/sqrt(pi)) · int_0^inf (1 − exp(−u²/γ̄))^L exp(−u²) du, which does not suffer from cancellation at high SNR.

AWGN limit: Pb = Q(sqrt(2γ)) = erfc(sqrt(γ)) / 2.

Outage CDFs, t = γth/γ̄: 1 branch 1 − e^(−t); SC (1 − e^(−t))^L; MRC 1 − e^(−t) sum_{k=0..L-1} t^k / k!.

Trade-off frontier (Zheng–Tse): piecewise linear through (k, (Nt−k)(Nr−k)). Capacity: C = log2 det(I + (ρ/Nt) H H^H).

How to use this demo

Treat the page as two labs. In Lab A (receive diversity) you compare a single Rayleigh-faded branch with selection combining and maximum-ratio combining over L independent branches. The BER view shows the average error rate against Eb/N0 per branch; the outage view shows how often the combined SNR drops below a threshold. In Lab B (multiplexing vs diversity) you pick the number of transmit and receive antennas and see the best diversity order available at each multiplexing gain, plus ergodic capacity versus SNR.

A quick sanity check: at 15 dB the single-branch BER is about 7.7×10−3; two-branch SC brings it to about 3.5×10−4 and two-branch MRC to about 1.8×10−4. The slope on the log plot steepens from 1 to L, which is exactly the diversity order. Open this preset to see it.

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FAQ

Why does QPSK give the same curves as BPSK? Gray-coded QPSK is two orthogonal BPSK streams, each seeing the same per-bit SNR, so the bit-error rate versus Eb/N0 is identical for coherent detection. Only Es/N0 differs (3.01 dB higher for QPSK at the same Eb/N0).

Why is MRC better than selection combining if both have diversity order L? The diversity order sets the slope; MRC also adds array gain because it sums the branch SNRs instead of keeping only the best one. At small outage thresholds SC behaves like tL while MRC behaves like tL/L!, a constant factor that is worth a few dB.

Are these curves simulated? The Lab A curves are exact closed forms (single branch, MRC) or an exact one-dimensional integral (SC). The capacity plot in Lab B is a seeded Monte-Carlo average of log det(I + ρ/Nt H HH) over 1000 channel draws, so it is repeatable but carries a small sampling error.

What assumptions are behind the diversity numbers? I.i.d. Rayleigh fading across branches, perfect channel knowledge at the receiver, coherent detection, and equal average SNR per branch. Correlated antennas, line-of-sight (Rician) fading, and imperfect channel estimates all reduce the gains shown here.

Why can’t I have full diversity and full multiplexing at the same time? Each antenna pair can either be used to repeat information (diversity) or to carry new information (multiplexing). The Zheng–Tse frontier shows the exact exchange rate: diversity falls to zero as you ask for all min(Nt, Nr) streams.