Ferkans — Interactive Telecom Tutor

Generalising Spatial Modulation

The point is that Spatial Modulation (§3) activates EXACTLY ONE antenna per symbol. Why not allow multiple? Why restrict to the space dimension — can we do the same with SUBCARRIERS in OFDM? With TIME SLOTS? With POLARISATIONS? The unified answer is INDEX MODULATION: use the INDEX of an activated resource — any resource — to carry bits.

Definition:
Generalized Spatial Modulation (GSM)

Generalized Spatial Modulation activates $n_a$ out of $n_t$ antennas per channel use (with $1 \le n_a \le n_t$ ). The activated ANTENNA SET carries $\lfloor \log_2 \binom{n_t}{n_a} \rfloor$ bits of index information; each active antenna additionally transmits a QAM symbol for $n_a \log_2 M$ more bits. Total: $R_{\rm GSM} = \lfloor \log_2 \binom{n_t}{n_a} \rfloor + n_a \log_2 M.$

Theorem: Optimal $n_a$ Maximises GSM Rate

For fixed $n_t$ and $M$ , the GSM rate $R_{\rm GSM}(n_a)$ is maximised at $n_a^* \approx n_t/2$ (approximately). The index bits peak at $n_a = n_t/2$ (binomial maximum), while the modulation bits grow linearly in $n_a$ — the two effects combine to give an interior optimum.

Proof

Binomial maximum

$\binom{n_t}{n_a}$ is maximised at $n_a = \lfloor n_t/2 \rfloor$ , reaching $\binom{n_t}{n_t/2} \approx 2^{n_t} / \sqrt{\pi n_t/2}$ (Stirling).

Modulation contribution

$n_a \log_2 M$ is linear in $n_a$ . For fixed $M$ , this adds $\log_2 M$ per additional active antenna.

Combined optimum

Taking the derivative of $R_{\rm GSM}$ with respect to $n_a$ : $dR/dn_a = d(\log_2 \binom{n_t}{n_a})/dn_a + \log_2 M$ . The first term is maximum 0 at $n_a = n_t/2$ (slope changes sign); the second is $\log_2 M > 0$ . Hence the optimum shifts slightly above $n_t/2$ toward larger $n_a$ for larger $M$ . For $M = 16$ : optimum $n_a^* \approx n_t \cdot 0.55$ . $\blacksquare$

GSM Bits Breakdown vs Active Antennas

Bar chart of bits per channel use from index (antenna-set selection) and modulation (per-antenna QAM). Total bits peak at an interior $n_a$ — the GSM sweet spot.

Parameters

Total Tx antennas

n_t

8

Example: GSM with 8 Tx Antennas

Compute the optimal number of active antennas $n_a^*$ and the resulting rate for GSM with $n_t = 8$ antennas and 16-QAM.

Solution

Index bits by $n_a$

$\log_2 \binom{8}{n_a}$ for $n_a = 1..8$ : $n_a = 1$ : 3, $n_a = 2$ : 4.8, $n_a = 3$ : 5.8, $n_a = 4$ : 6.1, $n_a = 5$ : 5.8, $n_a = 6$ : 4.8, ...

Total rate

$R(n_a) = \lfloor \log_2 \binom{8}{n_a} \rfloor + 4 n_a$ : $R(1) = 3 + 4 = 7$ ; $R(2) = 4 + 8 = 12$ ; $R(3) = 5 + 12 = 17$ ; $R(4) = 6 + 16 = 22$ ; $R(5) = 5 + 20 = 25$ ; $R(6) = 4 + 24 = 28$ ; $R(7) = 3 + 28 = 31$ ; $R(8) = 0 + 32 = 32$ .

Optimum

Maximum total rate is at $n_a = 8$ (full activation). For this case the "index" contribution collapses (only one way to activate all 8 antennas). GSM is beneficial for intermediate $n_a$ when RF-chain cost matters.

OFDM Index Modulation

The GSM idea extends to OFDM: activate a subset of subcarriers. The active subcarrier pattern carries index bits; each active subcarrier transmits its own QAM symbol. This is OFDM-IM. Similar generalisations exist for polarisation, time slot, and even relay index in cooperative systems. The unifying framework is "index modulation over any discrete resource".

🔧Engineering Note

Index Modulation in Practice

Index modulation schemes are primarily RESEARCH-STAGE:

SM / GSM: no production wireless standard adopts it. Challenges: increased decoder complexity, susceptibility to channel estimation errors, and lack of incentive in bandwidth- rich systems.
OFDM-IM: proposed for 5G-NR optional modes, not adopted.
Deployments: some IoT prototypes use SM for battery-powered sensors where RF chain count is a hard constraint. No mass-market deployment.
6G candidate: GSM is under study for mmWave sidelink in 3GPP Rel-19 / Rel-20.

Key Takeaway

Index modulation generalises Spatial Modulation: activate a SUBSET of a discrete resource (antennas, subcarriers, time slots, ...) and encode bits in the activated pattern. Total rate includes both index bits and per-resource modulation bits. Optimal active count is around $n_t/2$ . Research-stage, no mass deployment yet.

Index Modulation Generalisations