Spatial Modulation: Antenna Index as Information

Antenna Index as Information Dimension

The point is that massive-array MIMO has MANY more antenna indices than active RF chains. Spatial Modulation (SM, Mesleh et al. 2008) uses the ANTENNA INDEX itself as an information-bearing dimension — only ONE antenna is active per symbol, reducing RF chains to 1. The idea is elegant: log⁡2nt\log_2 n_t bits are encoded "for free" in the activated antenna index, adding to the log⁡2M\log_2 M bits from the transmitted QAM symbol.

Definition:

Spatial Modulation (SM)

A Spatial Modulation (SM) transmitter activates ONE of ntn_t transmit antennas per symbol; the activated antenna index carries log⁡2nt\log_2 n_t bits of information, and the transmitted QAM symbol on that antenna carries log⁡2M\log_2 M additional bits. Total rate: log⁡2nt+log⁡2M\log_2 n_t + \log_2 M bits/channel use with only ONE RF chain. The idle antennas are silent (no transmission).

Theorem: Spatial Modulation Rate Formula

Spatial Modulation with ntn_t transmit antennas and M-ary modulation per active antenna achieves rate RSM=log⁡2nt+log⁡2Mbits/ch.use.R_{\rm SM} = \log_2 n_t + \log_2 M \quad \text{bits/ch.use}. The antenna selection uses log⁡2nt\log_2 n_t bits; the QAM symbol uses log⁡2M\log_2 M bits. Only ONE RF chain is active at any time.

Proof
Encoder

The log⁡2nt+log⁡2M\log_2 n_t + \log_2 M input bits split: the first log⁡2nt\log_2 n_t bits select the antenna index i∈{1,...,nt}i \in \{1, ..., n_t\}; the remaining log⁡2M\log_2 M bits select the QAM symbol ss.

Transmit vector

x=s ei\mathbf{x} = s\,\mathbf{e}_i where ei\mathbf{e}_i is the ii-th standard basis vector in Cnt\mathbb{C}^{n_t}.

Rate

Per channel use, the information carried is exactly log⁡2nt+log⁡2M\log_2 n_t + \log_2 M bits — provided the receiver can reliably detect BOTH the antenna index and the transmitted symbol. ■\blacksquare

Spatial Modulation BER vs Conventional MIMO

BER comparison at matched rate between SM (antenna-index + QAM) and MIMO-multiplex (joint decoding of full antenna vector). SM trades diversity for hardware simplicity.

Parameters
4

Spatial Modulation: Antenna Index as Information

Animated encoder: info bits are split into antenna-index bits and QAM symbol bits. Only the selected antenna is activated; the rest are silent. A single RF chain carries all the transmission.

Example: SM Rate with 4 Tx + 16-QAM

Compute the rate of SM with nt=4n_t = 4 antennas and 16-QAM on each active antenna. Compare with full MIMO multiplexing at matched hardware (also 4 Tx antennas).

Solution
SM rate

RSM=log⁡24+log⁡216=2+4=6R_{\rm SM} = \log_2 4 + \log_2 16 = 2 + 4 = 6 bits/ch.use.

Full MIMO multiplex with 16-QAM

With nt=4n_t = 4 spatial streams each carrying 16-QAM: RMUX=4⋅log⁡216=16R_{\rm MUX} = 4 \cdot \log_2 16 = 16 bits/ch.use — much higher, but requires 4 active RF chains and joint ML detection.

Hardware cost

SM: 1 RF chain, nt=4n_t = 4 antennas. RF hardware cost ~ 11. MIMO-mux: 4 RF chains, 4 antennas. Cost ~ 44. SM is 75% cheaper for a 62% lower rate — potentially attractive for power- constrained IoT devices.

SM Has Diversity 1 at Full Rate

A common misconception: "SM uses ntn_t antennas, so it has diversity ntn_t." NO. SM transmits from ONE antenna at a time; the receiver cannot combine across antennas (the "used" antenna is different per symbol). Raw diversity is 1 (actually nrn_r at the receiver side, since all nrn_r receive antennas see the transmitted signal). To achieve diversity >nr> n_r, must add coding or averaging across multiple symbols.

Common Mistake: SM Decoder Must Detect Antenna Index AND Symbol

Mistake:

"The receiver just decodes 16-QAM on each antenna and picks the antenna with the strongest signal."

Correction:

The optimal ML receiver JOINTLY detects both the antenna index and the QAM symbol by searching over all nt⋅Mn_t \cdot M candidate hypotheses. Simpler "two-stage" detectors (detect antenna, then symbol) have ~3 dB loss at moderate SNR. This complexity is the hidden cost of SM.

Key Takeaway

Spatial Modulation uses antenna index as an information dimension, giving rate log⁡2nt+log⁡2M\log_2 n_t + \log_2 M bits/ch.use with only 1 RF chain. Trade-offs: (a) diversity is only nrn_r (not ntnrn_t n_r); (b) ML detection searches over nt⋅Mn_t \cdot M hypotheses. Best suited to power-constrained IoT and battery-powered sensor applications where RF hardware is the dominant cost.

1-Bit Quantised MIMO: Capacity and Constellation DesignIndex Modulation Generalisations