Ferkans — Interactive Telecom Tutor

Multiple Antennas and Capacity

Chapters 9 and 10 showed that multiple antennas provide diversity and array gain, reducing error probability at a fixed rate. This section asks the dual question: how much does adding antennas increase the capacity? The answer depends on whether the extra antennas are at the receiver (SIMO), the transmitter (MISO), or both (MIMO, previewed here and developed in Chapter 15).

Theorem: SIMO Channel Capacity

A single-input, multiple-output (SIMO) channel with $N_r$ receive antennas and channel vector $\mathbf{h} = [h_1, \ldots, h_{N_r}]^T$ has capacity

$C_{\text{SIMO}} = \log_2\!\left(1 + \|\mathbf{h}\|^2 \cdot \text{SNR}\right)$

where $\text{SNR} = P/(N_0 B)$ and $\|\mathbf{h}\|^2 = \sum_{i=1}^{N_r} |h_i|^2$ .

The optimal receiver is maximum ratio combining (MRC), which coherently combines the signals from all antennas using weights proportional to $h_i^*$ .

For i.i.d. Rayleigh fading ( $h_i \sim \mathcal{CN}(0,1)$ ), the ergodic capacity is

$C_{\text{SIMO}}^{\text{erg}} = E\!\left[\log_2\!\left(1 + \|\mathbf{h}\|^2 \cdot \text{SNR}\right)\right]$

where $\|\mathbf{h}\|^2 \sim \chi^2_{2N_r}/2$ (chi-squared with $2N_r$ degrees of freedom, scaled by $1/2$ ).

Each receive antenna provides an independent observation of the same signal. MRC coherently sums these observations, yielding an array gain of $N_r$ in SNR. The effective SNR is $\|\mathbf{h}\|^2 \cdot \text{SNR}$ , and on average $E[\|\mathbf{h}\|^2] = N_r$ , so the mean SNR scales linearly with the number of receive antennas.

Proof

Channel model

$\mathbf{y} = \mathbf{h} x + \mathbf{w}, \quad \mathbf{w} \sim \mathcal{CN}(\mathbf{0}, N_0 \mathbf{I}_{N_r})$ $After MRC:$ \tilde{y} = \mathbf{h}^H \mathbf{y} = |\mathbf{h}|^2 x + \mathbf{h}^H \mathbf{w} $. Effective noise:$ \tilde{w} = \mathbf{h}^H \mathbf{w} \sim \mathcal{CN}(0, |\mathbf{h}|^2 N_0)$.

Effective SNR

$\text{SNR}_{\text{eff}} = \frac{|\|\mathbf{h}\|^2|^2 P}{\|\mathbf{h}\|^2 N_0} = \|\mathbf{h}\|^2 \cdot \frac{P}{N_0} = \|\mathbf{h}\|^2 \cdot \text{SNR} \cdot B$ $Per-bandwidth capacity:$ C = \log_2(1 + |\mathbf{h}|^2 \cdot \text{SNR})$.

Optimality of MRC

MRC maximises the output SNR (by the Cauchy-Schwarz inequality applied to the beamforming vector). Since the scalar channel after MRC is AWGN, Gaussian input is optimal, and the capacity equals $\log_2(1 + \text{SNR}_{\text{eff}})$ . $\blacksquare$

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Theorem: MISO Channel Capacity

A multiple-input, single-output (MISO) channel with $N_t$ transmit antennas and channel vector $\mathbf{h} = [h_1, \ldots, h_{N_t}]^T$ :

Without CSIT (transmitter does not know $\mathbf{h}$ ):

$C_{\text{MISO}}^{\text{no CSIT}} = \log_2\!\left(1 + \frac{\|\mathbf{h}\|^2 \cdot \text{SNR}}{N_t}\right)$

The optimal strategy is equal power allocation across antennas ( $P/N_t$ each), with independent data streams.

With CSIT (transmitter knows $\mathbf{h}$ , beamforming):

$C_{\text{MISO}}^{\text{CSIT}} = \log_2\!\left(1 + \|\mathbf{h}\|^2 \cdot \text{SNR}\right)$

The optimal strategy is transmit beamforming: send along $\mathbf{v} = \mathbf{h}^*/\|\mathbf{h}\|$ , concentrating all power in the direction of the channel.

Without CSIT, the transmitter cannot beamform and must spread power equally. Each antenna gets $P/N_t$ , and the total received power is $\|\mathbf{h}\|^2 \cdot P/N_t$ , giving no array gain (only diversity gain in the fading statistics of $\|\mathbf{h}\|^2$ ).

With CSIT, beamforming coherently combines the transmit antennas, yielding array gain $N_t$ . The capacity with CSIT matches the SIMO capacity — the "reciprocity" of array gain.

Proof

Without CSIT

With equal power $P/N_t$ per antenna and independent signals:

$y = \mathbf{h}^T \mathbf{x} + w = \sum_i h_i x_i + w$

The mutual information is maximised by $\mathbf{x} \sim \mathcal{CN}(\mathbf{0}, (P/N_t)\mathbf{I})$ :

$I = \log_2\!\left(1 + \frac{\|\mathbf{h}\|^2 P}{N_t N_0 B}\right)$

With CSIT

Transmit $\mathbf{x} = \mathbf{v} \cdot s$ with $\mathbf{v} = \mathbf{h}^*/\|\mathbf{h}\|$ :

$y = \mathbf{h}^T \mathbf{v}\, s + w = \|\mathbf{h}\|\, s + w$

$\text{SNR}_{\text{eff}} = \|\mathbf{h}\|^2 P/(N_0 B)$ , giving the same capacity as SIMO with $N_r = N_t$ . $\blacksquare$

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Capacity Comparison: SISO vs SIMO vs MISO

Animated comparison of ergodic capacity for SISO, SIMO (MRC), MISO without CSIT, and MISO with beamforming as SNR sweeps from

-5

to

25

dB. Highlights the array gain of SIMO and the absence of array gain for MISO without CSIT.

SIMO and MISO with CSIT achieve identical array gain

N

, while MISO without CSIT tracks SISO (diversity only).

SIMO and MISO Capacity vs Number of Antennas

Compare the ergodic capacity of SISO, SIMO (MRC), MISO without CSIT, and MISO with CSIT (beamforming) as the number of antennas increases. Observe the array gain of SIMO and MISO with CSIT, and the absence of array gain for MISO without CSIT.

Parameters

Number of antennas4

SNR (dB)15

Show SISO reference

Example: Array Gain of 4-Antenna SIMO

A receiver has $N_r = 4$ antennas with i.i.d. Rayleigh fading. The per-antenna SNR is $\text{SNR} = 10$ dB.

(a) What is the instantaneous capacity if $\|\mathbf{h}\|^2 = 3.5$ (a typical realisation near the mean)?

(b) What is the array gain in dB compared to SISO?

(c) What is the approximate ergodic capacity?

Solution

Instantaneous capacity

$C = \log_2(1 + 3.5 \times 10) = \log_2(36) = 5.17$ bits/s/Hz.

Compare with SISO ( $|h|^2 = 1$ ): $C_{\text{SISO}} = \log_2(11) = 3.46$ bits/s/Hz.

Array gain

The effective SNR increases by factor $E[\|\mathbf{h}\|^2] = N_r = 4$ , corresponding to an array gain of $10\log_{10}(4) = 6.02$ dB.

Ergodic capacity

The ergodic capacity is $E[\log_2(1 + \|\mathbf{h}\|^2 \times 10)]$ .

Since $\|\mathbf{h}\|^2$ is chi-squared with 8 degrees of freedom (mean 4, variance 4), numerical evaluation gives $C_{\text{erg}} \approx 5.1$ bits/s/Hz.

Compared to SISO ergodic: $C_{\text{SISO}}^{\text{erg}} \approx 2.7$ bits/s/Hz.

The 4-antenna SIMO nearly doubles the ergodic capacity. $\blacksquare$

Preview of MIMO Capacity

When both transmitter and receiver have multiple antennas ( $N_t$ and $N_r$ ), the channel becomes a matrix $\mathbf{H} \in \mathbb{C}^{N_r \times N_t}$ . The MIMO capacity is

$C_{\text{MIMO}} = \log_2 \det\!\left(\mathbf{I} + \frac{\text{SNR}}{N_t}\mathbf{H}\mathbf{H}^{H}\right)$

This can scale as $\min(N_t, N_r) \cdot \log_2(\text{SNR})$ at high SNR — a linear increase in capacity with the number of antennas, far beyond what SIMO or MISO can achieve. MIMO does this by creating multiple spatial streams, each carrying independent data. The full treatment is in Chapter 15.

Quick Check

A MISO system with $N_t = 4$ antennas operates without CSIT. Compared to SISO at the same total transmit power, the MISO system provides:

Array gain of $10\log_{10}(4) = 6$ dB

Diversity gain (reduced fading variance) but no array gain

Both array gain and diversity gain

No benefit at all

Correction:

Diversity gain (reduced fading variance) but no array gain

Correct. Without CSIT, the transmitter cannot beamform, so the average received SNR is the same as SISO. However, $\|\mathbf{h}\|^2$ with $N_t = 4$ has less variance than $|h|^2$ with $N_t = 1$ , providing diversity gain that improves outage capacity.

Common Mistake: MISO Without CSIT Has No Array Gain

Mistake:

Claiming that adding $N_t$ transmit antennas increases the average received SNR by a factor of $N_t$ , regardless of whether the transmitter knows the channel.

Correction:

Without CSIT, the transmitter allocates $P/N_t$ per antenna with independent phases. The signals add incoherently, so the average received power is $E[\|\mathbf{h}\|^2] \cdot P/N_t = N_t \cdot P/N_t = P$ — the same as SISO. Array gain ( $N_t$ -fold increase in received SNR) requires CSIT and transmit beamforming. Without CSIT, the only benefit is diversity gain from the improved statistics of $\|\mathbf{h}\|^2$ .

SISO vs SIMO vs MISO Capacity Comparison

Configuration	Capacity (instantaneous)	Array gain	Diversity order
SISO ( $1 \times 1$ )	$\log_2(1+\|h\|^2 \cdot \text{SNR})$	1 (0 dB)	1
SIMO ( $1 \times N_r$ )	$\log_2(1+\\|\mathbf{h}\\|^2 \cdot \text{SNR})$	$N_r$	$N_r$
MISO ( $N_t \times 1$ ) no CSIT	$\log_2(1+\\|\mathbf{h}\\|^2 \cdot \text{SNR}/N_t)$	1 (0 dB)	$N_t$
MISO ( $N_t \times 1$ ) with CSIT	$\log_2(1+\\|\mathbf{h}\\|^2 \cdot \text{SNR})$	$N_t$	$N_t$

Why This Matters: Full MIMO Capacity in the MIMO Book

This section previews SIMO and MISO capacity. The full MIMO analysis — where both ends have multiple antennas — is developed in Chapter 15 of this book and treated comprehensively in the MIMO specialisation book:

MIMO capacity: $C = \log_2\det(\mathbf{I} + \frac{\text{SNR}}{N_t}\mathbf{H}\mathbf{H}^{H})$ and its decomposition via SVD into parallel eigenchannels
Spatial multiplexing: linear capacity scaling with $\min(N_t, N_r)$
Massive MIMO: capacity with $N_t \to \infty$ , channel hardening, and favourable propagation
Cell-free massive MIMO: user-centric architectures (CommIT contributions: Ngo, Caire et al.)
Near-field / XL-MIMO: spatial non-stationarity effects

The MIMO book also covers the interaction between capacity and spatial correlation, which is absent from the i.i.d. model used here.

Array Gain

The increase in average received SNR due to coherent combining of multiple antenna signals. SIMO with MRC achieves array gain $N_r$ ; MISO with beamforming achieves $N_t$ . Requires channel knowledge at the combining side.

Beamforming Gain

The capacity improvement from transmit beamforming in a MISO channel with CSIT. The transmitter concentrates all power in the direction of the channel vector, yielding array gain $N_t$ . Equivalent to the array gain of a SIMO system with $N_r = N_t$ .

Capacity with Diversity