SISO & MIMO Channel Capacity

SISO Channel Capacity

The capacity of a SISO (Single Input Single Output) channel represents the maximum achievable data rate that can be transmitted reliably over an AWGN channel.

Shannon Capacity Formula

C = B log₂(1 + SNR)

• C = channel capacity (bits/s)
• B = bandwidth (Hz)
• SNR = signal-to-noise ratio (linear scale)

Capacity per Unit Bandwidth

C = log₂(1 + SNR)   bits/s/Hz

Important Notes

• SNR must be converted from dB to linear: SNR_linear = 10^(SNR_dB/10)
• Capacity increases logarithmically with SNR
• High SNR approximation: C ≈ B log₂(SNR)
• Low SNR approximation: C ≈ (B / ln(2)) × SNR
log2​(1+SNR) ≈  SNR​ / ln(2) for SNR≪1 

Example Calculation

Given:

• B = 1 MHz
• SNR = 10 dB → 10 (linear)

Then:

C = 10⁶ × log₂(1 + 10) ≈ 3.46 Mbps

MIMO Channel Capacity (Telatar Formula)

For an \(M \times N\) MIMO system with channel matrix \(H\) and total transmit power, the Telatar capacity is:

\[ C = \log_{2} \det\left( I_N + \frac{\rho}{M} H H^{H} \right) \]

• \(\rho\) = SNR (linear)
• \(H\) = channel matrix (\(N \times M\))
• \(I_N\) = identity matrix
• Capacity is in bits/s/Hz

If the singular values of \(H\) are \(\lambda_1, \ldots, \lambda_r\):

\[ C = \sum_{i=1}^{r} \log_{2}\left( 1 + \frac{\rho}{M}\lambda_i^2 \right) \]

Example (2×2 MIMO)

\[ C = \log_{2} \det\left( I + \frac{\rho}{2} H H^{H} \right) \]

OFDM-Based MIMO Channel Capacity

OFDM divides the total bandwidth into \(K\) subcarriers, each with its own MIMO matrix \(H_k\). The total capacity is:

\[ C_{\text{OFDM}} = \frac{1}{K} \sum_{k=1}^{K} \log_{2} \det\left( I + \frac{\rho}{M} H_k H_k^{H} \right) \]

• \(K\) = number of OFDM subcarriers
• \(H_k\) = MIMO channel matrix for subcarrier \(k\)
• Result is capacity per Hz; multiply by bandwidth for bps

OFDM treats each subcarrier as an independent MIMO channel.

Further Reading

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