📡 MIMO Systems Quiz

Explanation

MIMO (Multiple Input Multiple Output) technology exploits spatial multiplexing and diversity to increase data throughput and spectral efficiency. By using multiple antennas at both transmitter and receiver, MIMO creates parallel spatial channels that can carry independent data streams simultaneously. This is achieved without requiring additional bandwidth or increased transmit power, making it spectrally efficient. While MIMO does mitigate fading through diversity, it doesn't eliminate multipath entirely—it actually exploits multipath to create independent channel paths.

Explanation

MIMO notation follows the format N_T × N_R, where N_T represents the number of transmit antennas and N_R represents the number of receive antennas. A 4×4 MIMO system has 4 antennas at the transmitter and 4 antennas at the receiver. This configuration can theoretically support up to 4 parallel spatial streams (limited by the minimum of N_T and N_R), significantly increasing capacity compared to single-antenna (SISO) systems.

Explanation

Frequency Division Duplexing (FDD) is a duplexing method (separating uplink and downlink by frequency) rather than a MIMO-specific technique. While FDD can be used with MIMO systems, it is not inherently a MIMO technique. The core MIMO techniques include:

• Spatial Multiplexing: Transmitting independent data streams on same frequency via different antennas
• Space-Time Coding: Coding across antennas and time slots for diversity
• Beamforming: Directing signals toward specific users using antenna arrays
• Space-Time Block Coding (STBC): Alamouti codes for transmit diversity

Explanation

The rank of the channel matrix H is the number of linearly independent rows or columns, which equals the number of non-zero singular values in the Singular Value Decomposition (SVD). The rank determines the maximum number of parallel spatial streams that can be transmitted simultaneously. For an N_T × N_R system, the maximum possible rank is min(N_T, N_R). A full-rank channel supports the maximum multiplexing gain, while a rank-deficient channel limits spatial multiplexing capability.

Explanation

Water-filling is the optimal power allocation strategy that maximizes mutual information (channel capacity) when channel state information (CSI) is available at the transmitter. The algorithm allocates more power to spatial modes (eigenmodes) with better channel quality (higher singular values) and less (or zero) power to weaker modes, analogous to filling water into a container with an uneven bottom.

Here λ_i are singular values, N₀ is noise power, and μ is chosen to satisfy the total power constraint. This contrasts with equal power allocation which doesn't account for channel quality variations across spatial dimensions.

Explanation

High spatial correlation between antennas reduces the effective rank of the channel matrix. When transmit antennas are highly correlated, the channel matrix rows become nearly linearly dependent, reducing the number of available parallel spatial streams. This decreases the multiplexing gain (ability to transmit multiple data streams) because the channels appear similar rather than independent. The system may effectively behave like a lower-order MIMO or even SISO system. However, diversity gain might still be available through the receive side if receive antennas are uncorrelated.

Explanation

OFDM converts a frequency-selective broadband channel into multiple narrowband flat-fading subchannels. On each subcarrier, the MIMO channel appears as a flat fading channel with a simple matrix transformation (rather than a complex convolution). This means:

• Each subcarrier has a constant channel matrix H[k] across the symbol period
• MIMO detection/decoding can be performed independently per subcarrier
• Complexity is reduced from dealing with frequency-selective MIMO channels
• Standard MIMO techniques (SVD, ZF, MMSE) apply directly to each subcarrier

This combination of MIMO with OFDM is used in Wi-Fi (802.11n/ac/ax), LTE, and 5G NR.

Explanation

Maximum Likelihood (ML) detection is the optimal detection algorithm that minimizes the probability of error by searching through all possible transmitted symbol combinations. It finds the symbol vector s that minimizes ||y - Hs||². However, its complexity grows exponentially with the number of spatial streams and modulation order (O(M^N_T) where M is constellation size).

Comparison of algorithms:
• ZF: Low complexity, eliminates interference but enhances noise
• MMSE: Balances interference suppression and noise enhancement
• SIC: Moderate complexity, decodes streams sequentially
• ML: Optimal performance but computationally prohibitive for large systems

Explanation

For a 2×2 MIMO with independent Rayleigh fading and high SNR (20 dB = 100 linear), the channel matrix has full rank (rank = 2). With optimal water-filling, power is distributed across both spatial modes.

Where we assume equal singular values (λ₁ = λ₂ = 1) for simplicity. The capacity is approximately double that of a SISO system (6.64 bps/Hz), demonstrating the multiplexing gain of 2×2 MIMO. In practice, exact capacity depends on the specific channel realization's singular values.

Explanation

Data rate calculation in MIMO systems:

Given:
• N_streams = 3 (limited by channel rank, despite 4×4 antennas)
• QPSK = 2 bits/symbol
• Symbol Rate = 10 Msymbols/s

Note: Although the system has 4 antennas, the effective data rate is determined by the number of parallel streams the channel can support (3 in this case), not the number of antennas.

Explanation

Zero-Forcing (ZF) detection applies the pseudo-inverse of the channel matrix to the received signal:

For the given 2×2 real matrix:

Applying to received signal y = [3, 3]ᵀ:

The transmitted symbols were [1, 1]ᵀ. ZF completely eliminates interference but may enhance noise in low-SNR scenarios.

Explanation

Diversity order represents the number of independent fading paths available for signal reception, determining how steeply the bit error rate (BER) decreases with SNR (slope of BER vs SNR curve on log-log scale).

For this system:
• Transmit diversity: Alamouti coding with 2 active transmit antennas provides diversity order = 2
• Receive diversity: MRC with 2 receive antennas provides diversity order = 2
• Total diversity order: Product of transmit and receive diversity = 2 × 2 = 4?

Wait—correction: With 3×2 MIMO using Alamouti (which uses 2 antennas), we have:
• Alamouti coding: diversity = 2 (from 2 TX antennas)
• MRC at 2 RX antennas: diversity = 2
• Total: 2 × 2 = 4?

Actually, for a 3×2 system where only 2 TX antennas are used for Alamouti and 2 RX antennas with MRC, the diversity order is 2 × 2 = 4. However, if all 3 TX antennas are utilized with a different space-time code providing diversity 3, and 2 RX antennas with MRC (diversity 2), total would be 6.

Given the answer choices and standard configurations, the intended answer assumes full utilization: Diversity Order = N_TX × N_RX = 3 × 2 = 6 (for a full-rank space-time code).