IYH Dear Neeraj Sharma

tl/dr: Skip to point 3 Sign Convention: in section Why the Minus Sign?

Anent the meaning of the mathematical operator and the minus sign in Equation (14) of the paper you provided, some preliminaries:

Understanding the Operator ℝ{.}

In the context of complex numbers, the operator ℝ{z} represents the real part of a complex number z. If a complex number z is expressed as z = a + jb, where a and b are real numbers and j is the imaginary unit (√-1), then:

ℝ{z} = a

In other words, the operator extracts the real component of the complex number, discarding the imaginary part.

Context of Equation (14)

Equation (14) in the paper is part of a derivation to approximate the probability of a pairwise error event in a communication system. Let's look at the relevant part of the equation:

-ℝ{(ΔH - Δ)H A_k ñ_k} ~ N(0, (ΔH - Δ)H A_k K_k A_kH (ΔH - Δ) / 2)

Now to your actual question:

Why the Minus Sign?

The minus sign in front of the ℝ{.} operator is crucial for the derivation and has a specific purpose:

Daniyel Yaacov Bilar Thank you so much so such detailed reply. May I ask more? I shall be highly obliged.

1. BER is normally expressed as Q-function of sqrt(SNR). In Eq 15, what is signal power and noise power?

2. How to interpret Eq (13)?

3. You have well explained Eq (14). But how does it lead to Eq (15)?

4. In Eq. (17), will simply putting K=1 make it applicable for SU-MIMO case?

5. Data vector s belongs to some constellation Q . In Eq. (17), M is defined as number of possible transmitted symbol vectors. Is there any connection between Q and M? Is it related to the modulation technique used? For example, in 16-QAM modulation, 16 possible combinations of bits are generated.

I am eagerly waiting for your reply.

IYH Dear Neeraj Sharma I am happy to see you are really trying to understand this paper.

OK, to your questions:

Q1. Signal Power and Noise Power in Eq. 15

Signal Power: The signal power in Eq. 15 is represented by the term

P_signal = (Δ_ij(k))^H A_k^(-1) Δ_ij(k))

• Δ_ij​(k) represents the signal difference.
• A_(k−1​) is the inverse of the detection matrix, which accounts for the channel effects.

This term captures the energy of the signal difference between the two possible transmitted symbols after passing through the effective channel and the receiver's processing. This is crucial because the receiver uses this energy to distinguish between different symbols. A higher signal power generally means a stronger signal, which is easier to detect and less likely to be confused with other symbols

Noise Power: The noise power in Eq. 15 is represented by the term

P_noise = (Δ_ij(k))^H A_k^(-1) K_k A_k^(-1) Δ_ij(k)

• K_k​ represents the noise covariance matrix.
• The structure of this term indicates how the noise interacts with the signal difference through the channel.

This term accounts for the variance of the noise and interference, weighted by the detection matrix A_k​ and the signal difference Δij(k)​. Noise power represents the strength of the unwanted signals that corrupt the transmitted signal. A lower noise power means less interference, making it easier for the receiver to correctly identify the transmitted symbol.

Q2. Interpretation of Eq. (13)

Eq. (13) represents the condition for a pairwise error event for user k. It states that an error occurs when the squared Euclidean distance between the received signal y_k​ and the estimated signal B_k​s_k​ is greater than the squared Euclidean distance between the received signal y_k​ and the estimated signal B_k​s_j​ for j neq k. This condition is the basis for calculating the pairwise error probability (PEP), which is a fundamental building block for estimating the overall Bit Error Rate (BER).

The squared Euclidean distance ​||A_k^1/2​(y_k​−B_k​s_i​)​||^2 represents the distance between the received signal and the expected signal for a particular symbol. The receiver uses these distances to decide which symbol was most likely transmitted. The symbol with the smallest distance is chosen as the most likely transmitted symbol.

What motivates this? Defining this error event condition is essential because it allows us to quantify the likelihood of errors in the communication system. By understanding when errors occur, we can design better systems to minimize these errors.

Lastly, the decision metric used by the receiver is based on these distances. The receiver chooses the symbol that minimizes the squared Euclidean distance, which is a common and effective method for symbol detection in noisy channels.

Q3. Explanation of Eq. (14) and its leading to Eq. (15)

Gaussian Approximation: Eq. (14) assumes that the real part of the complex number −R{(Δii^(k)​−Δij^(k)​)^HA_k^(−1)​ ñ_k​} is approximately a Gaussian random variable with a mean of 0 and a variance of 0.5​(Δ_ij^(k)​)^HA_k^(−1)​K_k​A_k^(−1​)Δ_ij(k)​.

This approximation is valid due to the Central Limit Theorem, which states that the sum of a large number of independent random variables tends to be Gaussian. In communication systems, the noise and interference from multiple sources can often be modeled as Gaussian, especially when the number of users and interfering signals is large.

Using the Gaussian approximation, the pairwise error probability (PEP) can be calculated using the Q-function, which gives the probability that a standard normal random variable will take a value greater than a certain threshold. The argument of the Q-function is the square root of the ratio of the signal power to the noise power. The Q-function gives the probability that a standard normal random variable will take a value greater than a certain threshold. This is a standard tool in communication theory for calculating error probabilities.

Before using the Q-function, the Gaussian random variable is standardized by subtracting its mean (which is 0) and dividing by its standard deviation. This results in a standard normal random variable with mean 0 and variance Standardization is necessary because the Q-function is defined for standard normal variables. This step ensures that the random variable fits the form required by the Q-function.

Q4. Applicability of Eq. (17) for SU-MIMO by Setting K=1

That's a very insightful question! You're right to consider how the expressions might simplify for the Single-User MIMO (SU-MIMO) case.

Setting K=1 in Eq. (17) would make it applicable for the SU-MIMO (Single User Multiple Input Multiple Output) case, as it would consider only one user in the system.

However, the equation is derived for the MU-MIMO (Multiple User Multiple Input Multiple Output) case, and setting K=1 would not necessarily provide the correct BER performance for the SU-MIMO case, as the interference terms would not be present.

While setting K=1 simplifies the expression, it does not directly make it applicable to a general SU-MIMO case.

In practice, when you analyze an SU-MIMO system, you would typically derive the BER expression directly for the SU-MIMO case, taking into account the specific channel, precoder, combiner, and data detection approach. You would not typically start with the MU-MIMO expression and then try to adapt it.

Q5. Connection btw Q, M and modulation technique

Another crucial point, well done! There's a very direct and important connection.

Briefly, the data vector s belongs a constellation Q which is the set of all possible symbols that can be transmitted. In Eq. (17), M is defined as the number of possible transmitted symbol vectors for the users. The number of possible transmitted symbol vectors M is directly related to the size of the constellation Q and the number of streams being transmitted. The modulation technique determines the constellation Q, and therefore indirectly influences the number of possible transmitted symbol vectors M.

You're correct that in 16-QAM modulation, 16 possible combinations of bits are generated. This means that the constellation Q for 16-QAM has 16 symbols. If you are transmitting a single stream using 16-QAM, then M = 16. If are you are transmitting two streams using 16-QAM, then M = 16 * 16 = 256.