Communication without Channel State Information at the Receiver (CSIR)

Introduction
In wireless communication, information-theoretic promises significant capacity gain when multiple antennas are used at the transmiter and the receiver. The capacity gain can be captured by either coherent communication schemes or non-coherent communication schemes. Coherent communication assumes the receiver knows the channel state information (CSIR), however, non-coherent communication does not assume CSIR is available at the receiver. In practice coherent schemes correspond to decoupling designs: first, training is used to acquire CSIR, and then information is transmitted. On the other hand, non-coherent shemes correspond to joint designs, where no explicit training is required to obtain CSIR, and therefore, achieve higher capacity than coherent schemes. Although for point-to-point MIMO channels the capacity gain of non-coherent over coherent is limited, this gain is considerable for channels changing fast or some multi-user communication scenarios. Nevertheless, due to its complexity the non-coherent communication is not as well-studied as coherent communication.

Unless otherwise specified, the channel model considered is as follows: a transmitter has $M$ antennas and a receiver has $N$ antennas, the channel between the transmitter and the receiver, denoted by $H$, follows a block-fading model with coherence time $T$. The receiver does not know $H$.

Capacity Results for Non-coherent Communication
The ergodic capacity for point-to-point MIMO channel without CSIR are first studied by Marzetta and Hochwald (1999 and 2000). The capacity results are quite different from coherent case. For any block length $T$, any number of receive antennas $N$ and SNR $\rho$, the capacity obtained with $M>T$ and $M=T$ are equal.
 * Capacity depends on $T$.

For any ρ, the capacity achieving signaling has the following structure: $X=\Phi \Sigma$, where $\Phi$ is a $T\times N$ unitary matrix, i.e., $\Phi^'\Phi=I$, and
 * Optimal signal structure.

$\Sigma = diag [\lambda_1,\cdots, \lambda_M]$.

In other words, the optimal signaling structure for non-coherent communication is a unitary matrix multiplying a diagonal matrix, which is completely different from coherent case that i.i.d. Gaussian signaling is capacity achieving.

When $T\rightarrow\infty$, the capacity tends to the coherent case where the perfect CSIR is assumed available. In addition, $\Sigma$ becomes identity matrix, therefore, all the information is carried by the direction of the vectors, and the norms do not convey any information.
 * Asymptotic capacity for $T\rightarrow\infty$.

Capacity at High SNR
Zheng and Tse (2002) studied the ergodic capacity for non-coherent communication in the high SNR region. They found

For large $\rho$, the optimal signal is a matrix with orthogonal columns whose distribution is isotropic, $X=[x_1,\cdots,x_M]\Phi$.
 * Optimal signal structure.

At the high SNR (large $\rho$), since the additive noise is negligible, we have $Y \approx H X$. Because $X$ is multiplied by a random and unknown $H$, the receiver cannot decode the particular $X$. Nevertheless, the multiplication of $H$ does not affect the subspace specified by $X$ because for any non-singular $H$, $Y$ spans the same row space as $X$. This indicates that the row space of $X$ can be used to carry information without knowing $H$, i.e., by letting row vectors of $X$ span different subspaces we can send different codewords.
 * The information is carried by subspaces.

Conveying information via subspaces can be viewed as communication on the Grassmann manifold where each distinct point in the manifold represents a different subspace. The set of all $k$-dimensional subspaces of $\mathcal{C}^n$ ($n>k$) constitutes the (complex) Grassmannian $\mathbb{G}(n,k)$, which is equivalently the quotient space of the Stiefel mannifold $\mathbb{F}(n,k)$ (the set of all $n\times k$ unitary matrices). The dimension of $\mathbb{G}(n,k)$ is $\dim\big(\mathbb{G}(n,k)\big) = k(n-k)$; each point in $\mathbb{G}(n,k)$ has a neighborhood that is equivalent (homeomorphic) to a ball in the Euclidean space of (complex) dimension $k(n-k)$. Intuitively, $k(n-k)$ complex parameters specify a $k$-dimentional subspace of $\mathcal{C}^n$. This dimension is equal to the maximum multiplexing gains of a non-coherent point-to-point MIMO channel.
 * Degrees of freedom

Capacity at Low SNR
In the low-SNR region, the results are different from the high-SNR case. If $\rho\rightarrow 0$, the coherent and non-coherent capacity are asymptotically the same (\approx N\rho). So the capacity gain promised by MIMO disappears and only power gain is attained by introducing more receive antennas.