The transmitted vector can be pilot symbol vector or data symbol vector where pilot symbol vector is used in channel estimation phase in pilot-adided channel estimation scheme ,these estimated channel cofficients will being used during the decoding of data symbol vector The entry of is the signal received at the receive antenna. Assuming a rich scattering environment, the entries of the channel matrix are often modeled as independent and identically distributed (i.i.d) (0, 1). Since the transmitter is power constrained, then, where is the total power available at the transmitter.

Also, where is the noise variance at each receive antenna. The average received signal-to-noise ratio (SNR) at each receive antenna is given by.

The MIMO signal detection can be classfied into two parts ,first part is the estimate of channel matrix(channel gain coffiecients) using the the received vector and given knowledge of the transmitted pilot symbol vector, determine H?c, an estimate of the channel gain matrix, the second phase is the detection of of the transmitted symbol vector given the received vector and knowledge of, determine X?C, an estimate of the transmitted symbol vector

MIMO Communication with Channel State Information at the Receiver (CSIR)-only:

In communication channels, error probability is one of the key performance indicators. In most communication systems channel coding schemes is used to decrease the probabilty of error, in order to achieve low probability of error, the transmission rate R must be strictly below the MIMO channel capacity. The MIMO channel capacity is dependent on and the transmit covariance matrix and is given by7:

there is two cases available first case is the availability of CSIR only (i.e., no CSIT) since the transmitter has no knowladge about the channel matrix it cannot adapt the transmission vector(Predcoding) with respect to the channel gainss so for fixed ? and rate R, the transmitter uses a fixed codebook, which does not change with changing channel gains. The transmitter codebook selection is very much dependent on whether the channel is slow fading or fast fading.