arts periodically by sending a single feedback that fully describes the precoder. Differential feedbacks follow the start and restart feedbacks. The start and restart feedbacks employ a codebook that is self- contained; e.g., the base codebook. Figure 10-40 illustrates the operation of differential feedback across time. An example differential feedback is transformation-based differential codebook. Let t, D(t), and V (t) denote the feedback index, the corresponding feedback matrix, and the corresponding precoder, respectively. The sequential index is reset to 0 at Tmax + 1. The index for the start and restart feedbacks are 0. Let U be a vector or a matrix and Qu be a rotation matrix determined by U. The precoders of the The non-uniform distribution causes large quantization error for uncorrelated channels Constant Uniform Transformed modulus codebook codebook codebook FIGURE 10-39 Distribution of the precoding vector coordinates in various transformation methods CHAPTER 10 The IEEE 802. 16m Physical Layer (Part II) Set of Ideal Beamforming Matrices V (t -1) Feedback Feedback Feedback Feedback Arrival Arrival Arrival Arrival FIGURE 10-40 An example illustration of differential feedback and effect on beamforming gain subsequent differential feedbacks are D(t) for t = 1,2, Tmax and the corresponding precoders are given as follows: V(t) ) =QV(+-1)D(t) = 0,1,2, Tmax (10-53) where the rotation matrix Qv(t-1) is a unitary N X N matrix derived from the previous precoder matrix V (t-1) - and N denotes the number of transmit antennas. The dimension of the feedback matrix D(t) is N t X Ns, where Ns denotes the number of spatial streams. The feedback matrix D(t) can be viewed as a description of time/frequency correlation properties which does not change over the entire frequency band and long periods of time, and Qv(t-1) can be viewed as a description of narrowband and short-term channel properties which correspond to a specific sub-band and subframe. Due to correlation between beamforming matrices across time and frequency