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Wireless communication systems usually use a training sequence for identifying the channel. The training sequence is sent during time-slots different from those of the signal. An advantage of this transmission scheme is that the channel estimation and the signal detection are independent. However, the training sequence occupies transmission bandwidth which would otherwise be available for transmission of the signal. This paper shows that in certain linearly precoded communication systems, such an independence can still be achieved without the training sequence consuming transmission bandwidth. A method for constructing such affine precoders, along with optimal training sequence design, and power allocation between the training sequence and the precoded signal are considered.