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The paper analyzes channel impulse response (CIR) estimation using aperiodic transmit sequences and a receiver which removes carrier frequency offset (CFO) after correlation. It is shown that when there are guard intervals between the transmit sequences longer than the CIR, a class of sequence sets previously introduced can be applied to achieve CIR estimation without any correlation sidelobes for an arbitrary CFO. As having these guard intervals may be viewed as a waste of on-air time, the same problem is solved when they are eliminated by introducing a new class of sequence blocks. Furthermore, for binary sequence blocks, it is shown that by careful construction of the base Hadamard matrices the minimum sizes of these blocks can be halved. The price to pay are the correlation sidelobes that are minimized but generally are not zero anymore, except for zero CFO and a couple of small Hadamard matrix sizes.