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Single-tone and multi-tone transmissions in the narrowband Internet of Things (NB-IoT) uplink system can be accomplished by employing different orthogonal basis. In this paper, we develop an uplink NB-IoT baseband system model and derive the corresponding analytical signal model based on the discrete cosine transform type-I (DCT-I) domain. Basically, the circular convolution and channel matrix diagonalization issues for designing DCT-based systems are addressed without exploiting any redundancy (a prefix and a suffix sequences, and zero-padding (ZP)) into each transmitted data symbol blocks. In our proposed DCT-I based transceiver, we employ a standard cyclic prefix (CP) and apply discrete Fourier transform (DFT) and inverse DFT (IDFT) operations directly to make the channel matrix circulant and diagonal, resulting in the use of one-tap frequency-domain channel equalizer without incorporating …