arrier. The subsymbol concept, which is introduced by GFDM, corresponds to a shift in the time-domain (TD). Therefore, a general time-frequency description of waveforms is useful to put different waveforms in a unified framework. 4.1.1 Time-Frequency Representation In SC waveform, a stream of data symbols d with the symbol spacing AT modulates a pulse shape g(t), where the baseband signal can be expressed as (4.1) Here, i denotes the symbol index, Z in the set of integer numbers, and * represents linear convolution operation. This is also interpreted as filtering of the random process - iAT), where di is a random variable and S(t) is the Dirac function. A multicar- rier waveform is a superposition of multiple SC signals, and each uses a pulse gk(t). Thus, the multicarrier waveform can be expressed as (4.2) where Kon is a finite set used for the subcarriers indexing. The original idea behind multicar- rier comes from FDM [2], where the pulses {gk(t)} are designed to occupy disjoint narrow bands. The idea is extended to allow overlapping among the subcarriers as in FMT [3]. 1 Prototype pulse and prototype filter are used exchangeably in this chapter. 4.1 Overview of Multicarrier Waveforms (M6 1)AT gk, 0 gk, 1 gk, 2 gk, 3 gk, 4 gk, 5 Non-active gk, 6 Overlap with next block Overlap with previous block Figure 4.1 Subsymbol concept. In this example M = 7, Mo = 3, and Ms=5. = Accordingly, gk(t) is generated by means of frequency shift of a prototype pulse g(t), such that gk(t) = g(t)ej2nk, where Af is the subcarrier spacing and Kon becomes the set of active subcarriers. Moreover, by expressing the symbol index using two indices (m, b) and an inte- ger Ms 1, such that =m+bM,,we get (4.3) where Ts = MSTT. The index m refers to the subsymbol index, and accordingly, AT defines the subsymbol spacing, and MS represents the number of active subsymbols. Therefore, 8k.m(t)=g(t- - mAT)ei2nkAf(1-mAT) In practice, the pulse g(t) is finite and causal. As shown in Figure 4.1, let M be the number of subsymbol spacing that g(t) spans,