# Appendix A

# Derivation of Time Domain Symbols of Localized FDMA and Distributed FDMA

In this appendix, we use the symbol notations described in Figure A.1: *x*_{m} (*m* = 0, 1, …, *M* − 1) represents modulated source symbols and *X*_{k} (*k* = 0, 1, …, *M* − 1) represents *M* samples of the DFT of *x*_{m}; *Y*_{l} (*l* = 0, 1, …, *N* − 1) represents the frequency domain samples after subcarrier mapping and *y*_{n} (*n* = 0, 1, …, *N* − 1) represents the transmitted time domain channel symbols that are obtained from the inverse DFT (IDFT) of *Y*_{n}.

## A.1 Time Domain Symbols of LFDMA

For LFDMA, the frequency samples after subcarrier mapping {*Y*_{l}} can be described as follows:

Let *n* = *Q* · *m* + *q*, where 0 ≤ *m* ≤ *M* − 1, 0 ≤ *q* ≤ *Q* − 1, and *N* = *Q* · *M*. Then

If *q* = 0, then

**Figure A.1** Symbol notations for SC-FDMA modulations

If *q* ≠ 0, since , then Equation (A.2) can be expressed as follows:

Thus,

## A.2 Time Domain Symbols of DFDMA

For DFDMA, the frequency ...