In general, the unipolar ACO-OFDM signal is generated by clipping the bipolar OFDM signal at zero, represented as in [14]

The clipping distortion that is resulted from clipping operation falls only on the even subcarriers, without affecting the data on the odd subcarriers. The ACO-OFDM signal in terms of data and clipping distortion is represented as

$${x}_{ACO,n}=\frac{1}{2}\left({x}_{n}+\left|{x}_{n}\right|\right)$$

2

$${x}_{ACO,n}={x}_{D,n}+{x}_{C,n}$$

3

where \({x}_{C,n}=\frac{1}{2}\left|{x}_{n}\right|\) (\(\left|.\right|\) denotes the absolute operator) is the clipping distortion falling on the even subcarriers, that does not affect the modulated data \({x}_{D,n}={\frac{1}{2}x}_{n}\) present on the odd subcarriers as shown in [9, 14].

Multiple layers of ACO-OFDM represented by Equation (3), are then combined and expressed in [12] as

$${x}_{LACO}\left(n\right)=\sum _{l=1}^{L}{x}_{ACO}^{l}\left(n\right)$$

4

$${x}_{ACO}^{l}={{x}^{l}}_{D,n}+{{x}^{l}}_{C,n}$$

5

where ‘l’ represents the layers, \({{x}^{l}}_{D,n}\) and \({{x}^{l}}_{C,n}\)represents the data and the clipping distortion of ‘l’ layers.

## 2.1 LACO-OFDM with TDDR

The TDDR was applied to each layer of LACO-OFDM in [12], represented as

\({x}_{LACO-TDDR}\left(n\right)\) =\(\left(1-\alpha \right)\text{*}{{x}^{l}}_{D}\left(n\right)+\alpha \text{*}\left(sign\right({{x}^{l}}_{D}\left(n\right)){\text{*}{x}^{l}}_{C}\)(n) (6)

where ‘\(\alpha\)’ is the combination factor that defines the relation between modulated data \({x}_{D,n}\) and the non-linear processing of \(sign\left({x}_{D,n}\right)\text{*}{|x}_{D,n}|\).

## 2.2 LACO-OFDM with TDDC

Similar to ACO-TDDR, the principle of operation remains the same in ACO-TDDC, represented as

\({x}_{LACO-TDDC}\left(n\right)=\left(1-\alpha \right)\text{*}{{x}^{l}}_{D}\left(n\right)+\alpha \text{*}\left(sign\right({{x}^{l}}_{D}\left(n\right)\left)\text{*}{{x}^{l}}_{D}\left(n\right)\right)\) (7) \(\)