The power spectral density (PSD) of the $n$-th subcarrier of the OFDM signal is given as $$ \left|S_{n}(f)\right|^{2}=\left|T_{\mathrm{s}}\cdot\operatorname{si}\left(\pi T_{\mathrm{s}}\cdot\left(f-n\cdot R_{\mathrm{s}}\right)\right)\right|^{2} $$ where

• $n$ denotes the subcarrier index
• $T_{\mathrm{s}}$ denotes the symbol duration
• $R_{\mathrm{s}}=\frac{1}{T_{\mathrm{s}}}$ denotes the symbol rate
• $\operatorname{si}\left(\cdot\right)$ is the sinus carialis function, $\operatorname{si}\left(x\right)=\frac{\sin\left(x\right)}{x}$

The resulting PSD of the sender signal $\left|X(f)\right|^{2}$ results from the sum of all activated subcarrier PSDs $$ \left|X(f)\right|^{2}=\frac{1}{\sqrt{N_{\mathrm{sub}}}}\cdot\sum_{n=0}^{N_{\mathrm{sub}}-1}\left|S_{n}(f)\right|^{2} $$

The input vector ${\mathbf{s}}_{k}$ with deactivated subcarriers from index $n_{\mathrm{F}}$ to $n_{\mathrm{L}}$ can be denoted as $\tilde{\mathbf{s}}_{k}$ $$ \tilde{\mathbf{s}}_{k}=\left(s_{k,0},\ldots,s_{k,n_{\mathrm{F}}-1},0,\ldots,0,s_{k,n_{\mathrm{L}}+1},\ldots,s_{k,N_{\mathrm{sub}}-1}\right)^{\mathrm{T}} $$ which results in the following PSD for the sender signal $$ \left|X(f)\right|^{2}=\frac{1}{\sqrt{N_{\mathrm{sub}}}}\left(\sum_{n=0}^{n_{\mathrm{F}}-1}\left|S_{n}(f)\right|^{2}+\sum_{n=n_{\mathrm{L}}+1}^{N_{\mathrm{sub}}-1}\left|S_{n}(f)\right|^{2}\right) $$

As we can see, simply deactivating subcarriers in the range where we want to suppress the spectrum is not sufficient due to the spectral sidelobes of the other subcarriers. Therefore advanced methods for spectral shaping of OFDM signals are investigated on the following pages.