### Windowing

The use of the IFFT within the OFDM modulator constitutes the
use of a rectangular pulse shape. This means that discontinuities
occur from one OFDM symbol to the next, resulting in out of band emissions.
(Alternatively, considering the frequency domain, the frequency response
of this rectangular pulse shape is a sinc pulse.) The discontinuities
between OFDM symbols can be reduced by using windowing, which smooths
the transitions between OFDM symbols. Within LTE System Toolbox, the
windowing is performed as follows:

For `Windowing`

= *N* samples,
the cyclic prefix added to the nominal OFDM symbol is further extended
by *N * samples.

This extended waveform is windowed by pointwise multiplication
in the time domain with a raised cosine window, which applies a taper
to the first *N* and last *N* samples,
with all other values being 1. The values, *y*, in
the first *N* samples are given by:

$$y=\frac{1}{2}\left(1-\mathrm{sin}\left(\pi \frac{N+1-2i}{2N}\right)\right),\text{\hspace{0.17em}}\text{where}\text{\hspace{0.17em}}i=1\dots N$$

The values in the last *N* samples are the
same values in reverse order.

The windowed OFDM symbols are then overlapped by commencing
transmission of each windowed OFDM symbol *N* samples
prior to the end of the previous OFDM symbol. This ensures that the
time between OFDM symbols is maintained as required by the standard.
Note that the taper at the start of the first OFDM symbol for transmission
is removed and is overlapped with the taper at the end of the last
OFDM symbol.

### Processing

The processing performed by this function is illustrated in
the following diagram.

The number of samples used for windowing depends on the cyclic
prefix length, normal or extended, and the number of resource blocks.
The number of samples is chosen in accordance with the *maximum* values
implied by tables F.5.3-1 and F.5.4-1 in [1].

Number of resource blocks (NRB) | Windowing
samples for normal cyclic prefix | Windowing samples for
extended cyclic prefix |

6 | 4 | 4 |

15 | 6 | 6 |

25 | 4 | 4 |

50 | 6 | 6 |

75 | 8 | 8 |

100 | 8 | 8 |

The number of windowing samples
is a compromise between the effective duration of cyclic prefix, and
therefore the channel delay spread tolerance, and the spectral characteristics
of the transmitted signal, not considering any additional FIR filtering.
For a larger amount of windowing, the effective duration of the cyclic
prefix is reduced but the transmitted signal spectrum has smaller
out-of-band emissions.