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# Guard interval and ISI-free OFDM transmission

This experiment illustrates the ISI-free transmission of OFDM systems under this condition:

The guard time exceeds the channel's maximum delay spread: ${T}_{G}\ge {\tau }_{\mathrm{max}}$ .

OFDM symbols are generated by an IFFT with size $N$. A guard interval (GI) is appended at the start of each symbol. This cyclic prefix comprises the last ${N}_{G}$ samples of each symbol.

## Start

In this example the guard time is ${T}_{G}=0.8\mu s$ . Further settings from IEEE 802.11g are used.

 Number of subcarriers, FFT size $N=64$ Discrete length of the guard interval ${N}_{G}=16$ Data subcarriers ${N}_{C}=52$ Pilot subcarriers ${N}_{P}=4$ Extended OFDM symbol period ${{T}^{\prime }}_{S}=4\mu s$ Guard interval duration ${T}_{G}=\frac{{N}_{G}{{T}^{\prime }}_{s}}{N+{N}_{G}}=0.8\mu s$

The simulation uses a single echo channel and starts with ${\tau }_{\mathrm{max}}=0.8\mu s$ (echo delay). This results in an ISI-free detection.

ISI free detection for ${\tau }_{\mathrm{max}}={T}_{G}=0.8\mu s$ .

## Experiment

Now, modify the echo delay. When restarting the simulation the channel estimation and equalizer initialization is performed.

Simulation - Setup (F12): modify “Echo delay” ( ${\tau }_{\mathrm{max}}$ ) and click on “OK”.

Approve the ISI free condition ${T}_{G}\ge {\tau }_{\mathrm{max}}$ for different echo delays.

ISI for ${\tau }_{\mathrm{max}}>{T}_{G}$ : ${\tau }_{\mathrm{max}}=0.85\mu s$ and ${\tau }_{\mathrm{max}}=1\mu s$

Note that the transmission sampling time is $0.05\mu s$ and thus the echo delay can be adjusted in this granularity, e.g. ${\tau }_{\mathrm{max}}=0.75,0.80,0.85,0.90\mu s$.

## Next steps

Modify the discrete length of the guard interval: Prefix length in Simulation ? Setup (F12).

Is this transmission ISI-free? Simulate and calculate!

Check the ISI-free condition ${T}_{G}\ge {\tau }_{\mathrm{max}}$ :

The initial setup uses 52 data QPSK symbols (2 bit) per OFDM symbol (4us). This yields to a source symbol duration of ${T}_{D}=769\mu s$  and a bit rate of 26 Mbit.

Note that in the simulation the source symbol duration and bitrate is constant if the discrete prefix length is modified. The new guard time is determined as follows:

 Guard interval duration ${T}_{G}={N}_{G}\cdot {t}_{s}$ with ${t}_{s}=\frac{{N}_{C}{T}_{D}}{N+{N}_{G}}$