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Guard interval and ISI-free OFDM transmission

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

The guard time is longer or equal to the channel’s maximum delay spread: T G τ max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaads fadaWgaaWcbaGaam4raaqabaGccqGHLjYScqaHepaDdaWgaaWcbaGa ciyBaiaacggacaGG4baabeaaaaa@3F80@ .

Ofdm
OFDM symbols are generated by an IFFT with size N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad6 eaaaa@37ED@ . A guard interval (GI) is appended at the start of each symbol. This cyclic prefix comprises the last N G MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad6 eadaWgaaWcbaGaam4raaqabaaaaa@38E5@ samples of each symbol.

Start

In this example the guard time is T G = 0.8 μ s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaads fadaWgaaWcbaGaam4raaqabaGccqGH9aqpcaaIWaGaaiOlaiaaiIda cqaH8oqBcaWGZbaaaa@3ED7@ . Further settings from IEEE 802.11g are used.

Number of subcarriers, FFT size N=64 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad6 eacqGH9aqpcaaI2aGaaGinaaaa@3A71@  
Discrete length of the guard interval N G =16 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad6 eadaWgaaWcbaGaam4raaqabaGccqGH9aqpcaaIXaGaaGOnaaaa@3B70@  
Data subcarriers N C = 52 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad6 eadaWgaaWcbaGaam4qaaqabaGccqGH9aqpcaaI1aGaaGOmaaaa@3B6C@
 (pilot subcarriers are used for data)
Extended OFDM symbol period T S =4μs MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqads fagaqbamaaBaaaleaacaWGtbaabeaakiabg2da9iaaisdacqaH8oqB caWGZbaaaa@3D7F@  
Guard interval duration T G = N G T s N+ N G =0.8μs MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaads fadaWgaaWcbaGaam4raaqabaGccqGH9aqpdaWcaaqaaiaad6eadaWg aaWcbaGaam4raaqabaGcceWGubGbauaadaWgaaWcbaGaam4Caaqaba aakeaacaWGobGaey4kaSIaamOtamaaBaaaleaacaWGhbaabeaaaaGc cqGH9aqpcaaIWaGaaiOlaiaaiIdacqaH8oqBcaWGZbaaaa@475F@  

The simulation uses a single echo channel and starts with τ max =0.8μs MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes 8a0naaBaaaleaaciGGTbGaaiyyaiaacIhaaeqaaOGaeyypa0JaaGim aiaac6cacaaI4aGaeqiVd0Maam4Caaaa@41CB@ (echo delay). This results in an ISI-free detection.

Signal
ISI free detection for τ max = T G =0.8μs MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes 8a0naaBaaaleaaciGGTbGaaiyyaiaacIhaaeqaaOGaeyypa0Jaamiv amaaBaaaleaacaWGhbaabeaakiabg2da9iaaicdacaGGUaGaaGioai abeY7aTjaadohaaaa@44AC@ .

Experiment

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

Simulation - Setup
Simulation - Setup (F12): modify “Echo delay” ( τ max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes 8a0naaBaaaleaaciGGTbGaaiyyaiaacIhaaeqaaaaa@3BDF@ ) and click on “OK”.

Approve the ISI free condition T G τ max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaads fadaWgaaWcbaGaam4raaqabaGccqGHLjYScqaHepaDdaWgaaWcbaGa ciyBaiaacggacaGG4baabeaaaaa@3F80@ for different echo delays.

Constellation1 Constellation2
ISI for τ max > T G MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes 8a0naaBaaaleaaciGGTbGaaiyyaiaacIhaaeqaaOGaeyOpa4Jaamiv amaaBaaaleaacaWGhbaabeaaaaa@3EC2@ : τ max = 0.85 μ s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes 8a0naaBaaaleaaciGGTbGaaiyyaiaacIhaaeqaaOGaeyypa0JaaGim aiaac6cacaaI4aGaaGynaiabeY7aTjaadohaaaa@428A@ and τ max =1μs MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes 8a0naaBaaaleaaciGGTbGaaiyyaiaacIhaaeqaaOGaeyypa0JaaGym aiabeY7aTjaadohaaaa@4058@

Note that the transmission sampling time is 0.05μs MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaic dacaGGUaGaaGimaiaaiwdacqaH8oqBcaWGZbaaaa@3CAD@ and thus the echo delay can be adjusted in this granularity, e.g. τ max =0.75,0.80,0.85,0.90μs MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes 8a0naaBaaaleaaciGGTbGaaiyyaiaacIhaaeqaaOGaeyypa0JaaGim aiaac6cacaaI3aGaaGynaiaacYcacaaIWaGaaiOlaiaaiIdacaaIWa GaaiilaiaaicdacaGGUaGaaGioaiaaiwdacaGGSaGaaGimaiaac6ca caaI5aGaaGimaiabeY7aTjaadohaaaa@4D57@ .

Next steps

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

ISI free
Is this transmission ISI-free? Simulate and calculate!

Check the ISI-free condition T G τ max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaads fadaWgaaWcbaGaam4raaqabaGccqGHLjYScqaHepaDdaWgaaWcbaGa ciyBaiaacggacaGG4baabeaaaaa@3F80@ :

The initial setup uses 52 data QPSK symbols (2 bit) per OFDM symbol (4us). This yields to a source symbol duration of T D =769μs MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaads fadaWgaaWcbaGaamiraaqabaGccqGH9aqpcaaI3aGaaGOnaiaaiMda cqaH8oqBcaWGZbaaaa@3EEA@  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 t s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaads fadaWgaaWcbaGaam4raaqabaGccqGH9aqpcaWGobWaaSbaaSqaaiaa dEeaaeqaaOGaeyyXICTaamiDamaaBaaaleaacaWGZbaabeaaaaa@4037@ with

t s = N C T D N+ N G MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaads hadaWgaaWcbaGaam4CaaqabaGccqGH9aqpdaWcaaqaaiaad6eadaWg aaWcbaGaam4qaaqabaGccaWGubWaaSbaaSqaaiaadseaaeqaaaGcba GaamOtaiabgUcaRiaad6eadaWgaaWcbaGaam4raaqabaaaaaaa@4180@