2x2 Channel Model

Abdul-Aziz .M Al-Yami Abdul-Aziz .M Al-Yami Khurram MasoodKhurram Masood

Channel Model and Simulation

Using Matlab  

Channel ModelChannel Model

• Discrete Multipath fading channel (2 paths)• Doppler filter

– Jake’s model– fd = 100 Hz

• Delay between paths = 8 samples = 0.5 *Ts

• Power of paths = [1 0.5]• Signal Bandwidth (Lowpass equivalent) Bs = 10 kHz

• Symbol time, Ts = 1/Bs = 0.1 msec• Data Rate = 10k sym/sec• Sampling rate = 160k samples/sec• Samples/symbol = 16

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Sampling and Doppler BandwidthSampling and Doppler Bandwidth

• An important aspect of the Tapped Delay Line (TDL) model is the sampling rate for simulations.

• In simulation we use sampled values which should be sampled at 8 to 32 times the bandwidth

• The doppler bandwidth, or the doppler spread, Bd, is the bandwidth of the doppler spectrum Sd(λ), and is an indicator of how fast the channel characteristics are changing (fading) as a function of time. If Bd is of the order of the signal bandwidth Bs (≈ 1/Ts), the channel characteristics are changing (fading) at a rate comparable to the symbol rate, and the channel is said to be fast fading. Otherwise the channel is said to be slow fading. Thus

– Bd << Bs ≈ 1/Ts (Slow fading channel)– Bd >> Bs ≈ 1/Ts (Fast fading channel)

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ParametersParameters

• Signal bandwidth = Bs = 10kHz

• Ts = 0.1 msec

• Maximum doppler frequency = fd = 100 Hz

• Sampling frequency = fs = 16*Bs = 160k samples/sec

• Simulation length = 5 / (fd) = 50 msec = 8k samples• Interpolation factor = 100• Delay between taps = 8 samples = 0.5 Ts

• Carrier – c(t) = exp[j2π(1000)t]

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Generation of Tap weightsGeneration of Tap weights

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Tap Input Process DataTap Input Process Data

• Two independent Gaussian random variables x1 and x2 are generated– X1,X2 ~ N(0,1)

• For a given Doppler Frequency fd and system symbol rate 1/Ts.

• The term fdTs is known as the fade rate. • Each I and Q components should have this fade rate.• The envelope should be Rayleigh distributed and the phase

should be uniformly distributed

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Doppler FilterDoppler Filter

• The models for doppler power spectral densities for mobile applications assume:– there are many multipath components– each multipath has different delays– all components have the same doppler spectrum.

• Each multipath component (ray) – made up of a large number of simultaneously arriving unresolvable

multipath components– angle of arrival with a uniform angular distribution at the receive

antenna.

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Jake’s ModelJake’s Model

• Jakes derived the first comprehensive mobile radio channel model for both doppler effects and amplitude fading effects

• The classical Jake’s doppler spectrum has the form

• where – fd is the maximum doppler shift

• The Jakes filter is implemented via FIR filter in time domain

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Doppler FilterDoppler Filter

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-100 -80 -60 -40 -20 0 20 40 60 80 1000

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PSD of Jakes filter with fD = 100 Hz

Frequency [Hz]

PS

D

Doppler spreadDoppler spread

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0 200 400 600 800 1000 1200 1400 1600 18000

0.5

1

Frequency (Hz)

Inpu

t P

SD

0 200 400 600 800 1000 1200 1400 1600 18000

0.5

1

1.5

Frequency (Hz)

Out

put

PS

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Linear InterpolationLinear Interpolation

• In generating the tap gain processes it should be noted that the bandwidth of the tap gain processes for slowly time-varying channels will be very small compared to the bandwidth of the signals that flow through them.

• In this case, the tap gain filter should be designed and executed at a slower sampling rate.

• Interpolation can be used at the output of the filter to produce denser samples at a rate consistent with the sampling rate of the signal coming into the tap.

• Designing the filter at the higher rate will lead to computational inefficiencies as well as stability problems.

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Processing of QPSK signal and carrierProcessing of QPSK signal and carrier

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Channel Input / OutputChannel Input / Output

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0 50 100 150 200 250 300 350 400 450 500-2

-1

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1

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Sample Index

Dire

ct I

nput

0 50 100 150 200 250 300 350 400 450 500-4

-2

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Sample Index

Dire

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utpu

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Envelope of outputEnvelope of output

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0 500 1000 1500 2000 2500 3000 35000

0.5

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1.5

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Sample Index

Env

elop

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agni

tude