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12-01-26 1 ECE 414 Tutorial 3 h(τ,t), H(f,t), H(τ,ν), H(f,ν) Examples

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ECE 414 Tutorial 3

h(τ,t), H(f,t), H(τ,ν), H(f,ν) Examples

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ECE 414 Tutorial 3

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ECE 414 Tutorial 3 l

l  delay power spectral density l  mean delay spread l  rms delay spread

l  delay spread à ISI/frequency selective?

Autocorrelation function of h(τ,t)

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ECE 414 Tutorial 3 l

l  time-frequency correlation function

l  Let Δt=0 à frequency correlation function

l  Side note: and are Fourier transform pair

à ISI/frequency selective?

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ECE 414 Tutorial 3 l

l  time-frequency correlation function

l  Let Δf=0 à time correlation function , which measures how fast the channel changes with time à Fast or slow fading

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ECE 414 Tutorial 3 l

l  is the FT of w.r.t. Δt

l  Let Δf=0 à Doppler power spectral density , the FT of

l  Side note:

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ECE 414 Tutorial 3 l  Mean Doppler shift:

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ECE 414 Tutorial 3

Slow and frequency selective

Fast and frequency selective

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ECE 414 Tutorial 3 l  Example 1

profileintensity multipath above theexhibiting channel For the

1) calculate the mean delay spread, τ

2) calculate the RMS delay spread, τσ

3) would this channel be suitable for AMPS (30kHz) or GSM (200kHz) without the user of equalizer?

0 1 5 20 )(usτ

0dB10−

dB20−

dB

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ECE 414 Tutorial 3 l  Solution:

usdd

h

h 02.1711.01.001.0

1201.051.0101.00)()(

)1 ≈+++

×+×+×+×==

∫∫

ττφ

τττφτ

usdd

dd

h

h

h

h 55.6)()(

)()()(

)22/1

222/12

≈⎥⎥⎦

⎢⎢⎣

⎡−=

⎥⎥⎦

⎢⎢⎣

⎡ −=

∫∫

∫∫ τ

ττφ

ττφτ

ττφ

ττφττστ

1 13) coherence bandwidth ( ) 152cm

f kHzT τσ

Δ ≈ = =

For AMPS, 30 ( ) frequency non-selectives cW kHz f= << Δ ⇒For GSM, 200 ( ) frequency selectives cW kHz f= >> Δ ⇒

AMPS: no ISI, no equalizer needed⇒neededequalizer ISI, :GSM

Here, the delay components are (0,1,5,20). What happens to the delay spread if the four component are (0,1,5,10) ?Question :

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ECE 414 Tutorial 3 l  Example 2

channel Verify the

0 ),cos()()/exp(),( ≥+Ω−= τθτττ tnTth

[ ]1 2 1 2

where and are constants, is a random variable uniformly distributed in , , and ( ) is a random process independent of , with [ ( )] and [ ( ) ( )] ( - ), is a WSSUSchannel?

n

Tn

E n E n n

θ

π π τ

θ τ µ τ τ δ τ τ

Ω

− +

= =

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ECE 414 Tutorial 3 0 ),cos()()/exp(),( ≥+Ω−= τθτττ tnTth

Refer to Tutorial 1

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ECE 414 Tutorial 3

Thus, h(τ,t) is WSSUS.

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ECE 414 Tutorial 3 l  Example 3 (P2-7)

l  The time variant transfer function of a WSSUS channel is given by

b. Derive the channel impulse response h(τ,t). c. Given that the transmitted signal is x(t), derive the

received signal r(t) in the absence of background noise. From the relation between x(t) and r(t), find the mean delay ( ) and delay spread ( ) of the channel.

( , ) ( )exp( ( )).H f t t j tα θ=

τ τσ

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ECE 414 Tutorial 3

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ECE 414 Tutorial 3

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ECE 414 Tutorial 3 l  Example 4 (P2-10)

l  A WSSUS channel: Multipath delay spread Tm = 1s, Doppler spread Bd = 0.02Hz, Channel bandwidth Ws = 5Hz, Symbol duration Ts = 10s.

a. Determine the coherence bandwidth and the coherence

time. b. Does this channel exhibit frequency selective fading?

Explain. c. Does this channel exhibit slow or fast fading? Explain.

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ECE 414 Tutorial 3 coherence bandwidth coherence time

slow (Δt)c>>Ts