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1 Fading multipath radio channels Narrowband channel modelling Wideband channel modelling Wideband WSSUS channel (functions, variables & distributions)

2 Low-pass equivalent (LPE) signal ( ) = Re ( ) s t RF carrier frequency { j2 f } ct z t e π Real-valued RF signal Complex-valued LPE signal ( ) = ( ) + ( ) = ( ) j z t x t jy t c t e φ ( t) In-phase signal component Quadrature component

3 Spectrum characteristics of LPE signal 0 magnitude f Real-valued time domain signal (e.g. RF signal) 0 phase f Signal spectrum is Hermitian Complex-valued LPE time domain signal Signal spectrum is not Hermitian

4 Radio channel modelling Narrowband modelling Calculation of path loss e.g. taking into account - free space loss - reflections - diffraction - scattering Basic problem: signal fading Wideband modelling Deterministic models (e.g. ray tracing, playback modelling) Stochastical models (e.g. WSSUS) Basic problem: signal dispersion

5 Signal fading in a narrowband channel magnitude of complex-valued LPE radio signal distance Tx propagation paths Rx fade <=> signal replicas received via different propagation paths cause destructive interference

6 Fading: illustration in complex plane quadrature phase component Received signal in vector form: resultant (= summation result) of propagation path vectors Rx Tx path delays are not important in-phase component Wideband channel modelling: in addition to magnitudes and phases, also path delays are important.

7 Propagation mechanisms A D C A: free space B: reflection C: diffraction D: scattering Transmitter B Receiver reflection: object is large compared to wavelength scattering: object is small or its surface irregular

8 Countermeasures: narrowband fading Diversity (transmitting the same signal at different frequencies, at different times, or to/from different antennas) - will be investigated in later lectures - wideband channels => multipath diversity Interleaving (efficient when a fade affects many bits or symbols at a time), frequency hopping Forward Error Correction (FEC, uses large overhead) Automatic Repeat request schemes (ARQ, cannot be used for transmission of real-time information)

9 Bit interleaving Transmitter Channel Receiver Bits are interleaved... Fading affects many adjacent bits After deinterleaving of bits, bit errors are spread!... and will be de-interleaved in the receiver Bit errors in the receiver (better for FEC)

10 Channel Impulse Response (CIR) Channel is assumed linear! time t delay spread T m h(τ,t) Channel presented in delay / time domain (3 other ways possible!) zero excess delay delay τ

11 CIR of a wideband fading channel The CIR consists of L resolvable propagation paths L 1 = i j () t i= 0 ( ) ( ) i h, t a t e φ ( ) τ δ τ τ i path attenuation path phase path delay LOS path τ

12 Received multipath signal Transmitted signal: = k k = () ( ) s t b p t kt complex symbol pulse waveform Received signal: () () () (, ) ( ) r t = h t s t = h τ t s t τ dτ L 1 jφ () t i() ( τ i) f () t δ ( t t0) dt = f ( t0) i= 0 i = a t e s t

13 Received multipath signal The received multipath signal is the sum of L attenuated, phase shifted and delayed replicas of the transmitted signal s(t) T : a e ( τ ) ( τ ) ( τ ) s t jφ0 0 0 ae st jφ1 1 1 a e s t jφ2 2 2 T m Normalized delay spread D = T m / T

14 Received multipath signal The normalized delay spread is an important quantity. When D << 1, the channel is - narrowband - frequency-nonselective -flat and thereis no intersymbolinterference(isi). When D approaches or exceeds unity, the channel is - wideband - frequency selective Important feature - time dispersive has many names!

15 BER vs. S/N performance In a Gaussian channel (no fading) BER <=> Q(S/N) erfc(s/n) BER Typical BER vs. S/N curves Frequency-selective channel (no equalization) Gaussian channel (no fading) Flat fading channel S/N

16 BER vs. S/N performance Flat fading (Proakis 7.3): BER ( ) ( ) = BER S N z p z dz z = signal power level BER Typical BER vs. S/N curves Frequency-selective channel (no equalization) Gaussian channel (no fading) Flat fading channel S/N

17 BER vs. S/N performance Frequency selective fading <=> irreducible BER floor BER Typical BER vs. S/N curves Frequency-selective channel (no equalization) Gaussian channel (no fading) Flat fading channel S/N

18 BER vs. S/N performance Diversity (e.g. multipath diversity) <=> improved performance BER Typical BER vs. S/N curves Gaussian channel (no fading) Frequency-selective channel (with equalization) Flat fading channel S/N

19 Time-variant transfer function Time-variant CIR: L 1 = i () t i= 0 ( ) ( ) j i h, t a t e φ ( ) τ δ τ τ i Time-variant transfer function (frequency response): L 1 jφi () t j2π fτi, =, = i i= 0 j2π fτ ( ) ( τ ) τ ( ) H f t h t e d a t e e L 1 In a narrowband channel ( ) = ( ) this reduces to: i j i H f, t a t e φ i= 0 () t

20 Example: two-ray channel (L = 2) jφ1 jφ2 ( τ ) = δ ( τ τ ) + δ ( τ τ ) h a e a e jφ1 j2π fτ1 jφ2 j2π fτ2 ( ) = + H f a e e a e e 1 2 At certain frequencies the two terms add constructively (destructively) and we obtain: ( constructive ) = 1+ 2 H f a a ( destructive ) = 1 2 H f a a f

21 Deterministic channel functions (Inverse) Fourier transform Time-variant impulse response h ( τ, t) H ( f, t ) d ( τ, ν ) D ( f, ν ) Doppler-variant transfer function Timevariant transfer function Time- Dopplervariant impulse response

22 Stochastical (WSSUS) channel functions Frequency time time correlation function φ H ( t) φh Δ T d ( Δf ) Channel intensity profile profile φ h τ; Δt ( ) φ ( Δ ; Δ ) S ( τ; ν ) H f t B m SH Δf; ν ( ) Channel Doppler spectrum h φh SH ( τ ) Scattering function ( ν ) T m σ τ B d

23 Stochastical (WSSUS) channel variables Maximum delay spread: T m Maximum delay spread may be defined in several ways. For this reason, the RMS delay spread is often used instead: φh T m ( τ ) τ σ τ h( ) ( ) ( ) ( ) 2 τ φ τ dτ τφh τ dτ = φh τ dτ φh τ dτ 2

24 Stochastical (WSSUS) channel variables Coherence bandwidth of channel: φ H ( Δf ) B m 1 T m B m Implication of coherence bandwidth: If two sinusoids (frequencies) are spaced much less apart than B m, their fading performance is similar. If the frequency separation is much larger than B m, their fading performance is different. 0 f

25 Stochastical (WSSUS) channel variables Maximum Doppler spread: B d SH ( ν ) The Doppler spectrum is often U-shaped (like in the figure on the right). The reason for this behaviour is the relationship (see next slide): V ν = cosα = fd cosα λ SH 0 B d ( ν ) p( ν ) ν Task: calculate p(ν) for the case where p(α) = 1/2π (angle of arrival is uniformly distributed between 0 and 2π).

26 Physical interpretation of Doppler shift arriving path α Rx V direction of receiver movement Doppler frequency shift V ν = cosα = fd cosα λ Maximum Doppler shift V = speed of receiver λ = RF wavelength Angle of arrival of arriving path with respect to direction of movement

27 Delay - Doppler spread of channel L = 12 components in delay-doppler domain delay τ B d 0 Doppler shift ν L 1 = i ( 2 t+ ) i= 0 ( ) ( ) j i i h, t a t e πν φ ( ) τ δ τ τ i

28 Fading distributions (Rayleigh) In a flat fading channel, the (time-variant) CIR reduces to a (time-variant) complex channel coefficient: ( ) ( ) ( ) p a a 2 2 a 2σ = e p ( φ ) σ 2 ( ) ( ) ( ) ( ) jφ t jφi t i i c t = a t e = x t + j y t = a t e When the quadrature components of the channel coefficient are independently and Gaussian distributed, we get: = 1 2π ( ) Rayleigh distribution Uniform distribution

29 Fading distributions (Rice) In case there is a strong (e.g., LOS) multipath component in addition to the complex Gaussian component, we obtain: () ( ) ( ) ( ) jφ t jφi t 0 0 i i c t = a + a t e = a + a t e From the joint (magnitude and phase) pdf we can derive: a ( a + a0 ) 2σ aa0 p( a) = e I σ σ ( ) Rice distribution Modified Bessel function of first kind and order zero

30 Representation in complex plane Complex Gaussian distribution is centered at the origin of the complex plane => magnitude is Rayleigh distributed, the probability of a deep fade is larger than in the Rician case iy p( x, y) Complex Gaussian distribution is centered around the strong path => magnitude is Rice distributed, probability of deep fade is extremely small iy a 0 x x Bell-shaped function

31 Countermeasures: wideband systems Equalization (in TDMA systems) - linear equalization - Decision Feedback Equalization (DFE) - Maximum Likelihood Sequence Estimation (MLSE) using Viterbi algorithm Rake receiver schemes (in DS-CDMA systems) Sufficient number of subcarriers and sufficiently long guard interval (in OFDM or multicarrier systems) Interleaving, FEC, ARQ etc. may also be helpful in wideband systems.

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