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Giacinto Gelli » 6.Multipath propagation


Complex representation of bandpass signals and systems
Multipath channel model
Delay spread and Doppler spread
Impulse response of the multipath channel
Narrowband and wideband fading channel models

Bandpass signals

Bandpass (BP) signals are common in electrical and optical communications:

  • bandpass modulations (analog/digital) with carrier frequency fc
  • white noise at the output of a BPF filter centered on fc

Frequency spectrum S(f) with one-sided bandwidth 2B centered on carrier frequency fc:

  • Hermitian symmetry with respect to f=0, no general symmetry with respect to fc

Representations of a bandpass signal

A bandpass signal can be represented in three equivalent forms.

Complex form: \[s(t) = \mbox{Re} \left\{ u(t) \, e^{j 2 \pi f_c t} \right\}\]

I-Q components: \[s(t) = s_I(t) \, \cos(2 \pi f_c t) - s_Q(t) \, \sin(2 \pi f_c t)\]
Envelope/phase: \[s(t) = a(t) \, \cos\left[ 2 \pi f_c t + \phi(t) \right]\]
Straightforward relationships hold between these three representations.

Complex envelope

The signal u(t) is the complex envelope of s(t):

\[u(t) = s_I(t) + j s_Q(t) = a(t) e^{j \phi(t)}\]

Lowpass (LP) signal with spectral support (-B,B).

Ps=Pu/2 (the powers of the BP and LP signals differ by a factor of two).

Relationship between spectra:

\[S(f) = \frac{1}{2} \left[ U(f-f_c) + U^{*}(f+f_c) \right]\]

Bandpass LTI systems

Let h(t) be the impulse response of a BP LTI system.
Since h(t) is BP, it can be represented as a BP signal:

  • \[h(t) = 2 \, \mbox{Re} \left\{ h_l(t) \, e^{j 2 \pi f_c t}\right\}\]
  • \[H(f) = H_l(f-f_c) + H_l^{*}(f+f_c)\]

where hl(t) is the complex envelope of h(t).
Note the extra factor 2, which allows one to simplify the input-output relationships (details on the following slide).

BP versus LP model

The advantage of the LP model over the BP one is elimination of the carrier frequency fc

useful also in simulations → reduction of the sampling frequency

Main features of the wireless channel

Multipath propagation:

  • due to the presence of reflectors/scatterers, the TX signal arrives at the RX through multiple paths with different lengths/delays
  • a transmitted single pulse is received as a train of pulses with different delays → time dispersion


  • due to relative motion between TX/RX and/or the reflectors/scatterers, the channel characteristics change with time
  • a transmitted sinusoid is received as multiple sinusoids with different carrier frequencies → frequency dispersion

Time/frequency dispersions are dual phenomena:

  • they are due to different causes

Scenario #1: free space, stationary

Example: transmission of a sinusoid

Assume that a simple sinusoid is transmitted:

\[s(t) = \mbox{Re}\{e^{j 2 \pi f_c t} \} = \cos(2 \pi f_c t)\]
\[r(t) = \mbox{Re}\{ v(t) e^{j 2 \pi f_c t} \} = \alpha \, \cos\left[ 2 \pi f_c \left( t - \frac{d_0}{c}\right) \right]\]

The received signal is a delayed sinusoid with the same frequency:

  • α is the attenuation introduced by the channel (α2 ∝ path loss)
  • τ0=d0/c is the propagation delay

This simple channel introduce only a time shift of the TX signal.

Scenario #2: free-space, moving RX

Doppler shift and Doppler spread

Assume that:

  • RX is moving toward TX with constant speed v → d(t)=d0-vt
  • a sinusoid is transmitted → u(t)=1

\[s(t) = \mbox{Re}\{e^{j 2 \pi f_c t} \} = \cos(2 \pi f_c t)\]
\[r(t) = \mbox{Re}\{ v(t) e^{j 2 \pi f_c t} \} =\alpha(t) \, \cos\left[ 2 \pi f_c \left( 1+\frac{v}{c}\right) t -  f_c \frac{d_0}{c} \right]\]

The received signal is not a simple delayed sinusoid, two new effects associated to channel variability:

  • frequency shift fD = fc v/c → Doppler shift
  • frequency dispersion (bandwidth expansion) due to amplitude modulation α(t) → Doppler spread

Effect of angle on Doppler shift

Assume that θ is the arrival angle of the received signal relative to the direction of motion.

The Doppler shift is:

\[f_D = \frac{v}{c} f_c \cos(\theta) = \frac{v}{\lambda} \cos(\theta)\]

where fD is maximum (in magnitude) when |cos(θ)|=1 → θ =0,π

Scenario #3: multipath, stationary

Example: transmission of a pulse

Assume that an ideal pulse is transmitted:

\[u(t) = \delta(t)\]
\[v(t) = \sum_{n=1}^{N} \alpha_n \delta(t-\tau_n) \, e^{- j 2 \pi f_c \tau_n}\]

The received signal is the superposition of N pulses, with different delays, amplitudes and phases.
A measure of the amount of time dispersion is the multipath delay spread Tm.

T_m = \max_{k \ne h} | \tau_k - \tau_h |\]

Scenario #4: multipath, moving RX

Time-varying impulse response

Response of the channel at time t to an impulse applied at time t-τ:

\[c(\tau,t) = \sum_{n=1}^{N} \alpha_n(t) e^{-j \phi_n(t)} \, \delta \left[ \tau - \tau_n(t) \right]\]


  • t is the time when the impulse response is observed
  • t-τ is the time when the impulse is applied to the channel
  • τ is the time difference between the observation time and the instant of application of the pulse
  • causality requires that c(τ,t)=0 for τ<0

The most general LTV system introduces time shift and time dispersion, as well as frequency shift and frequency dispersion.

Deterministic vs. statistical model

\[c(\tau,t) = \sum_{n=1}^{N} \alpha_n(t) e^{-j \phi_n(t)} \, \delta \left[ \tau - \tau_n(t) \right]\]

Deterministic channel description requires exact knowledge of the time-varying coefficients αn(t), phases φn(t), and delays τn(t).
Difficult to obtain in practice, it is common to model c(τ,t) as a random process → statistical channel description.
Each macroscopic path is generally a large sum of microscopic contributions → c(τ,t) can be modeled as a Gaussian random process due to the CLT.

Received signal characteristics

r(t) = \mbox{Re} \left\{ \sum_{n=1}^{N}\alpha_n(t) e^{-j \phi_n(t)} \, u \left[ t - \tau_n(t) \right] e^{j 2 \pi f_c t}\right\}

The RX signal is the sum of many multipath components.
Amplitudes αn(t) change slowly.
Phases φn(t) change rapidly, since:

\[\phi_n(t) = 2 \pi f_c \tau_n(t)\]

E.g. fc=1 GHz and τn = 50 ns → fcτn = 50 >> 1

Constructive and destructive addition of signal components.

Amplitude fading of received signal.

Narrowband vs. wideband fading

Narrowband fading:

  • when Tm << 1/Bu the multipath components are not resolvable and the temporal dispersion is negligible

Wideband fading:

  • when Tm >> 1/Bu the multipath components are resolvable and a significant temporal dispersion is observed

Different models for the two situations:

  • wideband model more general, includes narrowband one as a limiting case
Bu is the bandwith of u(t)
1/Bu  is the typical time variation of u(t)

Bu is the bandwith of u(t) 1/Bu is the typical time variation of u(t)

Bu  is the bandwith of u(t)
1/Bu  is the typical time variation of u(t)

Bu is the bandwith of u(t) 1/Bu is the typical time variation of u(t)


The wireless channel can be modeled as a linear time-varying (LTV) random system.
A wireless channel introduces both time dispersion and frequency dispersion:

  • time dispersion measured by delay spread
  • frequency dispersion measured by Doppler spread

The received signal exhibits amplitude fluctuations due to constructive/destructive interference between the paths.
Two different fading models (narrowband/wideband) depending on relation between delay spread Tm and inverse signal bandwidth 1/Bu.

I materiali di supporto della lezione

A. Goldsmith. Wireless Communications. Cambridge University Press, 2005 (app. A, chap. 3)

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