# Giacinto Gelli » 6.Multipath propagation

### Outline

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

Time-variability:

• 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

### 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.

### 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

### 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 |\]$

### Time-varying impulse response

Response of the channel at time t to an impulse applied at time t-τ: $$c(\tau,t) = \sum_{n=1}^{N} \alpha_n(t) e^{-j \phi_n(t)} \, \delta \left[ \tau - \tau_n(t) \right]$$

where:

• 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

### Conclusions

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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