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

- bandpass modulations (analog/digital) with carrier frequency f
_{c} - white noise at the output of a BPF filter centered on f
_{c}

Frequency spectrum S(f) with one-sided bandwidth 2B centered on carrier frequency f_{c}:

- Hermitian symmetry with respect to f=0, no general symmetry with respect to f
_{c}

A bandpass signal can be represented in three equivalent forms.

Complex form:

I-Q components:

Envelope/phase:

Straightforward relationships hold between these three representations.

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

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

P_{s}=P_{u}/2 (the powers of the BP and LP signals differ by a factor of two).

Relationship between spectra:

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:

where h_{l}(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).

The advantage of the LP model over the BP one is elimination of the carrier frequency f_{c
}

useful also in simulations → reduction of the sampling frequency

**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

Assume that a simple sinusoid is transmitted:

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

- α is the attenuation introduced by the channel (α
^{2}∝ path loss) - τ
_{0}=d_{0}/c is the propagation delay

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

Assume that:

- RX is moving toward TX with c
**onstant speed**v → d(t)=d_{0}-vt - a sinusoid is transmitted → u(t)=1

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

**frequency shift**f_{D }= f_{c}v/c →**Doppler shift****frequency dispersion**(bandwidth expansion) due to amplitude modulation α(t) →**Doppler spread**

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

The Doppler shift is:

where f_{D} is maximum (in magnitude) when |cos(θ)|=1 → θ =0,π

Assume that an ideal pulse is transmitted:

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 T _{m}**.

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

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 channel description** requires **exact** knowledge of the time-varying coefficients α_{n}(t), phases φ_{n}(t), and delays τ_{n}(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.

The RX signal is the sum of many multipath components.

Amplitudes α_{n}(t) change slowly.

Phases φ_{n}(t) change rapidly, since:

E.g. f_{c}=1 GHz and τ_{n} = 50 ns → f_{c}τ_{n} = 50 >> 1

Constructive and destructive addition of signal components.

Amplitude fading of received signal.

**Narrowband fading:**

- when T
_{m}<< 1/B_{u}the multipath components are not resolvable and the temporal dispersion is negligible

**Wideband fading:**

- when T
_{m}>> 1/B_{u}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

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 T_{m} and inverse signal bandwidth 1/B_{u}.

*3*. Current and emerging wireless systems

*5*. Shadowing

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

Progetto "Campus Virtuale" dell'Università degli Studi di Napoli Federico II, realizzato con il cofinanziamento dell'Unione europea. Asse V - Società dell'informazione - Obiettivo Operativo 5.1 e-Government ed e-Inclusion