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:
Frequency spectrum S(f) with one-sided bandwidth 2B centered on carrier frequency fc:
A bandpass signal can be represented in three equivalent forms.
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).
Ps=Pu/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 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).
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
Time/frequency dispersions are dual phenomena:
Assume that a simple sinusoid is transmitted:
The received signal is a delayed sinusoid with the same frequency:
This simple channel introduce only a time shift of the TX signal.
The received signal is not a simple delayed sinusoid, two new effects associated to channel variability:
Assume that θ is the arrival angle of the received signal relative to the direction of motion.
The Doppler shift is:
where fD 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 Tm.
Response of the channel at time t to an impulse applied at time t-τ:
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 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.
The RX signal is the sum of many multipath components.
Amplitudes αn(t) change slowly.
Phases φn(t) change rapidly, since:
E.g. fc=1 GHz and τn = 50 ns → fcτn = 50 >> 1
Constructive and destructive addition of signal components.
Amplitude fading of received signal.
Different models for the two situations:
The wireless channel can be modeled as a linear time-varying (LTV) random system.
A wireless channel introduces both time dispersion and frequency dispersion:
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.
A. Goldsmith. Wireless Communications. Cambridge University Press, 2005 (app. A, chap. 3)