Model Saleh Valenzuela

The signal at the input of the receiver in the Saleh-Valenzuela model

The Saleh – Valenzuela model is a theoretical model describing the multipath propagation of ultra-wideband signals indoors. In 2002-2003, it was adopted by the IEEE 802.15.4a working group as a standard model of an ultra-wideband channel.

Content

Description

The Saleh – Valenzuela model describes the propagation of an ultrashort pulse, which is represented by the Dirac delta function δ (t), in a limited enclosed space (for example, in an office room). The pulse can get from the transmitter to the receiver in various ways - either in a straight line (if the transmitter is directly observed from the receiving point), or reflected from various objects, possibly repeatedly. As a result, the signal entering the receiver is an aggregate of a large number of short pulses of various amplitudes arranged in different ways along the time axis.

Measurements made in 1987 by Adel Saleh and Reynaldo Valenzuela ^[1] showed that the pulses arrive in groups, which in the model are called “clusters”. Each cluster consists of a certain number of pulses, which in the model are called “rays” or “paths”. The cluster can be physically interpreted as reflection from an object, and rays as reflections from closely spaced parts of this object, including roughnesses and surface roughness.

Thus, the received signal is a burst of pulses (which may intersect in time), and each subsequent burst has an average lower amplitude than the previous one, and each individual pulse in the burst has a smaller amplitude compared to the previous pulse of this burst. The decrease in amplitude is manifested purely statistically, since the amplitude and delay of each pulse are a random variable.

Mathematical Description

The pulse transition function of the information transmission channel is a combination of a large number of delta functions of various amplitudes:

h(t)=\sum \limits _{l=0}^{\infty }\sum \limits _{k=0}^{\infty }\beta _{kl}\delta (t-T_{l}-\tau _{kl}),

{\ displaystyle h (t) = \ sum \ limits _ {l = 0} ^ {\ infty} \ sum \ limits _ {k = 0} ^ {\ infty} \ beta _ {kl} \ delta (t-T_ {l} - \ tau _ {kl}),}

Where

l

{\ displaystyle l}

- cluster number, for the first cluster l = 0;

k

{\ displaystyle k}

- pulse number in the cluster, for the first pulse in the cluster k = 0;

\beta _{kl}

{\ displaystyle \ beta _ {kl}}

Is the amplitude of the kth pulse in the lth cluster;

T_{l}

{\ displaystyle T_ {l}}

- delay of the l- th cluster (in the first pulse) relative to the transmitted pulse;

\tau _{kl}

{\ displaystyle \ tau _ {kl}}

- delay of the kth pulse in the lth cluster relative to the first pulse of the cluster.

The amplitude of the pulse in the cluster is a random variable, the mathematical expectation of the square of which falls exponentially in time of arrival of the cluster and time of arrival of the pulse relative to the beginning of the cluster:

{\overline {\beta _{kl}^{2}}}={\overline {\beta _{00}^{2}}}e^{-\Lambda \mathrm {T} _{l}}e^{-\lambda \tau _{kl}},

{\ displaystyle {\ overline {\ beta _ {kl} ^ {2}}} = {\ overline {\ beta _ {00} ^ {2}}} e ^ {- \ Lambda \ mathrm {T} _ {l }} e ^ {- \ lambda \ tau _ {kl}},}

Where

{\overline {\beta _{00}^{2}}}

{\ displaystyle {\ overline {\ beta _ {00} ^ {2}}}}

- mat. waiting for the square of the amplitude of the first pulse in the first cluster.

The time sequence of pulses is a double Poisson process: the time delays of the clusters relative to the previous cluster and the delay of pulses in the cluster relative to the previous pulse in the cluster are distributed according to Poisson. In other words, the time distribution function between neighboring clusters and neighboring pulses is given by the expressions

p(\Delta \mathrm {T} )=\Lambda e^{-\Lambda \cdot \Delta \mathrm {T} };

{\ displaystyle p (\ Delta \ mathrm {T}) = \ Lambda e ^ {- \ Lambda \ cdot \ Delta \ mathrm {T}};}

p(\Delta \tau )=\lambda e^{-\lambda \cdot \Delta \tau }.

{\ displaystyle p (\ Delta \ tau) = \ lambda e ^ {- \ lambda \ cdot \ Delta \ tau}.}

Notes

↑ Adel AM Saleh and Reinaldo A. Valenzuela. A statistical model for indoor multipath propagation. IEEE Journal on Selected Areas of Communications, SAC-5: 128–13, February 1987.

Sources

Qiyue Zou, Alireza Tarighat, Ali H. Sayed, Fellow. IEEEP Performance Analysis of Multiband OFDM UWB Communications With Application to Range Improvement. IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 6, NOVEMBER 2007.

[Saleh87-1] Adel AM Saleh and Reinaldo A. Valenzuela. A statistical model for indoor multipath propagation. IEEE Journal on Selected Areas of Communications, SAC-5: 128–13, February 1987.