In wireless telecommunications, multipath is the propagation phenomenon that results in radio signals' reaching the receiving antenna by two or more paths. Wireless communication is the transfer of information over a distance without the use of electrical conductors or " Wires quot Radio propagation is a term used to explain how Radio waves behave when they are Transmitted, or are propagated from one point on the Earth Radio is the transmission of signals by Modulation of electromagnetic waves with frequencies below those of visible Light. In Telecommunication, signalling (UK spelling or signaling (US spelling has the following meanings The use of signals for controlling communications An antenna is a Transducer designed to transmit or Receive electromagnetic waves In other words antennas convert electromagnetic waves into Causes of multipath include atmospheric ducting, ionospheric reflection and refraction, and reflection from terrestrial objects, such as mountains and buildings. In Telecommunication, an atmospheric duct is a horizontal Layer in the lower atmosphere in which the vertical Refractive index gradients are such that Ionospheric reflection: Of electromagnetic waves propagating in the Ionosphere, a redirection i Refraction is the change in direction of a Wave due to a change in its Speed. Reflection is the change in direction of a Wave front at an interface between two different media so that the wave front returns into the medium from which A mountain is a Landform that extends above the surrounding Terrain in a limited area with a peak In Architecture, Construction, Engineering and real estate development the word building may refer to one of the following Any man-made

The effects of multipath include constructive and destructive interference, and phase shifting of the signal. In physics interference is the addition ( superposition) of two or more Waves that result in a new wave pattern The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 In Telecommunication, signalling (UK spelling or signaling (US spelling has the following meanings The use of signals for controlling communications This causes Rayleigh fading, named after Lord Rayleigh. Rayleigh fading is a statistical model for the effect of a propagation environment on a Radio signal such as that used by Wireless John William Strutt 3rd Baron Rayleigh OM (12 November 1842 &ndash 30 June 1919 was an English Physicist who with William Ramsay, discovered The standard statistical model of this gives a distribution known as the Rayleigh distribution. In Probability theory and Statistics, the Rayleigh distribution is a Continuous Probability distribution.

Rayleigh fading with a strong line of sight content is said to have a Rician distribution, or to be Rician fading. This is about the phenomenon of loss of signal in telecommunications Line-of-sight propagation refers to Electro-magnetic radiation including light emissions traveling in a straight line In Probability theory and Statistics, the Rice distribution, named after Stephen O Rician fading is a Stochastic model for Radio propagation anomaly caused by partial cancellation of a radio signal by itself &mdash the

In facsimile and television transmission, multipath causes jitter and ghosting, seen as a faded duplicate image to the right of the main image. Fax (short for facsimile, from Latin fac simile, "make similar" i Television ( TV) is a widely used Telecommunication medium for sending ( Broadcasting) and receiving moving Images, either monochromatic In Telecommunications transmission is the process of sending propagating and receiving an analogue or digital information signal over a physical point-to-point or Jitter is an unwanted variation of one or more characteristics of a periodic signal in Electronics and Telecommunications. See also Television interference In Television, a ghost is an unwanted Image on the screen appearing Superimposed on the desired Ghosts occur when transmissions bounce off a mountain or other large object, while also arriving at the antenna by a shorter, direct route, with the receiver picking up two signals separated by a delay.

Radar multipath echoes from an actual target cause ghosts to appear.

In digital radio communications (such as GSM) multipath can cause errors and affect the quality of communications. GSM ( Global System for Mobile communications: originally from Groupe Spécial Mobile) is the most popular standard for Mobile phones in the The word error has different meanings and usages relative to how it is conceptually applied In the vernacular quality can mean a high degree of excellence (“a quality product” a degree of excellence or the lack of it (“work of average quality” or a property of The errors are due to Intersymbol interference (ISI). In Telecommunication, intersymbol interference ( ISI) is a form of Distortion of a signal in which one symbol interferes with Equalisers are often used to correct the ISI. Alternatively, techniques such as orthogonal frequency division modulation and Rake receivers may be used. Orthogonal frequency-division multiplexing ( OFDM) — essentially identical to Coded OFDM ( COFDM) and Discrete multi-tone modulation ( A rake receiver is a Radio receiver designed to counter the effects of Multipath fading.

In a Global Positioning System Receiver (GPSR), multipath signals can cause a stationary receiver's output to indicate as if it were randomly jumping about or creeping. Basic concept of GPS operation A GPS receiver calculates its position by carefully timing the signals sent by the constellation of GPS Satellites high above the Earth When the unit is moving the jumping or creeping is hidden, but it still degrades the displayed accuracy.

## Mathematical modeling

Mathematical model of the multipath impulse response.

The mathematical model of the multipath can be presented using the method of the impulse response used for studying linear systems. The impulse response of a system is its output when presented with a very brief input signal an impulse A linear system is a mathematical model of a System based on the use of a Linear operator.

Suppose to transmit a single, ideal Dirac pulse of electromagnetic power at time 0, i. The Dirac delta or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac. Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of e.

x(t) = δ(t)

At the receiver, due to the presence of the multiple electromagnetic paths, more than one pulse will be received (we suppose here that the channel has infinite bandwidth, thus the pulse shape is not modified at all), and each one of them will arrive at different times. Channel, in communications (sometimes called communications channel) refers to the medium used to convey Information from a In fact, since the electromagnetic signals travel at the speed of light, and since every path has a geometrical length possibly different from that of the other ones, there are different air travelling times (consider that, in free space, the light takes 3μs to cross a 1km span). In Classical physics, free space is a concept of Electromagnetic theory, corresponding to a theoretically "perfect" Vacuum, and sometimes Thus, the received signal will be expressed by

$y(t)=h(t)=\sum_{n=0}^{N-1}{\rho_n e^{j\phi_n} \delta(t-\tau_n)}$

where N is the number of received impulses (equivalent to the number of electromagnetic paths, and possibly very large), τn is the time delay of the generic nth impulse, and $\rho_n e^{j\phi_n}$ represent the complex amplitude (i. e. , magnitude and phase) of the generic received pulse. As a consequence, y(t) also represents the impulse response function h(t) of the equivalent multipath model.

More in general, in presence of time variation of the geometrical reflection conditions, this impulse response is time varying, and as such we have

τn = τn(t)

ρn = ρn(t)

φn = φn(t)

Very often, just one parameter is used to denote the severity of multipath conditions: it is called the multipath time, TM, and it is defined as the time delay existing between the first and the last received impulses

TM = τN − 1 − τ0

Mathematical model of the multipath channel transfer function.

In practical conditions and measurement, the multipath time is computed by considering as last impulse the first one which allows to receive a determined amount of the total transmitted power (scaled by the atmospheric and propagation losses), e. g. 99%.

Keeping our aim at linear, time invariant systems, we can also characterize the multipath phenomenon by the channel transfer function H(f), which is defined as the continuous time Fourier transform of the impulse response h(t)

$H(f)=\mathfrak{F}(h(t))=\int_{-\infty}^{+\infty}{h(t)e^{-j 2\pi f t} d t}=\sum_{n=0}^{N-1}{\rho_n e^{j\phi_n} e^{-j2 \pi f \tau_n}}$

where the last right-hand term of the previous equation is easily obtained by remembering that the Fourier transform of a Dirac is a complex exponential function, an eigenfunction of every linear system. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and In Mathematics, an eigenfunction of a Linear operator, A, defined on some Function space is any non-zero function f in

The obtained channel transfer characteristic has a typical appearance of a sequence of peaks and valleys (also called notches); it can be shown that, on average, the distance (in Hz) between two consecutive valleys (or two consecutive peaks), is roughly inversely proportional to the multipath time. The so-called coherence bandwidth is thus defined as

$B_C \approx \frac{1}{T_M}$

For example, with a multipath time of 3μs (corresponding to a 1 km of added on-air travel for the last received impulse), there is a coherence bandwidth of about 330 kHz.