نیمسال اوّل 92-91 افشین همّت یار دانشکده مهندسی کامپیوتر...

9191--9292نیمسال اّو�ل نیمسال اّو�ل یاریار افشین هم�تافشین هم�ت

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�ار ) �ار )مخابرات سی ((4040--626626مخابرات سی

سیمسیم کانال انتشار بیکانال انتشار بی

Propagation CharacteristicsPropagation Characteristics

Large scale )overall effects due to distance(:o Path loss )includes average shadowing(o Shadowing )due to obstructions(

Small scale )local effects due to reflection/diffraction(:o Multipath fadingo Signal fades 30-40dB by just moving tens of centimeters

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Path Loss ModelingPath Loss Modeling

Maxwell’s equations Complex and impractical

Free space path loss model Too simple

Ray tracing models Requires site-specific information

Empirical Models Don’t always generalize to other environments

Simplified power falloff models Main characteristics: good for high-level analysis

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Free Space LOS ModelFree Space LOS Model

Path loss for unobstructed LOS path )satellites, microwave links(:

P.L. = )4πd/λ(2

Power falls off: Proportional to d2 )20dB/dec( Proportional to f2 )f=c/λ( Usually power measured with respect to a reference distance d0:

Pr)d( = Pr)d0()d0/d(2

Where for example d0 = 1m )indoor( and about 100m )outdoor( 44

Ray Tracing ModelsRay Tracing Models

Three main effects modeled: 1. Reflection: Signal reflection from surfaces with sizes >> λ 2. Diffraction: Especially important at sharp edges

-> causes waves go behind direct obstacles 3. Scattering: Due to reflection with objects smaller than λ such as rough surfaces, small objects, trees,…Requires detailed geometry and dielectric properties of site - Similar to Maxwell, but easier math.Computer packages often used

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Two-Ray Reflection ModelTwo-Ray Reflection Model

Path loss for one LOS path and one reflected bounceo Good approximation for tall towers )over 50m(o Ground bounce approximately cancels LOS path above critical distance faster roll off o ∆ = d”-d’ ≈ 2hthr/d )for d >> hr+ ht( phase shift θ=2π∆/λ ≈ 4πhthr/(λd) P.L. = )4πd/λ(2/)GtGr)1-ejθ(( ≈ d4/)ht

2hr2GtGr(

Power falls off:o for small d is proportional to d2

o for large d is proportional to d4 and independent of λ66

Diffraction ModelDiffraction Model

Although much weaker than reflection and multipath, but main contributor when those not present, especially for receivers behind hills and mountains. A good model is the so-called “knife-edge” model

o For ν =0 )marginal direct sight, h=0(, diffraction loss =6db

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Empirical Models )1(Empirical Models )1(

Use experimental data based on the type of environment:

o Natural: Flat areas, hills, rivers, sea, foresto Man-made: Urban areas, suburban

Models extracted from real data for some typical scenarios.

Can be fine-tuned by adjusting various parameters for a specific region.

Commonly used in cellular system simulations

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Empirical Models )2(Empirical Models )2(

Durbin Modelo Use topographic data and LOS/diffraction)knife-edge( models.o Divide area into grids and use interpolation techniques.

Okumura modelo Empirically based )site/freq specific(o Awkward )uses graphs(

Hata modelo Analytical approximation to Okumura model

Cost 136 Modelo Extends Hata model to higher frequency )2 GHz(

Walfish/Bertoonio Cost 136 extension to include diffraction from rooftops.

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Okumura Model )1(Okumura Model )1(

o Completely experimental model )1968(o Based on extensive measurements in Tokyo areao Frequency range: 150-2000 MHzo Range: 1-100 Km

o Ht: 30-1000m

o Hr: 1-10m

o Propagation media: Urban, suburban, hilly1010

Okumura Model )2(Okumura Model )2(

Power Loss estimation:o Use free space estimate.o Adjust A using urban measurement.o Adjust for ht and hr )ref ht=200m, hr=3m(.o Adjust for various propagation areas.

Advantages:o Covers many scenarios.o Relatively reliable: Max. 10-14dB error

Disadvantages:o Only experimentalo Lots of curves requiredo Not very accurate for rural areas

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Hata ModelHata Model

Basically formulates Okumura data into a programmable formulation )for freq up to 1.5GHz( Gives three basic formulas for Urban, suburban and rural areas )1980(.

o For example, for urban area the model is as follows:

L)dB(=69.55+26.26log)fc(-13.82log)ht(-)hr( +)44.9-6.55log)ht((log)d(

d: T-R distance )1-20Km( -> not very accurate for small cells a)hr(: Correction factor for effective antenna height

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Simplified Path Loss Model )1(Simplified Path Loss Model )1(

Used when path loss dominated by reflection. Most important parameter is the path loss exponent n, determined empirically.

Lav)d( = Lav)d0( + 10nlog)d/d0( )in dB(

o d0 : ref distance > near field areas of antennao For cellular, d0 in Km and for microcellular, d0 in tens of meter rangeo n=2 for free space, n=4 for 2-Ray modelo In general, n can take values between 2 and 6.

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Path Loss Exponent for different environments

Free Space 2

Urban Area 2.7 to 3.5

In building LOS 1.6 to 1.8

Obstructed in building 4 to 6

Obstructed in factories 2 to 3

Simplified Path Loss Model )2(Simplified Path Loss Model )2(

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Shadowing )1(Shadowing )1(

Models attenuation from obstructions. Random due to random number and type of obstructions.

o Loss in passing through a wall of thickness d is s)d(=ce-ad where a is penetration factor of that wall.

o If total thickness of walls in an area is dt then: s)dt(=ce-adt

o Since dt can be modeled as Gaussian: log)s)dt(( = log)c( - adt

which is a log-normal distribution.1515


So, shadowing typically follows a log-normal distribution and combined path loss and shadowing can be modeled as:

L)d( = Lav)d( + Xσ )in dB(

= Lav)d0( + 10nlog)d/d0( + Xσ )in dB(

where X is a Gaussian random variable with mean zero and variance σ which is typically in the range of 4 to 12.

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Fit model to measurement data: Parameters to estimate: Lav)d0(, n, and σ

o “Best fit” line through dB datao Lav)d0( obtained from measurements at d0.o Exponent is MMSE estimate based on data

Shadowing varianceo Variance of data relative to estimated path loss model )straight line(


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Cell Coverage Area )1(Cell Coverage Area )1(

Path loss: circular cellso For a given radius, can estimate average power can simplified path loss model.o Then, using shadowing log-normal model can estimate percentage of power points in cell boundary with power than some threshold value:

o Next step: Estimate percentage of area that has power greater than the threshold:

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Cell Coverage Area )2(Cell Coverage Area )2(

Path loss + shadowing: amoeba cells

Outage probability: Probability of received power to be below given minimum in an area.

Cell coverage area: Percentage of cell locations at desired power, which increases as shadowing variance decreases.

For example, for n=4 and σ=8, if at boundary 75% of points are over threshold, then 90% of the area will have received power above threshold.

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Highly dependent on materials, partitioning, … Tables used for loss of various materials

Three main models:1( Log-normal: Similar to shadowing in cellular environments

L)d( = Lav)d0( + 10nlog)d/d0( + Xσ )in dB(

2( Piecewise liner model )Ericsson(: The loss exponent varies by distance and upper/lower

bounds are introduced.

3( Attenuation Factor: Take into account attenuation of floors and other obstacles

between transmitter and receiver:

L)d( = Lav)d0( + 10nSFlog)d/d0( + FAF + ΣPAF )in dB( )Same Floor n( )Floor Att.( )Partition Att.(

Indoor ModelsIndoor Models

2020

Large scale effects mainly important in overall design of network:

Cell radius, Tx power, Frequency planning, …

Small scale effects mainly important in TX/RX design:

Coding, Modulation, Diversity, Equalizer, …

Small Scale Effects )1(Small Scale Effects )1(

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In practice, signals arrive at receiver from multiple paths with different delays and attenuations.

Combination of these signals with different phases, results in an effect known as fading. Fading causes fast changes in received power, which is not related to overall path loss effect introduced earlier.


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Random Number of multipath components, each with

o Random amplitudeo Random phaseo Random Doppler shifto Random delay

Random components change with time. Leads to time-varying channel impulse response.

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Random change of received power )fading(

Random change of carrier frequency )frequency spread(

Random spread of signal in time )time spread/ISI(

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Fading: Random change of received power in spaceo Due to multipath, vector combination of multiple copies of

narrowband signal creates a location-varying distribution of power in space.

o This causes large changes in signal power in short distances.

o For a moving receiver, this means change of received power in time.

o Also, for a fixed receiver, this means that the receiver may get stuck inside a deep fade and never get an acceptable signal .

Use of antenna diversity discussed later.

o Fading depends on carrier frequency, so the two-dimension distribution of power is also a function of frequency.

Frequency response of channel at each point of space2525


Frequency Spread: Random change of carrier frequency

o Due to motion of receiver or moving objects in the environment, the frequency of received signal will also change randomly.

Doppler Effect: fd = v/λ cos)θ(

o Doppler effect causes spread of bandwidth of the received signal )frequency spread(.

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Time Spread: Random spread of signal in time

o Due to multipath, delayed versions of signal arrive at the receiver and are combined.

o Therefore, the received signal will also be spread in time.

o The time spread effect is not noticeable for narrowband signals and largely affects signal with larger bandwidths.

o For digital signals, this will result in ISI )Inter-Symbol Interference(.

Need for Equalizer

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Multipath channel characteristics, number of paths and their attenuation and delay, and its relation to BW of transmitted signal, will determine:

o Type of fading in frequency domain

o Time spread effects )ISI(

Velocity of movement of Tx, Rx, or surrounding objects, will determine:

o Rate of change of fading in time domain

o Amount of frequency spread

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Frequency Response of Channel )1(Frequency Response of Channel )1(

The multipath nature of wireless channel causes channel frequency response to be non-ideal. Although channel is in general time varying, for a given time, the channel can be considered approximately constant and we can plot frequency response of the channel. How the channel frequency response looks like, depends on number and overall delay of paths coming from different directions. What is important in terms of channel response is the relation between changes in channel response and signal bandwidth. So a channel can be close to ideal for one signal, but completely non-ideal for another one with a larger BW. 2929

Frequency Response of Channel )2(Frequency Response of Channel )2(

For a narrowband signal of BW=10KHz the above channel is close to ideal. For a GSM signal with BW=200KHz, channel is non-ideal. We will have ISI and equalization becomes essential. The term “Coherence BW” of a channel, BC, gives range of frequencies over which channel changes are small. )for the above channel, BC=40KHz( So, what is important is the relation between BC and signal BW.

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Time Response of Channel )1(Time Response of Channel )1( Time variation is wireless channel are due to relative velocity of

transmitter )or main obstacles( and the receiver. This relative motion causes spread of carrier frequency )Doppler Spread( proportional to the velocity. fd=v/λ cos)θ( frequency range: )fc – fd ( … )fc+fd(

for example: v= 1m/s , λ=10cm, θ=0 fd=10HzIn addition, motion causes change in channel frequency response )fading variation( in time/space.

Coherence time of a channel )Tc( is the time over which channel can be considered almost constant.

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Time Response of Channel )2(Time Response of Channel )2(

Baseband time response of channel at t to impulse at t-τ:

o t is time when impulse response is observed.o t-τ is time when impulse put into the channel.o τ is how long ago impulse was put into the

channel for the current observation. So for a general time varying channel, the amplitudes, delays and phases of paths will be different if you apply impulses at different times. 3232


Simplified Time Invariant Impulse Response

In practice, the attenuations and delays change with time, but changes are rather slow )compared with bit rate of signal(, so the above model is a good approximation. Question: What is a good statistical model for φn , an, and τn ? Answer: Easy one for φn is unif [0, 2π]. 3333


Statistical behavior of an

Even though an denotes the amplitude of signal arriving at τn, but an itself can still be the sum of many smaller paths arriving at “almost” the same time. In fact the resolution of a channel in separating different paths depends on its bandwidth, which in the best case is the same as the transmitted signal bandwidth )BW(. So delays with resolution )|τk1-τk2|( less than 1/BW will be combined together and form an. In other words each main path an will be formed from many “subpaths” added together:

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Statistical behavior of an Since anejφ is formed by sum of many random signals, its I and Q components will be Gaussian, therefore its envelop an will have a Rayleigh distribution:

where σ2 is the average power of the signal before envelop detection. Obviously, a Rayleigh random variable will have non-zero mean. Its mean is equal to 1.25σ. If an strong LOS path is available, the distribution will be slightly different and given by Rician distribution. In this case the random multipath will be added to an almost stationary LOS signal:

where A denotes the peak amplitude of the dominant signal and I0).( is the Bessel function of first kind and zero order.

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Rayleigh and Rician Distributions

3636


Statistical behavior of an Note that, path strength gets weaker as we go further from main path. Therefore, the paths with longer delays, τn, will have smaller E[an] Relative power density graphs as functions of delay. Also, in some scenarios paths arriving at different time delays might be correlated, so an can be somehow correlated for such delays. Measurements also support another distribution known as Nakagami distribution in some environments.

o Can model both Rician and Rayleigh, also can model worse than Rayleigh scenarios as well.o Better for closed form BER expressions.

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Statistical behavior of τn

The individual path delays can be modeled by Poisson distribution. In other words, the difference between path delays is given by exponential distribution. However, such models do not work very well for wireless channels that have “memory”. In practice, general behavior of path delays is expressed in terms of the so-called overall “delay spread”.

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Time Response of Channel )8(Time Response of Channel )8( Time Dispersion Parameters

o Mean Excess Delay:

o RMS delay spread:

where:

o Excess delay spread )for X dB signal(: The amount of delay when the signal power falls below X dB of strongest arriving path. 3939


Examples of RMS delay spread

Design for GSM is based on excess delays of upto 18us.

Frequency RMS Delay Notes

Urban 910 upto 3.5µS NYC

Urban 892 10-25µS SFO

Suburban 910 200-300nS Typical

Suburban 910 2µS Extreme

Indoor 1500 10-50nS Office

Indoor 850 270nS Office

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Delay Spread <1> Coherence BWDelay Spread <1> Coherence BW

As we discussed before, delay spread of channel impulse response in time domain determines the variations of channel response in frequency domain. Coherence BW of channel: Signals with frequencies apart more than BC are affected differently by the channel. We expect the Coherence BW of channel, BC, be dependent on channel delay spread. As channel delay spread increases we expect smaller BC and therefore faster channel variations in frequency domain.

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Delay Spread <2> Coherence BWDelay Spread <2> Coherence BW

The value of BC depends on how much correlation we specify between correlated carriers. For correlation above 0.5, we will have: BC ≅ 1/)5στ( For example, for στ = 20µsec , BC ≅ 10KHz

Signal BW >> BC ≡

Ch. delay spread )στ( >> Symbol duration )TS( ISI exists and Equalizer needed )Signal may be faded in some frequencies.(

Signal BW << BC ≡

Ch. delay spread )στ ( << Symbol duration )TS( No ISI and No equalizer needed )But signal may be totally faded.(

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Freq. Spread <1> Coherence TimeFreq. Spread <1> Coherence Time

So far we have discussed frequency variations of signal at a given point in space. The next question is how the channel changes over time as the mobile moves with speed of v m/sec? As we discusses before, these time variations are directly interconnected with space variations of signal, as the mobile moves through such space The parameter that we will use for this concept is the so-called “coherence time” of the channel, TC.

In order to find TC, an approximation for correlation function of received signal in time, is found.

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Consider following assumptions:o Narrowband signal with frequency fC

o Many paths arriving at receiver from a close to uniform distribution of angles in space

The received signal is given by:

r)t( = rI)t( cos)2πfCt( + rQ)t( sin)2πfCt(

where rI)t( = Σan cos)ϕn)t(( and rQ)t( = Σan sin)ϕn)t(( and ϕn)t( = 2π[fCτn + fDnτn – fDnt] rI)t( and rQ)t( will be jointly Gaussian random processes and since ϕn)t( is unif [- π, π], therefore, we will have:

E[rI)t(]=E[rQ)t(] = E[rI)t(rQ)t(] = 0 Our goal is to find autocorrelation of rI)t( and rQ)t(:

ArI)t,τ( = E[rI)t(rI)t+τ(]

4444


By definition:ArI

)t,τ(=E[rI)t(rI)t+τ(]=ΣE)an2(E[cos)ϕn)t ((cos)ϕn)t +τ((]

Now substituting ϕn)t( = 2π[fCτn + fDnτn – fDnt], we will have: E[cos)ϕn)t(ϕn)t +τ(] = 0.5E[cos2πfDnτ] + 0.5E[cos)4πfCτn + 4πfDnτn - 4πfDnt - 2πfDnτ(]

Since fCτn changes rapidly and has uniform distribution, the second term will be zero and thus: ArI

)t,τ( = 0.5 ΣE)an2(E[cos)2πfDnτ(]

= 0.5 ΣE)an2(Eθn

[cos)2πvτ cos)θn(/λC]

where we replaced Doppler frequency with fD = v/λ cos)θ( for angles of arrival θn at receiver.

Similarly, it can be shown that: ArI

)τ( = ArQ)τ( = ArI

)t,τ(

4545


Now in order to simplify the correlation functions we make an assumption about angles of arrivals:

If angles of arrival of multipath at receiver are uniformly distributed then: Eθn[cos)2πvτcos)θn(/λC]=J0)2πvτ/λC(=J0)2πfDτ( for fD= v/λC where J0 is a Bessel function of 0thorder

Correlation drops below 0.5 around λC/4 in space.

Usually λC/2 is considered the distance at which signals are uncorrelated.

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Autocorrelation also gives correlation in time:

vTC = λC/2 Tc = 0.5 λC/v = 1/)2fD(

For usual mobile channels:

fD = 100Hz TC = 5mS

For example for GSM: burst period = 0.5mS

therefore, channel almost constant during each burst.

4747


Note: The assumption about angles of arrival is valid for macrocells but not necessarily microcells. With this assumption:

ArI)τ( = ArQ

)τ( = ΩP/2 J0)2πfDτ(

Ar)τ( = ΩP/2 J0)2πfDτ( cos)2πfCτ(where ΩP = ΣE)an

2(

Also, it can be shown that the correlation of envelope of received signal r)t( will be proportional to J02)2πfDτ(.

Note: If spatial angles are not uniform, the above approximations are not valid. 4848


Doppler spread can be obtained by taking Fourier Transform of autocorrelation functions:

4949


Doppler effect usually negligible if fD is much smaller than signal BW:

fD << BW

Doppler PSD also used in implementing channel variations in computer simulations. Generate two independent white Gaussian noise sources with PSD of N0/2, and then pass them through a LPF with H)f( that satisfies:

SrI)f( = SrQ

)f( = N0/2 |H)f(|2

5050


Channel variations modeled by Rayleigh fading variations over time

5151


A simple way to understand the relation betweenDoppler shift and time variations: Consider time variations as a simple form of multiplying the carrier frequency by a single tone at fD:

In time domain: cos)fCt(.cos )fDt(

In frequency domain:

5252

ExampleExample

Typical Channel Parameters

5353

Small Scale Fading Channels )1(Small Scale Fading Channels )1(

Fading may be

Flat or Frequency Selective

Fast or Slowly changing

These two effects will happen independent from each other.

5454


Flat Fading Channel has constant gain and linear phase in a region greater than signal BW. Signal shape little change, but amplitude changes widely due to fading )Rayleigh model(. For flat fading the following equivalent relations should be valid:

BW << BC or TS >> στ

Please note that flat fading can change rapidly or slowly with time )depending on mobile speed(. Example: Narrowband signal and mobile moving with speed v m/sec

5555


Frequency Selective Fading Channel has constant gain and linear phase in a region smaller than signal BW. Signal shape may vary a lot in time domain due to ISI. For frequency selective fading the following equivalent relations should be valid:

BW > BC or TS < στ

In practice, if TS < 10στ, then channel is considered frequency selective. Please note that frequency selective fading can change rapidly or slowly with time )depending on mobile speed(.

5656


Fast Fading Fast fading if channel impulse response changes during symbol period:

TS > TC or BW < fD

One common system with fast fading characteristics is GPS:

o Doppler frequencies around 5KHzo Bit rate around 50Hz.

In other systems usually we have high data rates, TS is very small and channel will not be fast fading. Note that being a fast fading channel has no relation to whether channel is flat or frequency selective. Should use noncoherent receivers, and not coherent ones.

5757


Slow Fading Slow fading if channel impulse response does not change very much during symbol period:

TS << TC or BW >> fD

If we have high data rates, TS is very small and channel will be fast fading. Thus signal bursts are designed such that burst period is smaller than TC. During such bursts channel can be considered constant and equalizers )if needed( should be based on value of TC. Note that slow fading is a small-scale fading effect and should not be associated with large-scale channel effects )that change much more slowly!( Can use coherent receivers.

5858


Average Fade Duration (AFD): How long a signal stays below target R )SNR γ(.

Level Crossing Rate (LCR): Rate at which envelop normalized to rms level crosses threshold level R. Used in coding/interleaving design of wireless modem. For Rayleigh fading:

o Depends on )ρ(, the ratio of target threshold R to average rms level.o Inversely proportional to Doppler frequencyo Note that as fD increases, fade duration decreases; One good thing about moving fast!

5959

Summary )1(Summary )1(

We saw two different areas in addressing wireless channel effects:

1.Time/Frequency response of channel for a given point in space:

o Relation between coherence bandwidth of channel )BC( and signal bandwidth )BW(o Relation between delay spread of channel )στ( and symbol period )TS(

2. Time/space variations of a signal at a given carrier frequency and resulting frequency spread:

o Relation between coherence time of channel )TC( and symbol period )TS(o Relation between frequency spread of channel )fD( and signal bandwidth )BW(

6060

Summary )2(Summary )2(

How does fading affects broadband and narrowbandsignals? If a narrowband signal passes through )flat( fading channel, its instantaneous power varies heavily with time. If a wideband signal passes through a )frequency selective( fading channel, although each path may change with time, but received signal power will not change dramatically over time. For wideband signal, the effect of fading channel will be averaged over different frequencies. The average power of a wideband signal, will be almost the same as average power of a narrowband signal, averaged over time )if the receiver is moving!(

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نیمسال اوّل 92-91 افشین همّت یار دانشکده مهندسی کامپیوتر...

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