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1、l Linear time-variant channel modelChannel Impulse ResponseTime-Variant Transfer FunctionDoppler Spread Function Delay-Doppler spread functionImpulse responseDoppler spreadTransfer functionDelay-Doppler spreadExample 2.11 LTV Channel ModelConsider an LTV channel with impulse response given byWhere T
2、=0.1 ms,and .(a)Find the channel time-variant transfer function and the Doppler spread function(b)Given that the transmitted signal is Where T0=0.025ms,find the received signal in the absence of background noise.(c)Repeat part(b)if the transmitted signal isWhere T1=0.05ms(d)What do you observe from
3、the results of parts(b)and(c)?Consider an LTV channel with impulse response given byWhere T=0.1 ms,and .Example 2.11 LTV Channel ModelWhen the channel changes with time randomly,the channel functions are random processed and are difficult to characterize.Under the assumption that the random processe
4、s have zero mean,we are interested in the correlation functions of the random processes.For simplicity of analysis,we assume that(a)The channel impulse response is a wide-sense stationary(WSS)process(b)The channel impulses at and are uncorrelated if for any t.A channel under assumptions(a)and(b)is s
5、aid to be a wide-sense stationary uncorrelated scattering(WSSUS)channel.2.3.2 Frequency and Time Correlation Function 2.3.2 Frequency and Time Correlation Function 2.3.4 Examples 2.3.4 Examples 2.3.1 Delay Power Spectral Density 2.3.1 Delay Power Spectral Density 2.3.3 Doppler Power Spectral Density
6、 2.3.3 Doppler Power Spectral Density l The Definition of the Delay PSDl Statistics Describing the Delay PSDUnder assumption(a),the autocorrelation function of define as Which is a function of ,and ,and does not depend on t.The correlation function can be represented asUnder assumption(b),the autoco
7、rrelation function can be represented in the formor equivalentlywhereAt t=0,we defineThus,we haveAccording to the Wiener-Khintchine relations,the function represents power spectral density(PSD).It measures the average PSD at the channel output as a function of the propagation delay,and is therefore
8、called the delay PSD of the channel,also known as the multipath intensity profile.We observe that is the Fourier transform of the correlation function.The nominal width of the delay PSD pulse is called the multipath delay spread,denoted by TmThe nth moment of the delay,denoted by ,is given byThe mea
9、n propagation delay,denoted byThe RMS delay spread,denoted byIn calculating a value for the multipath delay spread,it is usually assumed thatExample 2.12 From example 2.9,find the Delay PSD Example 2.12 From example 2.9,find the Delay PSD Example 2.13calculate the rms delay spread for the two-path p
10、ower delay profile#1.t t0 0=0,=0,p0 0=0 dB.#2.t t1 1=1 s,p1 1=0 dBl Linear time-variant channel modelChannel Impulse ResponseTime-Variant Transfer FunctionDoppler Spread Function Delay-Doppler spread functionImpulse responseDoppler spreadTransfer functionDelay-Doppler spreadl Frequency correlation f
11、unctionl Time correlation functionFrom the relationship between and ,we have the following:a)If is WSS,then is also WSS with respect to t.As a result,we can define the autocorrelation function of the time-variant transfer function asFrom the relationship between and ,we have the following:whereTime-
12、frequency correlation functionb)The correlation function can be represented in term of the correlation function LettingThe nominal width of ,denoted by ,is called the channel coherence bandwidth.Fourier Trans.Since time and frequency have an inverse relationship,we haveLet Ws denote the bandwidth of
13、 the transmitted signal,the fading channel can be grossly categorized as follows:l If ,the channel is said to exhibit frequency selective fading which introduces severe ISI to the received signall If ,the channel is said to exhibit frequency nonselective fading or flat fading which introduces neglig
14、ible ISIIn the time-frequency correlation functionlettingIt characterizes,on average,how fast the channel transfer function changes with time at each frequency.The nominal width of ,denoted by ,is called the channel coherence time.l If the channel coherence time is much larger than the symbol interv
15、al of the transmitted signal slow fadingl If the channel coherence time is smaller than the symbol interval of the transmitted signal fast fadingThe correlation function of the Doppler spread function is defined asFor a WSSUS channel,the correlation function can be represented in the form can be obt
16、ained by Fourier transformation of the frequency-time correlation function with respect to .ThereforeAt ,we haveThe nominal width of ,denoted by ,is called the Doppler spread.The mean Doppler shift,denoted byThe RMS Doppler shift,denoted byRelationships between the channel correlation functions and
17、between the channel parametersFrequency-time correlationCoherenceBandwidthFrequencycorrelationTimecorrelationCoherenceTimeMultipath delay spreadDelayPSDDopplerPSDDopplerspreadl We can define 4 basic fading types2 basic fading types due to multipath delay spread:flat fading,frequency selective fading
18、2 basic fading types due to Doppler spread:fast fading,slow fadingl We can define 4 composite fading typesFlat/frequency selective are independent of fast/slow fadingSo as simple combination:fast flat fading,slow flat fading,fast frequency selective fading,slow frequency selective fading lFlat fadin
19、gChannel has constant gain&linear phase response over a bandwidth greater than the signal bandwidthReceived signal strength may be different,but no change in spectrum(waveform shape)Requirements:Example:when there is only one pathModeled as a single-tap channel:gain changes randomly,may cause deep f
20、adeslFlat fading illustration:time&frequency response of flat fading channellFrequency selective fadingChannel has different gains within the bandwidth of the signalRequirement:Multipath delay spread longer than symbol period introduce inter-symbol interference(ISI)ISI(time domain)different gains(fr
21、equency)Modeled as statistical linear filter channels with random,time varying coefficientslFrequency selective fading illustrationTime and frequency response of a frequency selective fading channellFast fading(time selective fading)Channel impulse response changes rapidly within each symbol duratio
22、nRequirement:Doppler spread causes frequency dispersion signal distortionOccurs usually for very low data rate,the change of channel is caused by motionlSlow fadingChannel is static over some symbol intervalsRequirements:Channel impulse response changes slower than symbolsDoppler spread is much less
23、 than baseband signal bandwidthDetermined by the mobile velocity and signallFading types experienced by a signal depend on the signals symbol rate or bandwidthFlat slow fadingFlat fast fadingFrequency selective slow fadingFrequency selective fast fadingFrequency selective fast fadingFrequency Select
24、ive slow fadingFlat fast fadingFlat slow fadingSymbol interval TsSymbol interval TsTcBaseband signal bandwidth BsBdBcBaseband signal bandwidth BsTsBsTsBsTsExample 2.14Consider a fading channel which exhibits a Doppler frequency shift uniformly distributed between-10Hz and 10Hz.Determinea.The mean Do
25、ppler shiftb.The RMS Doppler spread,andc.The coherence timeExample 2.15Consider a WSSUS channel whose time-variant impulse response is given byWhere T and are constant,is a random variable uniformly distributed in ,and is a real-valued random process independent of ,with and a.Calculate the delay ps
26、d and the multipath delay spreadb.Calculate the frequency correlation function and the channel coherence bandwidthc.Determine whether the channel exhibits frequency-selective fading for GSM systems with T=0.1 ms2.3.2 Frequency and Time Correlation Function 2.3.2 Frequency and Time Correlation Function 2.3.4 Examples2.3.4 Examples2.3.1 Delay Power Spectral Density 2.3.1 Delay Power Spectral Density 2.3.3 Doppler Power Spectral Density 2.3.3 Doppler Power Spectral Density l P2-10.