Small-Scale Fading I PROF. MICHAEL TSAI 2011/10/27
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1 Small-Scale Fading I PROF. MICHAEL TSAI 011/10/7
2 Multipath Propagation RX just sums up all Multi Path Component (MPC).
3 Multipath Channel Impulse Response An example of the time-varying discrete-time impulse response for a multipath radio channel h b (t,) Summed signal of all multipath components arriving at ~. t Excess delay: the delay with respect to the first arriving signal ( ) Maximum excess delay: the delay of latest arriving signal t (t 0 ) The channel impulse response when = (what you receive at the receiver when you send an impulse at time ) =0, andrepresents the time the first signal arrives at the receiver.
4 Time-Variant Multipath Channel Impulse Response h b (t,) Because the transmitter, the receiver, or the reflectors are moving, the impulse response is time-variant. t t 3 t (t ) (t 3 ) t 1 (t 1 ) t (t 0 ) 4
5 Multipath Channel Impulse Response The channels impulse response is given by: Summation over all MPC h b N 1 i= 0 ( t, ) = a ( t, ) exp[ j{ πf ( t ) + φ ( t, )}] δ ( t ( t) ) Amplitude change (mainly path loss) i If assumed time-invariant (over a small-scale time or distance): h b N 1 i= 0 ( ) = a exp[ jθ ] δ ( ) i Additional phase change due to reflections c i i i Phase change due to different arriving time i i
6 h b (t,) Two main aspects of the wireless channell t t 3 t (t ) (t 3 ) t 1 (t 1 ) t (t 0 ) Following this axis, we study how spread-out the impulse response are. (related to the physical layout of the TX, the RX, and the reflectors at a single time point) 6
7 h b (t,) Two main aspects of the wireless channell t t 3 t (t ) (t 3 ) t 1 (t 1 ) t (t 0 ) Following this axis, we study how spread-out the impulse response are. (related to the physical layout of the TX, the RX, and the reflectors at a single time point) 7
8 Power delay profile To predict h B () a probing pulse p(t) is sent s.t. ( ) Therefore, for small-scale channel modeling, POWER DELAY PROFILE is found by computing the spatial average of h B (t;) over a local area. ; h ; ( ) TX RX Average over several measurements in a local area 8
9 Example: power delay profile From a 900 MHz cellular system in San Francisco 9
10 Example: power delay profile In side a grocery store at 4 GHz 10
11 Time dispersion parameters Power delay profileis a good representation of the average geometry of the transmitter, the receiver, and the reflectors. To quantify how spread-out the arriving signals are, we use time dispersion parameters: Already taled about this Maximum excess delay: the excess delay of the latest arriving MPC Mean excess delay: the mean excess delay of all arriving MPC RMS delay spread: the standard deviation of the excess delay of all arriving MPC 11
12 Time dispersion parameters Mean Excess Delay RMS Delay Spread 1 = = P P a a ) ( ) ( First moment of the power delay profile = = P P a a ) ( ) ( ) ( σ = Second moment of the power delay profile Square root of the second moment of the power delay profile
13 Time dispersion parameters Maximum Excess Delay: Original version: the excess delay of the latest arriving MPC In practice: the latest arriving could be smaller than the noise No way to be aware of the latest Maximum Excess Delay (practical version): The time delay during which multipath energy falls to X db below the maximum. This X db threshold could affect the values of the timedispersion parameters Used to differentiate the noise and the MPC Too low: noise is considered to be the MPC Too high: Some MPC is not detected 13
14 Example: Time dispersion parameters 14
15 Coherence Bandwidth Coherence bandwidth is a statistical measure of the range of frequencies over which the channel can be considered flat a channel passes all spectral components with approximately equal gain and linear phase. Recall this: Transfer function
16 Coherence Bandwidth Bandwidth over which Frequency Correlation function is above 0.9 B c 1 50σ Bandwidth over which Frequency Correlation function is above 0.5 B c 1 5σ Those two are approximations derived from empirical results. 16
17 Typical RMS delay spread values 17
18 Signal Bandwidth & Coherence Bandwidth :signal bandwidth Transmitted Signal :symbol period 1 f t ( ; t
19 Frequency-selective fading channel f TX signal t ; Channel t These will become intersymbol interference! f RX signal 19
20 Flat fading channel f TX signal t ; Channel t No significant ISI RX signal f
21 Equalizer 101 An equalizer is usually used in a frequency-selective fading channel When the coherence bandwidth is low, but we need to use high data rate (high signal bandwidth) Channel is unnown and time-variant Step 1: TX sends a nown signal to the receiver Step : the RX uses the TX signal and RX signal to estimate the channel Step 3: TX sends the real data (unnown to the receiver) Step 4: the RX uses the estimated channel to process the RX signal Step 5: once the channel becomes significantly different from the estimated one, return to step 1. 1
22 Example P() 0dB -10dB -0dB -30dB Would this channel be suitable for AMPS or GSM without the use of an equalizer? Mean P( ) 5(1) + (0.1) + 1(0.1) + 0(0.01) Excess Delay = = = = 4. 38µ s P( ) P( ) (1)5 + (0.1) + (0.1)1 + (0.01)0 = = P( ) = 1.07µ s
23 Example Therefore: RMS Delay Spread = σ = ( ) = 1.07 (4.38) = 1. 37µs 1 1 Coherence Bandwidth = B C = = = 146KHz 5 5(1.37 µ ) σ Since B C > 30KHz, AMPS would wor without an equalizer. GSM requires 00 KHz BW > B C An equalizer would be needed.
24 h b (t,) Two main aspects of the wireless channell t t 3 t (t ) (t 3 ) t 1 (t 1 ) t (t 0 ) 4
25 Doppler Effect Difference in path lengths Phase change Frequency change, or Doppler shift, 5
26 Example Consider a transmitter which radiates a sinusoidal carrier frequency of 1850 MHz.For a vehicle moving 60 mph, compute the received carrier frequency if the mobile is moving 1. directly toward the transmitter.. directly away from the transmitter 3. in a direction which is perpendicular to the direction of arrival of the transmitted signal. Ans: Wavelength= 0.16 ( ) Vehicle speed =60 h= =. cos 0 =160.. =. cos = 160 ( ). 3. Since cos =0, there is no Doppler shift! = = 6
27 Doppler Effect If the car (mobile) is moving toward the direction of the arriving wave, the Doppler shift is positive Different Doppler shifts if different (incoming angle) Multi-path: all different angles Many Doppler shifts Doppler spread 7
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