Radio Propagation - Mobile Radio Channel

1
Radio Propagation - Mobile Radio Channel
2
Propagation - Mobile Radio Channel

• Difficult environment due to random, time-varying
  phenomena as a result of signal reflections,
  diffractions and scattering (multi-path),
  relative motion, shadowing.
• Main Impairments
• Path loss - - attenuation with distance
• Shadowing - - long-term variation of signal
  caused by
• (slow fading) obstructions
  (hills,buildings,mountain,foliage
• and for indoor
  wireless, walls, furniture
• Multipath - - short-term signal variations due
  to multiple
• (fast fading) reflections from buildings,
  walls and ground.

3
Path loss

• Prediction of average received signal strength at
  a given distance from transmitter. Hundred to
  thousands meters ? Large scale propagation model.
• Inverse power law

4
Path loss

• Reflections arise when the plane waves are
  incident upon a surface with dimension that are
  very large compared to the wavelength.
• Diffraction occurs according to Huygens
  principle when there is an obstruction between
  the transmitter and receiver antennas, and
  secondary waves are generated behind the
  obstructing bodies.
• Scattering occurs when the plane waves are
  incident upon an object whose dimensions are on
  the order of a wavelength or less, and causes the
  energy to be redirected in many directions.

5
Figure 4.1 Description of a mobile radio
environment
6
Path loss

• Base to mobile link length is usually lt 24 km.
• Path loss model not considering radio horizon
  effect (radio path loss attributable to curvature
  of earth) is available for distance up to 24 km.
• Local scatterers surrounding the mobile cause
  short term or fast fading.
• The radius of the active scatterer region at 850
  MHz was found to be around 100 ?2. The active
  scatterer region moves with the mobile as its
  centre. Some scatterers become inactive as the
  mobile drove away from them while some become
  active as the mobile approach them.
• When the operating frequency is lower, the
  propagation loss is smaller, the radius of the
  scatterer region becomes slightly larger.

7
Figure4.2 Schematic diagram of the propagation
loss
8
Path loss

• Assume the receiver is moving at a constant speed
  V, distant r from transmitter is r Vt.
• Variation of received signal strength or power
  with r can be viewed as variation with respect to
  time, t.

9
Path loss Mobile Radio Environment

• Propagation between base station and mobile unit
  not only by way of line of sight (LOS) route, but
  via many paths, by way of scattering, by
  reflections from or diffraction around buildings
  and terrain.
• Received signal by the mobile consists of a large
  number of plane waves whose amplitudes, phases,
  and angles of arrival relative to the direction
  of vehicle motion are random.
• These plane waves interfere and produce a varying
  field strength pattern with minima and maxima
  spaced on the order of a quarter wavelength.
• With the short wavelengths at the UHF and
  microwave frequencies, the received signal fades
  rapidly and deeply as the mobile moves through
  the interference pattern.

10
Path loss Multipath

• Typical mobile channels (outdoors and indoors),
  often there is no LOS
• path between transmitter and receiver.
  Received signal is the superposition of many
  (plane-wave) components of (approximately) equal
  power with random amplitude and phase that are
  independent.
• The resultant signal shows constructive (large
  amplitude) and destructive (small amplitude)
  patterns ? time varying signal amplitude/ fading.
  Sum varies widely ? 30 to 40 dB.
• Variability or rapid fluctuation of received
  signal strength in close
• proximity to a particular location or over
  very short travel distances (a few ? ) or short
  time durations (order of seconds) ? small scale
  ?fading model.

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Path loss Multipath
Figure4.3 Typical Profile of received signals
Raleigh fading envelope and phase.
Vehicular MS speed of 30mph,carrier frequency
of 900MHz
12
Doppler Shift

• Mobile moving at a constant speed V. Difference
  in path lengths travelled by wave from remote
  source S to the mobile at points X and Y is

13
Doppler Shift

• The phase change in the received signal due to
  the difference in path lengths is
• Where is the wave propagation
  constant.
• The apparent change in frequency or Doppler
  Shift is

14
Doppler Shift

• Mobile moving toward the direction of the arrival
  of the wave, Doppler Shift is positive, that is
  the apparent received frequency is increased.
  Mobile moving away from the direction of the
  arrival of the wave, Doppler shift is negative,
  that is the apparent received frequency is
  decreased.
• When , the Doppler shift is
  maximum. This leads to

15
Doppler ShiftExample

• A transmitter radiates a CW of 1800 MHz.
  Calculate the maximum Doppler Shift experienced
  by a receiver on a vehicle moving at 100km/hr.
  Then compute the received carrier frequency if
  the mobile is moving (a) directly towards the
  transmitter, (b) at 90o to the direction of
  arrival of the transmitted wave and (c) at 30o to
  the direction of arrival of the transmitted wave.

16
Doppler ShiftExample
17
Doppler ShiftTransmission Coefficient T(t) .

• Amplitude and phase of the received signal when a
  unit amplitude continuous wave (CW) signal is
  transmitted.
• Transmitted
• Mobile travels in the x-direction with speed V.
  Vehicle motion introduces a Doppler Shift in
  every wave

18
Doppler ShiftTransmission Coefficient T(t) .

• Assume the transmitted field is vertically
  polarized. The E field seen at the mobile is

Rices model of narrowband gaussian noise
19
Doppler ShiftTransmission Coefficient T(t) .

• is random phase of the n-th arriving wave,
  distributed uniformly over(0, 2p).
• Received
• Fading ? Decreases in the magnitude of T( t )
  with time as themobile moves through the
  interference pattern.

20
Doppler ShiftTransmission Coefficient T(t) .

• Variations in the phase of T( t ) , ,as time
  is varied ??
• ?random FM .
• Variations in the amplitude and phase of T( t ) ?
  as the frequency is varied are called the
  frequency selective fading and phase distortion
  of the channel, respectively. (to be dealt with
  in a later section).

21
Doppler ShiftStatistics of amplitude and phase
of T(t)

• T(t) is a complex stochastic process. With a
  given transmitted frequency fc, T(t) is the
  result of many received plane waves, each shifted
  in frequency by the Doppler Shift appropriate to
  the vehicle motion relative to the direction of
  the plane wave.
• Thus, the received signal is the sum of a large
  number of sinusoids of comparative amplitude and
  random phase, whose frequencies are confined to
  the Doppler spread around fc.
• This received signal conforms to the Rices
  model of narrowband Gaussian noise.

22
Doppler ShiftStatistics of amplitude and phase
of T(t)

• For N sufficiently large, by central limit
  theorem, both TC(t) and TS(t) are zero mean
  Gaussian random processes. They are uncorrelated
  and independent with

23
Doppler ShiftStatistics of amplitude and phase
of T(t)
24
Doppler ShiftStatistics of amplitude and phase
of T(t)
25
Doppler ShiftStatistics of amplitude and phase
of T(t)
26
Doppler ShiftStatistics of amplitude and phase
of T(t)
With Rayleigh pdf

• Tc and Ts are independent Gaussian random
  variables with zero means and variances.

27
Doppler ShiftStatistics of amplitude and phase
of T(t)

• Tc and Ts are independent because they are
  uncorellated.
• Their joint probability density is

28
Doppler ShiftStatistics of amplitude and phase
of T(t)
? Rayleigh pdf
29
Doppler ShiftStatistics of amplitude and phase
of T(t)
r and ? are independent random variables.
30
Doppler ShiftStatistics of amplitude and phase
of T(t)

• Variance of r,

31
Doppler ShiftCorrelation Functions of T(t)
32
Doppler ShiftStatistics of amplitude and phase
of T(t)

33
Doppler ShiftStatistics of amplitude and phase
of T(t)

• Consider
  as a wide sense
  stationary bandpass random process.

34
Doppler Shift Level crossing rate (LCR) and
average fade duration(AFD)
35
Doppler Shift Level crossing rate (LCR) and
average fade duration(AFD)
36
Doppler Shift Level crossing rate (LCR) and
average fade duration(AFD)

• Relates time rate of change, to level of received
  signal envelope, and to speed of mobile.
• Average number of level crossings per second at
  specified R

37
Doppler Shift Level crossing rate (LCR) and
average fade duration(AFD)
38
Doppler Shift Level crossing rate (LCR) and
average fade duration(AFD)

• NR is a function of mobile speed is apparent from
  the presence of fm in (1).
• Few crossings at both high and low level.
• NR is proportional to the product
• At high ?, NR is small because of
• At low ?, NR is small because of ?.

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Level Crossing Rates, NR

• Expected rate at which envelope, r , crosses a
  specified level, R.
• where the dot is time derivative and is
  the joint density function of r and at R r.
• Rice gives

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Level Crossing Rates, NR

• Integrating expression (2) over from 0 to 2p
  and
• from -8 to 8 , we get
• Derivation of Probability Density Function
• Recall that

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Level Crossing Rates, NR

• For N sufficiently large, by central limit
  theorem, both TC(t) and TS(t) are zero mean
  Gaussian random processes. For a fixed t, they
  are uncorrelated and independent zero mean
  Gaussian random variables with
• variance
• Now

42
Level Crossing Rates, NR

• For a fixed t, they are also uncorrelated and
  independent zero mean Gaussian random variables
  with variance.
• The joint pdf of multivariate Gaussian random
  variable
• is
• Where M is covariance matrix.

43
Level Crossing Rates, NR
44
Level Crossing Rates, NR
45
Level Crossing Rates, NR
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Level Crossing Rates, NR

• Substituting (3) into (1) we get

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Level Crossing Rates, NR

• Derivation of b0

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Level Crossing Rates, NR

• Derivation of b2

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Level Crossing Rates, NR
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Level Crossing Rates, NR
51
Level Crossing Rates, NR

• Signal envelopes experience deep fades only
  occasionally, but shallow fades are frequent.
• Maximum number of level crossings occurs at 3dB
  below rms level.

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Level Crossing Rates, NR
Fig. Fading RateLevel crossing rate of vertical
monopole
53
Level Crossing Rates, NR Average duration of
fade

• This is found by dividing the fading rate into
  the cumulative probability distribution

Fig. Average duration of fade
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Level Crossing Rates, NR Average duration of
fade
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Level Crossing Rates, NRTime Delay Spread and
Coherence Bandwidth

• The results on fading derived so far are based on
  the assumption of a CW signal (un-modulated
  carrier) and that there is no difference between
  the arrival times of the multipath waves
• In fact, differences exist in the multi-path
  delays.
• Consider a CW being transmitted to a mobile unit.

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Level Crossing Rates, NR Time Delay Spread and
Coherence Bandwidth
57
Level Crossing Rates, NR Time Delay Spread and
Coherence Bandwidth

• All scatterers associated with a certain path
  length can be located on an ellipse with the
  transmitter and receiver at its foci.
• TAR and TBR have the same arrival angle but
  different time delays.
• TBR and TCR have the same time delays but
  different angle of arrival.
• The received field is sum of a number of waves,

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Level Crossing Rates, NR Time Delay Spread and
Coherence Bandwidth

• nth wave arriving at an angle composed of M
  waves with propagation delay times Tnm. All these
  M waves experience the same Doppler shift,
• fn is maximum when
• Note that actually the argument of each cosine
  term in (1) should be

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Level Crossing Rates, NR Time Delay Spread and
Coherence Bandwidth

• Since
• Here, Cnm is determined from
• which is fraction of the incoming power within
  da of the angle a and within dT of the delay T,
  in the limit with N and M very large.

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Level Crossing Rates, NR Time Delay Spread and
Coherence Bandwidth

• Now consider propagation of signal that occupies
  a finite bandwidth.
• Consider two frequency components within the
  signal bandwidth.
• If the frequencies are close together, then the
  different propagation paths will have
  approximately the same electrical length for both
  components. Their amplitude and phase variations
  will be very similar.
• Provided the signal bandwidth is sufficiently
  small, all frequency
• components within it behave similarly and
  flat fading is said to exist.

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Level Crossing Rates, NR Time Delay Spread and
Coherence Bandwidth

• If the frequency separation is large enough, the
  behavior at one frequency tends to become
  uncorrelated with or independent from that at the
  other frequency, because the phase shifts along
  the various paths are different at the two
  frequencies.
• The maximum frequency difference for which the
  CWs are still strongly correlated is called the
  coherence bandwidth of the mobile transmission
  channel.
• Coherence Bandwidth is proportional to the
  inverse of the delay spread or the magnitude of
  the difference between the delay times.

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Level Crossing Rates, NR Time Delay Spread and
Coherence Bandwidth

• Typical spreads in time delays range from a
  fraction of a µ-seconds to many µ-seconds. Longer
  spreads in urban and shorter spreads in suburban
  areas.
• Signals that occupy a bandwidth greater than the
  coherence bandwidth will become distorted since
  the amplitudes and phases of the various spectral
  components in the received signal are not the
  same as they were in the transmitted
  signal.?Frequency selective fading.

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Level Crossing Rates, NR Envelope Correlation
Function or Envelope Correlation Coefficient

• In this section, we are interested to derive the
  Envelope Correlation Function or Envelope
  Correlation Coefficient between two CWs as a
  function of frequency separation and time
  separation.
• Two CWs at ?1 and ?2,

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Level Crossing Rates, NR Envelope Correlation
Function or Envelope Correlation Coefficient

• For large enough N and M, xi(t) s are Gaussian
  random processes. (By Central Limit Theorem).

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Level Crossing Rates, NR Envelope Correlation
Function or Envelope Correlation Coefficient

• Now we are interested in the correlation of the
  envelope of the CWs as a function of both time
  separation, ?t, and frequency separation,
• Define, for fixed t,
• These r.v. can also be written in terms of their
  envelopes and phases,

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Level Crossing Rates, NR Envelope Correlation
Function or Envelope Correlation Coefficient

• Moments of the r.v.s, ltxi,xj gt
• The average will vanish unless np and mq, which
  gives

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Level Crossing Rates, NR Envelope Correlation
Function or Envelope Correlation Coefficient

• In the limit as N, M ? 8
• By similar arguments

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Level Crossing Rates, NR Envelope Correlation
Function or Envelope Correlation Coefficient
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Level Crossing Rates, NR Envelope Correlation
Function or Envelope Correlation Coefficient
70
Level Crossing Rates, NR Envelope Correlation
Function or Envelope Correlation Coefficient

• Recall
• Similarly,

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Level Crossing Rates, NR Envelope Correlation
Function or Envelope Correlation Coefficient

• Now these moments are parameters in the joint pdf
  of
• Transforming the rvs to the
  amplitudes and phases, we get the corresponding
  pdf in terms of and .

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Level Crossing Rates, NR Envelope Correlation
Function or Envelope Correlation Coefficient

• Where
• Now assume an exponential distribution of the
  delay spreads and a uniform distribution in angle
  of the incident power.

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Level Crossing Rates, NR Envelope Correlation
Function or Envelope Correlation Coefficient

• Assume also that . The quantities in
  (17) may be worked out with the help of (10) to
  (15).
• Equation (15),

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Level Crossing Rates, NR Envelope Correlation
Function or Envelope Correlation Coefficient
75
Level Crossing Rates, NR Envelope Correlation
Function or Envelope Correlation Coefficient
76
Level Crossing Rates, NR Envelope Correlation
Function or Envelope Correlation Coefficient

• Using integration by parts on (19), can show that
• In summary,
• 
  (21)

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Level Crossing Rates, NR Envelope correlation
coefficient
78
Level Crossing Rates, NR Envelope correlation
coefficient

• where

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Level Crossing Rates, NR Frequency Selective
Fading

• Two CWs separated by a finite frequency range,
  propagating in a medium, do not observe the same
  fading.
• Frequency selective fading is closely related to
  the time delay spread, .
• Considering correlation in two frequencies but no
  time or space separation. That is

80
Level Crossing Rates, NR Frequency Selective
Fading

• Let as a criterion for determining
  the coherence bandwidth, we have from (27),
• 
  (28)
• Coherence bandwidth is inversely proportional to
  time delay spread.
• 
  

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Level Crossing Rates, NR Frequency Selective
Fading
82
Level Crossing Rates, NR Coherence Time

• Letting ?f 0,we have from (27)
• Again using as a criterion
  for correlatedness in time, we have

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Level Crossing Rates, NR Coherence Time
84
Level Crossing Rates, NR Spatial correlation
of the envelope

• Many mobile radio systems employ antenna
  diversity, where spatially separated antennas
  provide multiple faded replicas of the same
  information-bearing signal.
• What should be the antenna separation to provide
  uncorrelated antenna diversity branches?
• Consider two places separated by d. The mobile
  receiver is moving at speed V. The correlation
  function in terms of ?t is equivalent to that in
  terms of the distance d Vt.

85
Level Crossing Rates, NR Spatial correlation
of the envelope

• Assuming a uniform and noting that
  , we have
• The auto-covariance is zero at 0.38?, and is less
  than 0.3 for
• As a rule of thumb, uncorrelated diversity
  branches can be obtained at the mobile station by
  placing the antenna elements about a half
  wavelength apart.

86
Level Crossing Rates, NR Spatial correlation
of the envelope

• Base station space diversity

Figure 4.5 Analytical model for spatical
correlation at a base station
87
Level Crossing Rates, NR Spatial correlation
of the envelope

• The above result cannot usually be applied to a
  base station, since the uniform arrival angle is
  hardly satisfied.
• Because of the height of the base station
  antenna, there are few scattering objects around
  the base station.
• Since the arrival angle is not uniformly
  distributed, and the range of the arrival angle,
  correspondingly, the spread of the Doppler
  frequencies becomes smaller. Therefore, the time
  or spatial correlation at a base station becomes
  broad. (Recall that autocorrelation function is
  Fourier Transform of power spectrum).

88
Level Crossing Rates, NR Spatial correlation
of the envelope

• A spatial separation of the order of 10 ? is
  required for a base station diversity system.
• The fact that the active scatterers are moving
  with the mobile while the base station is
  standing still, can be viewed equivalently, as
  the active scatterers are standing still while
  the base station is moving at a speed V.

89
Level Crossing Rates, NR Coherence Bandwidth
and Power Delay Profile

• In the derivation of the envelope correlation
  coefficient ,it is assumed that
  the wave arrival angle and path delay are
  statistically independent. That is
• This assumption makes it possible to express both
  µ1 and µ2 as the product of a function of ?f only
  and a function of ?t only. That is

90
Level Crossing Rates, NR Coherence Bandwidth
and Power Delay Profile

• where

91
Level Crossing Rates, NR Coherence Bandwidth
and Power Delay Profile

• Similarly
• Now

92
Level Crossing Rates, NR Coherence Bandwidth
and Power Delay Profile
93
Level Crossing Rates, NR Coherence Bandwidth
and Power Delay Profile

• Exercise Find an expression for the coherence
  bandwidth of a mobile radio channel modelled by a
  two-path power delay profile

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Level Crossing Rates, NR Coherence Bandwidth
95
Level Crossing Rates, NR Coherence Bandwidth
96
Rician Distribution

• Rayleigh fading model is suitable for urban areas
  where high-rise building often block the line of
  sight (LOS) path between transmitter and
  receiver.
• When a LOS exists in addition to the multipath
  waves from scatterers
• ? Rician Fading Model.
• ? Suburban areas, micro or pico-cellular.

97
Rician Distribution

• Received signal y(t)

98
Rician Distribution

99
Rician Distribution
100
Rician Distribution
101
Rician Distribution

• Modified Bessel function of the first
  kind and order zero.
• For A0, I 0(0) 1, p(r) above reduces to the
  Rayleigh pdf.
• The pdf of the phase ? is given by

102
Rician Distribution
103
Rician DistributionLog-normal Shadowing

• Random shadowing effects which happen over a
  large number of measurement locations which have
  the same Transmitter-Receiver (T-R) separation
  but different level of clutter along the
  propagation path.
• Local mean signal strength (that is, the signal
  strength averaged over the Rayleigh fading) in an
  area at a fixed radius from a particular base
  station antenna is log-normally distributed.
• The received power at a mobile at distance d is

104
Rician DistributionLog-normal Shadowing

• r is a Rayleigh distributed r.v., e?accounts for
  the shadowing ( ? isGaussian with zero mean and
  variance ), Kd -? is the deterministic loss
  law, and PT is the transmitter power.
• Averaging the received signal strength over the
  Rayleigh fading, we get

105
Rician DistributionLog-normal Shadowing
106
Rician DistributionLog-normal Shadowing
107
Rician DistributionLog-normal Shadowing

• Example The local mean signal strength in areas
  at a fixed radius from a particular base station
  is log-normally distributed. Suppose the mean
  value of this local mean signal strength is 5 dBm
  and that the standard deviation is 6 dB. What is
  the particular local mean signal level, ? dB, so
  that the probability of it being exceeded is 10?
• Solution

108
Rician Distribution Log-normal Shadowing
109
Time-delay Spread modulation effects

• The concepts of time delay spread and coherence
  bandwidth have been dealt with in the last
  Section for CW signals only, that is without
  considering the effect of modulation. Now we
  consider modulation effects.
• Multipath channel causes delayed echoes of the
  transmitted signal, each with Rayleigh amplitude
  and uniform phase.
• If these delays are such that their spread is a
  significant fraction (gt50) of the symbol
  duration or exceeds symbol duration ? smearing
  of the transmitted signal, i.e. Inter-Symbol
  Interference (ISI).

110
Time-Delay Measurements
111
Delay Spread Function h(t, t)

• Transmitted signal x(t)
• complex envelope or low-pass equivalent
  signal ? containing the modulation.
• Received waveform from paths with delay
• This is already the result of superposition of
  many waves coming fromall directions, but with
  delays of the order of
• grouped together.

112
Delay Spread Function h(t, t)

• Note that and are random
  processes, which means that the effect of Doppler
  spread is already included.
• Relate
• The received signal is then
• where

For a large number of paths, can consider the
received signal as a continuum of multipath
components.
113
Delay Spread Function h(t, t)

• all between and are grouped
  together.

114
Delay Spread Function h(t, t)

• ? Delay Spread function or Impulse Response of
  channel.

115
Delay Spread Function h(t, t)

• ? Since
• For a linear time invariant system

h(t) Channel response to a impulse at t 0.
116
Delay Spread Function h(t, t)

• For a time-variant channel, the response to an
  impulse applied at t ? will not have the same
  shape as the response to an impulse applied at t
  0.
• Introduce channel response for an
  impulse applied at t ?, From (2), instead of
  we have , therefore

117
Delay Spread Function h(t, t)

• now let
• Recalling from (1) that
• we have

Channel response at t to an impulse applied at
time t - t,that is applied at t seconds in the
past.
118
Delay Spread Function h(t, t)
119
Delay Spread Function h(t, t)
120
Delay Spread Function h(t, t)
Fig. Examples of random channel impulse response
in two dimensions (a) time-variant channel
121
Delay Spread Function h(t, t)
Fig. Examples of random channel impulse response
in two dimensions (b) time-invariant channel
122
Delay Spread Function h(t, t)Channel
Classification

• In this section since we have included modulation
  in the analysis, we can relate Bc and Tc to Bs
  and fm .
• Frequency flat, multiplicative (time selective
  fading).
• Bs ltlt Bc,
• where, Bs signal bandwidth, Bc coherence
  bandwidth.

123
Delay Spread Function h(t, t)Channel
Classification

• All frequency components in U(f) undergo the same
  attenuation and linear phase shift through the
  channel.
• since U(f) has its
  frequency content concentratedin the vicinity of
  f 0.

124
Delay Spread Function h(t, t)Channel
Classification

• If rate of change of as(t ) with t is smaller
  than rate of change of u(t) with t, then shape of
  signal pulse is preserved. However it undergoes
  amplitude fading whenever dips.
• Frequency selective, time flat channels.
• Received signal duration (time during which
  signal is in flight) less than coherence time.
• Ts lt Tc ? Channel appears to the signal as
  time invariant.
• ? Time flat
  channels.

125
Delay Spread Function h(t, t)Channel
Classification

• However frequency
  selectivity means Bs gt Bc ,which implies that the
  signal spectrum U(f) will be modified by the
  multiplication with H(f). Shape of received
  waveform distorted.
• Also for digital transmission,
• Inter-Symbol-Interfe
  rence (ISI).

126
Delay Spread Function h(t, t)Classification of
Multi-path Fading

• 2 Channel parameters
• (1) Multi-path (rms) spread / coherence BW
• captures the multi-path channel conditions
  via delay-spread (in-time) or amplitude
  correlation (in frequency).

127
Delay Spread Function h(t, t)Classification of
Multi-path Fading

• (2) Doppler spread / Coherence Time

captures the rate of multi-path channel
variations via spread of carrier (in frequency)
or correlation of channel impulse response (in
time).
128
Delay Spread Function h(t, t)Two System Design
Parameters

• (1) Symbol Period Ts
• (2) Transmission BW Bs
• Narrowband (PSK / QAM) Wideband (Spread -
  Spectrum)

129
Delay Spread Function h(t, t)Classification
Based on Multipath Spread

• Flat (Freq. Non-select Fading) Freq.-Select
  Fading

Transmitted pulse shape is Transmitted pulse
shape (relatively) undisturbed, but is
distorted. amplitude fades with time
130
Delay Spread Function h(t, t) For Narrowband
Modulation

• ? Flat Fading ?? No Inter-symbol Interference
  (ISI).
• Frequency Selective Fading

131
Delay Spread Function h(t, t)For Wideband
Modulation

• Suppose (no ISI)
• However, it is possible that
• But ,Frequency-Selectivity.
• For wide-band signals, possible to have no ISI
  and frequency selectivity simultaneously.

132
Delay Spread Function h(t, t) Classification
Based on Doppler Spread

• Fast Fading Slow Fading
• Large Doppler Spread Small Doppler Spread
• ? Channel variations ? Channel variation
  slower
• faster than baseband than baseband signal
  variation
• signal, variations. channel is
  approximately invariant
• over
  several symbol duration.