This document discusses small scale fading in mobile radio propagation systems. It begins by introducing radio wave propagation and the factors that influence it, such as buildings, foliage, and motion. It then discusses small-scale fading models which characterize rapid fluctuations over short distances or times. Large scale propagation models are also introduced. The document goes on to discuss specific topics relating to small-scale fading like Doppler shift, multipath propagation, and modeling the mobile radio channel using the impulse response.

1. EE424 CommunicationEE424 Communication
SystemsSystems
Mobile Radio Propagation:
Small Scale Fading
HW: 5.6, 5.7,5.8, 5.13, 5.28,5 .30
Due Monday Dec 5

2. Introduction to Radio Wave Propagation
• The mobile radio channel places fundamental limitations
on the performance of wireless communication systems
• Mobile radio path is severely obstructed by buildings,
mountains, and foliage,……………….
• Radio channels are extremely random and do not offer
easy analysis
• The speed of motion impacts how rapidly the signal level
fades as a mobile terminals moves in the space
• Modeling radio channel is one of the most difficult part
and typically done in a statistical manner based on
measurements

6. Small-Scale models (fading models)
Propagation models that characterize rapid fluctuations of the
received signal strength over very short travel distances (few
wavelengths) or short time duration (on the order of seconds).
Large Scale Propagation Models
Propagation models are usually required to predict the average
received signal strength at a given distance from the transmitter
and estimating the coverage area (averaged over meters).
Introduction to Radio Wave Propagation

7. Small Scale and Large-Scale fading for an indoor radio

8. • Most radio propagation models are derived using a
combination of analytical (from a set of measured data) and
empirical methods. (based on fitting curves)
• All propagation factors through actual field measurements are
included.

• Some classical propagation models are now used to predict
large scale coverage for mobile communication systems
design.
Propagation models

9. Radio Propagation Mechanisms
Reflection
Conductors & Dielectric materials
Propagation wave impinges on an object which is large as compared to
wavelength
- e.g., the surface of the Earth, buildings, walls, etc.
Diffraction
Radio path between transmitter and receiver obstructed by surface with
sharp irregular edges
Waves bend around the obstacle, even when LOS (line of sight) does
not exist. (Huygen’s principal)
Scattering
The through which the wave travels consists of objects with dimensions
smaller than the wavelength and where the number of obstacles per unit
volume is large – rough surfaces, small objects, foliage, street signs,
lamp posts.
In mobile communication, the actual received signal is often stronger
than that is predicted by reflection and diffraction models.
Large Scale Path Loss

10. Practical Link Budget Design Using Path Models
• The empirical approach is based on fitting curves and
analytical expressions that recreate a set of measured data
• All propagation factors then considered
• It should not be directly used in other conditions such as
frequency, environment,… unless additional measured data is
achieved.
Most of the models are derived from combined
(i) analytical studies
(ii) experimental methods
Large Scale Path Loss

11. •In dB format:
(PL)dB = PL(do) + 10nlog(d/do)
•The ‘PL’ includes all possible average path losses.
•Bars denote the ensemble average of all possible path
loss values for a given d
•On a log-log scale plot, the modeled path loss is a
straight line with a slope equal to 10n dB per decade.
•do ~ 1 km for Large cell
•do ~ 1 to 100 m for microcell
Log Distance Path Loss Model
Large Scale Path Loss

12. Log-normal shadowing
• averaged received power in log distance model is inconsistent with
measured data
• The environmental conditions in Log-Distance model not necessarily to
be the same at two different locations having the same T-R separation.
• Measurement have shown that at any value of d, the path loss PL(d) at a
particular location is random and distributed log-normally about the
mean distance-dependent value.
Large Scale Path Loss

13. Log-normal shadowing
Thus, [PL(d)]dB = PL(d) + Xσ = PL(do) + 10nlog(d/do) + Xσ
where Xσ is Gaussian distributed random variable with zero mean
(in dB) and standard deviation σ (dB).
The log-normal distribution describes the random shadowing effects
which occur over a large number of measurement locations.
n and σ are computed from measured data
Log Normal Distribution - describes random shadowing effects
• for specific Tx-Rx, measured signal levels have normal distribution
about distance dependent mean (in dB)
• occurs over many measurements with same Tx-Rx & different
clutter standard deviation, σ (also measured in dB)
Large Scale Path Loss

14. Indoor Propagation Models
The indoor radio channel differs from the traditional radio channel in
two aspects:
1.The distances covered are much smaller.
2.The variability of the environment is much greater for a much smaller
range of T-R separation distances.
Propagation within building is strongly affected by:
1. The layout of the building .
2. The construction materials.
3. The building type.
The mechanisms are the same as outdoor models but the conditions
are much more variable
… Signal levels strongly affected whether the interior doors are open
or closed, antenna mounting, far field conditions…………
Large Scale Path Loss

15. Introduction
Large-scale fading represents the average signal power attenuation
or the path loss due to motion over large areas. This phenomenon is
affected by prominent terrain contours (hills, forests, billboards,
clumps of buildings, and so on) between the transmitter and the
receiver.
The receiver is often said to be “shadowed” by such prominences.
The statistics of large-scale fading provide a way of computing an
estimate of path loss as a function of distance. This is described in
terms of a mean-path loss (nth
-power law) and a log-normally
distributed variation about the mean.
Small Scale Fading

16. Fading is caused by interference between two or more
versions of the transmitted signal which arrive at the receiver at
slightly different times.
Fading is used to describe the rapid fluctuation of the
amplitude of the radio over a short period of time or
travel distance so that the large scale path loss effect
may be ignored.
Small-scale fading refers to the dramatic changes in signal
amplitude and phase that can be experienced as a result of small
changes (as small as a half wavelength) in the spatial positioning
between a receiver and a transmitter. Small-scale fading manifests
itself in two mechanisms: time-spreading of the signal (or signal
dispersion) and time-variant behavior of the channel.
Introduction
Small Scale Fading

17. Small Scale Multipath Propagation
Multipath in the radio channel creates small-scale
fading effects. The three most important effects are:
1.Rapid changes in signal strength over a small travel
distance or time interval.
2.Random frequency modulation due to varying Doppler
shifts on different multipath signals.
3.Time dispersion caused by multipath propagation
delays.

18. •Multi-path propagation
The presence of reflecting objects and scatterers in the
propagation path (buildings, signs, trees, fixed and moving vehicles)
•Speed of the Mobile
Random Frequency Modulation due to different Doppler
shifts of each of the multipath components
•Speed of the surrounding objects
Time varying Doppler shift on multipath components
If the surrounding objects move at a greater rate than the mobile ,
then this effect dominates the small scale fading
•The transmission bandwidth of the signal
If the transmitted radio signal bandwidth is greater than the
bandwidth of the multipath channel, the received signal will be
distorted
Factors Influencing Small Scale Fading

19. Doppler shift
 The shift in received signal frequency due to motion  is
directly proportional to the velocity and direction of motion of
the mobile with respect to the direction of arrival of the received
multipath wave.
Illustration of
Doppler effect
v is constant
Remote source
∆L is the difference in path length traveled by the
wave from source s to the mobile at points X and Y
∆ L = d cosθ = v ∆t cosθ
∆t is the time required for the mobile
to travel from point X to Y
The phase change in the received
signal due to the difference in path
length is ∆φ = 2π∆L/λ
The apparent change in frequency fd = ∆φ/2π∆t = v cosθ/λ

20. Doppler shift
• The Doppler shift is positive (i.e., the apparent received frequency is
increased), if the mobile is moving toward the direction of arrival of
the wave.
•  The Doppler shift is negative (i.e. the apparent received frequency
is decreased), if the mobile is moving away from the direction of
arrival of the wave.
•  Multipath components from a CW signal which arrive from
different directions contribute to Doppler spreading of the received
signal, thus increasing the signal bandwidth.

21. Example 5.1

22. Impulse Response Model of a Multipath Channel
• The small scale fading can be directly related to the impulse
response of the mobile radio channel.
• The impulse response is a wideband channel characteristics
• It contains all information necessary to analyze any type of
radio transmission through the channel
Mobile radio channel may be modeled as a linear
filter with a time varying impulse response, where
time variation is due to receiver motion in space.

23. Impulse Response….
• For a fixed position d, the channel between the transmitter and the
receiver can be modeled as a linear time invariant system.
• The different multipath waves have propagation delays which vary
over different spatial locations of the receiver. The impulse
response should be a function of the receiver position.
• Therefore the channel impulse response can be expressed as h(d,t)

24. x(t) transmitted signal
y(d,t) Received signal at d
For a causal system h(d,t) = 0 for t < 0
d = vt

25. Since v is constant y(vt,t) is just a function of t and then,
It is clear that the mobile radio channel can be modeled
as a linear time varying channel where the channel
changing with time and distance.

26. • v can be assumed constant over a short time or distance
interval
• x(t) represent the transmitted bandpass waveform
• y(t) the received signal waveform
• h(t,τ) the impulse response of the time varying multipath
radio channel
• t represents the time variations due to motion
• τ represents the channel multipath delay for a fixed
value of t

27. If the multipath channel is assumed to be a band limited bandpass
channel, then h(t,τ) may be equivalently described by a complex
baseband impulse response hb(t, τ) with the input and output being
the complex envelope representations of the transmitted and
received signals respectively
x(t) y(t)
c(t) r(t)
Baseband equivalent channel impulse response
Bandpass channel impulse response model

28. • The factor ½ is due to the properties of the complex
envelope in order to represent the passband radio
system at baseband.
• The lowpass characterization removes the high
frequency variations caused by the carrier.
• The average power of a bandpass signal x2
(t) is equal
to 0.5|c2
(t)| [Couch3]

29. Discretizing the multipath delay axis
• Discretize the multipath axis delay of the impulse
response into equal time delay segments called excess
delay bins
• Each bin has a time delay width ∆τ = τi+1 – τi
• τo = 0 (the first arriving signal at the receiver)
• τ1 = ∆τ, then τi = i∆τ i = 0 to N-1
• N is the total number of possible equally-spaced multipath
components.
• Quantizing the delay bins determines the time delay
resolution of the channel model
• The useful frequency span of the model = 2/∆τ
• The model can be used to analyze transmitted RF signals
having bandwidths which are less than 2/∆τ
• The maximum excess delay = N∆τ

30. The received signal consists of a series of
attenuated,
time delayed,
phase shifted replicas of the transmitted signal
The baseband impulse response of a multipath channel can be
expressed as
ai(t,τ) the amplitude of the ith
multipath component
τi the excess delay of the ith
multipath component
2πfcτi (t) the phase shift due to free space
propagation of the ith
multipath component
φi(t, τ) any additional phase shifts which are
encountered in the channel.

31. and δ(τ - τi(t)) is the unit impulse function which determines
the specific multipath bins that have components at time t
and excess delays τi
In general, the phase term can be simply represented by a single
variable θ(t, τi)

32. Discretize the multipath delay axis

33. If the channel impulse response is assumed to be time
invariant, then the channel impulse response may be
simplified as
When measuring or predicting hb(τ) a probing pulse p(t) which
approximates a delta function is used at the transmitter.
That is,
p(t) ≈ δ(t-τ)

34. Relationship Between Bandwidth and Received Power
(1) Consider a pulsed, transmitted RF signal of the form
p(t) is a repetitive baseband pulse train with very narrow
pulse width Tbb and repetition period TREP which is much
greater than the maximum measured excess delay τmax
Such a wideband pulse will produce an output that approximates
hb(t,τ)
We will consider two extreme channel sounding cases as a
means of demonstrating how the small-scale fading behaves
quite differently for two signals with different bandwidths in
the identical multipath channel.

35. Let
= 0 elsewhere
The low pass channel output r(t) closely approximates the
impulse response , It is given by
r(t) = p(t) ⊗ (1/2)hb(t,τ)

36. Note that if all the multipath components are resolved by the
probe p(t), |τj-τi| > Tbb for all j ≠ i then

37. For a wideband probing signal p(t):
Tbb is smaller than the delays between multipath
components in the channel
•Equation (5.18):
The total received power is simply related to the
sum of the powers in the individual multipath components, and is
scaled by the ratio of the probing pulse’s width and amplitude, and
the maximum observed excess delay of the channel.
Assuming that the received power from the multipath
components forms a random process where each component has
a random amplitude and phase at any time , the average small-
scale received power for the wideband probe is found from
Equation (5.17) as

38. In Equation (5.19), Ea,θ[•] denotes the ensemble average over
all possible values of ai and θi in a local area, and the overbar
denotes sample average over a local measurement area
small-scale received power is simply the sum of the average
powers received in each multipath component.
In practice, the amplitudes of individual multipath components
do not fluctuate widely in a local area. Thus, the received
power of a wideband signal such as p(t) does not fluctuate
significantly when a receiver is moved about a local
area[Rap89].

39. 2) Now, instead of a pulse, consider a CW signal which is
transmitted into the exact same channel, and let the complex
envelope be given by c(t) = 2.
Then, the instantaneous complex envelope of the received
signal is given by the phasor sum
and the instantaneous power is given by

40. As the receiver is moved over a local area, the channel induces changes on
r(t), and the received signal strength will vary at a rate governed by the
fluctuations of ai and θi. As mentioned earlier, ai varies little over local
areas, but θi will vary greatly due to changes in propagation distance over
space, resulting in large fluctuations of r(t) as the receiver is moved
over small distances (on the order of a wavelength).
The average received power over a local area is then given
by

41. Note that when cos (θi - θj) = 0 and/or rij = 0,
then the average power for a CW signal is equivalent
to the average received power for a wideband signal in a
small-scale region. This is seen by comparing Equation (5.19)
and Equation (5.24).
Measured wideband and
narrowband received
signals over a 5λ (0.375 m)
measurement
Wideband (Tbb = 10 ns)
Narrowband (CW 4GHz)
Carrier 4GHz
Local average is the same

42. Example 5.2
Assume a discrete channel impulse response is used to model urban RF
radio channels with excess delays as large as 100 μs and microcellular
channels with excess delays no larger than 4 μs. If the number of multipath
bins is fixed at 64, find (a) ∆τ and (b) the maximum RF bandwidth which the
two models can accurately represent. Repeat the exercise for an indoor channel
model with excess delays as large as 500 ns. As described in section 5.7.6,
SIRCIM and SMRCIM are statistical channel models based on Equation (5.12)
that use parameters in this example.

43. Example 5.3
Assume a mobile traveling at a velocity of 10 m/s receives two multipath
components at a carrier frequency of 1000 MHz. The first component is
assumed to arrive at τ = 0 with an initial phase of 0° and a power of
–70 dBm, and the second component which is 3 dB weaker than the first
component is assumed to arrive at τ = 1μs, also with an initial phase of
0°. If the mobile moves directly toward the direction of arrival of the first
component and directly away from the direction of arrival of the second
component, compute the narrowband instantaneous power at time intervals of 0.1
s from 0 s to 0.5 s. Compute the average narrowband power received over this
observation interval. Compare average narrowband and wideband received
powers over the interval, assuming the amplitudes ofthe two multipath
components do not fade over the local area.
Solution
Given v = 10 m/s, time intervals of 0.1 s correspond to spatial intervals of
1 m. The carrier frequency is given to be 1000 MHz, hence the wavelength
of the signal is

44. The narrowband instantaneous power can be computed using
Equation (5.21).
Note –70 dBm = 100 pW. At time t = 0, the phases of both
multipath components are 0°, hence the narrowband instantaneous
power is equal to
Now, as the mobile moves, the phase of the two multipath
components changes in opposite directions.

45. Since the mobile moves toward the direction of arrival of the first
component, and away from the direction of arrival of the second
component, θ1 is positive, and θ2 is negative.
Therefore, at t = 0.1 s, θ1 = 120°, and θ 2 = –120°, and the
instantaneous power is equal to
Similarly, at t = 0.2 s, θ1 = 240°, and θ 2 = –240°, and the
instantaneous power is equal to

46. It follows that at t = 0.4 s, |r (t )|2
= 79.3pW, and at t = 0.5 s, |r (t )|2
= 79.3 pW.
The average narrowband received power is equal to
Similarly, at t = 0.3 s, θ1
= 360° = 0°, and θ 2
= –360° = 0°,
and the instantaneous power is equal to

47. Using Equation (5.19), the wideband power is given by
As can be seen, the narrowband and wideband received power are
virtually identical when averaged over 0.5 s (or 5 m). While the
CW signal fades over the observation interval, the wideband
signal power remains constant over the same spatial interval.

48. Small-Scale Measurements
1. Direct pulse measurements
2. Spread spectrum sliding correlator measurements
3. Swept frequency measurement

49. Direct RF Pulse System
A simple approach to determine rapidly the power delay
profile of any channel
A wideband pulsed bistatic radar
Transmits a repetitive pulse of width Tbb
Receiver with a wide bandpass filter = 2/Tbb
The signal is amplified,
detected and displayed
Immediate measurement of the square of the channel impulse response
convolved with the probing pulse. If the oscilloscope is set on averaging mode,
the system can provide a local average power delay profile.

50. • The minimum resolvable delay between multipath
components is equal to the probing pulse width Tbb.
Disadvantages
• It is subject to interference and noise due to the wide
passband filter required for multi-path time resolution
• If the first arriving signal is blocked or fades, severe
fading occurs. (triggering oscilloscope with the first
arrival signal is important)
• The phases of the individual multipath components are
not received due to the use of an envelope detector

51. Spread Spectrum Sliding Correlated Chanel Sounding
Can be detected using a narrowband receiver
preceded by a wideband mixer
Improving the dynamic range of the
system compared to the direct RF pulse
system.
Wideband probing signal
A carrier signal is spread over a large bandwidth by mixing it with a
binary pseudo-noise PN sequence having a chip duration Tc and a
chip rate Rc equal to 1/Tc Hz. The power spectrum envelope of the
transmitted spread signal is given by
The null-to-null RF bandwidth (BW) = 2Rc

52. Spread Spectrum Channel Impulse Response Measurement System

53. • The spread spectrum signal is received, filtered, and despread
using a PN sequence generator identical to that used at the
transmitter.
• PN sequence are identical
• The transmitter chip clock is run at a slightly faster rate than the
receiver chip clock
• Mixing the chip sequence in this fashion implements a sliding
correlator.
• When the PN code of the faster chip clock catches up with the
PN code of the slower chip clock, the two chip sequences will
virtually identical aligned (maximal correlation)

54. When the two sequences are not maximally correlated, mixed
signals will spread into large bandwidth. Then narrow band filter
can reject almost all the incoming signal power.
Processing gain (PG) = 2 Rc/Rbb = 2Tbb/Tc
Tbb = 1/Rbb is the period of the baseband information.
The time resolution (∆τ) of multipath components using a spread
spectrum system with sliding correlation is
∆τ = 2 Tc = 2/Rc
The system can resolve two multipath components as long as they
are equal to or greater than 2Tc s a part.

55. The time between maximal correlations is

57. Frequency Domain Channel Sounding

58. The number and spacing of the frequency steps impact the
time resolution of the impulse response measurement.
For each frequency step, the S-parameter test set:
transmits a known signal level at port 1
monitors the received signal level at port 2
Frequency domain is then converted to Time domain using
inverse discrete Fourier transform given a band limited
version of the impulse response
The technique is useful in very close measurements
It is also non real time nature of the measurement.

59. For time varying channels, the channel frequency
response can change rapidly, giving an erroneous impulse
response measurement. This effect can be overcome by
fast sweep time.
The technique has been used successfully for indoor
propagation studies.

60.  Many multipath channel parameters are derived from the
power delay profile:
Power delay profiles are generally represented as plots of relative
received power as a function of excess delay with respect to a fixed
time delay reference.
Power delay profiles are found by averaging instantaneous power
delay profile measurements over a local area in order to determine
an average small-scale power delay profile.
5.4 Parameters of Mobile Multipath Channels

61. Depending on the time resolution of the probing pulse and
the type of multipath channels studied
From 450 MHz to 6 GHz
sample at spatial separations of λ/4
over receiver movements < 6 m in outdoor channels
< 2 m in indoor channels
Avoids averaging bias in the resulting small-scale statistics


62. Typical power delay profile plots
Outdoor channels, determined from a large
number of closely sampled instantaneous profiles
Measured multipath power delay
profile at 900 MHz cellular
system (San Francisco)

63. indoor channels, determined from a large number of
closely sampled instantaneous profiles.
Typical power delay profile plots
Inside store at 4 GHz

64. Time Dispersion Parameters
The mean excess delay
RMS delay spread
These delays are measured relative to the first detectable
arriving at the receiver at τo = 0
Power is plotted in relative amplitudes

65. Typical measured values of RMS delay spread

66. Maximum excess delay
The maximum excess delay of the power delay profile is
defined to be the time delay during which multipath energy
falls to X dB below the maximum.

67. Coherence Bandwidth
Coherence Bandwidth Bc
It is a statistical measure of the range of frequencies over which
the channel can be considered flat (A channel which passes all
spectral components with approximately equal gain and linear
phase.
Two sinusoids with frequency separation greater than Bc are
affected quite differently by the channel.
If the coherence bandwidth is defined as the bandwidth over
which the frequency correlation
function is above 0.9, then
If the definition is relaxed so that the frequency
correlation function is above 0.5 then

68. Calculate the mean excess delay, rms delay
spread, and the maximum excess delay (10dB) for
the multipath profile given in the figure below.
Estimate the 50% coherence bandwidth of the
channel. Would this signal to be suitable for AMPS
and GSM service without the use of equalizer
Example 5.5

69. The mean excess delay
rms delay spread
Coherence bandwidth

70. 5.4.3 Doppler Spread and Coherence Time
Delay Spread and Coherence bandwidth are parameters which describe
the time dispersive nature of the channel.
Doppler Spread and Coherence Time are parameters which describe the
time varying nature of the channel in small-scale region.
Doppler spread BD is a measure of a spectral broadening caused by the
rate of change of the mobile radio channel and is defined as the range of
the frequencies over which the received Doppler spectrum is essentially
non-zero.
If the baseband signal bandwidth is much greater than BD, the effects of
Doppler spread are negligible at the receiver. This is a slow fading
channel

71. Coherence time Tc is the time domain dual of Doppler spread and is
used to characterize the time varying nature of the frequency
dispersiveniss of the channel in the time domain.
Tc = 1/fm (fm is the maximum Doppler shift and given by v/λ)
Coherence time is actually a statistical measure of the time duration
over which the channel impulse response is essentially invariant and
quantifies the similarity of the channel response at different times.
If the coherence time is defined as the time over
which the time correlation function is above 0.5, then
In modern digital communication Tc is defined as

72. Two signals arriving with a time separation greater than
Tc are affected differently by the channel.
Vehicle traveling 60 mph (~ 95 km/h) using 900 MHz
carrier
Tc= 2.22 ms from
1/Tc = 545 bps
If the symbol rate is greater than 545 b/s the channel
will not cause distortion due to motion. However,
channel may cause distortion from multipath time delay
spread (depending on the channel impulse response)

73. Example 5.6

74. Types of Small-Scale Fading
Based on Multipath Time Delay Spread
Flat Fading
1.BW of the signal < BW of channel
2.Delay spread < Symbol period
Frequency Selective Fading
1.BW of the signal > BW of channel
2.Delay spread >Symbol period
Based on Doppler Spread
Fast Fading
1.High Doppler Spread
2.Coherence Time < Symbol period
3.Channel Variation Faster than
baseband Signal Variations
Slow Fading
1.Low Doppler Spread
2.Coherence Time > Symbol period
3.Channel Variation Slower than
baseband Signal Variations

75. Flat Fading Due to Multipath Time Delay Spread
1. The mobile radio channel has constant gain and
linear phase response over a bandwidth which is
greater than the bandwidth of the transmitted signal
2. It is the most common type of fading described in
the literature.
3. The strength of the received signal changes with
time, due to fluctuations in the gain of the channel
caused by multipath

76. Flat Fading Channel Characteristics
The BW of the signal is narrow compared to the channel flat fading BW
Flat fading channel, amplitude varying channels, narrow band channels
Typical flat fading channels cause deep fade and thus may require 20 to 30
dB transmitter power increase to achieve low bit error rates during times of
deep fade as compared to systems operating over non-fading channels

77. The distribution (such as Rayleigh ) of the instantaneous gain of flat fading
channels is important for designing radio links

78. Frequency Selective Fading
The channel possesses a constant gain and linear phase response
over a bandwidth that is smaller than the bandwidth of the
transmitted signal
The channel impulse response has a multipath delay spread which
is greater than the reciprocal bandwidth of the transmitted message
waveform.
It is due to time dispersion of the transmitted symbols within the
channel.
Bs > Bc
Ts < στ

79. • The received signal includes multiple versions of the transmitted
waveform which are attenuated (faded) and delayed in time, and
hence the received signal is distorted.
• The channel induces intersymbol interference (ISI).
• Certain frequency components in the received signal spectrum
have greater gains than others.

80. Fading Effects Due to Doppler Spread
Fast Fading
Rate of change due to motion
How rapidly the transmitted baseband signal changes as
compared to the rate of change of the channel?
The channel impulse response changes rapidly within
the symbol duration Ts > Tc (Bs<BD)
Frequency Dispersion
(signal distortion)

81. In the case of flat fading channel we can approximate the
impulse response to be simply a delta function
Flat fading, fast fading channel
is a channel in which the amplitude of the delta function varies
faster than the rate of change of the transmitted baseband signal.
Frequency selective, fast fading channel
The amplitudes, phases, and time delays, of any one of the
multipath components vary faster that the rate of change of the
transmitted signal.
In practice fast fading is only occurs for very low data rate

82. Slow fading
In slow fading channel, the channel impulse response
changes at a rate much slower than the transmitted
baseband signal s(t).
The channel may be assumed to be static over one or
several reciprocal bandwidth intervals.