## Fading Basics—Narrow Band, Wide Band and Spatial Channels

The concept of RF signal fading is a familiar one to engineers who deal with wireless RF, but today’s focus on advanced radio techniques (e.g., MIMO, spatial channel modeling) begs for more than a superficial understanding of these complex subjects.

Spirent’s Doug Reed has authored a trio of thoughtful yet accessible white papers that will provide a better understanding of the RF challenges faced by every engineer working in the wireless industry.

This white paper is the first of the series, guiding a reader through the basic concepts of fading and into some critical RF-related topics such as:

- Fading Models
- Flat Fading
- Frequency-Selectivity
- Multi-path Delays and Delay Spread
- Multiple antennas
- Wide band Channels
- Shadowing
- Channel Modeling and MIMO Capacity

### There are two more White Papers in this series available for download:

Rev. X mm/08
fading basics
narrow band, Wide band, and spatial channels
White Paper 101
White Paper
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fading basics
narrow band, Wide band, and spatial channels
White Paper 101
SPIRENT WhITE PAPER • i
Table of Contents
1.1. Introduction .......................................... 6
1.2. Fading Models ........................................ 9
1.2.1. Flat Fading ........................................... 9
1.3. Why is a Signal Frequency-Selective?..................... 11
1.4. Multi-path Delays and Delay Spread ..................... 15
1.5. Multiple Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6. Wide-band Channels .................................. 18
1.7. Shadowing .......................................... 19
1.8. Channel Modeling and MIMO Capacity ................... 20
6 • SPIRENT WhITE PAPER
Fading Basics
Narrow Band, Wide Band, and Spatial Channels
1.1. introduction
Radio technologies have undergone increasingly rapid evolutionary changes
in the recent past. The first cellular phones used narrow-band FM modulation,
which was soon replaced by digital modulation in second & third generation
devices. Today, multiple-antenna systems are being employed to increase data
rates. These provide improved quality while decreasing operational costs.
As technology progresses to take advantages of more complex channel
characteristics, the channel modeling required to emulate the radio environment
for testing becomes both more critical and more complex. For instance, when
bandwidths are increased (to support higher data rates) receivers become more
susceptible to Inter-Symbol Interference (ISI). To ensure that measurements
in the lab accurately correlate to the quality of the user’s experience, channel
models must account for all the aspects of the radio environment.
Figure 1: The Wireless Propagation Environment
Figure 1 illustrates the modern radio propagation environment, consisting of a
number of different components. Referring to the figure, a signal transmitted
from the base station to a subscriber is shown to consist of a number of paths.
This is generally referred to as multi-path propagation. For purposes of system
design and testing, a limited number of paths, usually 4-24, are used to model
the radio channel.
The radio bandwidth determines the number of paths required to produce an
adequate model. Each path is made up of a number of sub-paths, representing
individual plane waves received from nearby reflections. Multiple sub-paths are
closely associated with a single path and may not be observable in the received
signal. In reality, each sub-path arrives at the receiver with a slightly different
time delay and Angle of Arrival (AoA). These sub-paths characteristics cause
each path to have its own characteristic Delay Spread (DS) and Angle Spread
(AS). Due to bandwidth limitations, the Path DS is usually considered to be zero.
This means it is assumed that all the sub-paths arrive at the same time.
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Narrow Band, Wide Band, and Spatial
For narrow-band channels (such as early FM radios operating at 25-30 khz
bandwidth), a receiver cannot resolve the different paths. In this case, the
receiver sees a single composite signal which is the vector sum of all the multi-
path components.
This is illustrated in Figure 2, which further describes the effects of bandwidth
on a receiver’s ability to resolve multi-path components. Wide-band
measurements indicate that each individual path tends to be received from a
particular direction and with a limited AS [ ]. Figure 2 is a conceptual description
of a signal received on a wide band radio. The powers of the five multi-path
components are shown with their peak normalized to zero dB. The paths are
represented by the colors: red, blue, black, magenta, and green, which are used
to denote power received at each of the delays. Each path has a unique power-
angular distribution or Power Azimuth Spectrum (PAS) at its given delay.
The five paths shown are made up of many distinct sub-path plane-waves,
received at slightly different AoAs. It is assumed that all sub-paths for a given
path are received as a cluster and arrive at the same time.
Figure 2: Conceptual Power Azimuth Spectrum for a Wide Bandwidth Signal
When the signal in Figure 2 is received by a narrow band receiver, all multi-path
components are indistinguishable and are combined together at the antenna.
The Power Azimuth Spectrum is nearly uniform for this case, and results in
classical Rayleigh Fading.
In general, as the bandwidth increases, so does the ability of the receiver to
resolve the different paths, thus increasing the number of paths required by
an accurate channel model. As the number of observable paths increase, the
statistical characteristics of fading change; while the narrow-band model treats
multiple paths as a single composite path, the wide-band model requires
multiple paths.
8 • SPIRENT WhITE PAPER
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Narrow Band, Wide Band, and Spatial Channels
Referring back to Figure 1, increased bandwidth (BW) distinguishes a multi-path
“group” from a cluster. A cluster is defined to be the source of a reflected path
that can not be separated into additional paths by increasing the bandwidth.
A multi-path group, however, will become resolvable into separate paths with
increased bandwidth. In the limit case with infinite bandwidth, every “path
component”, no matter how insignificant, is resolvable, producing thousands of
paths. however, actual bandwidths filter our ability to resolve different paths and
lead to lower and more practical numbers of paths we use in today’s models.
Each path is shown to depart the antenna with an angle spread (AS) expressed in
degrees. In reality, signals leave even highly-directional antennas in all directions,
but only certain paths reach the mobile station (MS) with receivable levels of
power. Only these paths are used in modeling the channel.
When the transmit frequency is the same at the base station (BS) and MS (for
example, in Time Division Duplex [TDD] systems), the path is identical in either
direction. This bi-directional equivalence at a given frequency is called reciprocity.
This principal can be used to understand how paths behave, and why only
certain paths are modeled. In Frequency Division Duplex (FDD) systems, different
frequencies are used in each direction. Since the frequency is not the same, the
paths are not reciprocal, but their average powers are still highly correlated.
Figure 3: Angle Spread at the Subscriber
The AS of an individual path is different at the BS and the MS due to scattering
near the antenna. Since the subscriber is near the ground and in the presence of
more clutter, there are reflections near the MS antenna, which leads to a larger AS
than what is observed at the BS. Figure 3 presents a top view of a subscriber in
an urban area, where a path is arriving at the subscriber antenna. The individual
“rays” depicted in the figure are sub-paths, and are the components that make
up a path.
Because the small differences in the length of travel for these sub-paths are non-
resolvable in typical channel bandwidths, these sub-paths act together as a path.
The sub-paths are combined at the receive antenna and produce a faded signal
due to the vector sum of sinusoids of varying phases.
The arrival of sub-paths from a variety of directions causes the path to have
an AS. Sub-paths that appear from “behind” the MS are generally weak since a
reflection with a high angle of incidence produces a weaker signal than those
with low angles of incidence. This produces the tendency towards narrow angle
spreads (as is observed in measured data). The strongest receive paths tend to
be the most narrow, because they receive a dominant signal, and reflections
contribute less to the result.
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When all paths are combined, a “composite” angle spread can be calculated,
(not to be confused with the path AS described above.) The composite angle
spread is different at the base station and at the subscriber’s location, due to
the unique propagation effects present at each end of the radio link.
The average angle associated with the angle spread represents the Angle
of Departure (AoD) or Angle of Arrival (AoA) of the signal. When a path is
completely specified at each end, the channel can be described as a “double
directional” channel.
1.2. fading models
1.2.1. flat fading
Since multiple paths are not resolvable in narrow bandwidths, the RF
environments seen in narrow-band FM technologies such as AMPS, NAMPS,
TACS, can be modeled by “flat” fading channels. These frequency-flat channels
fade the same amount across the frequency band and are easily modeled by
single-path fader models such as the Jakes Fader, JTC Fader, or others.
Figure 4: Sample Rayleigh-Faded Narrow Band 30 kHz Signal
The flat fading signal illustrated in Figure 4 represents a signal path faded due
to reflections produced by localized clutter. Technically, this signal and the
fading associated with it are due to multi-path reception. however, the term
multi-path is more commonly applied to delayed paths leading to delay spread,
and this single-path fading behavior is the result of local scattering.
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Narrow Band, Wide Band, and Spatial Channels
In this example, a 30m radius of scattering is used, producing many reflections
of the signal in close proximity to the subscriber. These reflected signals are
equivalent to the sub-paths described earlier and are received at various levels
and phases and combined at the subscriber’s antenna. Because the bandwidth
of 30 khz is so small, there is virtually no difference between the fades across
the band. The flat fading characteristic remains true even with the addition of
multiple delayed paths, even with fairly lengthy delays (tens of µs), since these
are not resolvable in narrow band channels.
1.2.2. frequency selective fading
Digital radio technologies, including CDMA, WCDMA, UMB, LTE, and WiMAX,
transmit digital signals in a bandwidth larger than the coherence bandwidth
of the channel. This means that the channel no longer looks “flat” across the
frequency band; rather, the fading is “frequency-selective” with different signal
strengths present at different frequencies across the band. Figure 5 illustrates
a single-path channel with the same 30m scattering radius as shown in Figure
4. This channel represents a single strong path that is faded due to localized
clutter, with many reflections in close proximity to the subscriber. Note from the
Figure that the frequency is stepped across the 5 Mhz bandwidth in 0.5 Mhz
steps (shown by various color traces) and illustrates how the fading changes with
frequency.
Figure 5: Sample Rayleigh-Faded 5 MHz Wide Signal
Typically, more than one strong path is received, each having a delay based on
the distance the signal travels. This is called multi-path propagation. As each
delayed path arrives at the receiver, it is scattered by local clutter. When multiple
paths are added together at the receiver, each with progressively longer delays;
the combined signal exhibits the same frequency-selective fading behavior as the
locally scattered path, but the change with frequency is much more rapid. The
channel looks coherent (the same) over a much smaller bandwidth when multi-
path is present, and has a lower coherence bandwidth than the single-path case.
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1.3. Why is a signal frequency-selective?
Consider an example of 2 paths (phasors) having equal power but different path
lengths d1 & d2, as shown in Figure 6. Each signal path has a phase measured
at the receiver equal to:
Φ
1
=
Φ
2
=
Figure 6: Sample Frequency-Selective Fading 2-Path
The relative phase difference ΔΦ = Φ2 - Φ1 is a function of frequency and the
differential path length d2-d1.
ΔΦ =
At the given receiver location, ΔΦ increases in phase as the frequency is
increased. Every time ΔΦ rotates by 2π, there is a phasor addition and a phasor
cancellation of the two path summation, i.e. at 0 and π. This interaction between
the two paths produces frequency selective fading.
This can be seen by the phase relationship:
ΔΦ = = π · (2n–1)
where n is an integer number representing phase differences that are odd
multiples of π, which represent the frequency selective fades.
Thus: λ = =
Or: f =
d
1
· 2 π
λ
2 π(d
2
–
d
1
)
λ
d
2
· 2 π
λ
2 π(d
2
–
d
1
)
λ
2 (d
2
–
d
1
)
2n–1
(2n–1)C
2(Δd)
2Δd
2n–1
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Narrow Band, Wide Band, and Spatial Channels
This formula means that every time the frequency is an integer multiple of
C/(2 Δd), there is a frequency-induced fade in the receiver
Consider the frequency over a bandwidth: f
2
–f
1
= Δf where n2 = n1 + m, and m is
an integer representing m fades across the band.
f
1
=
f
2
=
Now let n2 = n1 + m, where m is an integer representing m fades across the
band.
f
2
– f
1
= Δf = – =
Therefore,
ΔfΔd = mC
Where Δf = the bandwidth in hz
Δd = the path length difference in meters
m = an integer number representing the number of fades across the band
C = speed of light
Consider the path differences required to observe a fade in a given receiver
bandwidth; for example, achieving an odd multiple of π phase shift between the
two paths.
1 Note that when m is a non-integer, there may be a counting ambiguity where depending on the
starting phases (which is a function of the path difference) the BW may contain m or m+1 fades.
Since the equation was derived assuming m is an integer then we should only obtain m fades, so
this should not be a problem.
C(2n
1
–1)
2(Δd)
C(2n
2
–1)
2(Δd)
C(2n
2
–1)
2(Δd)
mC
Δd
C(2(n
1=
+m)–1)
2(Δd)
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Table 1: Frequency Selective Fading Sample for a 2 Path Model
Figure 7 illustrates two of the examples shown in Table 1, where a path delta of
15 and 60 meters are shown to experience 1 and 4 fades respectively in a 20
Mhz BW.
Figure 7: Frequency Selective Fading for Some Path Length Differences
Notice that narrow-band radio will probably not have significant problems with
frequency-selective fading. This is because it would require path differences
of at least 10 km with roughly equal path power levels (since a null requires
cancellation). For typical path differences, only a small fraction of a fade can be
seen across the band in the narrow band case.
For 20 Mhz-wide band radios, frequency selective fading is quite evident. For
typical path delays, there may be 20-40 or more fades across the band.
Receiver
Bandwidth
Path Difference for
one fade across the
band
# of fades for a 60 m
path difference
# of fades for a 600
m path difference
30 kHz 10 km 0.006 0.06
20 Mhz 15 m 4 40
Δf Δd = m = m =
mC
Δd
ΔfΔd
C
ΔfΔd
C
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Narrow Band, Wide Band, and Spatial Channels
This concept is also illustrated in Figure 8 and Figure 9, where a path difference
is shown at two separate measurement frequencies. Because the wavelength
changes with frequency, so does the number of sine-wave repetitions within the
fixed path difference. The phase difference of the two signals (resulting from the
path difference), is dependent on frequency, and each sine-wave repetition that
occurs within the path difference represents a frequency selective fade in the
band.
Now at Frequency 1 in Figure 8, the phases between the two paths nearly align,
so the signals are adding constructively. At Frequency 2 in Figure 9, the phase
difference has increased and the signals from the two paths are out of phase
and are cancelling. Therefore, the fading produced by a two-path signal is
frequency selective.
Figure 8: Two Path Sample at Frequency 1
Figure 9: Two Path Sample at Frequency 2
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Narrow Band, Wide Band, and Spatial
1.4. multi-path delays and delay spread
Frequency-selective channels result from multi-path propagation. As shown in
Figure 5, local scattering can exhibit frequency selectivity, but it is more evident
from the combination of distinct paths. Since the delays between paths are
much larger than the delays within a path (i.e. the small intra-path delays due to
local scattering,) these latter delays are typically ignored and assumed to have
zero delay spread. Therefore each path will experience flat fading. The frequency
selectivity of the channel is then only a function of the relative path delays.
These delayed paths are illustrated in Figure 10, again assuming no intra-path
delay. Each path has a unique delay time and relative power, as shown.
Figure 10: Sample Power Delay Profile from the ITU Vehicular A Channel Model
1.5. multiple antennas
When multiple antennas are used at the transmitter, the receiver, or both,
significant improvements in performance can be obtained. Multiple Antenna
techniques vary and include simple diversity selection and combining schemes,
and more complex approaches like beam forming, and Multiple-Input-Multiple-
Output (MIMO) systems.
Modeling multiple antenna approaches requires a fading channel with the
proper correlation between antenna branches. high correlation means that
the signals are very similar, so both branches may experience a strong or
weak signal at the same time, making it harder to withstand a given fade. Low
correlation means that the signals are more random, such that a fade on one
branch might be mitigated by a stronger signal on the other branch.
Early models set the correlation between antenna branches to an average
value for evaluating simple diversity receivers. Today, more complex models
are required since new air interfaces are designed to adapt between different
techniques based on the dynamics of the channel. These may include a variety
of multiple antenna techniques including Beam Forming, MIMO, Space Division
Multiple Access (SDMA), and scheduling approaches like frequency-selective
scheduling. Multiple antennas are usually expressed as an MxN combination,
where M is the number of antennas at the transmitter, and N is the number of
antennas at the receiver. Typical configurations may include 1x2, 2x2, 4x1, 4x2,
4x4, 1x4, and others.
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Figure 11: 2x2 Multiple Antenna Configuration
A 2x2 example is shown in Figure 11, where a total of 4 connections are present
between antenna elements. These connections are indicated by the h11, h21,
h12, and h22, each representing a connection between the base and the
subscriber. Each connection has a complex path gain (for example, amplitude
and phase) measured with respect to a normalized average power. These terms
are grouped together to form an h matrix as shown in Figure 12. There is a
unique h matrix for each delayed path. For example, a six-path channel will have
six h matrices that will be updated quickly enough to track the Rayleigh fading
of each path.
Figure 12: Complex Channel Matrix H
The signals at the transmitter and receiver antenna elements are correlated,
not random. Extensive measurements have shown that the correlation is not
constant, but varies significantly over a geographic area or drive route. The
correlation between antenna elements is a mathematical function related to the
make-up of the local scattering and is a function of the signal AS, its AoA, and
the subscriber’s direction of travel (DoT).
The Power Azimuth Spectrum (PAS) of each path is typically modeled by a
Laplacian distribution wherein the signal drops off exponentially (linearly in dB)
as the angle increases in magnitude from the average direction of arrival.
Figure 13 illustrates the complex correlation that results from the Laplacian PAS
when a 2º AS is specified for BS antennas separated by four wavelengths. The
magnitude indicates that the correlation between antenna elements is quite
high, ranging from about 0.7 to 1.0 (where a value of 0 represents no correlation
and a value of 1.0 represents perfect correlation).
H =
h
11
h
12
h
21
h
22
h
11
h
22
h
21
h
12
TX1 RX1
TX2 RX2
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Narrow Band, Wide Band, and Spatial
Figure 13: Base Station Antenna Correlation
Figure 14 illustrates the correlation between subscriber antennas separated
by half a wavelength. While the antennas are closer together, the correlation is
somewhat lower due to the larger AS (35°). Using such an array at the subscriber
is assumed for simplicity. Multiple antenna configurations may actually include
polarized antennas to obtain low correlation with antenna elements in close
proximity. For both the base station and subscriber, the antenna correlation is a
function of the path angle.
Figure 14: Subscriber Antenna Correlation
2 It should be noted that averaging the complex correlation across angles of arrival 0-2π will
result in exactly the correlation of the uniform ρ = -0.304.
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Narrow Band, Wide Band, and Spatial Channels
Correlation can be plotted in another way, as shown in Figure 15. This is based
on the separation of antenna elements or the distance between samples. These
two interpretations are exactly the same, and indicate how much the signal is
changing versus distance.
Figure 15: Autocorrelation
Note that using the narrow angle spread, selected to match field measurements,
increases correlation as compared to the uniform or classical Doppler
assumption. This result causes a reduced fading rate which changes as a
function of AoA.
1.6. Wide-band channels
As described earlier, when channel bandwidth increases so does the ability
to resolve multi-path. In the narrow-band case, it is typically assumed that all
path sub-components arrive at the same time, i.e. the path delay spread is
zero, leading to frequency-selective fading by the interaction between paths.
however, for extended bandwidths (≥ 20 Mhz) it is desirable to have either more
paths, or some intra-path delay spread to enhance the modeling of frequency
selectivity.
Most channel models to date have been limited to a small number of paths
since channels were only a few Mhz wide. With wider bandwidths, the Spaced-
Frequency Correlation Function[ ] exhibits periodic oscillation across frequency
and describes how different frequencies are correlated across the band. Figure
16 shows the result for the Vehicular A channel model, described earlier in
Figure 10. The oscillations in correlation are due to the limited number of paths,
wherein the differences in path lengths contribute a different amount of phase
at each frequency. As the frequency is increased, the phases advance and
produce the oscillation. When the complex path components are combined,
there are some frequencies in which paths cancel and other frequencies in
which paths add constructively. When fading is added to these paths, the fading
is correlated across frequency based on the phase relationships between paths.
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Narrow Band, Wide Band, and Spatial
Figure 16: Spaced-Frequency Correlation Function
A Spaced-Frequency correlation function exhibiting this oscillatory behavior,
shown in Figure 16, is sometimes undesirable, such as when modeling
frequency-selective schedulers. Wide-band measurements indicate that the
spaced-frequency correlation of actual channels drops to a low level and
remains low. To reduce the level of oscillation and improve the wide-band
characteristics of the channel, various improvements can be made by adding
additional paths or by splitting one or more existing paths into multiple delayed
paths, sometimes called mid-paths.
1.7. shadowing
Shadow fading (SF), also called slow fading or log-normal shadowing is
the variation in average received power from one location to another. This
log-normally distributed parameter is generally independent of path loss;
for example, the distance from the BS, and represents the variation due to
shadowing or “blockage” from clutter on the ground.
SF is typically correlated in two ways. First, it is correlated with distance, where
the shadow fading value changes slowly with movement of the subscriber. This
distance, referred to as a de-correlation distance, is typically tens of meters
in urban areas and a few hundred meters in suburban and rural areas. This
distance is descriptive of the size of the clutter that obstructs the path to the
subscriber. Individual buildings may be the main component of the clutter
in urban areas, whereas city blocks or terrain changes may be the clutter in
suburban areas.
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Narrow Band, Wide Band, and Spatial Channels
Shadow fading can also be correlated with respect to the angle of the
paths between different BS sites. There will be a common component of
the shadowing present at the subscriber’s location, along with a difference
component of the shadowing for the path to each base station. This difference
component is present since each path is unique and sees clutter that is unique.
The common component of the shadow fading produces a correlation and this
is called site-to-site correlation. This is often modeled with a constant 50%
correlation.
1.8. channel modeling and mimo capacity
It is important to have channel models that correctly emulate real-world
conditions in order to adequately simulate multiple antenna performance. Since
algorithms will be compared and optimized based on channel models, the
models must include proper correlation between antenna elements. Adding
correlation diminishes the ability of a MIMO transceiver to spatially separate the
channel into orthogonal components in order to support additional transmission
streams. Thus, high degrees of correlation limit the potential capacity of MIMO,
and therefore must be included in a Spatial Channel model.
The narrow angle spreads described in Figure 14 and Figure 15 were selected to
account for this increased correlation between antenna elements.
The effect of the correlation is apparent in how the MIMO capacity is described
in the following equation[iii]. This equation gives the instantaneous capacity
value, as shown in Figure 17.
C = log
2
det I+( – )HH
H
bps/hz Instantaneous Capacity
Where:
H is the channel matrix of complex path gains,
I is the identity matrix,
Φ is the average SNR,
m is the number of transmit antennas, and
H
H
is the complex conjugate transpose.
This equation is quite interesting. It is very similar to the Shannon capacity
formula but the (1+SNR) is replaced with a matrix equation.
The determinant of a matrix is always the product of eigenvalues of the matrix:
det(HH
H
) = λ
1
λ
2
· · · λ
N
.
Adding an identity matrix shifts the eigenvalues of a matrix:
det(I+HH
H
) = (1+λ
1
) (1+λ
2
) · · · (1+λ
N
).
Φ
m
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Taking the log of a product is equivalent to summing the individual capacities on
each eigenvalue C = Σlog
2
(1+ α
i
SNR) , where: α
i
= λ
i
–
Figure 17: Instantaneous MIMO Capacity
The average capacity for the MIMO system is given by:
C = E log
2
det I = – HH
H
bps/hz Average Capacity
Where:
E is the expected value over the random channel H.
When correlation is added to the path gains, the terms in h become less
random. This results in a diminished ability to spatially separate the channel
into its constituent eigenvalues, which are the orthogonal components of the
channel capable of supporting a transmission. Therefore, the highest capacity
would be possible when each element of the h matrix is i.i.d. Rayleigh faded
signal representing no correlation between elements. Once the correlation
between antenna elements is included, the capacity is reduced. If the Spatial
Channel Model (SCM) is used instead of ideal (i.e. Rayleigh) fading, the
correlation can be much higher, significantly reducing the ideal capacity of a
MIMO system.
Φ
m
Φ
m
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Narrow Band, Wide Band, and Spatial Channels
Because the correlation predicted by the SCM is a function of antenna
orientation, AoA, AS, DoT, etc; it is important to analyze the capacity in terms of
a system simulation. This is shown in Figure 18 below.
Figure 18: Average MIMO Capacity
COST-259 Final Report, Wireless Flexible Personalised Communications, COST 259: European Co-operation in Mobile
Radio Research, Edited by Luis M. Correia
John G. Proakis, Digital Communications, 3rd Edition, McGraw-hill, 1995.
G.J. Foschini and M.J. Gans, “On limits of wireless communications in a fading environment when using multiple
antennas,” Wireless Personal Communications., vol. 6, no. 3, pp. 311-335, Mar. 1998.
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