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MODELLING THE WIDEBAND INDOOR CHANNEL COMPLEX IMPULSE RESPONSE
"Robert J. Achatz
U.S. Department of Commerce
V
National Telecommunications and Information Administration
Institute for Telecommunication Sciences
$325 Broadway
Boulder, Colorado 80303 USA
!Telephone: (303) 4973498
Y#(Rev. 11/19/92)
INDOOR CHANNEL MODELS
The accuracy of a wideband indoor channel model is evaluated by its ability to predict
detection errors for various modulation methods in lineofsight (LOS) links, obstructed lineofsight (OLOS) links, or both. The model must be able to simulate the complex impulse response
of the channel for these predictions. Current wideband indoor channel models include the
geometric model, the stochastic Gaussian widesense stationary uncorrelated scattering
(GWSSUS) model, and the statistical power delay profile model. This paper will describe each
modelling method briefly, point out how the complex impulse is derived from each one, and
discuss each model's advantages and disadvantages.
I.GEOMETRIC MODEL
The indoor channel can be modeled using geometric optics, physical optics, and the geometrical theory of diffraction (GTD). Geometrical optics is characterized by its ray tracing
techniques of reflection and refraction. The reflection can be specular or diffuse. Physical
optics and GTD are used to predict shadow phenomenon such as diffraction around edges.
Lawton [Lawton,1991] and McKown [McKown,1992] have presented geometric models for the
indoor channel. Allen [Allen,1991] has presented a geometric model for the urban canyon
channel.
v#0*$$SPECULAR REFLECTION OFF CONDUCTIVE SURFACES
The simplest indoor channel to model is an empty rectangular room with smooth and
conductive wall, ceiling, and floor surfaces. Room openings such as doors and windows are not
included. Transmit and receive antennas are described by an antenna pattern and location in
the room. Since the room is empty, a direct ray exists between the transmitter and receiver.
Single reflections include a reflection from each wall, the ceiling, and the floor. Multiple reflections can be included to increase model accuracy.
REFLECTION AND REFRACTION OFF COMMON BUILDING MATERIALS
Incident rays divide between reflected and refracted components when the walls, ceiling,
and floor have lossy dielectric properties.
The attenuation of the reflected ray is determined by a reflection coefficient. The reflection coefficient has vertical and horizontal components that predict the attenuation of the
normal and tangential E fields respectively. The reflection coefficient is determined by the
dielectric properties of the building material and the angle of incidence. Some building
materials reflect very little power.
Power that is not reflected is refracted and transmitted into the adjoining room or outside the building. To return to the receiver, the ray will experience at least one more reflection
and refraction before reentering the room.
Allen simplified the urban canyon model by ignoring contributions from refracted paths
and attenuating the reflected path's amplitude with fixed horizontal and vertical reflection coefficients that were independent of the angle of incidence [Allen,1991].
DIFFUSE REFLECTION
The reflection will be diffuse if the surfaces are not smooth. Diffuseness is accounted for
by introducing a reflection coefficient "reduction factor" that is dependent upon the angle of
incidence, rms height of the surface irregularities, and wavelength [Boithias,1987]. Rayleigh'sr#0*$$criterion is used to determine if a reduction factor is necessary [Brown,1973]. Using Rayleigh's
criterion, it can be shown that indoor wall, ceiling, or floor surfaces are not likely to support
diffuse reflection at low microwave frequencies (i.e., 1.5 GHz).
LOST RAYS DUE TO ROOM OPENINGS
If door and window openings are added to the model, a possibility of "lost rays" exists.
Lost rays leave a room through an opening and never return. When modeling the urban canyon
with geometric optics, Allen found that lost rays caused by cross streets degraded model performance. This degradation was eliminated by assuming the rays were not lost but attenuated.
Allen reasoned that urban canyons are dominated by long and narrow streets. This geometry
caused rays to have angles of incidence near 90 degrees. With a rough building exterior, the
lost ray's first Fresnel zone is an ellipsoidal area. The cross street subtracted from the Fresnel
zone's surface area and thus attenuated the ray rather than "lose" it. This reasoning may not
apply to the indoor channel since wall, ceiling, and floor surfaces are smooth and angles of
incidence are not always near 90 degrees. Thus, for the indoor channel, lost rays may exist.
DIFFRACTION
The prediction of diffraction effects due to the addition of doors and room partitions
into the model requires the use of physical optics or GTD.
The physical optics solution to "knife edge" diffraction is given by Jordon [Jordon, 1968],
for example. Jordon describes the ratio of the magnitude of the obstructed path's electric field
to the magnitude of the unobstructed path's electric field in terms of the Fresnel integral. The
value of the Fresnel integral is determined by a parameter that is a function of wavelength and
the geometry of the obstructed path. Diffraction attenuation decreases with increasing wavelength.
"0*$$GTD postulates "diffracted rays" result when rays strike edges, tips, and corners. The
initial conditions of the ray are determined by diffraction coefficients. GTD diffraction coefficients have been determined for several "canonical" objects such as an edge or corner
[Keller,1962]. After diffraction, the diffracted ray obeys laws of geometrical optics such as
reflection and refraction.
SCATTERING AND RADAR CROSS SECTION
When objects such as desks, file cabinets, shelves, and machinery clutter the room, the
diffraction problem becomes considerably more complex. Radar cross section principles are
useful for describing the behavior of rays incident on complex objects. The bistatic radar equation can predict the received ray magnitude given the object's radar cross section. Ray phase
information is lost with this method.
POLARIZATION EFFECTS IN THE INDOOR CHANNEL
The indoor channel model may need to account for polarization effects. Diffraction and
oblique reflections depolarize the rays. Depolarization from oblique reflections can be
predicted by using accurate horizontal and vertical reflection coefficients which are dependent
on the angle of incidence. Crosspolarization is the ratio of magnitude of the electric field in the
copolarized wave to the magnitude of the electric field in the crosspolarized wave. Cox
[Cox,1986] has reported median crosspolarization factors at 800 MHz of 2.5 dB in residential
houses to 1 dB in large buildings.
NARROWBAND COVERAGE PREDICTION
The multipath rays are assumed to be cw signals when predicting narrowband signal
coverage. The instantaneous amplitude and phase of each multipath ray is vectorially added at
each location. The power in each ray is assumed to attenuate with the square of the distance.
Losses from the antenna pattern as a function of ray departure angle and arrival angle are inr#0*$$Ԯcluded. Frequency selective fading statistics can be constructed from an ensemble of closely
spaced ray amplitudes.
SIMULATION OF THE COMPLEX IMPULSE RESPONSE
Because narrowband coverage predictions assume a cw signal, the time delay due to path
length is not used. Frequency selective predictions of the complex channel impulse response are
possible using the method outlined in narrowband coverage predictions if path time delay is
taken into account. With path time delay, the impulse response can be built by summing amplitudes and phases at each time delay for each receiver location. The fading statistics for multipath components at each time delay can be determined from the ensemble of multipath
components at that time delay in nearby impulse responses.
II.THE GWSSUS MODEL
The Gaussian Widesense Stationary Uncorrelated Scattering (GWSSUS) model was
developed by P.A. Bello [Bello, 1963] who was studying multipath in the ionospheric and troposcatter channels. The model describes the channel as a function of channel dynamics and frequency selectivity. The channel dynamics can be shown to be in units of time or Doppler
frequency. The frequency selectivity can be depicted in units of time delay or spaced frequency.
Although all channels can be described as a function of both channel dynamics and frequency selectivity, most channels can be simplified by elimination of one of the variables. If the
channel is narrowband, it is subject to power fading caused by channel dynamics and weakly
affected by intersymbol interference (ISI) caused by frequency selectivity. Intuitively this makes
sense since as the symbol bandwidth decreases the symbol time increases. The increase in symbol period decreases the possibility of ISI, but increases the chance that the channel will change
within the longer symbol period. Bello called this special case the "frequency flat" or the "time
varying frequency nonselective" model. The impulse response for a narrowband channel is a
single, complex value that is a function of time only.r#0*$$ԌOn the other hand, if the channel is wideband, it is likely to have ISI caused by
frequency selectivity, but unlikely to be affected by power fading caused by channel dynamics.
This also is intuitive since as the symbol bandwidth increases, symbol period decreases. The
decrease in symbol period increases the chance of ISI, but decreases the possibility of a change
in the channel during the short symbol period. Bello called this special case the "time flat" or
"time invariant frequency selective" model. The wideband impulse response is a complex function that is dependent on time and time delay.
THREE ASSUMPTIONS OF THE GWSSUS
The three assumptions of the GWSSUS can be inferred from its acronym. The first
assumption is that the inphase and quadrature components of the complex impulse response at
any time delay are defined by independent, identically distributed, zero mean Gaussian random
variables. This assumption rules out the presence of specular components, i.e., those having a
preferred phase at any time delay in the impulse response.
Any change that generates new inphase and quadrature components of the complex
impulse response can drive the process defined by these random variables. If the channel is
dynamic, the process is randomized by time. If the channel is static, the process can be randomized by spatial displacement.
The widesense stationarity (WSS) of the process assures that the complex impulse
response's expectation is a constant that is independent of starting time, and its autocorrelation
is a constant dependent only on time difference. For static channels, starting time and time
difference can be replaced by starting location and distance difference. Spatial WSS is assumed
to apply only over small areas (less than 5 wavelengths) for the indoor channel
[Devasirvatham,1987].
The assumption of uncorrelated scattering assures that the autocorrelation of the channel's impulse response is zero for all delta time delays but zero. The assumption of uncorrelated scattering is experimentally validated by observing the Doppler spectrum as a function ofr#0*$$time delay. If the Doppler spectrum is different at each time delay, it can be assumed that
different (independent) objects were illuminated by the measurement system transmitter.
GWSSUS IMPULSE RESPONSE
The GWSSUS impulse response has a discrete and continuous component. The discrete
component is an FIR filter with a finite number of taps. The location of the filter taps represent the delay times of the resolved paths. The amplitudes and phases of the taps represent
each resolved path's amplitude and phase. The continuous component of the impulse response
represents a continuum of uncorrelated multipath components. Mathematically it is represented
by a positive function of time multiplied by white noise.
GWSSUS POWER SPECTRAL DENSITY
The wideband indoor channel is frequently modeled as a time invariant frequency selective channel. Because the impulse response is approximated as a WSS process, it follows that
its transfer function can also be approximated as WSS. The WienerKhintchine theorem states
that the power spectral density of a WSS process can be computed by taking the Fourier transform of the process's autocorrelation function. The resulting function is always real. The real
power spectral density of the wideband channel is called the power delay profile (PDP).
Bello stated that the PDP is equivalent to the averaged impulse response magnitude
squared and can be easily measured using a pulsed transmitter and a square law detector
receiver. For dynamic channels, time averaging can be used. For static channels, spatial averaging can be used. In either case, the averaging is done over a time or space for which the transfer function is WSS.
SIMULATION OF THE COMPLEX IMPULSE RESPONSE
Using the assumptions given above, it can be shown that the PDP represents twice the
variance of the zero mean Gaussian random variable, which defines the complex impulser#0*$$response at that time delay. This fact is frequently used to simulate the complex impulse
response from PDP measurements. To simulate the value of the complex impulse response at
each time delay, a mean of zero and a variance equal to half the PDP value at that time delay is
entered into a Gaussian random number generator twice. One number represents the inphase
component, the other represents the quadrature component. The impulse response is
completely simulated when this has been done for a finite number of time delays [Chuang,1987].
PREDICTION OF BER WITHOUT THE COMPLEX IMPULSE RESPONSE
Bello's bit error rate (BER) estimates are dependent upon the PDPs. Studies have
shown [Winters, 1985] that the exact shape of the PDP (or complex impulse response) is not as
important in BER estimates as the PDP's rms delay spread (standard deviation of the PDP)
provided the ratio of the rms delay spread to the symbol period is less than 0.1. Chuang
[Chuang,1986] has shown how closed form BER expressions for Gaussian noise can be used for
predicting multipath BER by replacing the signaltonoise power ratio with signal to intersymbol
interference power ratio.
III.PDP STATISTICAL MODEL
Statistical models capable of simulating power delay profile (PDP) measurements have
been used to characterize the wideband indoor channel. These models describe the behavior of
discrete multipath components within a PDP. Rappaport [Rappaport,1991], Ganesh
[Ganesh,1989], and Saleh [Saleh,1987] have published PDP statistical models.
Like the GWSSUS model, PDP statistical models assume the indoor channel's complex
impulse response can be modeled as a linear FIR filter with a finite number of taps. PDP measurements provide a bandlimited estimate of the strength and delay of these taps by convolving
a pulse with the channel's impulse response. Phase information is lost because the measurements are square law detected."0*$$The behavior of the multipath component is derived from PDP measurements made at
many different locations. Time or spatial averaging is not always performed on the PDP measurements before fitting to the model.
POWER DELAY PROFILE STATISTICAL PARAMETERS
Rappaport's statistical model is capable of predicting the dynamic "shape" of the PDP as
the transceiver is moved in a room. The model is built from a large ensemble of PDP profile
measurements performed at widely dispersed locations. At each location, 19 PDP measurements were made 1/4 wavelength apart. The closely spaced measurements are used to predict
changes in the PDP due to small changes in position. The model divides the statistical
parameters between intraPDP parameters and interPDP parameters. The behavior of a multipath component is conditioned on that of nearby multipath components in the same PDP and
on multipath components at the same time delay in adjacent PDPs.
IntraPDP parameters are (1) the number of multipath components, (2) the probability
of multipath component arrival at any time delay, and (3) the power of the multipath component. Relationships between multipath components within the PDP are (1) the correlation with
respect to a difference in time delay of multipath component arrival and (2) the correlation with
respect to a difference of time delay of multipath component power.
InterPDP parameters include (1) fading statistics for multipath components over a large
area, (2) fading statistics for multipath components over a small area, and (3) the correlation
with respect to distance of multipath component powers at the same delay.
Ganesh used a modified Poison distribution to describe multipath component arrival
time. The modified Poisson distribution allows the multipath component arrival rate to increase
if the previous time delay bin had a multipath component present. Multipath component amplitudes at each delay were found to fit lognormal distributions. Average multipath component
powers at each delay showed an exponential decrease with delay time. The exponential" 0*$$decrease established a correlation between closely spaced multipath components. These relationships were verified for LOS and OLOS paths.
In Saleh's model, ray clusters determine the PDP shape. The clusters are attributed to
the "direct" path, reflections from internal walls (if constructed with reflective materials), and
reflections from external walls. The clusters arrive at a Poisson distribution rate. The average
power within each cluster decays exponentially with increasing time delay. Within a cluster, the
rays also arrive at a Poisson distribution rate and average ray power decreases exponentially
with time delay. Poisson rates and exponential time constants of clusters and rays are independent. The ensemble of normalized multipath component amplitudes for all time delays were fit
to a Rayleigh distribution.
SIMULATION OF COMPLEX IMPULSE RESPONSE FROM THE SIMULATED POWER
DELAY PROFILE
Complex impulse responses can be generated from any of the simulated PDPs using
techniques described in the GWSSUS sectionprovided measurements without specular components were used to build the statistical model. Saleh pointed out that he did not use LOS measurements in building his model so that the GWSSUS method of generating the complex
impulse response could be used.
Rappaport's method of generating complex impulse responses from PDPs differs
markedly from the GWSSUS method because it deterministically predicts phase from an arbitrary channel geometry as the receiver is moved. Rappaport proposes that this is a reasonable
method if the data used to build the PDP have the time delay resolution necessary to isolate
specular multipath components with preferred (instead of random) phases. Measurement data
have confirmed that the model is built from specular multipath components by showing that
multipath component fading over small distances has significantly smaller standard deviations
compared to multipath component fading over large distances.
"
0*$$At the start of a simulation, using Rappaport's model, a PDP is generated from the statistical model. As the transceiver is moved through the room, successive PDPs are generated
following the rules of multipath component behavior over small distances.
To generate the complex impulse response from wideband PDPs, phase from a uniform
distribution is assigned to each multipath component when it first appears in a PDP. In subsequent PDPs, the phase of the multipath component is determined from an arbitrary channel
geometry. This geometry is constructed by assigning each path a single reflection point and a
length that accounts for the time delay. The change of phase of the multipath component is
determined from the change in the length of this reflected path as the transceiver is moved.
Tests have shown that this method generates accurate narrowband fading statistics [Seidel,1991].
IV.CONCLUSIONS
Geometrical, stochastic GWSSUS, and statistical PDP modeling methods have been discussed. It was shown how all three modeling methods can be used to predict the complex
impulse response. The geometric model can simulate complex impulse responses for LOS and
OLOS links from architectural drawings. This eliminates the need for laborious channel measurements. Accurate predictions of mobile transceiver performance are possible because the
geometric model's complex impulse responses are dependent upon location in a room. Performance predictions for different antenna locations and antenna patterns can be quickly executed
on a computer. Transceiver designs that mitigate ISI with direction diversity are easily
evaluated.
Predictions from geometric models are criticized because of their inability to account for
rays that leave the building, reflect, and reenter via room openings such as windows and doors.
The predictions are also questioned because estimated values of reflection coefficients of common building materials are used.
Bello cautioned that the stochastic GWSSUS model not be used for channels with specular components in the received signal. For this reason, most indoor channel studies that rely onr#0*$$the GWSSUS model have not used LOS measurements. This may not be a disadvantage since
worst case estimates would not assume a specular component. The well behaved statistical
functions used in the GWSSUS model make some computations easier. For example, Bello was
able to derive expressions for BER analytically for various modulation methods using only the
channel's PDP.
The GWSSUS model is incapable of linking changes in the complex impulse response to
small spatial displacements. Instead the GWSSUS model can be used to differentiate performance between two different rooms or two different floors in a building.
The statistical PDP model can be applied to LOS and OLOS links. If the simulated
PDP's are assumed to be free of specular components, they can be averaged and used as variances to a GWSSUS complex impulse response generator. Rappaport's method of deriving
phase from simulated PDPs allows simulation of complex impulse responses to be linked to
small spatial displacements. This method does not require the absence of specular components.
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