Measures of the Spectrum Utilization
R.H.M. Hafez and G.K. Chan
*
Carleton University, Ottawa, Canada
Industry Canada, Ottawa, Canada
**
Abstract
In this paper, we summarize the known definitions of
the spectrum utilization and discuss their applicability
to existing radio services and their practical usefulness.
An important parameters that limits the full utilization
of the available radio spectrum is the non-uniformity of
the user density. We illustrates that when users are
clustered in small areas, the available spectrum
resources in that area actually shrink. This leads to
viewing the spectrum availability as a random process
that changes with time and space. The randomness of
spectrum availability is rooted in the randomness in the
topology and demographic of the service area.
I. Introduction
It is widely recognized that the radio spectrum is
defined along three dimensions: space, frequency and
time [3]. The term "spectrum utilization" refers to the
amount of information (measured in bits) that is being
carried by a spectrum unit (measured in m2. Hz. Sec.).
Therefore, an appropriate "theoretical" measure for the
spectrum utilization is the average bits/m2/Hz/Sec. The
maximum achievable rate of information per unit
spectrum depends on many factors ranging from the
physical propagation conditions to the state of
technology and system design.
In general, the measures of spectrum utilization fall
into two categories: (1) absolute measures and (2)
relative measures. Absolute measures help spectrum
managers and government administrators in assessing
the overall level of spectrum utilization across a wide
rage of radio services. Such measures tend to be either
over simplified or very complex. Relative measures, on
the other hand, are apply to specific services with
known network parameters. They are often expressed
in terms of system capacity and are used to guide the
system designers in achieving a higher level of
spectrum utilization efficiency.
In this paper, we summarize the known definitions of
the spectrum utilization and discuss their applicability
to existing radio services and their practical usefulness.
We also discuss the tools required to monitor and
evaluate the spectrum utilization. In most of the
discussions presented in this paper, we view the
spectrum as a national resource that should be used
efficiently and shared fairly among all citizens and
organizations. This is the typical view of government
administrators trying to access the commercial value of
the radio spectrum and allocate it in a way that ensure
the highest level of utilization.
The rest of the paper is organized as follow: Section 2
contains some of the known definitions of the spectrum
utilization. In section 3, we examine the practicality of
proposed definitions by applying them to some known
radio services. In section 4, we view the spectrum
assignment as a random process and shows the
stochastic nature of the spectrum availability. The
paper is concluded in section 5.
2.
Definitions
The radio spectrum is available in "Hz. m2. Sec.".
These spectrum units can be viewed as containers that
are filled with useful signals and interference.
Therefore, RF signal emitted from a transmitter
partially fills many spectrum units with RF energy.
This energy can be either "useful signal" or "undesired
interference" depending on the presence or absence of
an intended receiver. When an intended receiver is
present at the location of the spectrum unit, the RF
energy received from the desired transmitter is viewed
as "useful signal" and all other RF energies of other
transmitters are considered " interference". The signalto-interference ratio encountered at each of the said
spectrum unit can be translated into information rate.
The utilization level of a spectrum unit is measured by
the amount of information it carries.
A) Absolute Measure
One generally accepted definition of the spectrum
utilization is based on "The average amount of
information per spectrum unit" as follow [1]:
SUE 
M
(1)
W  A T
where: SUE = Spectrum Utilization Efficiency. M =
amount of information transferred (in bits). W =
1
frequency bandwidth. A = geometric space (usually
area). T = time. The product: "B.S.T" denotes the total
spectrum space under consideration. The definition
stated in (1) is in absolute form. In practice, there
could be many problems in interpreting the meaning of
the absolute SUE as explained in section 2.
B) Relative Measure
The "relative" approach is to compare the utilization of
a given system with another system as follow [1]
SUE a
(2)
RSE 
SUE s
where: RSE = Relative Spectral utilization, which is
the ratio between the SUE of the actual system SUEa
and that of a reference or standard system SUEs.
A candidate standard system that was mentioned in [1]
is an idealized system with a capacity form similar to
that of an additive white Gaussian channel.
Co  Fo  log e (1   o )
(3)
where Fo is the bandwidth of the wanted
communications and o is the signal to noise (plus
interference) ratio at the receiver output.
3.
Spectrum Use by Two Radio Services
To illustrate the applicability of the two definitions
given above, we will apply them to three different
types of radio services.
A) Specialized Mobile Radio (SMR)
SMR is one of the oldest types of mobile
communications. SMR licenses are granted to small
businesses individually and give them the right to use
one or few frequencies on an exclusive or shared basis.
In a given band, the primary frequencies (channels)
have fixed bandwidth and fixed frequency spacing.
Some frequencies could be assigned at half the
frequency spacing between two primary carriers.
Those frequencies are called interstitial channels.
SMR systems are used mainly for voice dispatch
applications and operate in an autonomous mode
(except for shared licenses).
N
defined as : Occ   C k . N is the number of
k 1
(2)
voice channels in the area and Ck is the fraction
of time the voice channel # k is active
The denominator (W.A) is interpreted as the total
spectrum units Hz. m2/s in the area of interest.
The spectrum occupancy varies across the area of
interest. In [2], the spectrum occupancy was estimated
as follow:
- Divide the total area of interest into M small subareas (cells),
- Form a matrix of sub-areas and voice channels,
- Estimate the presence of each of the voice channels
in each of the cells
- Estimate the occupancy of each voice channel by
accumulating its presence in all cells
- Substitute the results in the occupancy definition
given above.
The maximum number of assignments, N, depends on
the space distribution of the individual radio systems
and the frequency assignment strategy. Figure 1
graphically illustrates a typical interference model
applicable to SMR. Within the area of interest, each
frequency is assigned once. Frequency reuse within a
city is very rare. The single cell system creates three
circles of interference limits. The large circle is the cochannel interference limit, a smaller circle represents
the interference limit for interstitial channel and the
Fn
Fn+1
Adjacent
channel
interferenc
e
Fn
Co-channel
interference
(interstitial)
Fn+1/2
Co-channel
interference
(primary)
Coordination area
In order to apply the SUE model of Eq. (1) to the SMR
service, one must interpret the information rate and
spectrum units in a way compatible with the SMR
system model. This can be done as follows:
M
M /T
Occ
(4)
SUE 


W  AT W  A W  A
(1) The information rate (M/T) is converted into
spectrum occupancy, Occ, which is further
smallest circle sets the adjacent interference limit.
Figure 1- A model for SMR interference areas
B) PCS Cellular Systems
PCS follow a different spectrum utilization model. In
PCS, a band of frequencies is assigned to an operator.
2
The government responsibility is to coordinate the
spectrum usage among different services sharing the
band or operating in adjacent bands. The service
provider, on the other hand, refers to the spectrum
utilization efficiency as the system capacity and is
measured in terms of how many customers (users) can
simultaneously use the system. In a voice-only system,
the SUE is interpreted as:
N
(5)
Cellular _ SUE 
WA
Where N is the number of voice connections, W is the
total bandwidth and A is the total area. Notice that
according to this definition, the control and signaling in
the network lower the utilization, and increasing the
complexity of the infrastructure (i.e. more cells and/or
sectors) increases the utilization. Notice also that the
area topology affects the utilization. For example,
indoor cellular systems have much higher utilization
than outdoor systems.
4.
The Effect of User Density
The demand for radio services is never uniform in
space. Radio users tend to be clustered in given areas
that are usually referred to as “hot spots”. The
operating transmitters within a “hot spot” create high
interference field that eliminates other users from using
the assigned frequencies as well as adjacent
frequencies. The eliminated channels are referred to as
the “denied spectrum” which is larger than the
spectrum being utilized. In cellular systems, the
channels are assigned in space in an interleaved way
that ensure low co-channel and adjacent channel
interference. Unfortunately, the spectrum resources
that are available outside the “hot spots” have less
practical value simply because there is little demand
for it. This means that while spectrum resources
continue to be available in low demand areas, the band,
cell or system has already been saturated.
Therefore, the available spectrum is a random variable
that varies from one location to location. The SUE
definition given in (1) does not account for this fact. In
(1), we assume that the total available spectrum
(W.A.T) is constant, and we compute the utilization
efficiency as a fraction of this “constant” amount of
resources. A more accurate definition is:
Actual _ SUE 
M /T
W ( x, y) dxdy
(6)
The double integration in the demodulator replaces the
constant frequency-space resources. In the expression
the available bandwidth is explicitly expressed as a
function of location as it should be. In voice-only
systems, M/T can be replaced by the number of voice
connections, N, and in a heterogeneous network (voice,
data and other forms of information), M/T can be
replaced by the aggregate throughput data rate. The
denominator in (5) less than or equal the nominal
spectrum [W.A] and the actual spectrum utilization can
be written as:
Actual _ SUE 
M /T
g W  A
 SUE (7)
where g is a factor ( 1) dependent on the variance of
user density in the area of interest. It is important to
note that g is not constant. It is a non-linear function of
the traffic load as well as the user distributions. To
illustrate the significance of this modification we
constructed a frequency assignment simulation for
areas of different user distribution densities. The
simulation details are described in the next section.
(A) Simulation Model
We consider the problem of assigning 50 channels
(consecutive frequencies) to different locations within
a square area of 100 km x 100 km. Within that area,
each frequency is assigned once (SMR model). The
area resolution is one sq. km. The adjacent channel
interference area is 12 km x 12 km centered at the
position of the transmitter. There are no interstitial
channel assignments and the frequency assignments are
made at random provided that they satisfy the adjacent
channel interference criterion.
The user distribution, f(x,y),was modeled as a
truncated, bi-variate Gaussian random variable with the
variance as a parameter:
f ( x, y) 
e ( x
2
 y2 )
; x  b ; y  b (8)
 2b 
4  L 2 
 
where 2b is the side length of a squared service, s is the
standard deviation of the model probability density
function and
2
x
L( x )   e t
2
/2
dt
0
The simulation is conducted as follow:
(1) A user location is generated at random according to
the density of (8). This will point to (x,y) location in
the service area.
(2) A frequency is selected at random from the set of
un-assigned frequencies [each frequency is assigned
once in the area]
(3) The selected frequency is checked for the ACI
condition. If it passes the test , it gets selected and
removed from the set of remaining frequencies;
otherwise, another frequency is selected. If all
3
The purpose of this simulation is to confirm the
hypothesis that the spectrum resources in an area with
skewed user distribution (small variance) is less than
the nominal value of the resources. In this model, the
assignable frequencies are less than the total available
of 50. If we simply view the utilization as the fraction
of assigned frequencies (relative to the total available),
we encounter saturation at very low level of utilization
when the user density function has a small variance.
Some simulation results are shown in Figures 2 and 3.
In Figure 2, the probability of rejecting a request for
assignment is plotted against the fraction of assigned
frequencies. The dashed line is an upper bound on this
probability and the solid line is a lower bound. The
bounds are obtained from [4]. The circle points are the
simulation results when the standard deviation is 7.1.
The simulation results confirm the view that the
available spectrum resource is a random variable that is
related to the variance of the user distribution. For
example, Figure 2 suggests that the spectrum is nearly
saturated when only about 50% of the frequencies are
assigned. This may vary somewhat with different
assignment strategies, but the fact remains that not all
frequencies can be assigned. In other words, at 50%
assignment level, the spectrum is fully utilized.
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Fraction of assigned frequencies
0.7
0.8
0.9
Figure 3- Probability of rejection s=14.3
The same argument can be extended to other factors.
For example, an area with high attenuation and dividers
can support higher level of frequency re-use and
therefore the spectrum can be utilized at higher level
than in flat areas.
1
0.9
Probability of rejection st. dev. = 7.1
(B) Discussion
Probability of rejection st. dev. = 14.3
candidate frequencies are tested and rejected, the
request for assignment at (x,y) is rejected.
(4) After n assignments are made, we stop an further
assignments and keep generating new requests to see if
any of them will be rejected. Using that technique we
can estimate the probability of rejection (saturation) as
a function of the spectrum utilization level.
0.8
0.7
0.6
0.5
5.
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Fraction of assigned frequencies
0.7
0.8
0.9
Conclusions
The ratio of the total information communicated to the
total available bandwidth is universally acceptable as a
measure for the spectrum utilization. The
communicated information can be expressed as bits/sec
of Erlang units for voice only networks.
Figure 2- Probability of rejection. =7.1
Figure 3 illustrates the case when the standard
deviation has a higher value of 14.3. In this case the
users are more widely spread out and as a results it was
less probable to reject a new request.
Simulation runs were also conducted for a much higher
standard deviation of 27.3. In that case, the users were
almost uniformly distributed throughout the area and as
a result, there were no measurable probability of
rejection and all 50 frequencies were successfully
assigned.
In This paper we illustrated briefly how can this simple
concept be applied to practical systems. We also argue
that the amount of spectrum resources available in any
given area is a random variable that depends on the
user distribution as well as the topological characters of
the area of interest. In some places it becomes
impossible to attain full utilization. But if we modify
the utilization definition to be the ratio between
information and the actual spectrum availability, we
would have a more accurate view of the spectrum
utilization level.
References
[1] Radio Communication Study Group, "Definition of
Spectrum Use and Efficiency of Radio Systems",
4
International Telecommunication Union (ITU).
Document 1/4-E February 1996.
[2] G.K. Chan and B.A. Brown, "A measure of
Spectrum Utilization in Land Mobile Bands",
Canadian Conference on Electrical and Computer
Engineering, 1996
[3] CCIR Report Rep.662-1 (Mod.F),. Dubrovnik,
1986.
[4] H.M. Hafez, S.Towaij and R.A. Goubran, "The
Effect of User's Geographical Distribution on the
Utilizayion of Radio Spectrum", IEEE Trans. on
Electromagnetic Compatibility, vol. 32, No.4,
November 1990
5