GHz-Wide Sensing and Decoding Using the Sparse Fourier Transform
Haitham Hassanieh Lixin Shi Omid Abari Ezzeldin Hamed Dina Katabi
Massachusetts Institute of Technology
{haitham, lixin, abari, ezz, dina}@csail.mit.edu
Abstract– We present BigBand, a technology that can capture
GHz of spectrum in realtime without sampling the signal at
GS/s –i.e., without high speed ADCs. Further, it is simple
and can be implemented on commodity low-power radios.
Our approach builds on recent advances in the area of sparse
Fourier transforms, which show that it is possible to reconstruct
a sparse signal without sampling it at the Nyquist rate. To
demonstrate our design, we implement it using 3 software
radios, each sampling the spectrum at 50 MS/s, producing
a device that captures 0.9 GHz — i.e., 6× larger digital
bandwidth than the three software radios combined. Finally,
an extension of BigBand can perform GHz spectrum sensing
even in scenarios where the spectrum is not sparse.
1 INTRODUCTION
The rising popularity of wireless communication and the
potential of a spectrum shortage have motivated the FCC
to take steps towards releasing multiple bands for dynamic
spectrum sharing [1]. The government’s interest in re-purposing
the spectrum for sharing is motivated by the fact that the actual
utilization of the spectrum is sparse in practice. For instance,
Fig. 1 from the Microsoft Spectrum Observatory [2] shows
that, even in urban areas, large swaths of the spectrum remain
underutilized. To use the spectrum more efficiently, last year,
the President’s Council of Advisors on Science and Technology
(PCAST) [3] has advocated dynamic sharing of much of the
currently under-utilized spectrum, creating GHz-wide spectrum
superhighways “that can be shared by many different types
of wireless services, just as vehicles share a superhighway by
moving from one lane to another.”
Motivated by this vision, this paper presents BigBand, a
technology that enables realtime GHz-wide spectrum sensing
and reception using low-power radios, similar to those in WiFi
devices. Making GHz-wide sensing (i.e. the ability to detect
occupancy) and reception (i.e. the ability to decode) available
on commodity radios enables new applications:
• In particular, r ealtime GHz sensing enables highly dynamic
spectrum access, where secondary users can detect short sub-
millisecond spectrum vacancies and leverage them, thereby
increasing the overall spectrum efficiency [4].
• Further, a cheap low-power GHz spectrum sensing tech-
nology enables the government and the industry to deploy
thousands or such sensors in a metropolitan area for large-
scale realtime spectrum monitoring. This will enable a
better understanding of spectrum utilization, identification
and localization of breaches of spectrum policy, and a more-
informed planning of spectrum allocation.
• Beyond sensing, the ability to decode signals in a GHz-
wide band enables a single radio to receive concurrent
transmissions from diverse parts of the spectrum. This would
0
20
40
60
80
100
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Occupancy %
Frequency (GHz)
Microsoft Observatory Seattle Monday 01/14/2013 10-11am
Fig. 1—Spectrum Occupancy: The figure shows the average spec-
trum occupancy at the Microsoft spectrum observatory in Seattle on
Monday January 14, 2013 during the hour between 10 am and 11 am.
The figure shows that between 1 GHz and 6 GHz, the spectrum is
sparsely occupied.
enable future cell phones to use one radio to concurrently
receive Bluetooth at 2.4 GHz, GSM at 1.9 GHz, and CDMA
at 1.7 GHz.
Realtime GHz signal acquisition, however, is challenging.
For example, existing methods for spectrum sensing, like those
used in the Microsoft spectrum observatory [2], do not work
in realtime. They rely on sequential hopping from one channel
to the next, acquiring only tens of MHz at a time [5], [6]. As
a result, each band is monitored only occasionally, making it
easy to miss short lived signals ( e.g., radar).
The key difficulty in capturing GHz of bandwidth in realtime
stems from the need for high-speed analog-to-digital converters
(ADCs), which are costly, power hungry, and have a low bit
resolution [7], [8]. Compare typical low-speed ADCs used in
WiFi or cellular phones with the very high speed ADCs needed
to capture GHz of bandwidth. A 100 MS/s ADC, like in Wi-Fi
receivers, costs a few dollars, consumes a few milli Watts, and
has a 12 to 16-bit resolution [8], [9], [10]. In contrast, a high
speed ADC that can take multiple giga-samples per second may
cost hundreds of dollars, consume multiple orders of magnitude
more power, and have only 6 to 8-bits resolution [7], [8], [9].
In this paper, we explore how one can achieve the best of
both worlds. Specifically, we would like to capture GHz of
spectrum but using only a few ADCs that each samples the
signal at tens of MS/s.
We introduce BigBand, a technology that can acquire GHz
of signal using a few (3 or 4) low-speed ADCs. BigBand can do
more than spectrum sensing – the action of detecting occupied
bands. It can also decode the signal (i.e., obtain the I and Q
components). To achieve its goal, BigBand builds on advances
in the area of sparse Fourier transform [11], [12], [13], which
permit signals whose frequency domain representation is sparse
to be recovered using only a small subset of their samples –
i.e., we can recover GHz of spectrum without sampling it at
the Nyquist rate.
Some past work has proposed using compressive sensing
978-1-4799-3360-0/14/$31.00
c
2014 IEEE
to acquire GHz signals at sub-Nyquist rate [14], [15], [16],
[17]. BigBand builds on this work but differs f rom it sub-
stantially. Approaches based on compressive sensing require
random sampling of the signal which cannot be done simply
by using standard low-speed ADCs. It needs analog mixing at
Nyquist rates [14], [16] and expensive processing to recover
the original signal. Such a design is quite complex and could
end up consuming as much power as an ADC that samples
at the Nyquist rate [18], [19]. Like the compressive-sensing
approaches, BigBand can acquire a wideband signal without
sampling it at the Nyquist rate. Unlike compressive sensing,
however, BigBand does not need analog mixing or random
sampling and can work using commodity radio and s tandard
low-speed ADCs. Further, it computes the Fourier transform
of a sparse signal faster than the FFT, reducing baseband
processing.
We have built a working prototype of BigBand using USRP
software radios. Our prototype uses three USRPs, each of
which can capture 50 MHz bandwidth to produce a device
that captures 0.9 GHz –i.e., 6× larger bandwidth than the
digital bandwidth of the three USRPs combined. We have
used our prototype to sense the spectrum between 2 GHz and
2.9 GHz, a 0.9-GHz stretch used by diverse technologies [2].
Our outdoor measurements reveal that, in our metropolitan
area,
1
the above band has an occupancy of 2–5%. These results
were verified using a spectrum analyzer are in sync with similar
measurements conducted at other locations [2]. We further use
our prototype to decode 30 transmitters that are simultaneously
frequency hopping in a 0.9 GHz band, hence demonstrating that
BigBand decodes the signals, not only senses their power.
Finally, we have extended BigBand to perform spectrum
sensing (not decoding) even when the spectrum utilization
is not sparse. To do so, we leverage the idea that even if
the spectrum itself i s densely occupied, only a small fraction
of the spectrum is likely to change its occupancy over short
intervals of a few milliseconds. We build on this basic idea to
sense densely occupied spectrum using sub-Nyquist sampling.
We also evaluate our design empirically showing that it can
detect frequency bands that change occupancy even when the
spectrum is 95% occupied.
2 RELATED WORK
This paper builds on recent theoretical advances in sparse
Fourier sampling [11], [12], [13], [20], [21]. In contrast to
past work however, this paper focuses on the practical problem
of realtime low-power GHz-wide spectrum acquisition, and
presents the first practical system that adapts the sparse FFT
algorithms to address this application. It also implements its
design and empirically evaluates it, demonstrating that it ad-
dresses ADC speed, wireless channels, and radio related issues.
The paper is also the first to adapt the sparse FFT for spectrum
sensing in scenarios with a dense spectrum occupancy.
Our work is related to signal acquisition via digital and
analog compressive sensing [14], [15], [16], [17], [22], [23],
[24]. However, compressive sensing needs random sampling
and analog mixing at Nyquist rates [14], [16], [24]. These
approaches cannot be built using commodity radios and ADCs
with regular sampling; they require a custom design and could
1. MIT campus, Cambridge MA, USA.
end up consuming as much power as an ADC that samples at
the Nyquist rate [18], [19].
Our work is also related to theoretical work in signal
processing on co-prime sampling [25], [26], [27]. In [25],
[26], co-prime sampling patterns are utilized to sample sparse
spectrum. These methods however require k ADCs with co-
prime sampling patterns, where k is the number of occupied
frequencies. In contrast, the analysis of our sparse FFT algo-
rithm shows that our approach requires only a constant small
number of ADCs [13]. Our approach is further implemented
and shown to work in practice. In [27], co-prime sampling is
used to sample linear antenna arrays [27]. This work however
assumes the presence of a second dimension where signals can
be fully sampled and cross-correlated and hence cannot be used
for spectrum acquisition.
Also relevant to our work is the theoretical work on using
multicoset sampling to capture the signals in a wideband sparse
spectrum with a small number of low speed ADCs [28],
[29]. However, in order to recover the original signals f rom
the samples, these techniques require prior knowledge of the
locations of occupied frequencies in the spectrum and hence
are not useful for spectrum sensing. In contrast, our approach
recovers both the locations of the occupied frequencies and
the signals in these fr equencies and thus can be used for both
spectrum sensing and decoding.
Some proposals for test equipment reconstruct wideband
periodic signals by undersampling [30], [31]. These approaches
however assume that the signal is periodic –i.e., the same signal
keeps repeating for very long time – which allows them t o take
one sample during each period until all samples are recovered
and rearranged in the proper order. Though this requires one
low speed ADC, it is only applicable to test equipment where
the same signal is repeatedly transmitted [30].
Finally, there is also significant literature about spectrum
sensing. Most of this work focuses on narrowband sensing [32],
[33], [34]. It includes techniques for detecting the signal’s
energy [33], its waveform [32], its cyclostationarity [35], or
its power variation [34]. In contrast, we focus on wideband
spectrum sensing, an area that is significantly less explored.
A recent system called QuickSense [36] senses a wideband
signal using a hierarchy of analog filters and energy detectors.
BigBand differs f rom QuickSense in that it can recover the
signal (obtain the I and Q components) as opposed to only
detecting spectrum occupancy. Second, for highly utilized
spectrum (i.e. not sparse), the approach in [36] reduces to se-
quentially scanning the spectrum whereas BigBand’s extension
for the non-sparse case provides a fast sensing mechanism.
3 BIGBAND
BigBand is a receiver that can recover a sparse signal
with sub-Nyquist sampling using low-power commodity radios.
Thus, BigBand can do more than spectrum sensing – the action
of detecting occupied bands. BigBand provides the details of
the signals in those bands (I’s and Q’s of wireless symbols),
which enables decoding those signals.
BigBand adapts the sparse FFT algorithm for spectrum
acquisition using low speed ADCs. We use x and
b
x to denote
a time signal and its Fourier transform respectively. We also
use the t erms: the value of a frequency and its position in
the spectrum to distinguish
b
x
f
and f . BigBand discovers the
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01
23
Alias
Frequencies
11
Buckets
0
Fig. 2—Bucketization using aliasing filter: Sub-sampling a signal by
3× in the time domain, results in the spectrum aliasing. Specifically,
the 12 frequency will alias into 4 buckets. Frequencies that are equally
spaced by 4 (shown with the same color) end up in the same bucket.
occupied frequency positions f and estimates their values
b
x
f
.
Once
b
x is computed, it can recover the time signal x and decode
the wireless symbols. BigBand’s design has three components:
frequency bucketization, estimation, and collision resolution.
Below we explain these components. (Further details of Big-
Band can be found in our technical report [37] and the complete
analysis of the algorithm and proofs of correctness can be found
in [13].)
3.1 STEP 1: Frequency Bucketization
BigBand starts by hashing the frequencies in the spectrum
into buckets. Since the spectrum is sparsely occupied, many
buckets will be empty and can be simply discarded. BigBand
then focuses on the non-empty buckets, and computes the
values of the frequencies in those buckets in what we call the
estimation step.
So how can we hash frequencies into buckets? Recall the
following basic property of the Fourier transform: sub-sampling
in the time domain causes aliasing in the frequency domain.
Formally, let x be a time signal of bandwidth BW, and
b
x its
frequency representation. Let b be a sub-sampled version of x,
i.e., b
i
= x
i·p
where p is the sub-sampling factor. Then,
b
b, the
FFT of b is an aliased version of
b
x, i.e.:
b
b
i
=
p−1
X
m=0
b
x
i+m(BW/p)
. (1)
Thus, an aliasing filter is a form of bucketization in which
frequencies equally spaced by an interval BW/p hash to
the same bucket, i.e., frequency f hash to bucket i = f
mod BW/p, as shown in Fig. 2. Further, the value in each
bucket is the sum of the values of only the frequencies that hash
to the bucket as shown in Eq. 1. Most importantly, aliasing is
naturally done by sampling the signal using a low-speed ADC
slower than the Nyquist rate.
Now that we hashed the frequencies into buckets, we can
leverage the fact that the spectrum of interest is sparse and
hence most buckets have noise and no signal. BigBand com-
pares the energy (i.e., the magnitude square) of a bucket with
the receiver’s noise level and considers all buckets whose
energy is below a threshold to be empty. It then focuses on
the occupied buckets and ignores empty buckets.
3.2 STEP 2: Frequency Estimation
Next, for each of the occupied buckets we want to identify
which frequencies created the energy in these buckets, and what
are the values of these frequencies. If we can do that, we then
have r ecovered a complete representation of the frequencies
with non-zero signal values, i.e., we acquired the full signal in
the Fourier domain.
Recall that our spectrum is sparse; thus, as mentioned earlier,
when hashing frequencies into buckets many buckets are likely
to be empty. Even for the occupied buckets, the sparsity of the
spectrum means that many of these buckets will likely have a
single non-zero frequency hashing into them, and only a small
number will have a collision of multiple non-zero (or occupied)
frequencies. In the next section, we present a mechanism
to detect whether a bucket has a collision and resolve such
collisions. In this section, we focus on buckets wit h a single
non-zero fr equency and estimate the value and the position of
this non-zero frequency, i.e.,
b
x
f
and the corresponding f .
In the absence of a collision, the value of the occupied
frequency is the value of the bucket. Said differently, the value
of a bucket after aliasing,
b
b
i
is a good estimate of the value
of the occupied frequency
b
x
f
in that bucket, since all other
frequencies in the bucket have zero signal value (only noise).
Although we can easily find the value of the non-zero
frequency in a bucket, we still do not know its frequency
position f , since aliasing mapped multiple frequencies to the
same bucket. To compute f , we leverage the phase-rotation
property of the Fourier transform, which states that a shift in
time domain translates into phase rotation in the frequency do-
main [38]. Specifically, we perform the process of bucketization
again, after shifting the input signal by τ. Since a shift in time
translates into phase rotation in the frequency domain, the value
of the bucket of changes from
b
b
i
=
b
x
f
to
b
b
(τ)
i
=
b
x
f
· e
2π j·f ·τ
.
Hence, using the change in the phase of the bucket, we can
estimate our frequency of interest and we can do this for all
buckets that do not have collisions.
Two points are worth noting:
• First, recall that the phase wraps around every 2π. Hence,
the value of τ has to be small to avoid the phase wrapping
around for large values of f . In particular, τ should be on
the order of 1/BW where BW is the bandwidth of interest.
For example, to acquire one GHz of spectrum, τ should be
on the order of a nanosecond.
2
• Second, to sample the signal with a τ shift, we need a second
low-speed ADC that has the same sampling rate as the ADC
in the bucketization step but whose samples are delayed by
τ. This can be achieved by connecting a single antenna to
two ADCs using different delay l ines (which is what we do
in our implementation). Alternatively, one can use different
delay lines to connect the clocks to the two ADCs.
3.3 STEP 3: Collision Detection and Resolution
We still need to address two questions: how do we distin-
guish the buckets that have a single non-zero frequency from
those that have a collision? and in the case of a collision, how
do we resolve the colliding frequencies?
3.3.1 Collision Detection
Again we use the phase rotation property of the Fourier
transform to determine if a collision has occurred. Specifically,
if the bucket contains a single non-zero frequency, i.e., no
collision, then performing the bucketization with a time shift
τ causes only a phase rotation of the value in the bucket but
2. In fact, one can prove a looser version of this constraint where large τ
are fine. Formally, for τ larger than 1/BW, the FFT window size must be a
non-integer multiple of τ .
the magnitude of the bucket does not change –i.e., with or
without the time shift, k
b
b
i
k = k
b
b
(τ)
i
k = k
b
x
f
k. In contrast,
consider the case where there is a collision between, say, two
frequencies f and f
′
. Then the value of the bucket without a
time-shift is
b
b
i
=
b
x
f
+
b
x
f
′
while its value with a time-shift
of τ is
b
b
(τ)
i
=
b
x
f
· e
2π j·f τ
+
b
x
f
′
· e
2π j·f
′
τ
. Since the colliding
frequencies rotate by different phases, the overall magnitude
of the bucket will change. Thus, we can determine whether
there is a collision or not by comparing the magnitudes of the
buckets with and without the time-shift.
3
3.3.2 Collision Resolution
To reconstruct the full spectrum, we need to resolve the
collisions –i.e., for each non-zero frequency in a collision
we need to estimate its value
b
x
f
and position f . We present
two approaches for resolving collisions which may also be
combined in case the spectrum is less sparse.
A. Resolving Collisions with Co-prime Aliasing Filters
One approach to resolve collisions is to bucketize the spec-
trum multiple times using aliasing filters with co-prime sam-
pling rates. Co-prime aliasing filters guarantee (by the Chinese
remainder theorem) that any two frequencies that collide in
one bucketization will not collide in the other bucketizations.
To better understand this point, consider the example in Fig. 3.
The first time we bucketize, we use an aliasing filter that sub-
samples the time signal by a factor of 3. In this case, the two
frequencies labeled in red and blue collide in a bucket whereas
the frequency labeled in green does not collide, as shown in the
figure. The second time we bucketize, we use an aliasing filter
that sub-samples by 4. This time the blue and green frequencies
collide whereas the red frequency does not collide. Now we can
resolve collisions by iterating between the two bucketizations.
For example, we can estimate the green frequency from the first
bucketization, where it does not collide. We subtract the green
frequency from the colliding bucket in the second bucketization
to obtain the blue frequency. We then go back to the first
bucketization and subtract the blue frequency from the bucket
where it collides to obtain the red frequency.
Thus, by using co-prime aliasing filters to bucketize and
iterating between the bucketizations –i.e., estimating frequen-
cies from buckets where they do not collide and subtracting
them from buckets where they do collide– we can recover the
spectrum. This suggests that to capture a spectrum bandwidth
BW, we can use two ADCs that sample at rates BW/p
1
and BW/p
2
where p
1
and p
2
are co-prime. For example, to
recover a 1 GHz spectrum, we can use a 42 MHz ADC [?]
along with a 50 MHz ADC. The combination of these two
ADCs can capture a bandwidth of 1.05 GHz because 42 MHz
= 1.05 GHz/25 and 50 MHz = 1.05 GHz/21, where 21 and
25 are co-prime. Note that we also repeat each of these co-
prime bucketization with a time shift (as explained in §3.2,
which requires a total of 4 low-speed ADCs.
B. Resolving Collisions without Co-prime Aliasing Filters
Co-prime aliasing filters are an efficient way to resolve
collisions, but they are not necessary. Here, we show how to
3. Even if one occasionally falsely detects a collision when there is a single
frequency, BigBand can still correct this error. This is because the collision
resolution step described next will estimate the values of the presumed colliding
frequencies to zero.
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Frequencies
110
01
23
1
st
Bucketization
(Sub-sample by 3)
01
2
2
nd
Bucketization
(Sub-sample by 4)
Fig. 3—Resolving collisions with co-prime filters: Using 2 co-prime
aliasing filters, we ensure the frequencies that collide in one filter will
not collide in the second. For example, frequencies 5 and 9 collide in
the first filter. But frequency 5 dies not collide in the second which
allows us to estimate it and subtract it.
resolve collisions while still using ADCs that sample at the
same rate. This means that one can use one type of ADCs for
building the whole system.
4
We use one type of aliasing filter. However, we perform it
for more than twice using multiple different time shifts. To see
how this can help resolve collisions, consider again the case
where two frequencies f and f
′
collide in a bucket. If we use
two time shifts τ
1
and τ
2
, we get three values for each bucket.
For the bucket where f and f
′
collide, these values are:
b
b
i
=
b
x
f
+
b
x
f
′
b
b
(τ
1
)
i
=
b
x
f
· e
2π j·f τ
1
+
b
x
f
′
· e
2π j·f
′
τ
1
b
b
(τ
2
)
i
=
b
x
f
· e
2π j·f τ
2
+
b
x
f
′
· e
2π j·f
′
τ
2
(2)
If we know the positions of f and f
′
, the above becomes an
overdetermined system of equations where the only unknowns
are
b
x
f
,
b
x
′
f
. Since only few frequencies hash into each bucket,
there is a limited number of possible values of f and f
′
. For
each of these possibilities, the above over-determined system
can be solved to find
b
x
f
,
b
x
′
f
. Hence, we can solve overdeter-
mined system for the possible (f , f
′
) pairs and choose the pair
that minimizes the mean square error. While the above does
not guarantee that the solution is unique, in case multiple pairs
(f ,f
′
) satisfy the equations, BigBand can detect that event and
report to the user that the values of these f requencies remain
unresolved.
5
Our empirical results (in §7.3) show however that
for practical spectrum sparsity (which is about 5%) 3 shifted
bucketizations are enough to uniquely resolve the colliding
frequencies.
We note that though this method requires more digital
computation, we only need to do this for the few buckets that
have a collision, and we know the number of collisions is small
due to the sparsity of the spectrum.
We also note that this method can be combined with the co-
prime approach to deal with less sparse spectrum. In this case,
one uses this method to resolve collisions of two frequencies
while iterating between the co-prime filters.
4 CHANNEL ESTIMATION AND CALIBRATION
The earlier description of BigBand assumes that the different
ADCs can sample exactly the same signal at different time-
4. This makes it possible to build BigBand using only USRPs [39].
5. Note that theoretically, for a collision of k frequencies, 2k samples can
guarantee a uniqu e solution in the absence of noise.
-3
-2
-1
0
1
2
3
4
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4
Unwraped Phase
Frequency Range in GHz
∆
φ
1-2
∆
φ
1-3
Fig. 4—Phase rotation vs frequency: The figure shows that the phase
rotation between the 3 USRPs is linear across the 900 MHz frequency
spectrum and can be used to estimate the time shifts.
shifts. However, because the signals experience different chan-
nels, they will be scaled differently and the ADCs will not be
able to sample exactly the s ame signal.
To better understand this problem, let us consider the case
where we resolve collisions without the co-prime sub-sampling.
In this case, we will have 3 ADCs each sampling a signal that
is delayed by a time shift. In this case, consider a non-zero
frequency f whose value is
b
x
f
. If f hashes to bucket i and does
not collide, then the value of the bucket at each of the ADCs
can be written as:
b
b
i
= h
w
(f ) · h
1
(f ) ·
b
x
f
b
b
(τ
1
)
i
= h
w
(f ) · h
2
(f ) ·
b
x
f
· e
2π j·f τ
1
b
b
(τ
2
)
i
= h
w
(f ) · h
3
(f ) ·
b
x
f
· e
2π j·f τ
2
(3)
where h
w
(f ) is the channel on the wireless medium,
h
1
(f ), h
2
(f ), h
3
(f ) are the hardware channels on each of the
radios, and ·(f ) indicates that these parameters are frequency
dependent. We can ensure that h
w
(f ) is the same in all three
bucketizations by connecting the RF frontends to the same
antenna. As a result, h
w
(f ) cancels out once we take the ratios,
b
b
(τ
1
)
i
/
b
b
i
and
b
b
(τ
2
)
i
/
b
b
i
of the buckets. However, the hardware
channels are different for the different bucketizations. We need
to estimate them and compensate for them in order to perform
frequency estimation and also resolve the collisions.
Furthermore, though it is simple to create time-shifts be-
tween the three ADCs as explained in §3.2, we need to
know the values of these time-shifts τ
1
, τ
2
in order to perform
frequency estimation based on phase rotation. Hence, we also
need a way to estimate these time-shifts.
4.1 Estimating the Channels and Time-Shifts
To estimate the channels and the time shifts, we divide the
total bandwidth BW that BigBand captures into p consecutive
chunks. We then transmit a known signal in each chunk, one by
one. Since we only transmit in one chunk at a time, there are
no collisions at the receiver after aliasing. We then use Eq. 3 to
estimate the ratios h
2
(f )·e
2π j·f τ
1
/h
1
(f ) and h
3
(f )·e
2π j·f τ
2
/h
1
(f )
for each frequency f in the spectrum.
Now that we have the ratios, we need to compute h
2
(f )/h
1
(f )
for each frequency f , and the delay τ
1
. We can estimate this as
follows: Both the magnitude and phase of the hardware channel
ratio will be different for different frequencies. The magnitude
differs with frequency because different frequencies experience
different attenuation in the hardware. The phase varies linearly
with frequency because all frequencies experience the same
delay τ
1
, and the phase rotation of a frequency f is simply
0
0.2
0.4
0.6
0.8
1
1.2
1.4
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4
Magnitude
Frequency Range in GHz
|h
1
/h
2
|
|h
1
/h
3
|
Fig. 5—Hardware channel magnitude: The relative channel magni-
tudes |h
1
(f )/h
2
(f )| and |h
1
(f )/h
3
(f )| are not equal to 1 and are not
flat across the frequency spectrum. Hence, we need to compensate for
these estimates to be able to detect and solve collisions.
2πf τ
1
. We can therefore plot the phase of the ratio as a function
of frequency, and compute the delay τ
1
from the slope of the
resulting line.
Fig. 4 shows the phase result of this estimation performed
on the USRP software radios used in our implementation
described in §6. As expected, the phase is linear across
900 MHz. Hence, by fitting the points in Fig. 4 to a line we
can estimate the shifts τ
1
, τ
2
and the relative phases of the
hardware channels (i.e.
6
h
1
(f )/h
2
(f ) and
6
h
1
(f )/h
3
(f )). Fig. 5
also shows the relative magnitudes of the hardware channels
on the USRPs (i.e. |h
1
(f )/h
2
(f )| and |h
1
(f )/h
3
(f )|) over the
900 MHz between 3.05 GHz and 3.95 GHz. These hardware
channels and time shifts are stable. For our implementation,
we estimated them only once at the set up time.
5 SENSING NON-SPARSE SPECTRUM
We extend BigBand’s algorithm to sense a non-sparse spec-
trum. The key idea is that although the spectrum might not be
sparse, changes in spectrum usage are typically sparse, i.e., over
short intervals, only a small percentage of the frequencies are
freed up or become occupied. This makes it possible to estimate
the occupancy without sampling the signal at the Nyquist rate.
We refer to sparse changes as differential sparsity, and call the
extension that deals with such non-sparse spectrum D-BigBand.
We note however that unlike in the case where the spectrum
is sparse, in the non-sparse setting we only perform spectrum
sensing but we cannot recover the I and Q components of the
signal. Below we explain the two components of D-BigBand.
A. Frequency Bucketization: D-BigBand also bucketizes
the s pectrum using sub-sampling filters. However, since the
spectrum is not sparse, it is very likely that all buckets will
be occupied. Thus, D-BigBand tries to detect changes in the
occupancy of frequencies that hash to each buckets. To do
so, D-BigBand computes the average power of the buckets
over two consecutive time windows TW by performing the
bucketization multiple times during each time window.
6
Since
the changes in spectrum occupancies are sparse, only the
average power of few buckets would change between the two
time windows. D-BigBand can then focus only on the few
buckets where the average power changes.
B. Frequency Estimation: Now that we know in which
buckets the average power has changed, we need to estimate
6. The number of times D-BigBand can average is = TW/T where T is the
FFT window time.