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Some Fundamental Limitations
for Cognitive Radio
Anant Sahai
Wireless Foundations, UCB EECS
Joint work with Niels Hoven and Rahul Tandra
Work supported by the NSF ITR program
November 1 at BWRC Cognitive Radio Workshop
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Outline
1. Why cognitive radios?
2. Fundamental need to detect very weak primary signals
3. Knowledge of modulation does not help but knowledge of pilot
signals does
4. Receiver uncertainty and quantization’s impact on detection
5. Conclusions
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Apparent spectrum allocations
• Traditional spectrum allocation picture
• Apparent spectrum scarcity
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Apparent spectrum usage
Actual measurements show that < 20% of spectrum is used, but:
• Some users listen for very weak signals
– GPS
– Weather radar and remote sensing
– Radio astronomy
– Satellite communications
• Spectrum use can vary with space and time on all scales.
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Cognitive radio justification
Wireless interference is primarily a local phenomenon.
“If a radio system transmits in a band and nobody else is
listening, does it cause interference?”
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Ambitious Goal
Would like to take advantage of plentiful spectrum without requiring
a lot of regulation or assumed coordination among users.
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Objectives
• Protect primary users of the spectrum
– Socially important services may deserve priority on band
– Legacy systems may not be able to change
• Allow for secondary users to use otherwise unused bands
– Not the UWB approach: “speak softly but use a wide band”
– May have to coordinate/coexist with other secondary users
See what happens for the case of a single secondary user first.
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Primary decodability region
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Primary protected region
B
A
Users at the very edge of the currently decodable region will do
worse under any changes to the exclusive-use model.
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“No talk” zones
B
A
Mice can get close...
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“No talk” zones
B
A
But keep the lions far away!
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If the protected region nears the decodability
bound...
A
B
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The “no talk” zones grow dramatically
A
B
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Union of “no talk” zones
A
B
Need to be able to detect an undecodable transmission if either the
protected radius or the allowed secondary power is large.
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Fundamental Tradeoffs
Assume 1d2 attenuation of signals:
rcensored =
√
√
√
√
Ps
Pp
(22R−1)r2
protected
− σ2
Ps = r2censored
[
Pp
(22R − 1)r2protected
− σ2
]
• To allow secondary power to increase, we must be able to
detect weak primary signals in order to protect primary
receivers from interference.
• To protect primary receivers with already marginal reception,
the censored radius must grow and so even weaker primary
signals must be able to be detected.
November 1 at BWRC Cognitive Radio Workshop
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Censored radius vs. interferer power and
protected radius
1000 1500 2000 2500 3000 3500 4000 4500 50000
5
10
15
20
25
Protected radius (m)
Allo
wa
ble
in
terf
ere
r p
ow
er
(W)
Effects as protected radius nears decodability bound (censored radius = 10 meters)
0 1000 2000 3000 4000 50000
5
10
15
20
25
30
Protected radius (m)
Ce
nso
red
ra
diu
s (
m)
Effects as protected radius nears decodability bound (200 mW secondary user)
0 200 400 600 800 10000
5
10
15
20
25
30
Interferer power (mW)
Ce
nso
red
ra
diu
s (
m)
Censored zones for mice and lions (4.5 km from transmitter)
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Shadowing
C
A secondary user might be in a local shadow while his
transmissions could still reach an unshadowed primary receiver.
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Shadowing
C
1
2
Secondary user can not distinguish between positions (1) and (2)
— must be quiet in both: must detect even weaker primary
signals.
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Cognitive radio is still potentially useful
Even while protecting primary users, a large geographic area may
still be available for secondary users in any given band.
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Lessons so far
• “Don’t transmit if you can decode” is a poor rule
• Could do much better if we could detect undecodable signals
– Better protect primary users
– Allow longer range and higher rate secondary uses
• How hard is this?
November 1 at BWRC Cognitive Radio Workshop
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Model
• Hypothesis testing problem: is the primary signal out there?
H0 : Y [n] = W [n]
Hs : Y [n] = W [n] + x[n]
• Moderate Pfa, Pmd targets.
• Potentially very low SNR at the detector: will need many
samples to distinguish hypotheses.
• Proxy for difficulty: How long must we listen?
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Assume perfect knowledge
• x[n] known exactly at receiver
• Optimal detector is a matched filter
N∑
k=1
y[n]x[n]
‖x‖Hs
≷H0
‖x‖2
The power of processing gain: we only require O(1/SNR) samples
November 1 at BWRC Cognitive Radio Workshop
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Assume minimal knowledge
• Only know power and signal is like white Gaussian noise
• No processing gain available
• Optimal detector is an energy detector (radiometer)
N∑
k=1
y[n]2Hs
≷H0
N
(
σ2 +P
2
)
• We require O(1/SNR2) samples
November 1 at BWRC Cognitive Radio Workshop
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Energy detector vs. Coherent detector
−60 −55 −50 −45 −40 −35 −30 −25 −202
4
6
8
10
12
14
SNR (in dB)
log 10
N
Energy and the Coherent detectorEnergy DetectorCoherent Detector
Slope = −2
Slope = −1
November 1 at BWRC Cognitive Radio Workshop
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Undecodable BPSK
What if we had a little more information?
• Power is low.
• Modulation scheme (BPSK) is known
• Assume perfect synchronization to both the carrier frequency
and symbol timing.
H0 : Y [n] = W [n]
Hs : Y [n] = W [n] + X[n]√
P
X[n] ∼ iid. Bernoulli(1/2)
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Undecodable BPSK cont.
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.50.5 f
1(x|a =−0.25)
0.5 f1(x|a= 0.25)
f1(x)
f0(x)
SNR ≈ −12 dB
P−1/2
P−1/2
• Optimal detector turns out to be like the energy detector at
low SNR
• We require O(1/SNR2) samples
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Numerical plots for number of required samples..
−60 −50 −40 −30 −20 −10 00
2
4
6
8
10
12
14
SNR (in dB)
log 10
NEnergy DetectorUndecodable BPSKCoherent Detector
BPSK −− Detector Performance
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General symbol constellations
• Is the story bad only for BPSK?
H0 :−→Y [n] =
−→W [n]
Hs :−→Y [n] =
−→W [n] +
−→X [n]
−→x [n] = −→c i , i ∈ {1, 2, . . . , 2LR}−→w [n] ∼ N (0, σ2IL)
• Assumptions
– Short symbols ~ci of length L.
– 2LR symbols known to the receiver
– Symbol constellation is zero-mean
– Symbols independent
– Very little energy in any individual symbol
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Examples of zero-mean symbol constellations
p = 1/4
p = 1/4 p = 1/4
p = 1/4
p = 1/5
p = 1/5
p = 1/5
p = 1/5
p = 1/5
p = 1/3
p = 1/3
p = 1/3 p = 6/7 p = 1/7
[−1 , 0 ] [6 , 0]
Also includes symbols multiplied by short PRN sequences, short
OFDM packets, etc.
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Result
• Optimal detector is like an energy detector at low SNR
• We require O(1/SNR2) samples
−60 −50 −40 −30 −20 −10 010
2
104
106
108
1010
1012
1014
SNR (dB)
log
10 N
Asymmetric Constellation − Detector Performance
Optimal DetectorEnergy Detector
November 1 at BWRC Cognitive Radio Workshop
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The importance of the zero-mean assumption
p2
p1
[−1 , 0 ] [2 , 0]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3x 10
6
p1
N
Energy DetectorOptimal DetectorDeterministic Detector
The peak occurs when the constellation has zero mean.
November 1 at BWRC Cognitive Radio Workshop
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Assume a weak pilot signal alongside the BPSK
−60 −50 −40 −30 −20 −10 00
2
4
6
8
10
12
14
SNR (in dB)
log 10
NEnergy Detector Undecodable BPSKBPSK with Pilot signal Sub−optimal schemeDeterministic BPSK
BPSK −− Detector Performance
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The story so far
• Without help from the primary user, the secondary users
require a long time to detect free bands.
– Less agility
– Overhead in searching for unused bands
• It gets a whole lot worse.
– Noise uncertainty
– Quantization
November 1 at BWRC Cognitive Radio Workshop
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Receiver chain structure
down−converter
Frequency Intermediatefrequencyamplifier Converter
A/Damplifier
Low−noise
Demodulator
Receivingantenna
November 1 at BWRC Cognitive Radio Workshop
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Noise uncertainty
• Let residual noise uncertainty be x dB within the band.
• Receiver faces an SNR within
[SNRnominal, SNRnominal + x]
• Pnoise ∈ [Pnominal, α · Pnominal] , α = 10(x/10)
• Therefore, the energy detector fails if:
Pnoise ≥ Pnominal + Psignal
⇒ SNRnominal ≤ 10 log10 [10(x/10)− 1]
November 1 at BWRC Cognitive Radio Workshop
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Energy detector with noise uncertainty
−40 −35 −30 −25 −20 −15 −10 −5 00
2
4
6
8
10
12
14
Nominal SNR
log
10 N
x = 0.1 dBx = 0.001 dB x = 1 dB
November 1 at BWRC Cognitive Radio Workshop
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“Wall” position as a function of uncertainty
SNRwall = 10 log10 [10(x/10)− 1]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−40
−35
−30
−25
−20
−15
−10
−5
SNR sensitivity (in dB)
Det
ecta
bilit
y th
resh
old
(in d
B)
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Quantization: Our abstraction
Secondary
DemodulationData
SignalDetection
SSampler Quantizer
Q
Noise
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SNR loss from quantization
−45 −44 −43 −42 −41 −40 −39 −38 −37 −36 −35
8.5
9
9.5
10
10.5
SNR (in dB)
log
10 N
non−coherent detector6−bin non−coherent detector4−bin non−coherent detectorContinuous Energy detector
−45 −44 −43 −42 −41 −40 −39 −38 −37 −36 −354.6
4.8
5
5.2
5.4
5.6
5.8
SNR (in dB)
log
10 N
Coherent detector 2−bin Coherent detector4−bin Coherent detector6−bin Coherent detectorContinuous Coherent detector
Quantization SNR loss SNR loss
bins (coherent detector) (non-coherent detector)
2 bins 2 dB ∞
4 bins 0.5 dB 1.4 dB
6 bins 0.3 dB 0.7 dB
November 1 at BWRC Cognitive Radio Workshop
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Noise uncertainty under quantization
• Detection can be absolutely impossible for 2-bit quantizer
under noise variance uncertainty alone.
– Can make the distributions identical under both hypotheses
if
Q
(
d1
σ0
)
=1
2
[
Q
(
d1 +√
P
σ1
)
+ Q
(
d1 −√
P
σ1
)]
– σ2i is noise variance under hypothesis i
– d1 is the quantization bin boundary
• Wall always exists for any detector.
November 1 at BWRC Cognitive Radio Workshop
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Quantized detector with noise uncertainty:
2-bit quantizer
−40 −35 −30 −25 −20 −15 −10 −5 00
2
4
6
8
10
12
14
Nominal SNR
log
10 N
x=0.1 dB x=1 dBx=0.001 dB
November 1 at BWRC Cognitive Radio Workshop
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Conclusions
• Cognitive radios must be able to detect the presence of
undecodable signals
– Just knowing the modulation scheme and codebooks is
nearly useless: stuck with energy detector performance.
– Even small noise uncertainty causes serious limits in
detectability.
– Quantization makes matters even worse.
• Primary users should transmit pilot signals or sirens.
• If not, some serious “infrastructure” will be needed to support
cognitive radio deployment.
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Key future questions
• Multiuser situations
– Control channel use and coordination
– Distributed reliable environmental proofs
– Efficiency and robustness
• How to ensure forward compatibility
– Can future computational capabilities help systems
engineered today?
– Poorly engineered systems today will hinder future systems.
• Congestion is still potentially possible in the future.
November 1 at BWRC Cognitive Radio Workshop