Some Fundamental Limitations for Cognitive Radio Anant Sahai Wireless Foundations, UCB EECS sahai@eecs.berkeley.edu Joint work with Niels Hoven and Rahul Tandra Work supported by the NSF ITR program
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
Apparent spectrum allocations Traditional spectrum allocation picture Apparent spectrum scarcity
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.
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?
Ambitious Goal Would like to take advantage of plentiful spectrum without requiring a lot of regulation or assumed coordination among users.
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.
Primary decodability region
Primary protected region A B Users at the very edge of the currently decodable region will do worse under any changes to the exclusive-use model.
No talk zones A B Mice can get close...
No talk zones A B But keep the lions far away!
If the protected region nears the decodability bound... A B
The no talk zones grow dramatically A B
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.
Assume 1 d 2 Fundamental Tradeoffs attenuation of signals: r censored = [ P s = r 2 censored P p P s (2 2R 1)r 2 protected P p (2 2R 1)r 2 protected σ 2 σ 2 ] 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.
Censored radius vs. interferer power and protected radius Effects as protected radius nears decodability bound (censored radius = 10 meters) 25 20 Allowable interferer power (W) 15 10 5 0 1000 1500 2000 2500 3000 3500 4000 4500 5000 Protected radius (m) Effects as protected radius nears decodability bound (200 mw secondary user) 30 30 Censored zones for mice and lions (4.5 km from transmitter) 25 25 Censored radius (m) 20 15 10 Censored radius (m) 20 15 10 5 5 0 0 1000 2000 3000 4000 5000 Protected radius (m) 0 0 200 400 600 800 1000 Interferer power (mw)
Shadowing C A secondary user might be in a local shadow while his transmissions could still reach an unshadowed primary receiver.
Shadowing 1 2 C Secondary user can not distinguish between positions (1) and (2) must be quiet in both: must detect even weaker primary signals.
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.
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?
Model Hypothesis testing problem: is the primary signal out there? H 0 : Y [n] = W [n] H s : Y [n] = W [n] + x[n] Moderate P fa, P md targets. Potentially very low SNR at the detector: will need many samples to distinguish hypotheses. Proxy for difficulty: How long must we listen?
Assume perfect knowledge x[n] known exactly at receiver Optimal detector is a matched filter N k=1 y[n] x[n] x H s H 0 x 2 The power of processing gain: we only require O(1/SN R) samples
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] 2 H s H 0 N We require O(1/SNR 2 ) samples ( σ 2 + P 2 )
Energy detector vs. Coherent detector 14 Energy and the Coherent detector Energy Detector Coherent Detector 12 log 10 N 10 8 Slope = 2 6 4 Slope = 1 2 60 55 50 45 40 35 30 25 20 SNR (in db)
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. H 0 : Y [n] = W [n] H s : Y [n] = W [n] + X[n] P X[n] iid. Bernoulli(1/2)
Undecodable BPSK cont. 0.5 0.45 0.4 0.5 f 1 (x a = 0.25) 0.5 f 1 (x a= 0.25) f 1 (x) f 0 (x) 0.35 0.3 SNR 12 db 0.25 0.2 0.15 0.1 0.05 0 5 4 3 2 1 0 P 1/2 1 P 1/2 2 3 4 5 Optimal detector turns out to be like the energy detector at low SNR We require O(1/SNR 2 ) samples
Numerical plots for number of required samples.. 14 12 BPSK Detector Performance Energy Detector Undecodable BPSK Coherent Detector 10 log 10 N 8 6 4 2 0 60 50 40 30 20 10 0 SNR (in db)
General symbol constellations Is the story bad only for BPSK? H 0 : Y [n] = W [n] H s : Y [n] = W [n] + X [n] x [n] = c i, i {1, 2,..., 2 LR } w [n] N (0, σ 2 I L ) Assumptions Short symbols c i of length L. 2 LR symbols known to the receiver Symbol constellation is zero-mean Symbols independent Very little energy in any individual symbol
Examples of zero-mean symbol constellations p = 1/4 p = 1/4 p = 1/5 p = 1/5 p = 1/5 p = 1/4 p = 1/4 p = 1/5 p = 1/5 p = 1/3 p = 1/3 p = 6/7 p = 1/7 [ 1, 0 ] [6, 0] p = 1/3 Also includes symbols multiplied by short PRN sequences, short OFDM packets, etc.
Result Optimal detector is like an energy detector at low SNR We require O(1/SNR 2 ) samples 10 14 10 12 Asymmetric Constellation Detector Performance Optimal Detector Energy Detector 10 10 log 10 N 10 8 10 6 10 4 10 2 60 50 40 30 20 10 0 SNR (db)
The importance of the zero-mean assumption 3 x 106 Energy Detector Optimal Detector Deterministic Detector 2.5 2 N 1.5 p 2 p 1 [ 1, 0 ] [2, 0] 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 p 1 The peak occurs when the constellation has zero mean.
Assume a weak pilot signal alongside the BPSK 14 12 BPSK Detector Performance Energy Detector Undecodable BPSK BPSK with Pilot signal Sub optimal scheme Deterministic BPSK 10 log 10 N 8 6 4 2 0 60 50 40 30 20 10 0 SNR (in db)
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
Receiver chain structure Receiving antenna Low noise amplifier Frequency down converter Intermediate frequency amplifier A/D Converter Demodulator
Noise uncertainty Let residual noise uncertainty be x db within the band. Receiver faces an SNR within [SNR nominal, SNR nominal + x] P noise [P nominal, α P nominal ], α = 10 (x/10) Therefore, the energy detector fails if: P noise P nominal + P signal SNR nominal 10 log 10 [10 (x/10) 1]
Energy detector with noise uncertainty 14 12 10 x = 0.001 db x = 0.1 db x = 1 db log 10 N 8 6 4 2 0 40 35 30 25 20 15 10 5 0 Nominal SNR
Wall position as a function of uncertainty SNR wall = 10 log 10 [10 (x/10) 1] 5 10 Detectability threshold (in db) 15 20 25 30 35 40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SNR sensitivity (in db)
Quantization: Our abstraction Noise Secondary Data Demodulation S Sampler Q Quantizer Signal Detection
SNR loss from quantization 10.5 non coherent detector 6 bin non coherent detector 4 bin non coherent detector Continuous Energy detector 5.8 5.6 Coherent detector 2 bin Coherent detector 4 bin Coherent detector 6 bin Coherent detector Continuous Coherent detector 10 5.4 log 10 N 9.5 log 10 N 5.2 9 5 4.8 8.5 45 44 43 42 41 40 39 38 37 36 35 SNR (in db) 4.6 45 44 43 42 41 40 39 38 37 36 35 SNR (in db) 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
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 ( ) [ ( d1 Q = 1 d 1 + ) ( P d 1 )] P Q + Q σ 0 2 σ 1 σ 1 σ 2 i is noise variance under hypothesis i d 1 is the quantization bin boundary Wall always exists for any detector.
Quantized detector with noise uncertainty: 2-bit quantizer 14 12 x=0.001 db x=0.1 db x=1 db 10 log 10 N 8 6 4 2 0 40 35 30 25 20 15 10 5 0 Nominal SNR
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.
Multiuser situations Key future questions 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.