Wideband Spectrum Sensing at Sub-Nyquist Rates
WWideband Spectrum Sensing at Sub-Nyquist Rates
Moshe Mishali and Yonina C. Eldar
Spectrum sensing refers to the task of identifying the frequency support of a given in-put signal. Standard radio-frequency (RF) lab equipment can provide this functionality. Anexample is a spectrum analyzer ( e.g.,
HP-8563E) which sweeps the center frequency of ananalog bandpass filter and draws the in-band signal energy. The frequency support thenconsists of those spectrum intervals in which the signal power exceeds the noise floor. Re-cently, there has been growing interest in spectrum sensing for mobile cognitive radio (CR)receivers [1], which aim at utilizing unused frequency regions on an opportunistic basis.Commercialization of CR technology necessitates a spectrum sensing mechanism that re-acts in real time to cognitive decisions. A mobile device, however, cannot embed solutionsbased on standard lab equipment, due to size, weight, power and cost limitations. Sensingin CR mobiles must be performed using minimal hardware and software resources. There-fore, enabling widespread use of CRs calls for innovative spectrum sensing techniques.In this paper, we present a mixed analog-digital spectrum sensing method that is espe-cially suited to the typical wideband setting of CRs. The next section briefly summarizesexisting approaches to CR sensing. The advantages of our system with respect to currentarchitectures are threefold. First, our analog front-end is fixed and does not involve scan-ning hardware. Second, both the analog-to-digital conversion (ADC) and the digital signalprocessing (DSP) rates are substantially below Nyquist. Finally, the sensing resources areshared with the reception path of the CR, so that the lowrate streaming samples can be usedfor communication purposes of the device, besides the sensing functionality they provide.Combining these advantages leads to a real time map of the spectrum with minimal useof mobile resources. Our approach is based on the modulated wideband converter (MWC)system [2], which samples sparse wideband inputs at sub-Nyquist rates. We report on re-sults of hardware experiments, conducted on an MWC prototype circuit [3], which affirmfast and accurate spectrum sensing in parallel to CR communication. This can help alleviateone of the current main bottlenecks in wide-spreading deployment of CRs. ——— Cognitive Radios and Spectrum Sensing ———
Traditional communication, such as television, radio stations, mobile carriers and airtraffic control is carried over predetermined frequency bands. Over the years, govern-ment agencies allocated the majority of the spectrum to legacy users, reserving a particular1 a r X i v : . [ c s . A R ] S e p Table 1] SPECTRUM SENSING APPROACHES FOR COGNITIVE RADIO.
Analog ADC/DSP Shared withApproach front-end rate CR reception P a r a m e t r i c Analog pilot detection / matched-filtering scanning n/a (cid:55)
Digital pilot detection / matched-filtering (cid:55)
Nyquist (cid:51)
Cyclostationary feature extraction (cid:55)
Nyquist (cid:51)
Waveform-based sensing (cid:55)
Nyquist (cid:51)
Radio identification (cid:55)
Nyquist (cid:51) G e n e r i c Analog energy detection scanning n/a (cid:55)
Digital energy detection (cid:55)
Nyquist (cid:51)
Multi taper spectrum estimation (cid:55)
Nyquist (cid:51)
Filter bank spectrum sensing (cid:55)
Nyquist (cid:51)
This paper fixed sub-Nyquist (cid:51) [FIG1]
A cognitive radio aims at sensingthe available frequency holes in consec-utive time intervals. frequency interval for each owner. This resource allocation strategy has led to spectrumcongestion, to such a point that, today, the increasing demand for transmission bands canrarely be satisfied by a permanent allocation. Fortunately, studies conducted by the Fed-eral Communications Commission (FCC) in the United States and by similar agencies inother countries indicate that the spectrum is underutilized; In a given geographical loca-tion and time duration, only a small number of legacy users transmit concurrently. Thislow frequency utilization, illustrated in Fig. 1, is what drives CR technology.The idea behind CR is to exploit temporarily available spectrum holes belonging to in-active primary users. Spectrum sensing therefore takes place whenever the CR searches foravailable transmission holes. After a certain frequency band is chosen, the CR continuouslymonitors the spectrum in order to detect any change in the activity of the primary users.Once a primary user becomes active, the CR must choose another working band, or tailorits transmission to reduce in-band power. Quick and efficient spectrum sensing is evidentlyan essential component of CR functionality.A special issue of the
IEEE Signal Processing Magazine from November 2008 reviewsexisting CR technology [4, 5]. Current approaches for spectrum sensing are briefly sum-marized in Table 1 according to [6, 7]. From a bird’s eye view, previous methods can becategorized into either fully hardware or fully software solutions. Known analog methodsimitate the scanning mechanism used in lab equipments, thereby requiring tunable circuits,independent of the CR reception hardware. The software solutions assume that the inputis sampled at the Nyquist rate f NYQ = f max , (1)that is twice the highest wideband frequency f max . No analog preprocessing is needed andthe samples can be shared with the subsequent CR stages. However, since CR typicallyoperates in a wideband environment, the sampling rate f NYQ can be prohibitively large.Consequently, utilizing these sensing algorithms requires premium ADC and DSP devicesthat can accommodate high-rate streaming data.2 he Modulated Wideband Converter
RF front-endHigh B.W. Lowrate ADCsLow B.W.
Overlayedenergy
AM FM QPSKQAM
Coarse support recovey(Continuous-to-Finite, CTF)FrameConst. Sparse detection
Light computational loadShort delay
Sub-Nyquist Spectrum Sensing & CR reception • Carrier & baseband recovery • Digital signal processing • Analog reconstruction (Real time)
Analog ``message’’Digital bitsAvailable frequency``holes’’(eq. 10) [FIG2]
Block diagram of the modulated wideband converter. The digital recovery algorithm recovers the multiband input.
Table 1 distinguishes between parametric and generic approaches. Parametric methodsrely on a specific structure that the input signal is assumed to obey. For example, matchedfiltering requires the exact transmission shape of the primary user. Other parametric ap-proaches incorporate knowledge on preambles, midambles, synchronization bits, cyclosta-tionarity, modulation format etc. In contrast, the generic methods avoid assumptions onthe underlying signal content. Sensing based on the MWC, introduced below, belongs tothe generic family of methods, and possesses additional unique features: fixed hardware,sub-Nyquist ADC and DSP rates, and shared acquisition resources between sensing and CRreception. Mixed analog-digital system design is the enabling factor behind these uniquebenefits. ——— Sub-Nyquist Sampling: Modulated Wideband Converter ———
Consider a signal consisting of several concurrent transmissions. In order to avoid sam-pling at the high Nyquist rate, the common practice in engineering is demodulation. Thesignal is multiplied by the carrier frequency of a band of interest, so as to shift the desiredcontents to the origin, where filtering and sampling at a low rate take place. When the bandpositions are unknown, e.g., in a CR receiver, standard demodulation cannot be used.The MWC treats multiband signals when knowledge of the carrier frequencies is presentor absent. The only assumption is that the spectrum is concentrated on N frequency inter-vals with individual widths not exceeding B . The sampling rate is proportional to the ef-fective spectrum occupation NB rather than f NYQ . Typically, the spectrum is underutilizedso that NB (cid:28) f NYQ . A digital algorithm detects the spectral support and enables eithersignal reconstruction or lowrate processing of the individual band contents. In this article,3e take advantage of the MWC for a slightly different task – instead of aiming at the in-formation bands, our goal is to detect the inactive support. This complementary viewpointallows optimizing the MWC design for holes detection at the expense of the tasks that arenot required in the CR settings, namely reconstruction and processing of the primary trans-missions. The resulting MWC-based spectrum sensing is categorized under the “Generic”rubric of Table 1, since no assumption is made on the signal shape of the legacy users ortheir specific modulation techniques. Nonetheless, the sampling rate is comparable withthat of a demodulator who knows the exact carrier of each transmission.We now explain the MWC sampling stage, as depicted in Fig. 2. The system consistsof a front-end of m channels. In the i th channel, the input signal x ( t ) is multiplied by aperiodic waveform p i ( t ) with period T p , lowpass filtered by h ( t ) , and then sampled at rate f s = T s . The basic MWC configuration has f p = T p ≥ B , T p = T s , m ≥ N . (2)The parameter choice (2) results inSampling rate = m f s ≈ NB , (3)which in general is far below f NYQ . In practice, an advanced configuration which we de-scribe in the sequel is used in our hardware experiments, allowing to reduce the number ofbranches m at the expense of increasing the sampling rate f s on each channel so that overall m f s ≈ NB .To derive an expression for the i th sequence of samples y i [ n ] we note that since each p i ( t ) is periodic, it has a Fourier expansion p i ( t ) = ∞ ∑ (cid:96) = − ∞ c i (cid:96) e j π f p (cid:96) t , (4)for some coefficients c i (cid:96) . Denote by z (cid:96) [ n ] the sequence that would have been obtained ifthe signal was mixed by a pure sinusoid e j π f p (cid:96) t and lowpass filtered. This sequence corre-sponds to uniform samples at rate f p of a section of x ( t ) , conceptually obtained by bandpassfiltering an f p -width interval around (cid:96) f p and demodulating to the origin. Since the systemis linear, modulating by p i ( t ) and lowpass filtering is equivalent to summing the weightedcombinations of all the sequences z (cid:96) [ n ] : y i [ n ] = L ∑ (cid:96) = − L c i (cid:96) z (cid:96) [ n ] , (5)where the sum limits − L ≤ (cid:96) ≤ L represent the range of coefficients c i (cid:96) with non-negligibleamplitudes. It follows that the number of spectrum intervals that are aliased to the origin is4 = L +
1. In principle, any periodic function with high-speed transitions within the pe-riod T p can be used to obtain this aliasing. One possible choice for p i ( t ) is a sign-alternatingfunction, with M sign intervals within the period T p [2]. Popular binary patterns, e.g., theGold or Kasami sequences, are especially suitable for the MWC [8].Mixing by periodic waveforms aliases the spectrum to baseband, such that each fre-quency interval of width f p = T p receives a different weight. The energy of the variousspectral intervals is overlayed at baseband, as visualized in Fig. 2. At first sight, the se-quences y i [ n ] seem corrupted due to the deliberate aliasing. Nonetheless, the fact that onlya small portion of the wideband spectrum is occupied, together with the different weightsin the different channels, permits the recovery of x ( t ) . The next section explains the dig-ital computations of Fig. 2, and in particular how the spectrum sensing functionality isachieved. ——— Computationally-Light Software Algorithm ——— Mathematically, the analog mixture boils down to the linear system [2] y [ n ] = Cz [ n ] , (6)where the vector y [ n ] = [ y [ n ] , · · · , y m [ n ]] T collects the measurements at t = nT s . Thematrix C consists of the coefficients c i (cid:96) and z [ n ] consists of the values of z (cid:96) [ n ] arrangedin vector form. From (2) and the definition of z (cid:96) [ n ] , it follows that at most 2 N sequences z (cid:96) [ n ] are active, namely contain signal energy [2]. The spectrum sensing functionality is,therefore, tantamount to finding the index set S = { (cid:96) | z (cid:96) [ n ] (cid:54) = } (7)which reveals the spectrum support of x ( t ) at a resolution of f p Hz. The choice f p ≥ B in (2) implies a minimal resolution which should match the expected bandwidth of legacytransmissions. Achieving smaller resolution f p < B is discussed in the next section.Detecting S by inverting C in (6) is not possible, since the m × M matrix C is under-determined; the MWC uses m (cid:28) M to reduce the sampling rate below Nyquist. Under-determined systems have in general infinitely many solutions. Nonetheless, under the pa-rameter choice (2), and additional mild conditions on the waveforms p i ( t ) , a sparse z [ n ] with at most 2 N nonzero entries is unique and can be recovered in polynomial time [2] byrelying on results in the field of compressed sensing. Further simplification of the DSP canbe obtained by noting that z [ n ] are jointly sparse over time, namely the index set S doesnot depend on the time index n . Therefore, S can be estimated from several consecutivesamples, which increases the robustness of the estimate.5upport recovery is performed in the continuous-to-finite (CTF) block in Fig. 2, whichis the heart of the MWC reconstruction algorithm. The CTF builds a frame (or a basis) fromthe measurements using y [ n ] Frame construct −−−−−−−−−→ Q = ∑ n y [ n ] y H [ n ] Decompose −−−−−−−→ Q = VV H , (8)where the (optional) decomposition allows removal of the noise space. The active spectrumslices are detected from the sparse solution of the following underdetermined system V = CU . (9)It is proven in [9] that (9) has a unique solution matrix U with minimal number of non-identically zero rows, and that the locations of these rows coincide with the support set S of x ( t ) . This is the point where the CR device can decide how to allocate its energy, sinceSpectrum holes = (cid:91) (cid:96) / ∈ S (cid:20) l f p − f p l f p + f p (cid:21) . (10)Additional steps on the same sample sequences y i [ n ] enable processing and reconstructionof any input transmission. We refer to [2] for a detailed description of these recovery steps.Note that among the transmissions in x ( t ) , some belong to primary users while otherscan be CR communications. The fact that the same samples enable reconstruction of CRcommunications is highly important – it enables the CR to both sense the spectrum withthe system of Fig. 2 and intercept communications as a standard receiver. Although beyondthe current scope, we note that the CTF has a major role in signal reconstruction, beyondthe robustness in estimating S . The CTF isolates the support recovery to a single executionof a polynomial-time algorithm. Once S is known, real time processing and reconstructionis possible, that is at the (low) speed of the streaming measurements y [ n ] [10].In the next section, we describe the circuit prototype of the MWC [3], which is used inour experiments. ——— Efficient Hardware Realization ——— The basic configuration (2) has m ≥ N channels, which may be too large to fit into aCR device. In addition, the waveforms p i ( t ) need to be different so as to capture linearlyindependent mixtures of the spectrum, which results in additional hardware per channel.To moderate the physical size, we constructed an advanced MWC configuration, proposedin [2]. In this MWC version • the number of channels m is collapsed by a factor q > FIG3]
A hardware realization of the MWC consisting of two circuit boards. The left pane implements m = sampling channels, whereasthe right pane provides four sign-alternating periodic waveforms of length M = , derived from different taps of a single shift-register. • a single shift-register provides a basic periodic pattern, from which m periodic wave-forms are derived using delays, that is by tapping m different locations of the register.Technically, this configuration allows to collapse channels all the way down to a singlesampling channel at the same sub-Nyquist rate.A board-level design of the MWC using this advanced configuration to treat a multi-band model with N = B =
20 MHz was reportedin [3]. The RF stage covers a wideband range of inputs with f NYQ = NB =
120 MHz. An aliasing resolution of f p =
20 MHz in conjunction witha sampling rate of 1/ T s =
70 MHz results in a collapsing factor of q =
3. Using m = f p can be improved by setting f p < B =
20 MHz. In the original MWC scheme [2], this choice is avoided since it in-creases the computations needed for signal reconstruction when a transmission occupiesmore than two sequences z (cid:96) [ n ] . The CR settings permit f p < B , as only the support set S isneeded for sensing; Reconstructing the primary transmissions is not of interest. Since Fig. 2is also used for CR reception, the resolution f p needs only to exceed the bandwidth of theCR communication, rather than the expected bandwidth B of the primary users, which canin general be higher.The nonordinary RF design that stems from sub-Nyquist sampling is described in [3].For instance, lowcost analog mixers are specified for a pure sinusoid in the oscillator port,whereas the MWC requires simultaneous mixing with the many sinusoids comprising thewaveforms p i ( t ) . Another circuit challenge pertains to generating p i ( t ) with 2 GHz alter-nation rates. The severe timing constraints involved in this logic are overcome in [3] byoperating commercial devices beyond their datasheet specifications. The reader is referredto [3] for further technical details. 7
00 200 300 400 500 600 700 800 900 1000 110010020030040050060070080090010001100 Input carrier frequency (MHz) S en s ed c a rr i e r f r eqeun cy ( M H z ) + + FM @ 631.2 MHz AM @ 807.8 MHz Sine @ 981.9 MHz Overlayed sub-Nyquist aliasing around 6.171 MHz = FM @ 631.2 MHz AM @ 807.8 MHz
Experiment
Experiment . M H z
10 kHz 100 kHz [FIG4]
Results of hardware experiments demonstrating accurate spectrum sensing combined with signal reception, both accomplishedat a sub-Nyquist rate. ——— Spectrum Sensing Demonstration ———
To verify the sensing potential of the MWC in a wideband environment, we conductedtwo experiments. In the first experiment, an HP-E4432B signal generator inputs a sinusoidto the MWC hardware. The four output channels were recorded using an Agilent Infiniium54855A four-channel scope. All digital computations were carried out in Matlab. The puresinusoid represents a challenging scenario of a legacy user with extremely narrow band-width. We varied the sinusoid frequency from 100 MHz to 1100 MHz in steps of 5 MHz.The CTF outputs the spectral support at resolution f p =
20 MHz. We also executed the ad-ditional recovery blocks of Fig. 2 so that the algorithm estimates the input carrier frequencyas well. The results, in the left pane of Fig. 4, demonstrate that out of these 200 experiments,there are only 2 outliers, which means 99% correct support and carrier estimation.It is important to understand the reason for the outliers in Fig. 4. It is well known thatfinding the sparse solution of an underdetermined system, such as (9), is NP-hard. In prac-tice, we solve (9) using polynomial-time algorithms which coincide with the true solutionover a wide range of possible inputs, 99% of the cases in our experiments. The detectionperformance could have been improved for a higher number of sampling channels, say m =
5. Our design choice of a 4-channel prototype [3] represents a customary engineer-ing compromise; saving the extra 25% in hardware size and digital computations of the m = C . The small dimensions of C , 12 ×
111 inour prototype, is what makes the MWC sensing practically feasible from the computationalperspective. We point out that this sensing duration is negligible with respect to cognitiveprotocols. For instance, the IEEE 802.22 standard for CR devices and networks, which isstill under development, specifies a sensing duration of 30 seconds [11].Interestingly, Cordeiro et al. summarized the IEEE 802.22 standard for CR in [12] in 2006and envisioned that the sensing procedure would probably be carried out in two steps.First, a coarse and fast support detection, as does the CTF with a spectral resolution of f p =
20 MHz. Then, a finer estimation, if needed. In the experiments of the previoussection, the carrier recovery algorithm of [10] obtains carrier estimates within 10 kHz of thetrue input frequencies.In the second experiment we exemplify the resource sharing of sensing and reception.Fig. 4 depicts the setup of three signal generators that were combined at the input terminalof the MWC prototype: an amplitude-modulated (AM) signal at 807.8 MHz with 100 kHzenvelope, a frequency-modulation (FM) source at 631.2 MHz with 1.5 MHz deviation at10 kHz rate and a pure sine waveform at 981.9 MHz. Together, this scenario represents amixture of primary and cognitive transmitters. The carrier positions were chosen so thattheir aliases overlay at baseband, as the photos in Fig. 4 demonstrate. The digital recov-ery algorithm was executed and detected the correct support set S (CTF) and input carrierpositions. In addition, the figure demonstrates correct reconstruction of the AM and FMsignal contents, affirming the potential of standard signal reception combined with spec-trum sensing.A video recording of these experiments and additional documentation are available on-line on the authors’ websites: , http://webee.technion.ac.il/Sites/People/YoninaEldar/Info/hardware.html .In addition, we prepared a graphical package to demonstrate the MWC numerically,which is also available on our websites. The software guides the user through a four-stageflow: defining the multiband signal model N , B , f max , adjusting the MWC parameters andsub-Nyquist sampling, CTF support recovery, and signal reconstruction. A screenshot isshowed in Fig. 5. 9 FIG5]
Graphical user interface of the MWC system. ——— Outlook ———
The proliferation of wireless devices necessitates flexible and efficient use of the spec-trum. To render CR a widespread reality, the spectrum sensing bottleneck must be resolved.The sensing task is a crucial step in the CR life-cycle – it precedes all other cognitive deci-sions. Besides identifying available frequency holes, continuous monitoring is needed inorder to detect appearance of primary users, an event which has immediate implicationson the CR transmissions.Cognitive communication is still a dream to come true. Research on CR is rapidly de-veloping, providing sophisticated solutions for the multitude of challenges this technologytriggers. Of the various goals, spectrum sensing is unique. In contrast to other cognitivedecisions, e.g., spectrum collaboration and CR networking, which are performed at higherapplication layers, sensing is the only task that involves the analog hardware as it beginsin the analog domain. Traditionally, scanning a wide span of the spectrum is done usinglab equipment, which is not constrained by size, power, cost, volume, etc. The CR era callsfor innovative solutions for miniaturizing the sensing core into a mobile device, that hasmany other functionalities to perform in parallel. We therefore foresee much of the futureresearch and development devoted to improving and innovating in the sensing stage.We have demonstrated a low-rate, efficient spectrum sensing mechanism which can reli-ably determine inactive bands over a wide span of the spectrum, within a few milliseconds.With respect to existing sensing strategies, our approach proposes a mixed analog-digitaldesign, with a simple and fixed analog front-end. Besides sensing, the system also servesas the CR reception path. Our design is efficiently realized in hardware and introducesonly light computational loads. Hardware experiments report fast and accurate spectrumsensing due to the low sampling rate. The contribution of this work is in outlining the prac-tical considerations for CR sensing, and providing an initial circuit-level proof of feasibility.Future work should address various hardware-related aspects, including how to miniature10he design into a chip, so that it can later be embedded into existing mobile platforms.
AUTHORSMoshe Mishali ([email protected]) is a PhD student of electrical engineeringat the Technion, Institute of Technology, Haifa, Israel.
Yonina Eldar ([email protected]) is a professor of electrical engineering at theTechnion, Institute of Technology, Haifa, Israel. She is also a Research Affiliate with theResearch Laboratory of Electronics at MIT, and a visiting professor at Stanford, California,United States. R
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