Research Article A Simplified Multiband Sampling and...

Research ArticleA Simplified Multiband Sampling and Detection MethodBased on MWC Structure for Mm Wave Communications in5G Wireless Networks

Min Jia Xue Wang Xuemai Gu and Qing Guo

School of Electronics and Information Engineering Harbin Institute of Technology Harbin 150001 China

Correspondence should be addressed to Min Jia jiaminhiteducn

Received 14 August 2015 Accepted 15 October 2015

Academic Editor Wei Xiang

Copyright copy 2015 Min Jia et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The millimeter wave (mm wave) communications have been proposed to be an important part of the 5G mobile communicationnetworks and it will bring more difficulties to signal processing especially signal sampling and also cause more pressures onhardware devices In this paper we present a simplified sampling and detection method based on MWC structure by usingthe idea of blind source separation for mm wave communications which can avoid the challenges of signal sampling broughtby high frequencies and wide bandwidth for mm wave systems This proposed method takes full advantage of the beneficialspectrum aliasing to achieve signal sampling at sub-Nyquist rate Compared with the traditional MWC system it provides theexact quantity of sampling channels which is far lower than that of MWC In the reconstruction stage the proposed methodsimplifies the computational complexity by exploiting simple linear operations instead of CS recovery algorithms and providesmore stable performance of signal recovery Moreover MWC structure has the ability to apply to different bands used in mmwavecommunications by mixed processing which is similar to spread spectrum technology

1 Introduction

In wireless communication the most common means ofsignal transmission is to modulate the information signals bythe high carrier frequency Taking current situations of lowerfrequency spectrum into consideration which are crowdedand about to run out at present the next generation (5G)mobile network not only exploits the reintegration of theoriginal spectrum but also is towards higher frequencies

The explosive growth of consumer demand leads tohigher transmission rateThe contradiction between capacitydemand and spectrum shortage has become increasinglyprominent Bandwidth is considered a bottleneck whichrestricts the development of 5G mobile communicationnetworks And the mm wave communications have beenproposed to be an important part of 5G mobile networks[1 2] The application of mm wave communications will bebound to bring new challenges Many countries such as theUnited States Japan and South Korea have opened up tothe 60GHzmillimeter wave frequency band because there is

an abundance of available spectrum surrounding 60GHz tosupport the high-rate wireless communications [3] Howeverthere exists the severe oxygen absorption in the 60GHzband Fortunately the frequency bands 71ndash76GHz and 81ndash86GHz collectively called the E-band have been released toprovide broadband wireless services with low atmosphericattenuation but E-band propagation loss is severe [4] Tofurther increase the data rate and transmission distance theMIMO technique has been widely adopted to bring hightransceiver complexity in such large MIMO systems witha large number of antennas To reduce the complexity ofsystems more advanced antenna arrays have been used suchas uniform circular array which is studied in [5 6] On theother hand researches on MIMO channel over such bandshave become more important [7] And input and outputsignals of MIMO channel are considered multiband signalmodel which describes that a group of several transmissionsignals are modulated by high carrier frequencies From theperspective of frequency domain this kind of signal only hasvalues within several continuous intervals spreading over a

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2015 Article ID 873673 10 pageshttpdxdoiorg1011552015873673

2 International Journal of Antennas and Propagation

wide frequency spectrum Such feature is called the sparsestructure which is used to achieve signal sampling at low rate

As sampling technology expands nonuniform periodicsampling was considered namely coset sampling [8] Forone-dimensional multiband signals with arbitrary frequencysupport it could be sampled without loss arbitrarily close tothe theoretically minimum rate in the scenario of nonuni-form sampling [9] In [10] a universal sampling pattern andcorresponding reconstruction algorithms were developedAnd it could guarantee well-conditioned reconstruction of allmultiband signals with a given spectrum occupancy boundwithout prior knowledge of spectral support Exploiting theconditions of exact reconstruction an explicit reconstructionformula has been derived [11] In [12] there was an iterativealgorithm for finding the minimum sampling frequency formultiband signals even when the ordering of replicas wasconstrained Another method of reconstructing multibandsignals with arbitrary spectrum support has been presentedallowing the use of low sampling rates close to the Landaurate the theoretically lowest sampling rate that still permitsperfect reconstruction of the sampled signal [13]

With the rapid development of Compressed Sensing (CS)theory it brought new ideas for processing of multibandsignal [14 15] in order to solve the problems of largebandwidth and big data A novel Analog-to-InformationConverter (AIC) architecture has been developed [16] inwhich the multiband signal would pass a wideband pseu-dorandom demodulator then integrated and sampled ata low rate With the sampling below the Nyquist rate itpresents promising reconstruction results In [17] by thesolution of one-finite-dimensional problem a method forjoint recovery of the entire set of sparse vectors has beendeveloped and it takes the continuous problem into a finite-dimensional one In [18] it has proved that the recovery of anarbitrary number of jointly sparse vectors was equivalent tothe recovery of a finite set of sparse vectors In [19] undermild conditions on the sparsity and measurement matrixthe analysis of average-case performance of 119897

12recovery of

multichannel signals has been given The spectral supportwithout any information in the reconstruction stage a perfectreconstruction scheme from point-wise sub-Nyquist ratesamples for multiband signals has been proposed whichcould ensure the perfect reconstruction formultiband signalssampled at the minimal rate [20 21] At present the widelyusedmethod of multiband signals with wide band is a systemcalled the Modulated Wideband Converter (MWC) whichwas proposed in [22] This system could be used to processwideband sparsemultiband signals with no prior informationon the transmitter carrier positions The multiband signalwas firstly multiplied by a bank of periodic waveforms Theproduct was then low-pass filtered and sampled uniformly ata low rate which could reduce the sampling rate significantlyTo realize the proposedMWC the circuit has been presentedwhich could sample multiband signals according to theiractual bandwidth occupation [23 24] A technique to tackleconventional analog mismatch errors in direct conversionreceivers has been presented including the mathematicalderivation about robustness under similar error environ-ments [25] In [26] it has developed performance limits of

sparse signals support recovery when Multiple MeasurementVectors (MMV) were available and the proposed methodol-ogy also had the potential to address other theoretical andpractical issues associated with sparse signal recovery Inorder to solve the problemof joint sparse recovery five greedyalgorithms designed for the Single Measurement Vector(SMV) sparse approximation problem have been extended totheMMVproblem [27] Another novel approach to obtainingthe solution to a sequence of SMV problems with a jointsupport has been presented which could be adaptive in thatit was solved as a sequence of weighted SMV problems ratherthan collecting the measurement vectors and solving theMMV problem [28]

In order to achieve higher transmission rate the trans-mission signal will become the broadband signal So itwill bring more pressure to sampling and storage devicesespecially for hardware implementation As a result moreand more attention is focused on the broadband multibandsignal sampled at sub-Nyquist rate Due to the capabilityof processing the broadband signal MWC system seems tobe the best choice to process multiband signals with sparsespectrum structure which can not only be used in thescenario of arbitrary frequency support but also achieve thesignal reconstruction without any prior information aboutthe spectral support [29] FurthermoreMWCtechnology hasbeen widely used in the field of cognitive radio to achievewideband spectrum sensing [30 31] However MWC runs byadopting the idea of CS which contains many restrictionsin which the number of observation times is the mostimportant problem The number of sampling channels isequivalent to the MWC system which makes effects on thereconstruction performance Present researches cannot offeran explicit solution so it leads to the unsatisfactory andunstable performance of signal reconstruction followed bythe present principles and it also brings enormous difficultiesto the realization of hardware

In this paper we propose a simplified multiband sam-pling and detection method based on the traditional MWCstructure which can avoid the challenges of signal samplingbrought by high frequencies and wide bandwidth for mmwave systems For the scenario of signals with arbitraryfrequency support only MWC has the ability to reconstructthem without any prior knowledge but its performance isnot ideal Based on MWC structure this proposed methodtakes full advantage of the beneficial spectrum aliasing toachieve signal sampling at sub-Nyquist rate Compared withthe traditional MWC system it provides the exact quantityof sampling channels which is far lower than that of MWCIn the reconstruction stage the proposed method simplifiesthe computational complexity by exploiting simple linearoperations instead of CS recovery algorithms and providesmore stable performance of signal recovery Moreover MWCstructure has the ability to apply to different bands used inmm wave communications by mixed processing which issimilar to spread spectrum technology

The remainder of this paper is organized as followsSection 2 describes the principle of MWC system and someproblemsWepresent amethodofmmwave communications

International Journal of Antennas and Propagation 3

p1(t)

pi(t)

pm(t)

h(t)

h(t)

h(t)

t = nTs

t = nTs

t = nTs

y1[n]

yi[n]

ym[n]

x(t)

Figure 1 The sampling part of MWC structure

based on MWC structure in Section 3 Simulation resultsdiscussed in Sections 4 and 5 conclude the paper

2 MWC System and Problem Statements

MWC system aims at efficient hardware implementation andlow computational loads on the digital processing In thereconstruction stage the equation of CS theory ingeniously iscombined to obtain the frequency support which is the keyto reduce the complexity of signal recovery and allow the low-rate processing [22] The principles of MWC are as follows

21 MWC Structure and Principle The inspiration of MWCstructure comes from the thought of AIC architecture andconventional parallel data processing methods For MWCstructure an analog mixing front-end achieves the spectrumalignment whose goal is to make a spectrum portion fromeach band appear in baseband It is the most important partto realize the low-rate sampling

An analog mixing front-end consists of several channelsand uses the mixing function 119901

119894(119905) to obtain different mixing

of an analog multiband signal 119909(119905) which includes severaltransmission signals modulated by high carrier frequenciesAnd the mixing function 119901

119894(119905) is similar to pseudorandom

code in spread-spectrum techniques which is chosen asa piecewise constant function that alternates between twolevels

Considering the limitation of CS theory the number 119898

of processing channels should reach a certain amount torecover such a sparse multiband signal The sampling partof MWC structure is depicted in Figure 1 After mixing themixtures are truncated by the same low-pass filterswith cutoff1(2119879

119904) Then the filtered signal is sampled at low rate 119879

119904

relative to the carrier frequencies of the input signal by usingcommon sampling theory

According to [22] we know that the input of LPF ℎ(119905) is alinear combination of 119891

119901-shifted copies of 119883(119891) where 119891

119901is

the frequency of the mixing function 119901119894(119905) and 119883(119891) stands

for the Fourier transform of the multiband signal 119909(119905) Afterlow-pass filtering only the part in baseband is retainedwhichincludes information from each band And then the filteredsignal is sampled uniformly at rate 119879

119904which is matched to

Construct Support informationReconstruct joint support

y[n] VSolve V = AU forsparsest matrix U

S

S = ⋃i

supp(Ui)frame V

Figure 2 Continuous to finite block

the cutoff frequencies of LPF Finally the channel output119910119894[119899]

can be obtained and the total samples y[119899] are used to recoverthe signal The relation between the sample sequences 119910

119894[119899]

and the input signal 119909(119905) is expressed as in frequency domain

119884119894(1198901198952120587119891119879

119904) =

infin

sum

119899=minusinfin

119910119894 [119899] 119890minus1198952120587119891119899119879

119904

=

infin

sum

119897=minusinfin

119888119894119897119883(119891 minus 119897119891

119901)

(1)

where 119888119894119897represents the Fourier coefficients of the mixing

function 119901119894(119905)

Rewrite (1) in matrix form as

y (119891) = Az (119891) (2)

where y(119891) is made up of the element 119910119894(119891) the Discrete-

Time Fourier Transform (DTFT) of 119894th sequence 119910119894[119899]

MatrixA contains the coefficients 119888119894119896and z(119891) consists of 119891

119901-

shifted copies of 119883(119891) which has sparse structureConsidering (2) it is similar to the typical problem in

CS theory So in the reconstruction stage CS is integratedto acquire the carrier frequency support Solving (2) directlyis regarded as the IMV problem which is so difficult In thissystem another method has been proposed in [15 21] Thismethod has a two-step flow which recovers the frequencysupport set from a finite-dimensional system at first and thenrecovers the signal For the first step the algorithm constructsa finite frame V from the sample sequence which can beobtained by computing [15]

Q = int119891isin119865119904

119910 (119891) 119910119867

(119891) d119891 =

+infin

sum

119899=minusinfin

119910 [119899] 119910119879[119899] (3)

And then any matrix V comes from Q = VV119867 119865119904denotes

the frequency range 119865119904= [minus119891

1199042 1198911199042] with 119891

119904= 1119879

119904

In the second step it finds the unique solution U to theMMV system V = AU that has the fewest nonzero rows andthe frequency support ofU equals that of themultiband signal119909(119905) The whole operations are gathered in a block namedcontinuous to finite (CTF) depicted in Figure 2

In this part the recovery algorithm under CS theoryis exploited to obtain the spectrum support taking fulladvantage of spectral sparsity It is also the key of the wholesystem However CS recovery algorithms have their owncomputation complexity mostly from iterative procedureAfter finding band position themultiband signal is recoveredby the position index information


22 Problem Statement As mentioned above the biggestfeature of MWC system is to apply CS theory to constructmultiband signals with sparse structure at low sampling ratewell below the Nyquist rate

CS theory is to solve the problembased on the probabilityIn order to achieve a high probability of reconstructing signalcertain conditions must be met in the process of sampling inother words the requirements for sampling times of signalshould be up to a certain amount Though sampling rateunder CS theory can be far below the Nyquist rate there stillexists a theoretical lower bound value which is not beingapplied directly because of a big difference to the practicalapplication In fact the quantity used in practice is larger thanthat sometimes far larger than that which is still less than therequirements of Nyquist sampling theorem

In MWC system combined with the typical equationunder CS theory all the mixing functions make up the mea-surementmatrix and the number of sampling channels is rownumber of matrices in other words the times of observationIn order to recover the sparse signal in the reconstructionstage the amount of sampling channels should be moreenough In [22] the relationship between the number ofbands 119873 and the quantity of sampling channels 119898 in theoryis given namely 119898 ge 119873 for nonblind reconstruction or119898 ge 2119873 for blind reconstruction However the simulationresults which is described later in Section 41 show that justmeeting the relationship is far from enough

Because the requirements of observation times are notclear and taking into account the fact that there are the greatdifferences between the theory and the actual situation whenfacing the unknown number of bands 119873 the clear basisof the sampling channel selection in practical application isnot given Just taking successful recovery as standard thereference value of the sampling channel amount is offeredfrom the perspective of experience Even so facing theunknown signal in application a certain error of referencevalue still exists The number of bands 119873 has great effectson the selection of sampling channels In order to ensure thesignal recovery it is easy to think of the effective methodsto increase the number of sampling channels However suchincreasing numbers will bring more difficulties to hardwareimplementations whichmeans thatmore components wouldbe needed adding the costs of the system as well as extraburden on calculation process due to more data

The most striking feature of MWC system is to handlethe multiband signal with sparse structure at a low rate farbelow the Nyquist rate The total sampling rate of MWC isthe product of the channel number and the actual samplingrate of each channel Therefore increasing the number ofsampling channels is undoubtedly to raise the total samplingrate which reduces the advantages of MWS system greatlyAlthough a method to reduce the actual physical channelby using collapse factor with the costs of improving thesampling rate is given [22] under the existing technologysuch problem is not solved radically On the other hand evenif the sampling channel is enough it leads to the instabilityrecovery performance which is even not satisfied because ofthe unstable property of CS recovery algorithm and is shownlater in Section 41

The expectation is to fundamentally solve the problemsof the uncertain quantity of sampling channels and theunstable performance of recovery besides new ideas andmethods should be taken into account In this paper based oncomprehensive considerations of the above issues and gettinginspiration from MWC structure as well as its ability to dealwith the signal which has arbitrary frequency support a sim-plified multiband sampling and detection method based onMWC structure for mm wave communications is presentedmain thought is to exploit the beneficial spectrum aliasing asshown in Section 42 Starting from the characteristics of thealiasing spectrum the frequency supports about each carrierfrequencies are acquired by calculation Then each bandis separated to obtain the integrated information from thealiasing spectrum to avoid the uncertainty results brought byCS More detail about this method is described in Section 3

3 A Simplified Multiband MethodBands Based on MWC Structure forMm Wave Communications

Considering the ability to process the multiband signalwith arbitrary frequency support MWC technology canbe used for signals in mm wave communications whichmay be common in 5G mobile networks However MWCsystem is not applied directly because of the problems ofthe uncertain conditions of sampling channel quantity andunstable reconstruction results

A simplifiedmultiband sampling and detectionmethod isproposed in this paper taking MWC structure into accountand exploiting the beneficial spectrum aliasing After mixingand low-pass filtering the results present the superpositionof shifted copies from each band modulated by the Fouriercoefficients of the mixing functions depicted in Figure 3However these coefficients that belong to any channel aredifferent from each other based on the frequency supportand it is also different for the same band in different samplingchannel because of the different mixing functions Thereexists a fact that for the same band in different channels theeffects brought by the spectrum support are the same Theseare also the basis of this method

31 Calculation Process To achieve the spectrum aliasingthis method also exploits the spread-spectrum techniques tomultiple the inputs by119879

119901-periodic waveforms 119901

119894(119905) Based on

MWC structure which is shown in Figure 2 aftermixing themixtures can be expressed as 119909

119894(119905) = 119909(119905)119901

119894(119905)

Considering any 119894th channel the Fourier transform of themixing functions is as follows

119888119894119897=

1

119879119901

int

119879119901

0

119901119894(119905) 119890minus119895(2120587119879

119901)119897119905d119905 (4)

where

119901119894(119905) =

infin

sum

119897=minusinfin

119888119894119897119890119895(2120587119879

119901)119897119905 (5)


0

fp B

k1fp k2fp k3fp fmax f

cik1cik2

cik3

clk3

clk2clk1

The spectrum of yi[n] The spectrum of yl[n]

The ith channel The lth channel

The spectrum of the multiband signal x(t)

Figure 3 Results of mixing and low-pass filtering

So the mixture has a Fourier expansion

119883119894(119891) = int

infin

minusinfin

119909119894(119905) 119890minus1198952120587119891119905d119905

= int

infin

minusinfin

119909 (119905) (

infin

sum

119897=minusinfin

119888119894119897119890119895(2120587119879

119901)119897119905)119890minus1198952120587119891119905d119905

=

infin

sum

119897=minusinfin

119888119894119897int

infin

minusinfin

119909 (119905) 119890minus1198952120587(119891minus119897119879

119901)119905d119905

=

infin

sum

119897=minusinfin

119888119894119897119883(119891 minus 119897119891

119901)

(6)

It represents a linear combination of 119891119901-shifted copies of

119883(119891) After low-pass filtering only the part in baseband isretained so it includes small copies from each band Theiramplitudes are determined by the mixing functions and thefrequency support Obviously if the spectrum support canbe acquired it is possible to deduce which copies for eachband are retained which is the key of this method In orderto achieve this purpose first the spectrum can be divided intofrequency intervals and code each interval According to thenumber of frequency intervals the locations of each band canbe determined Later in Section 32 there are some detailsabout parameter selection especially the relation betweenfrequencies of the mixing functions and the LPF cutoffAssuming that filtered signal contains only one copy of eachband and is guaranteed by the parameter selection for themultiband signal with119873 subsignals the sampled signal of 119894thchannel can be expressed as

119910119894(119891) =

119873

sum

119896=1

119888119894119897119896

119883119896(119891119896minus 119897119896119891119901) (7)


119910119894(119891) = [119888119894119897

1

1198881198941198972

sdot sdot sdot 119888119894119897119873

]

[[[[[[[

[

1198831(1198911minus 1198971119891119901)

1198832(1198912minus 1198972119891119901)

119883119873

(119891119873

minus 119897119873119891119901)

]]]]]]]

]

(8)

where 119910119894(119891) is the DTFT of the 119894th sequence 119910

119894[119899] 119891

119896

represents the frequency support of each band and 119883119896(119891)

is the spectrum of each signal band 119897119896stands for the label

number of shifted copies of119883119896(119891) So119883

119896(119891minus 119897119896119891119901) is 119897119896th 119891119901-

shifted copies of each band The Fourier coefficients 119888119894119897119896

aredetermined by the mixing functions 119901

119894(119905) in (5) and 119897

119896

And 119897119896is defined as follows

119897119896=

119891119896

119891119901

(9)

For the different sampling channels the parameters 119897119896of

each band are the same Suppose 119897119896is a constant and derive

the expression of 119883119896(119891 minus 119897

119896119891119901) For all 119898 sampling channels

get the equation

[[[[[[

[

1199101(119891)

1199102(119891)

119910119898

(119891)

]]]]]]

]

=

[[[[[[[

[

11988811198971

11988811198972

sdot sdot sdot 1198881119897119873

11988821198971

11988821198972

sdot sdot sdot 1198882119897119873

d

1198881198981198971

1198881198981198972

sdot sdot sdot 119888119898119897119873

]]]]]]]

]

[[[[[[[

[

1198831(1198911minus 1198971119891119901)

1198832(1198912minus 1198972119891119901)

119883119873

(119891119873

minus 119897119873119891119901)

]]]]]]]

]

(10)

orY (119891) = AXlowast (11)

To solve the equations with 119873 unknown numbers 119873

equations are required namely119898 = 119873 With reference to themethod of solving linear equations the expression of 119883

119896(119891)

is

(A | Y) = (

(

11988811198971

11988811198972

sdot sdot sdot 1198881119897119898

| 1199101(119891)

11988821198971

11988821198972

sdot sdot sdot 1198882119897119898

| 1199102(119891)

d

|

1198881198981198971

1198881198981198972

sdot sdot sdot 119888119898119897119898

| 119910119898

(119891)

)

)

(12)

By transforming the augmentedmatrix we can obtain thefollowing form

(A | Y) 997888rarr (

1 0 sdot sdot sdot 0 | 1198891(119891)

0 1 sdot sdot sdot 0 | 1198892(119891)

d

|

0 0 sdot sdot sdot 1 | 119889119898

(119891)

) (13)


where 119889119894(119891) is the linear combination of 119910

119894(119891) and equal to

119883119896(119891119896minus 119897119896119891119901) Then the expression119883

119896(119891119896minus 119897119896119891119901) is obtained

containing other119873 unknown numbers of 119897119896 which demands

other 119873 equationsTo acquire the other 119873 equations we need to carry out

the above operation again and connect the two correspondingequations with the bridge of the expression 119883

119896(119891119896minus 119897119896119891119901) It

means that completing the signal recovery needs totally 2119873

equationsAs we know in MWC system a theoretical conclusion

that is 119898 ge 119873 for nonblind reconstruction or 119898 ge 2119873 forblind reconstruction is given The proposed method has anadvantage that it can provide the explicit quantity of samplingchannels and solve the reconstruction stability problem Asdiscussed above completing the signal recovery needs totally2119873 equations while119873 equations are demanded to obtain theexpression119883

119896(119891119896minus119897119896119891119901) and another119873 equations are needed

to get the frequency support 119897119896 According to the conditions

of sampling channels quantity the proposed method haslower sampling rate than MWC system and can reduce thehardware implementation complexity with few channels

To simplify the analysis we consider that the multibandsignal has two subsignals 119873 = 2

For the first two channels we can get the equations

[1199101(119891)

1199102(119891)

] = [11988811198971

11988811198972

11988821198971

11988821198972

][

[

1198831(1198911minus 1198971119891119901)

1198832(1198912minus 1198972119891119901)

]

]

(14)

Solving (14)

1198831(1198911minus 1198971119891119901) =

11988821198972

1199101(119891) minus 119888

11198972

1199102(119891)

11988811198971

11988821198972

minus 11988811198972

11988821198971

(15a)

1198832(1198912minus 1198972119891119901) =

11988821198971

1199101(119891) minus 119888

11198971

1199102(119891)

11988811198972

11988821198971

minus 11988811198971

11988821198972

(15b)

And the same procedure may be easily adapted to obtainanother two channels

1198831(1198911minus 1198971119891119901) =

11988841198972

1199103(119891) minus 119888

31198972

1199104(119891)

11988831198971

11988841198972

minus 11988831198972

11988841198971

(16a)

1198832(1198912minus 1198972119891119901) =

11988841198971

1199103(119891) minus 119888

31198971

1199104(119891)

11988831198972

11988841198971

minus 11988831198971

11988841198972

(16b)

Simultaneous equations (15a) with (16a) and (15b) with(16b) get the equations

11988821198972

1199101(119891) minus 119888

11198972

1199102(119891)

11988811198971

11988821198972

minus 11988811198972

11988821198971

=11988841198972

1199103(119891) minus 119888

31198972

1199104(119891)

11988831198971

11988841198972

minus 11988831198972

11988841198971

11988821198971

1199101(119891) minus 119888

11198971

1199102(119891)

11988811198972

11988821198971

minus 11988811198971

11988821198972

=11988841198971

1199103(119891) minus 119888

31198971

1199104(119891)

11988831198972

11988841198971

minus 11988831198971

11988841198972

(17)

The solutions of (16a) and (16b) are the spectrum support119897lowast

119896 119896 = 1 2 Due to the effect of bandwidth the sequence

numbers of frequency intervals where the bands are occupiedmay be 119897

lowast

119896and 119897lowast

119896plusmn1 written as Bindex = 119897

lowast

119896minus1 119897lowast

119896 119897lowast

119896+1The

goal of this operation is to avoid the effect resulting from thebands that occupies two consecutive intervals due to arbitraryfrequency support

Once Bindex is found we can get the submatrix ABindexwhich contains the columns ofA indexed by Bindex Recoverthe information of multiband signal 119911

119897[119899] as follows

zBindex [119899] = A+Bindexy [119899]

119911119897 [119899] = 0 119897 notin Bindex

(18)

where A+Bindex = (A119867BindexABindex)minus1A119867Bindex is the pseudoin-

verse of ABindex and 119911119897[119899] is the inverse-DTFT of 119911

119897(119891)

This process is a conventionalmatrix processing Itmeansthat if we can recover the frequency support successfully wewill get the original signal

32 Parameters Selection and Performance Analysis Thecopyof 119883119896(119891) retained after low-pass filtering is the one that

locates around zero frequency in baseband so the number offrequency intervals119871 should be oddwhich is also determinedby the frequencies of mixing functions 119891

119901 in other words 119871

stands for the number of 119891119901-shifted copies of each band

119871 =119891max119891119901

(19)

where119891max represents the highest frequency of themultibandsignal

This method exploits the beneficial spectrum aliasingfrom different bands To avoid the aliasing from the singleband the frequencies of mixing functions119891

119901should be larger

than the maximum bandwidth of all bands namely 119891119901

ge 119861The output of LPF contains only one copy of each bandmatching the shifting effect so the cutoff of LPF 119891

119904should

be equal to 119891119901 So it is easily seen that only if 119891

119901is chosen as

119861 the minimum sampling rate can be achievedThe basis of the selection of main parameters is to

offer the beneficial spectrum aliasing and this is the key ofthe proposed method The choice of the mixing functionfrequencies can make sure that after low-pass filtering thereare only small copies in baseband retained to avoid aliasingthemselves The method exploits the beneficial spectrumaliasing and samples the low-pass filtered signal at a low rateto obtain the information about each band of the multibandsignal Combined with signal superposition principle thespectrum support about each carrier frequency is acquiredby calculating the finite samples through a certain quantityof sampling channels And the amount of channels is equalto twice of the number of subsignals After that each bandis separated from the spectrum aliasing and the multibandsignal recovery is completed The proposed method replacesthe partial processing steps in MWC system and eliminatesthe uncertainty factors For the frequency support recoverythis method exploits simple linear operations instead of CSrecovery algorithm which can improve the performance ofsignal recovery This is also the greatest advantage of themethod Meanwhile MWC system has the unstable andunsatisfactory reconstruction performance More details ofthe simulation results will be presented in next part


4 Simulation Results

We now demonstrate several performance aspects of MWCsystem and our approach by using the simulation resultsIn this paper taking multiband signals in mm wave com-munications as the research object a simplified multibandsampling and detection method based on MWC structureis presented to solve the problem of uncertain quantity ofsampling channels in MWC system The multiband signalmodel is described as follows

119909 (119905) =

119873

sum

119894=1

radic119864119894119861119894sinc (119861

119894(119905 minus 120591119894)) cos (2120587119891

119894(119905 minus 120591119894)) (20)

There are 119873 pairs of bands (or 119873 subsignals) becausetwo symmetrical couples stand for one subsignal and 2119873 isthe number of the total bands 119861

119894stands for the width of

each band 119864119894represents the energy coefficients and 120591

119894is the

time offsets For each subsignal the carrier frequencies 119891119894are

chosen uniformly at random in the interested rangeThe proposed method starts with the situation of non-

clear conditions for the sampling channel quantity and theimperfect performance of signal reconstruction as shown inthe first part Then the method exploits the MWC structureespecially the beneficial aliasing spectrum depicted in thenext part Finally the advantage of the proposed method willbe demonstrated

41 Reconstruction Performance of MWC System In thereconstruction stage of MWC system CS theory is integratedinto the process The mixing functions play the role of themeasurement matrix In order to recover the multibandsignal the quantity of sampling channels should be enoughHowever in fact the theoretical value is far from enough Asmentioned above we know the key of the signal processingis to recover the frequency support which decides the finalresults Thus let the rate of correct support recovery be thereference object to measure the reconstruction performance

The carrier frequency119891119894is chosen uniformly at random in

the range [60GHz 70GHz] and the value 119861 is chosen from20MHz to 300MHz The number of subsignals 119873 is equalto 1 2 and 3 Simulation process is repeated 1000 times toshow the performance of signal recovery And the numberof sampling channels ranges from 10 to 100 The simulationresults are demonstrated in Figures 4 5 and 6

In these three figures it is easily seen that for the same119873 there are different results with different bandwidthsOn the whole the results with 119873 = 1 have the bestperformance since the number of bands grows the correctsupport recovery rate decreases However in one figure thisrate is not decided by the bandwidth simply especially forFigures 5 and 6 the smallest bandwidth 119861 = 20MHz has thelowest success rate The vast majority of success rate is lowerthan 90 even for the case of one subsignal there are alsoseveral results below 90

On the other hand when the success rate of supportrecovery can reach up to 90 in the case of one subsignalmostly more than ten channels are needed and in other two

20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channelsC

orre

ct su

ppor

t rec

over

y ra

te

Number of Subsignal = 1

B = 300MHzB = 280MHzB = 260MHzB = 240MHzB = 220MHzB = 200MHzB = 180MHz

B = 160MHzB = 140MHzB = 120MHzB = 100MHzB = 80MHzB = 60MHzB = 40MHzB = 20MHz

Figure 4 Recovery performance with one subsignal

cases even if the number of sampling channels is larger thanten times of 119873 the simulation results are still unsatisfactory

Conclusively the simulation results show that the signalrecovery is unstable and mostly unsatisfied just since CStheory is to solve the problem based on the probabilityand nonclear condition of sampling channels Because ofthe property of instability it cannot play a guiding role inpractical application

42 Result of the Mixing Stage The method exploits thebeneficial spectrum aliasing After low-pass filtering onlythe part in baseband is retained which includes informationfrom each band And we sample the low-pass filtered signalat a low rate to obtain the information about every band ofthe multiband signalThese are also the bases of this methodFor simplifying we consider the case of two subsignals Thecarrier frequency 119891

119894is chosen uniformly at random in the

range [71GHz 76GHz] and the bandwidth 119861 is 500MHzThe positive frequency part of signal spectrum is shown inFigure 7 Here the sampling rate 119891

119904is slightly larger than 119861

In Figure 8 there are the sampled signals of the outputof LPF from channel 1 and 2 It is easily seen that theonly difference among each channel is the magnitude effectscaused by the Fourier coefficients of the mixing functions


20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channels

Cor

rect

supp

ort r

ecov

ery

rate




Figure 5 Recovery performance with two subsignals

We can also see that the locations of shifted copies whichbelong to each band are the same in baseband due to the samefrequency support and the same band in different samplingchannels are different caused by the different mixing func-tions In baseband several copies from different bands aliastogether including all information of bands and it is called thebeneficial spectrum aliasing And this method is put forwardjust based on these characteristics

43 System Sampling Rate Comparison As discussed aboveMWC system has unstable effects on signal recovery And thesimulation results with 119873 = 1 have the best performance asthe number of bands grows the correct support recovery ratedecreases In order to compare the whole system samplingrate between MWC system and the proposed one we choosethe case of one subsignal For two methods the samplingrate of each channel is in both chosen slightly larger thanbandwidth 119861 As shown in Figure 9 we only consider thecases that the correct support recovery rate is more than 90

The system sampling rate is the product of the channelquantity and the actual sampling rate of each channelSince this method provides the exact conditions of samplingchannels quantity which is equal to the minimum limittheoretical value of MWC system it is possible to handle the

20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channelsC

orre

ct su

ppor

t rec

over

y ra

te




Figure 6 Recovery performance with three subsignals

signal with the lower sampling rate than MWC system It isseen in Figure 9 whose ordinate is the ratio of the samplingrate of MWC to that of the proposed one that the proposedmethod has lower sampling rate

5 Conclusion

In this paper a simplified multiband sampling and detectionmethod based on MWC structure is proposed for mm wavecommunications MWC structure which is multichannelparallel provides the beneficial spectrum aliasing Startingfrom it the low-pass filtered signal is sampled at a low rateto obtain the information about each band of the multibandsignal and to acquire the spectrum support by calculating thefinite samples using a certain quantity of sampling channelsThen each copy can be separated from the aliasing spectrumand on this basis to recover the multiband signal Comparedwith the traditional MWC technology the proposed methodprovides the exact conditions on sampling channels quantitywhich is smaller than that of traditional MWC system Andit means that the sampling rate of the whole system is muchlower Moreover the main idea of the proposed method isto get the spectrum support Compared with the traditionalMWC systems integrated CS theory the proposed method


71 715 72 725 73 735 74 745 75 755 760

05

1

15

2

25

3Spectrum of original signal

Frequency (GHz)

Mag

nitu

de

times106

Figure 7 Spectrum of original multiband signal

0 05 1 15 20

100

200

300

Frequency (Hz)

Mag

nitu

de

0

100

200

300

400

Frequency (Hz)

Mag

nitu

de

times108

times108

DigitalSignalSamples signal (m = 2)


minus2 minus15 minus1 minus05

0 05 1 15 2minus2 minus15 minus1 minus05

Figure 8 The beneficial spectrum aliasing from two channels

20 40 60 80 100 120 140 160 180 200 220 240 260 2802

3

4

5

6

7

8

9

10

11System sampling rate comparison

Bandwidth (MHz)

Ratio

=ra

teof

MW

Cpr

opos

ed ra

te

Figure 9 System sampling rate comparison

simplifies the computational complexity in the reconstruc-tion stage by using simple linear operations instead of CSrecovery algorithm And it also can avoid the instability andimprove the performance of signal recovery due to the certaincondition of sampling channels quantity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China under Grants nos 61201143 and91438205 and the Fundamental Research Funds for theCentral Universities (Grant no HIT IBRSEM 201309)

References

[1] M Elkashlan T Q Duong and H-H Chen ldquoMillimeter-wavecommunications for 5G fundamentals part I [guest editorial]rdquoIEEE Communications Magazine vol 52 no 9 pp 52ndash54 2014

[2] M Elkashlan T Q Duong and H-H Chen ldquoMillimeter-wavecommunications for 5G-Part 2 Applicationsrdquo IEEE Communi-cations Magazine vol 53 no 1 pp 166ndash167 2015

[3] R C Daniels and R W Heath ldquo60GHz wireless communica-tions emerging requirements and design recommendationsrdquoIEEE Vehicular Technology Magazine vol 2 no 3 pp 41ndash502007

[4] PWang Y Li L Song andBVucetic ldquoMulti-gigabitmillimeterwave wireless communications for 5G from fixed access tocellular networksrdquo IEEE Communications Magazine vol 53 no1 pp 168ndash178 2015

[5] L Zhou and Y Ohashi ldquoLow complexity linear receivers formmWave LOS-MIMO systems with uniform circular arraysrdquo inProceedings of the IEEE 80th Vehicular Technology Conference(VTC Fall rsquo14) pp 1ndash5 Vancouver Canada September 2014

[6] L Zhou andYOhashi ldquoPerformance analysis ofmmWave LOS-MIMO systems with uniform circular arraysrdquo in Proceedings ofthe 81st IEEE Vehicular Technology Conference (VTC Spring rsquo15)pp 1ndash5 IEEE Glasgow Scotland May 2015

[7] PWang Y Li X Yuan L Song and B Vucetic ldquoTens of gigabitswireless communications over E-band LoS MIMO channelswith uniform linear antenna arraysrdquo IEEE Transactions onWireless Communications vol 13 no 7 pp 3791ndash3805 2014

[8] B Foster and C Herley ldquoExact reconstruction from periodicnonuniform samplesrdquo in Proceedings of the 20th InternationalConference on Acoustics Speech amp Signal Processing vol 2 pp1452ndash1455 IEEE Detroit Mich USA May 1995

[9] C Herley and P W Wong ldquoMinimum rate sampling of signalswith arbitrary frequency supportrdquo in Proceedings of the IEEEInternational Conference on Image Processing (ICIP rsquo96) pp 85ndash88 Lausanne Switzerland September 1996

[10] P Feng and Y Bresler ldquoSpectrum-blind minimum-rate sam-pling and reconstruction of multiband signalsrdquo in Proceedingsof the IEEE International Conference on Acoustics Speech andSignal Processing (ICASSP rsquo96) vol 3 pp 1688ndash1691 IEEEComputer Society Atlanta Ga USA May 1996


[11] R Venkataramani and Y Bresler ldquoPerfect reconstruction for-mulas and bounds on aliasing error in sub-Nyquist nonuniformsampling of multiband signalsrdquo IEEE Transactions on Informa-tion Theory vol 46 no 6 pp 2173ndash2183 2000

[12] Y-P Lin Y-D Liu and S-M Phoong ldquoA new iterativealgorithm for finding the minimum sampling frequency ofmultiband signalsrdquo IEEE Transactions on Signal Processing vol58 no 10 pp 5446ndash5450 2010

[13] D Qu and J Zhou ldquoA novel sparse multiband signal recon-struction method by using Periodic Nonuniform Samplingrdquo inProceedings of the 5th International Congress on Image and SignalProcessing (CISP rsquo12) pp 1412ndash1416 IEEE Chongqing ChinaOctober 2012

[14] J N Laska S Kirolos M F Duarte T S Ragheb R GBaraniuk and Y Massoud ldquoTheory and implementation of ananalog-to-information converter using randomdemodulationrdquoin Proceedings of the IEEE International Symposium on Circuitsand Systems (ISCAS rsquo07) pp 1959ndash1962 IEEE NewOrleans LaUSA May 2007

[15] J A Tropp J N Laska M F Duarte J K Romberg andR G Baraniuk ldquoBeyond Nyquist efficient sampling of sparsebandlimited signalsrdquo IEEE Transactions on InformationTheoryvol 56 no 1 pp 520ndash544 2010

[16] S Kirolos J Laska M Wakin et al ldquoAnalog-to-informationconversion via random demodulationrdquo in Proceedings of theIEEEDallasICASWorkshop onDesign Applications Integrationand Software pp 71ndash74 IEEE Richardson Tex USA October2006

[17] M Mishali and Y C Eldar ldquoThe continuous joint sparsity priorfor sparse representations theory and applicationsrdquo in Proceed-ings of the 2nd IEEE International Workshop on ComputationalAdvances in Multi-Sensor Adaptive Processing pp 125ndash128 StThomas Virgin Islands USA December 2007

[18] M Mishali and Y C Eldar ldquoReduce and boost recoveringarbitrary sets of jointly sparse vectorsrdquo IEEE Transactions onSignal Processing vol 56 no 10 pp 4692ndash4702 2008

[19] Y C Eldar and H Rauhut ldquoAverage case analysis of multichan-nel sparse recovery using convex relaxationrdquo IEEE Transactionson Information Theory vol 56 no 1 pp 505ndash519 2010

[20] M Mishali and Y C Eldar ldquoSpectrum-blind reconstruction ofmulti-band signalsrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo08) pp 3365ndash3368 IEEE Las Vegas Nev USA April 2008

[21] MMishali and Y C Eldar ldquoBlindmultiband signal reconstruc-tion compressed sensing for analog signalsrdquo IEEE Transactionson Signal Processing vol 57 no 3 pp 993ndash1009 2009

[22] M Mishali and Y C Eldar ldquoFrom theory to practice sub-Nyquist sampling of sparse wideband analog signalsrdquo IEEEJournal on Selected Topics in Signal Processing vol 4 no 2 pp375ndash391 2010

[23] M Mishali Y C Eldar O Dounaevsky and E Shoshan ldquoSub-Nyquist acquisition hardware for wideband communicationrdquo inProceedings of the IEEE Workshop on Signal Processing Systems(SiPS rsquo10) pp 156ndash161 San Francisco Calif USA October 2010

[24] M Mishali Y C Eldar O Dounaevsky and E ShoshanldquoXampling analog to digital at sub-Nyquist ratesrdquo IET CircuitsDevices amp Systems vol 5 no 1 pp 8ndash20 2011

[25] C Choudhuri A Ghosh U Mitra and S Pamarti ldquoRobustnessof xampling-based RF receivers against analog mismatchesrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo12) pp 2965ndash2968IEEE Kyoto Japan March 2012

[26] Y Jin and B D Rao ldquoSupport recovery of sparse signals in thepresence of multiple measurement vectorsrdquo IEEE Transactionson Information Theory vol 59 no 5 pp 3139ndash3157 2013

[27] J D Blanchard M Cermak D Hanle and Y Jing ldquoGreedyalgorithms for joint sparse recoveryrdquo IEEE Transactions onSignal Processing vol 62 no 7 pp 1694ndash1704 2014

[28] R Amel and A Feuer ldquoAdaptive identification and recovery ofjointly sparse vectorsrdquo IEEE Transactions on Signal Processingvol 62 no 2 pp 354ndash362 2014

[29] H Sun W-Y Chiu J Jiang A Nallanathan and H V PoorldquoWideband spectrum sensing with sub-Nyquist sampling incognitive radiosrdquo IEEE Transactions on Signal Processing vol60 no 11 pp 6068ndash6073 2012

[30] H Sun A Nallanathan S Cui and C-X Wang ldquoCooperativewideband spectrum sensing over fading channelsrdquo IEEE Trans-actions on Vehicular Technology 2015

[31] H Sun A Nallanathan C-X Wang and Y Chen ldquoWidebandspectrum sensing for cognitive radio networks a surveyrdquo IEEEWireless Communications vol 20 no 2 pp 74ndash81 2013

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



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Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



wide frequency spectrum Such feature is called the sparsestructure which is used to achieve signal sampling at low rate

As sampling technology expands nonuniform periodicsampling was considered namely coset sampling [8] Forone-dimensional multiband signals with arbitrary frequencysupport it could be sampled without loss arbitrarily close tothe theoretically minimum rate in the scenario of nonuni-form sampling [9] In [10] a universal sampling pattern andcorresponding reconstruction algorithms were developedAnd it could guarantee well-conditioned reconstruction of allmultiband signals with a given spectrum occupancy boundwithout prior knowledge of spectral support Exploiting theconditions of exact reconstruction an explicit reconstructionformula has been derived [11] In [12] there was an iterativealgorithm for finding the minimum sampling frequency formultiband signals even when the ordering of replicas wasconstrained Another method of reconstructing multibandsignals with arbitrary spectrum support has been presentedallowing the use of low sampling rates close to the Landaurate the theoretically lowest sampling rate that still permitsperfect reconstruction of the sampled signal [13]

With the rapid development of Compressed Sensing (CS)theory it brought new ideas for processing of multibandsignal [14 15] in order to solve the problems of largebandwidth and big data A novel Analog-to-InformationConverter (AIC) architecture has been developed [16] inwhich the multiband signal would pass a wideband pseu-dorandom demodulator then integrated and sampled ata low rate With the sampling below the Nyquist rate itpresents promising reconstruction results In [17] by thesolution of one-finite-dimensional problem a method forjoint recovery of the entire set of sparse vectors has beendeveloped and it takes the continuous problem into a finite-dimensional one In [18] it has proved that the recovery of anarbitrary number of jointly sparse vectors was equivalent tothe recovery of a finite set of sparse vectors In [19] undermild conditions on the sparsity and measurement matrixthe analysis of average-case performance of 119897

12recovery of

multichannel signals has been given The spectral supportwithout any information in the reconstruction stage a perfectreconstruction scheme from point-wise sub-Nyquist ratesamples for multiband signals has been proposed whichcould ensure the perfect reconstruction formultiband signalssampled at the minimal rate [20 21] At present the widelyusedmethod of multiband signals with wide band is a systemcalled the Modulated Wideband Converter (MWC) whichwas proposed in [22] This system could be used to processwideband sparsemultiband signals with no prior informationon the transmitter carrier positions The multiband signalwas firstly multiplied by a bank of periodic waveforms Theproduct was then low-pass filtered and sampled uniformly ata low rate which could reduce the sampling rate significantlyTo realize the proposedMWC the circuit has been presentedwhich could sample multiband signals according to theiractual bandwidth occupation [23 24] A technique to tackleconventional analog mismatch errors in direct conversionreceivers has been presented including the mathematicalderivation about robustness under similar error environ-ments [25] In [26] it has developed performance limits of

sparse signals support recovery when Multiple MeasurementVectors (MMV) were available and the proposed methodol-ogy also had the potential to address other theoretical andpractical issues associated with sparse signal recovery Inorder to solve the problemof joint sparse recovery five greedyalgorithms designed for the Single Measurement Vector(SMV) sparse approximation problem have been extended totheMMVproblem [27] Another novel approach to obtainingthe solution to a sequence of SMV problems with a jointsupport has been presented which could be adaptive in thatit was solved as a sequence of weighted SMV problems ratherthan collecting the measurement vectors and solving theMMV problem [28]

In order to achieve higher transmission rate the trans-mission signal will become the broadband signal So itwill bring more pressure to sampling and storage devicesespecially for hardware implementation As a result moreand more attention is focused on the broadband multibandsignal sampled at sub-Nyquist rate Due to the capabilityof processing the broadband signal MWC system seems tobe the best choice to process multiband signals with sparsespectrum structure which can not only be used in thescenario of arbitrary frequency support but also achieve thesignal reconstruction without any prior information aboutthe spectral support [29] FurthermoreMWCtechnology hasbeen widely used in the field of cognitive radio to achievewideband spectrum sensing [30 31] However MWC runs byadopting the idea of CS which contains many restrictionsin which the number of observation times is the mostimportant problem The number of sampling channels isequivalent to the MWC system which makes effects on thereconstruction performance Present researches cannot offeran explicit solution so it leads to the unsatisfactory andunstable performance of signal reconstruction followed bythe present principles and it also brings enormous difficultiesto the realization of hardware

In this paper we propose a simplified multiband sam-pling and detection method based on the traditional MWCstructure which can avoid the challenges of signal samplingbrought by high frequencies and wide bandwidth for mmwave systems For the scenario of signals with arbitraryfrequency support only MWC has the ability to reconstructthem without any prior knowledge but its performance isnot ideal Based on MWC structure this proposed methodtakes full advantage of the beneficial spectrum aliasing toachieve signal sampling at sub-Nyquist rate Compared withthe traditional MWC system it provides the exact quantityof sampling channels which is far lower than that of MWCIn the reconstruction stage the proposed method simplifiesthe computational complexity by exploiting simple linearoperations instead of CS recovery algorithms and providesmore stable performance of signal recovery Moreover MWCstructure has the ability to apply to different bands used inmm wave communications by mixed processing which issimilar to spread spectrum technology

The remainder of this paper is organized as followsSection 2 describes the principle of MWC system and someproblemsWepresent amethodofmmwave communications


p1(t)

pi(t)

pm(t)

h(t)

h(t)

h(t)

t = nTs

t = nTs

t = nTs

y1[n]

yi[n]

ym[n]

x(t)














119904




119901is






S

S = ⋃i

supp(Ui)frame V




119894[119899]


119884119894(1198901198952120587119891119879

119904) =

infin

sum

119899=minusinfin

119910119894 [119899] 119890minus1198952120587119891119899119879

119904

=

infin

sum

119897=minusinfin

119888119894119897119883(119891 minus 119897119891

119901)

(1)


function 119901119894(119905)


y (119891) = Az (119891) (2)




119901-



Q = int119891isin119865119904

119910 (119891) 119910119867

(119891) d119891 =

+infin

sum

119899=minusinfin

119910 [119899] 119910119879[119899] (3)



1199042 1198911199042] with 119891

119904= 1119879

119904















119894(119905) Based on


119894(119905) = 119909(119905)119901

119894(119905)


119888119894119897=

1

119879119901

int

119879119901

0

119901119894(119905) 119890minus119895(2120587119879

119901)119897119905d119905 (4)

where

119901119894(119905) =

infin

sum

119897=minusinfin

119888119894119897119890119895(2120587119879

119901)119897119905 (5)


0

fp B


cik1cik2

cik3

clk3

clk2clk1






119883119894(119891) = int

infin

minusinfin

119909119894(119905) 119890minus1198952120587119891119905d119905

= int

infin

minusinfin

119909 (119905) (

infin

sum

119897=minusinfin

119888119894119897119890119895(2120587119879

119901)119897119905)119890minus1198952120587119891119905d119905

=

infin

sum

119897=minusinfin

119888119894119897int

infin

minusinfin

119909 (119905) 119890minus1198952120587(119891minus119897119879

119901)119905d119905

=

infin

sum

119897=minusinfin

119888119894119897119883(119891 minus 119897119891

119901)

(6)



119910119894(119891) =

119873

sum

119896=1

119888119894119897119896

119883119896(119891119896minus 119897119896119891119901) (7)


119910119894(119891) = [119888119894119897

1

1198881198941198972

sdot sdot sdot 119888119894119897119873

]

[[[[[[[

[

1198831(1198911minus 1198971119891119901)

1198832(1198912minus 1198972119891119901)

119883119873

(119891119873

minus 119897119873119891119901)

]]]]]]]

]

(8)


119894[119899] 119891

119896




119896(119891minus 119897119896119891119901) is 119897119896th 119891119901-



119894(119905) in (5) and 119897

119896


119897119896=

119891119896

119891119901

(9)





get the equation

[[[[[[

[

1199101(119891)

1199102(119891)

119910119898

(119891)

]]]]]]

]

=

[[[[[[[

[

11988811198971

11988811198972

sdot sdot sdot 1198881119897119873

11988821198971

11988821198972

sdot sdot sdot 1198882119897119873

d

1198881198981198971

1198881198981198972

sdot sdot sdot 119888119898119897119873

]]]]]]]

]

[[[[[[[

[

1198831(1198911minus 1198971119891119901)

1198832(1198912minus 1198972119891119901)

119883119873

(119891119873

minus 119897119873119891119901)

]]]]]]]

]

(10)

orY (119891) = AXlowast (11)



119896(119891)

is

(A | Y) = (

(

11988811198971

11988811198972

sdot sdot sdot 1198881119897119898

| 1199101(119891)

11988821198971

11988821198972

sdot sdot sdot 1198882119897119898

| 1199102(119891)

d

|

1198881198981198971

1198881198981198972

sdot sdot sdot 119888119898119897119898

| 119910119898

(119891)

)

)

(12)


(A | Y) 997888rarr (

1 0 sdot sdot sdot 0 | 1198891(119891)

0 1 sdot sdot sdot 0 | 1198892(119891)

d

|

0 0 sdot sdot sdot 1 | 119889119898

(119891)

) (13)



119894(119891) and equal to






119896(119891119896minus 119897119896119891119901) It









[1199101(119891)

1199102(119891)

] = [11988811198971

11988811198972

11988821198971

11988821198972

][

[

1198831(1198911minus 1198971119891119901)

1198832(1198912minus 1198972119891119901)

]

]

(14)

Solving (14)

1198831(1198911minus 1198971119891119901) =

11988821198972

1199101(119891) minus 119888

11198972

1199102(119891)

11988811198971

11988821198972

minus 11988811198972

11988821198971

(15a)

1198832(1198912minus 1198972119891119901) =

11988821198971

1199101(119891) minus 119888

11198971

1199102(119891)

11988811198972

11988821198971

minus 11988811198971

11988821198972

(15b)


1198831(1198911minus 1198971119891119901) =

11988841198972

1199103(119891) minus 119888

31198972

1199104(119891)

11988831198971

11988841198972

minus 11988831198972

11988841198971

(16a)

1198832(1198912minus 1198972119891119901) =

11988841198971

1199103(119891) minus 119888

31198971

1199104(119891)

11988831198972

11988841198971

minus 11988831198971

11988841198972

(16b)


11988821198972

1199101(119891) minus 119888

11198972

1199102(119891)

11988811198971

11988821198972

minus 11988811198972

11988821198971

=11988841198972

1199103(119891) minus 119888

31198972

1199104(119891)

11988831198971

11988841198972

minus 11988831198972

11988841198971

11988821198971

1199101(119891) minus 119888

11198971

1199102(119891)

11988811198972

11988821198971

minus 11988811198971

11988821198972

=11988841198971

1199103(119891) minus 119888

31198971

1199104(119891)

11988831198972

11988841198971

minus 11988831198971

11988841198972

(17)




lowast



lowast


119896 119897lowast

119896+1The



119897[119899] as follows


119911119897 [119899] = 0 119897 notin Bindex

(18)



119897(119891)






119871 =119891max119891119901

(19)






119904should


119901is chosen as






119909 (119905) =

119873

sum

119894=1

radic119864119894119861119894sinc (119861

119894(119905 minus 120591119894)) cos (2120587119891

119894(119905 minus 120591119894)) (20)




119894is the









20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channelsC

orre

ct su

ppor

t rec

over

y ra

te













20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channels

Cor

rect

supp

ort r

ecov

ery

rate








20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channelsC

orre

ct su

ppor

t rec

over

y ra

te






5 Conclusion



71 715 72 725 73 735 74 745 75 755 760

05

1

15

2

25


Frequency (GHz)

Mag

nitu

de

times106


0 05 1 15 20

100

200

300

Frequency (Hz)

Mag

nitu

de

0

100

200

300

400

Frequency (Hz)

Mag

nitu

de

times108

times108






20 40 60 80 100 120 140 160 180 200 220 240 260 2802

3

4

5

6

7

8

9

10


Bandwidth (MHz)

Ratio

=ra

teof

MW

Cpr

opos

ed ra

te





Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










p1(t)

pi(t)

pm(t)

h(t)

h(t)

h(t)

t = nTs

t = nTs

t = nTs

y1[n]

yi[n]

ym[n]

x(t)














119904




119901is






S

S = ⋃i

supp(Ui)frame V




119894[119899]


119884119894(1198901198952120587119891119879

119904) =

infin

sum

119899=minusinfin

119910119894 [119899] 119890minus1198952120587119891119899119879

119904

=

infin

sum

119897=minusinfin

119888119894119897119883(119891 minus 119897119891

119901)

(1)


function 119901119894(119905)


y (119891) = Az (119891) (2)




119901-



Q = int119891isin119865119904

119910 (119891) 119910119867

(119891) d119891 =

+infin

sum

119899=minusinfin

119910 [119899] 119910119879[119899] (3)



1199042 1198911199042] with 119891

119904= 1119879

119904















119894(119905) Based on


119894(119905) = 119909(119905)119901

119894(119905)


119888119894119897=

1

119879119901

int

119879119901

0

119901119894(119905) 119890minus119895(2120587119879

119901)119897119905d119905 (4)

where

119901119894(119905) =

infin

sum

119897=minusinfin

119888119894119897119890119895(2120587119879

119901)119897119905 (5)


0

fp B


cik1cik2

cik3

clk3

clk2clk1






119883119894(119891) = int

infin

minusinfin

119909119894(119905) 119890minus1198952120587119891119905d119905

= int

infin

minusinfin

119909 (119905) (

infin

sum

119897=minusinfin

119888119894119897119890119895(2120587119879

119901)119897119905)119890minus1198952120587119891119905d119905

=

infin

sum

119897=minusinfin

119888119894119897int

infin

minusinfin

119909 (119905) 119890minus1198952120587(119891minus119897119879

119901)119905d119905

=

infin

sum

119897=minusinfin

119888119894119897119883(119891 minus 119897119891

119901)

(6)



119910119894(119891) =

119873

sum

119896=1

119888119894119897119896

119883119896(119891119896minus 119897119896119891119901) (7)


119910119894(119891) = [119888119894119897

1

1198881198941198972

sdot sdot sdot 119888119894119897119873

]

[[[[[[[

[

1198831(1198911minus 1198971119891119901)

1198832(1198912minus 1198972119891119901)

119883119873

(119891119873

minus 119897119873119891119901)

]]]]]]]

]

(8)


119894[119899] 119891

119896




119896(119891minus 119897119896119891119901) is 119897119896th 119891119901-



119894(119905) in (5) and 119897

119896


119897119896=

119891119896

119891119901

(9)





get the equation

[[[[[[

[

1199101(119891)

1199102(119891)

119910119898

(119891)

]]]]]]

]

=

[[[[[[[

[

11988811198971

11988811198972

sdot sdot sdot 1198881119897119873

11988821198971

11988821198972

sdot sdot sdot 1198882119897119873

d

1198881198981198971

1198881198981198972

sdot sdot sdot 119888119898119897119873

]]]]]]]

]

[[[[[[[

[

1198831(1198911minus 1198971119891119901)

1198832(1198912minus 1198972119891119901)

119883119873

(119891119873

minus 119897119873119891119901)

]]]]]]]

]

(10)

orY (119891) = AXlowast (11)



119896(119891)

is

(A | Y) = (

(

11988811198971

11988811198972

sdot sdot sdot 1198881119897119898

| 1199101(119891)

11988821198971

11988821198972

sdot sdot sdot 1198882119897119898

| 1199102(119891)

d

|

1198881198981198971

1198881198981198972

sdot sdot sdot 119888119898119897119898

| 119910119898

(119891)

)

)

(12)


(A | Y) 997888rarr (

1 0 sdot sdot sdot 0 | 1198891(119891)

0 1 sdot sdot sdot 0 | 1198892(119891)

d

|

0 0 sdot sdot sdot 1 | 119889119898

(119891)

) (13)



119894(119891) and equal to






119896(119891119896minus 119897119896119891119901) It









[1199101(119891)

1199102(119891)

] = [11988811198971

11988811198972

11988821198971

11988821198972

][

[

1198831(1198911minus 1198971119891119901)

1198832(1198912minus 1198972119891119901)

]

]

(14)

Solving (14)

1198831(1198911minus 1198971119891119901) =

11988821198972

1199101(119891) minus 119888

11198972

1199102(119891)

11988811198971

11988821198972

minus 11988811198972

11988821198971

(15a)

1198832(1198912minus 1198972119891119901) =

11988821198971

1199101(119891) minus 119888

11198971

1199102(119891)

11988811198972

11988821198971

minus 11988811198971

11988821198972

(15b)


1198831(1198911minus 1198971119891119901) =

11988841198972

1199103(119891) minus 119888

31198972

1199104(119891)

11988831198971

11988841198972

minus 11988831198972

11988841198971

(16a)

1198832(1198912minus 1198972119891119901) =

11988841198971

1199103(119891) minus 119888

31198971

1199104(119891)

11988831198972

11988841198971

minus 11988831198971

11988841198972

(16b)


11988821198972

1199101(119891) minus 119888

11198972

1199102(119891)

11988811198971

11988821198972

minus 11988811198972

11988821198971

=11988841198972

1199103(119891) minus 119888

31198972

1199104(119891)

11988831198971

11988841198972

minus 11988831198972

11988841198971

11988821198971

1199101(119891) minus 119888

11198971

1199102(119891)

11988811198972

11988821198971

minus 11988811198971

11988821198972

=11988841198971

1199103(119891) minus 119888

31198971

1199104(119891)

11988831198972

11988841198971

minus 11988831198971

11988841198972

(17)




lowast



lowast


119896 119897lowast

119896+1The



119897[119899] as follows


119911119897 [119899] = 0 119897 notin Bindex

(18)



119897(119891)






119871 =119891max119891119901

(19)






119904should


119901is chosen as






119909 (119905) =

119873

sum

119894=1

radic119864119894119861119894sinc (119861

119894(119905 minus 120591119894)) cos (2120587119891

119894(119905 minus 120591119894)) (20)




119894is the









20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channelsC

orre

ct su

ppor

t rec

over

y ra

te













20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channels

Cor

rect

supp

ort r

ecov

ery

rate








20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channelsC

orre

ct su

ppor

t rec

over

y ra

te






5 Conclusion



71 715 72 725 73 735 74 745 75 755 760

05

1

15

2

25


Frequency (GHz)

Mag

nitu

de

times106


0 05 1 15 20

100

200

300

Frequency (Hz)

Mag

nitu

de

0

100

200

300

400

Frequency (Hz)

Mag

nitu

de

times108

times108






20 40 60 80 100 120 140 160 180 200 220 240 260 2802

3

4

5

6

7

8

9

10


Bandwidth (MHz)

Ratio

=ra

teof

MW

Cpr

opos

ed ra

te





Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





















119894(119905) Based on


119894(119905) = 119909(119905)119901

119894(119905)


119888119894119897=

1

119879119901

int

119879119901

0

119901119894(119905) 119890minus119895(2120587119879

119901)119897119905d119905 (4)

where

119901119894(119905) =

infin

sum

119897=minusinfin

119888119894119897119890119895(2120587119879

119901)119897119905 (5)


0

fp B


cik1cik2

cik3

clk3

clk2clk1






119883119894(119891) = int

infin

minusinfin

119909119894(119905) 119890minus1198952120587119891119905d119905

= int

infin

minusinfin

119909 (119905) (

infin

sum

119897=minusinfin

119888119894119897119890119895(2120587119879

119901)119897119905)119890minus1198952120587119891119905d119905

=

infin

sum

119897=minusinfin

119888119894119897int

infin

minusinfin

119909 (119905) 119890minus1198952120587(119891minus119897119879

119901)119905d119905

=

infin

sum

119897=minusinfin

119888119894119897119883(119891 minus 119897119891

119901)

(6)



119910119894(119891) =

119873

sum

119896=1

119888119894119897119896

119883119896(119891119896minus 119897119896119891119901) (7)


119910119894(119891) = [119888119894119897

1

1198881198941198972

sdot sdot sdot 119888119894119897119873

]

[[[[[[[

[

1198831(1198911minus 1198971119891119901)

1198832(1198912minus 1198972119891119901)

119883119873

(119891119873

minus 119897119873119891119901)

]]]]]]]

]

(8)


119894[119899] 119891

119896




119896(119891minus 119897119896119891119901) is 119897119896th 119891119901-



119894(119905) in (5) and 119897

119896


119897119896=

119891119896

119891119901

(9)





get the equation

[[[[[[

[

1199101(119891)

1199102(119891)

119910119898

(119891)

]]]]]]

]

=

[[[[[[[

[

11988811198971

11988811198972

sdot sdot sdot 1198881119897119873

11988821198971

11988821198972

sdot sdot sdot 1198882119897119873

d

1198881198981198971

1198881198981198972

sdot sdot sdot 119888119898119897119873

]]]]]]]

]

[[[[[[[

[

1198831(1198911minus 1198971119891119901)

1198832(1198912minus 1198972119891119901)

119883119873

(119891119873

minus 119897119873119891119901)

]]]]]]]

]

(10)

orY (119891) = AXlowast (11)



119896(119891)

is

(A | Y) = (

(

11988811198971

11988811198972

sdot sdot sdot 1198881119897119898

| 1199101(119891)

11988821198971

11988821198972

sdot sdot sdot 1198882119897119898

| 1199102(119891)

d

|

1198881198981198971

1198881198981198972

sdot sdot sdot 119888119898119897119898

| 119910119898

(119891)

)

)

(12)


(A | Y) 997888rarr (

1 0 sdot sdot sdot 0 | 1198891(119891)

0 1 sdot sdot sdot 0 | 1198892(119891)

d

|

0 0 sdot sdot sdot 1 | 119889119898

(119891)

) (13)



119894(119891) and equal to






119896(119891119896minus 119897119896119891119901) It









[1199101(119891)

1199102(119891)

] = [11988811198971

11988811198972

11988821198971

11988821198972

][

[

1198831(1198911minus 1198971119891119901)

1198832(1198912minus 1198972119891119901)

]

]

(14)

Solving (14)

1198831(1198911minus 1198971119891119901) =

11988821198972

1199101(119891) minus 119888

11198972

1199102(119891)

11988811198971

11988821198972

minus 11988811198972

11988821198971

(15a)

1198832(1198912minus 1198972119891119901) =

11988821198971

1199101(119891) minus 119888

11198971

1199102(119891)

11988811198972

11988821198971

minus 11988811198971

11988821198972

(15b)


1198831(1198911minus 1198971119891119901) =

11988841198972

1199103(119891) minus 119888

31198972

1199104(119891)

11988831198971

11988841198972

minus 11988831198972

11988841198971

(16a)

1198832(1198912minus 1198972119891119901) =

11988841198971

1199103(119891) minus 119888

31198971

1199104(119891)

11988831198972

11988841198971

minus 11988831198971

11988841198972

(16b)


11988821198972

1199101(119891) minus 119888

11198972

1199102(119891)

11988811198971

11988821198972

minus 11988811198972

11988821198971

=11988841198972

1199103(119891) minus 119888

31198972

1199104(119891)

11988831198971

11988841198972

minus 11988831198972

11988841198971

11988821198971

1199101(119891) minus 119888

11198971

1199102(119891)

11988811198972

11988821198971

minus 11988811198971

11988821198972

=11988841198971

1199103(119891) minus 119888

31198971

1199104(119891)

11988831198972

11988841198971

minus 11988831198971

11988841198972

(17)




lowast



lowast


119896 119897lowast

119896+1The



119897[119899] as follows


119911119897 [119899] = 0 119897 notin Bindex

(18)



119897(119891)






119871 =119891max119891119901

(19)






119904should


119901is chosen as






119909 (119905) =

119873

sum

119894=1

radic119864119894119861119894sinc (119861

119894(119905 minus 120591119894)) cos (2120587119891

119894(119905 minus 120591119894)) (20)




119894is the









20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channelsC

orre

ct su

ppor

t rec

over

y ra

te













20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channels

Cor

rect

supp

ort r

ecov

ery

rate








20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channelsC

orre

ct su

ppor

t rec

over

y ra

te






5 Conclusion



71 715 72 725 73 735 74 745 75 755 760

05

1

15

2

25


Frequency (GHz)

Mag

nitu

de

times106


0 05 1 15 20

100

200

300

Frequency (Hz)

Mag

nitu

de

0

100

200

300

400

Frequency (Hz)

Mag

nitu

de

times108

times108






20 40 60 80 100 120 140 160 180 200 220 240 260 2802

3

4

5

6

7

8

9

10


Bandwidth (MHz)

Ratio

=ra

teof

MW

Cpr

opos

ed ra

te





Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

fp B


cik1cik2

cik3

clk3

clk2clk1






119883119894(119891) = int

infin

minusinfin

119909119894(119905) 119890minus1198952120587119891119905d119905

= int

infin

minusinfin

119909 (119905) (

infin

sum

119897=minusinfin

119888119894119897119890119895(2120587119879

119901)119897119905)119890minus1198952120587119891119905d119905

=

infin

sum

119897=minusinfin

119888119894119897int

infin

minusinfin

119909 (119905) 119890minus1198952120587(119891minus119897119879

119901)119905d119905

=

infin

sum

119897=minusinfin

119888119894119897119883(119891 minus 119897119891

119901)

(6)



119910119894(119891) =

119873

sum

119896=1

119888119894119897119896

119883119896(119891119896minus 119897119896119891119901) (7)


119910119894(119891) = [119888119894119897

1

1198881198941198972

sdot sdot sdot 119888119894119897119873

]

[[[[[[[

[

1198831(1198911minus 1198971119891119901)

1198832(1198912minus 1198972119891119901)

119883119873

(119891119873

minus 119897119873119891119901)

]]]]]]]

]

(8)


119894[119899] 119891

119896




119896(119891minus 119897119896119891119901) is 119897119896th 119891119901-



119894(119905) in (5) and 119897

119896


119897119896=

119891119896

119891119901

(9)





get the equation

[[[[[[

[

1199101(119891)

1199102(119891)

119910119898

(119891)

]]]]]]

]

=

[[[[[[[

[

11988811198971

11988811198972

sdot sdot sdot 1198881119897119873

11988821198971

11988821198972

sdot sdot sdot 1198882119897119873

d

1198881198981198971

1198881198981198972

sdot sdot sdot 119888119898119897119873

]]]]]]]

]

[[[[[[[

[

1198831(1198911minus 1198971119891119901)

1198832(1198912minus 1198972119891119901)

119883119873

(119891119873

minus 119897119873119891119901)

]]]]]]]

]

(10)

orY (119891) = AXlowast (11)



119896(119891)

is

(A | Y) = (

(

11988811198971

11988811198972

sdot sdot sdot 1198881119897119898

| 1199101(119891)

11988821198971

11988821198972

sdot sdot sdot 1198882119897119898

| 1199102(119891)

d

|

1198881198981198971

1198881198981198972

sdot sdot sdot 119888119898119897119898

| 119910119898

(119891)

)

)

(12)


(A | Y) 997888rarr (

1 0 sdot sdot sdot 0 | 1198891(119891)

0 1 sdot sdot sdot 0 | 1198892(119891)

d

|

0 0 sdot sdot sdot 1 | 119889119898

(119891)

) (13)



119894(119891) and equal to






119896(119891119896minus 119897119896119891119901) It









[1199101(119891)

1199102(119891)

] = [11988811198971

11988811198972

11988821198971

11988821198972

][

[

1198831(1198911minus 1198971119891119901)

1198832(1198912minus 1198972119891119901)

]

]

(14)

Solving (14)

1198831(1198911minus 1198971119891119901) =

11988821198972

1199101(119891) minus 119888

11198972

1199102(119891)

11988811198971

11988821198972

minus 11988811198972

11988821198971

(15a)

1198832(1198912minus 1198972119891119901) =

11988821198971

1199101(119891) minus 119888

11198971

1199102(119891)

11988811198972

11988821198971

minus 11988811198971

11988821198972

(15b)


1198831(1198911minus 1198971119891119901) =

11988841198972

1199103(119891) minus 119888

31198972

1199104(119891)

11988831198971

11988841198972

minus 11988831198972

11988841198971

(16a)

1198832(1198912minus 1198972119891119901) =

11988841198971

1199103(119891) minus 119888

31198971

1199104(119891)

11988831198972

11988841198971

minus 11988831198971

11988841198972

(16b)


11988821198972

1199101(119891) minus 119888

11198972

1199102(119891)

11988811198971

11988821198972

minus 11988811198972

11988821198971

=11988841198972

1199103(119891) minus 119888

31198972

1199104(119891)

11988831198971

11988841198972

minus 11988831198972

11988841198971

11988821198971

1199101(119891) minus 119888

11198971

1199102(119891)

11988811198972

11988821198971

minus 11988811198971

11988821198972

=11988841198971

1199103(119891) minus 119888

31198971

1199104(119891)

11988831198972

11988841198971

minus 11988831198971

11988841198972

(17)




lowast



lowast


119896 119897lowast

119896+1The



119897[119899] as follows


119911119897 [119899] = 0 119897 notin Bindex

(18)



119897(119891)






119871 =119891max119891119901

(19)






119904should


119901is chosen as






119909 (119905) =

119873

sum

119894=1

radic119864119894119861119894sinc (119861

119894(119905 minus 120591119894)) cos (2120587119891

119894(119905 minus 120591119894)) (20)




119894is the









20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channelsC

orre

ct su

ppor

t rec

over

y ra

te













20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channels

Cor

rect

supp

ort r

ecov

ery

rate








20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channelsC

orre

ct su

ppor

t rec

over

y ra

te






5 Conclusion



71 715 72 725 73 735 74 745 75 755 760

05

1

15

2

25


Frequency (GHz)

Mag

nitu

de

times106


0 05 1 15 20

100

200

300

Frequency (Hz)

Mag

nitu

de

0

100

200

300

400

Frequency (Hz)

Mag

nitu

de

times108

times108






20 40 60 80 100 120 140 160 180 200 220 240 260 2802

3

4

5

6

7

8

9

10


Bandwidth (MHz)

Ratio

=ra

teof

MW

Cpr

opos

ed ra

te





Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119894(119891) and equal to






119896(119891119896minus 119897119896119891119901) It









[1199101(119891)

1199102(119891)

] = [11988811198971

11988811198972

11988821198971

11988821198972

][

[

1198831(1198911minus 1198971119891119901)

1198832(1198912minus 1198972119891119901)

]

]

(14)

Solving (14)

1198831(1198911minus 1198971119891119901) =

11988821198972

1199101(119891) minus 119888

11198972

1199102(119891)

11988811198971

11988821198972

minus 11988811198972

11988821198971

(15a)

1198832(1198912minus 1198972119891119901) =

11988821198971

1199101(119891) minus 119888

11198971

1199102(119891)

11988811198972

11988821198971

minus 11988811198971

11988821198972

(15b)


1198831(1198911minus 1198971119891119901) =

11988841198972

1199103(119891) minus 119888

31198972

1199104(119891)

11988831198971

11988841198972

minus 11988831198972

11988841198971

(16a)

1198832(1198912minus 1198972119891119901) =

11988841198971

1199103(119891) minus 119888

31198971

1199104(119891)

11988831198972

11988841198971

minus 11988831198971

11988841198972

(16b)


11988821198972

1199101(119891) minus 119888

11198972

1199102(119891)

11988811198971

11988821198972

minus 11988811198972

11988821198971

=11988841198972

1199103(119891) minus 119888

31198972

1199104(119891)

11988831198971

11988841198972

minus 11988831198972

11988841198971

11988821198971

1199101(119891) minus 119888

11198971

1199102(119891)

11988811198972

11988821198971

minus 11988811198971

11988821198972

=11988841198971

1199103(119891) minus 119888

31198971

1199104(119891)

11988831198972

11988841198971

minus 11988831198971

11988841198972

(17)




lowast



lowast


119896 119897lowast

119896+1The



119897[119899] as follows


119911119897 [119899] = 0 119897 notin Bindex

(18)



119897(119891)






119871 =119891max119891119901

(19)






119904should


119901is chosen as






119909 (119905) =

119873

sum

119894=1

radic119864119894119861119894sinc (119861

119894(119905 minus 120591119894)) cos (2120587119891

119894(119905 minus 120591119894)) (20)




119894is the









20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channelsC

orre

ct su

ppor

t rec

over

y ra

te













20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channels

Cor

rect

supp

ort r

ecov

ery

rate








20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channelsC

orre

ct su

ppor

t rec

over

y ra

te






5 Conclusion



71 715 72 725 73 735 74 745 75 755 760

05

1

15

2

25


Frequency (GHz)

Mag

nitu

de

times106


0 05 1 15 20

100

200

300

Frequency (Hz)

Mag

nitu

de

0

100

200

300

400

Frequency (Hz)

Mag

nitu

de

times108

times108






20 40 60 80 100 120 140 160 180 200 220 240 260 2802

3

4

5

6

7

8

9

10


Bandwidth (MHz)

Ratio

=ra

teof

MW

Cpr

opos

ed ra

te





Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119909 (119905) =

119873

sum

119894=1

radic119864119894119861119894sinc (119861

119894(119905 minus 120591119894)) cos (2120587119891

119894(119905 minus 120591119894)) (20)




119894is the









20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channelsC

orre

ct su

ppor

t rec

over

y ra

te













20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channels

Cor

rect

supp

ort r

ecov

ery

rate








20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channelsC

orre

ct su

ppor

t rec

over

y ra

te






5 Conclusion



71 715 72 725 73 735 74 745 75 755 760

05

1

15

2

25


Frequency (GHz)

Mag

nitu

de

times106


0 05 1 15 20

100

200

300

Frequency (Hz)

Mag

nitu

de

0

100

200

300

400

Frequency (Hz)

Mag

nitu

de

times108

times108






20 40 60 80 100 120 140 160 180 200 220 240 260 2802

3

4

5

6

7

8

9

10


Bandwidth (MHz)

Ratio

=ra

teof

MW

Cpr

opos

ed ra

te





Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channels

Cor

rect

supp

ort r

ecov

ery

rate








20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Sampling channelsC

orre

ct su

ppor

t rec

over

y ra

te






5 Conclusion



71 715 72 725 73 735 74 745 75 755 760

05

1

15

2

25


Frequency (GHz)

Mag

nitu

de

times106


0 05 1 15 20

100

200

300

Frequency (Hz)

Mag

nitu

de

0

100

200

300

400

Frequency (Hz)

Mag

nitu

de

times108

times108






20 40 60 80 100 120 140 160 180 200 220 240 260 2802

3

4

5

6

7

8

9

10


Bandwidth (MHz)

Ratio

=ra

teof

MW

Cpr

opos

ed ra

te





Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










71 715 72 725 73 735 74 745 75 755 760

05

1

15

2

25


Frequency (GHz)

Mag

nitu

de

times106


0 05 1 15 20

100

200

300

Frequency (Hz)

Mag

nitu

de

0

100

200

300

400

Frequency (Hz)

Mag

nitu

de

times108

times108






20 40 60 80 100 120 140 160 180 200 220 240 260 2802

3

4

5

6

7

8

9

10


Bandwidth (MHz)

Ratio

=ra

teof

MW

Cpr

opos

ed ra

te





Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article A Simplified Multiband Sampling and...

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Transcript of Research Article A Simplified Multiband Sampling and...