Erick L. Oberstar Fall 2001 Project: Sidelobe Canceller & GSC 1. Advanced Digital Signal Processing Sidelobe Canceller (Beam Former)

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1 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 1 Advaced Digital Sigal Proceig Sidelobe Caceller (Beam Former) Erick L. Obertar 001

2 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC Sidelobe caceller (Beam Former): A complete aalyi/dicuio of reult i icluded. Aumptio: 1. A liear equally paced array of thirtee eor paced by oe half wavelegth.. Plae wave propagatio. 3. All igal/iterferece are arrowbad. (Aumig for coveiece a ormalized frequecy of π.) 4. All arrival agle are defied with repect to the lie perpedicular to the axi of the eor array. The beampatter i defied a 10log 10 ( r(θ,π) ), where r(θ,π) =w H d(θ,π). Here w i the vector of beamformer weight ad d(θ,π) i the teerig vector or array repoe vector for directio θ ad frequecy π.aume the data coit of a igal arrivig from θ = 5.74 degree with power, a iterferer arrivig from θ I = degree with power I,ad ucorrelated (white) oie of power. The igal, iterferer, ad oie are all tatitically idepedet. The project i orgaized accordig to problem umber, e.g., 1, a, b, etc. For each, a ummary that highlight the reult obtaied ad cotai dicuio commet i icluded. Problem Statemet: A. Sidelobe Caceller #1 #10... W p + - Σ Θ i = Θ =-5.74 #11 w 11 #1 λ/ w 1 Σ #13 w 13

3 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 3 1. Aume the firt 10 eor are ued to form the primary atea output. Fid the weight vector (wp) of miimum orm (wp H wp) that ha uit repoe to igal arrivig from θ degree ( r(θ,π) = 1). Plot the beampatter for wp.. Aume I 0 ad that a adaptive weight i placed i eor chael 11 (1-13 are et to zero). Fid a expreio for the gai to the igal10log 10 ( r(θ,π) ), a a fuctio of /. Plot thi expreio over a rage of of / from 10 3 to Plot beampatter for / = 10, 1, Aume 0. Fid a expreio for the gai to the iterferer, 10log 10 ( r(θ I,π) ) a a fuctio of I / ad the umber of adaptive weight (either 1,, or 3 weight with uued weight et equal to zero). Plot thi expreio over a rage of I / from 10 3 to 10 5 for 1,, ad 3 adaptive weight. Alo plot the iterferece output power, 10log 10 ( I r(θ I,π) ), a a fuctio of I (over a rage 10 3 to 10 5, let = 1). Plot beampatter for I / = 10, 1, 10, 10 4 for a igle adaptive weight. Do multiple weight provide igificat icreae i iterferece cacellatio relative to a igle weight? 4. Aume I / = Plot the gai to the igal, 10log 10 ( r(θ,π) ), a a fuctio of / (over a rage 10 3 to 10 3 ) for oe (chael 11), two (chael 11,1), ad three (chael 11-13) adaptive weight. Commet. B. Geeralized Sidelobe Caceller We deire a beamformer that miimize the output power ubject to the igal repoe cotrait r(θ,π) = 1. Utilize all thirtee eor. 1. Determie w q ad C i the GSC repreetatio w = w q C w. Plot the beampatter of w q. Alo plot the beampatter of each colum of C (all o the ame graph).. Coider replacig C by C T = C T where T i a oigular matrix. Show that w i idepedet of T for ay data covariace matrix R. Commet. 3. Aume I / = Fid a expreio for the gai to the igal, 10log 10 ( r(θ,π) ), a a fuctio of /. Plot thi fuctio. 4. Fid a expreio for the gai to the iterferer, 10log 10 ( r(θ I,π) ),a a fuctio of I /. Plot thi expreio over a rage of I / from 10 3 to Note that: 1 H 1 H 1 1 ( A u)( v A ) ( A+ uv ) = A H v A u where all matrice are aumed oigular, u ad v are colum vector. 5. Aume I / = 10 4 ad that the igal i arrivig from 4 degree itead of θ (ot chagig the cotrait to thi directio). Evaluate ad plot the gai to the igal, 10log 10 ( r(4,π) ), a a fuctio of / (over a rage 10 3 to 10 3 ). Plot the beampatter for / = 10. Commet. 6. Aume the data i due to a igal at θ degree with power 10, ucorrelated oie with

4 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 4 power 1, ad five iterferer with the followig directio ad power: 1) 40 degree at 10 4 ) 30 degree at ) 0 degree at ) 45 degree at ) 70 degree at 10 4 Plot the beampatter. Compute the array gai, defied a the ratio of the SNR at the beamformer output to the SNR at a igle eor (i thi defiitio oie refer to both iterferece ad ucorrelated oie). PART A.1 Tap/(receiver) 1-10 are multiplied by a weight vector w p to atify the cotrait of paig deired igal (Θ,π) (The igal from directio Θ at frequecy π.) with uit gai. The cotrait i : M 1 k= 0 w k e jkφ 0 = g i.e. r( Θ, π ) = 9 k = 0 w e k jkφ = 1 The Steerig vector d(θ,π) = d( Θ, π ) = [1 e jφ e jφ j3φ j4φ j5φ j6φ j7φ j8φ j9φ e e e e e e e ] T where φ = i( Θ ) - the time delay the igal take to get to a adjacet atea after hittig oe. The tap weight vector wp wa calculated uig 1 g * R d( Θ, π ) wp = H 1 With R = 1 ice due to white oie oly o the iput without d ( Θ, π ) R d( Θ, π ) H -1 d( Θ, π ) the igal of iteret. Thi yield wp = d( Θ, π )(d ( Θ, π )d( Θ, π )) = 10 Plottig 10log 10 ( r(θ,π) ) where r(θ,π) = w H p d(θ,π)

5 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 5 Notice that the igal gai at Θ = i 1. I.e. it pae the deired igal without ditortio. Thi plot wa doe by a_1.m

6 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 6 PART A. The igal gai a a fuctio of igal to oie power wa foud. ( ) m m r 1 1, π + = Θ Where m i the umber of auxiliary tap. The above equatio wa plotted with m = 1 auxiliary tap. Thi plot i how below: The beam patter plot for = i how below:

7 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 7 The beam patter plot for = 1. 0 i how below: The beam patter plot for = i how below:

8 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 8 It become apparet a the igal to oie ratio get better more igal of iteret leak through the auxiliary tap cauig the Weier filter to kill the igal itelf i order to miimize the mea quared error. Thee plot were doe by a_.m

9 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 9 PART A.3 A expreio wa derived for the gai of the iterferer i term of the iterferece power to oie power ratio i : ( ) m m r i i i i 1 1, π + = Θ where m i the umber of auxiliary tap. Thi gai wa plotted over the rage of < < i 10 for 1,, ad 3 auxiliary tap. Thee plot are how below:

10 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 10 It i worthwhile to ote that the iterferece gai tart droppig at a lower INR a the umber of auxiliary tap icreae. 10 i i π The iterferece power, log ( r( Θ, ) ) i oie ratio follow below: 10 wa plotted a a fuctio of the iterferece to 3 i 5 over the rage of 10 < < 10 for 1,, ad 3 auxiliary tap. Thee plot

11 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 11

12 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 1 i 4 The beam patter for = 10,1,10,10 were plotted by weepig the teerig vector over the rage 90 to 90. Thee plot are how below: The above plot with INR = 0.01, doe ot have a ull at Θ i =

13 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 13 The above plot with INR = 1.0 i tartig to develop a ull at Θ i = The above plot with INR = 1.0 how a troger ull at Θ i = due to two additioal auxiliary tap.

14 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 14 The above plot with INR = 100 how a trog ull at Θ i = The above plot with INR = 10 4 how a trog ull at Θ i = However thi ull i ot igificatly troger tha the ull geerated whe the INR i two order of magitude maller.

15 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 15 The above plot with INR = 10 4 how a trog ull at Θ i = with two additioal tap. Thi ull i ot improved by addig the two additioal tap. I ummary iterferece cacellatio i related to both INR ad the umber of auxiliary tap. However it i mot eitive to the INR. The plot for part A.3 were geerated by a_3.m.

16 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 16 PART A.4 With the iterferece to oie power ratio i fixed at 10 4 the igal gai 10 log10 ( r( Θ, π ) ) wa 3 plotted while varyig the igal to oie ratio over the rage 3 10 < < 10 for the umber of auxiliary tap varyig from oe to three tap. Thee plot follow below ad are labeled correpodig to umber of additioal tap:

17 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 17 ` The ame primary problem exit with the beamformer. A the umber of auxiliary chael icreae, the amout of deired igal leakig through the auxiliary chael icreae. Whe thi happe the filter doe what it i uppoed to: miimize mea quared error, which maifet by reducig the filter output by the auxiliary tap gai time the igal of iteret. The plot for part A.4 were geerated by a_4.m.

18 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 18 PART B.1 The cotrait matrix C wa determied. Uig Matlab, I determied the orthogoal bai to C which i C a a well a the auxiliary chael weight w a. C a ca be iterpreted a a Blockig FIR Filter Bak cetered at Θ = Show below i the gai plot v.. igal agle for w q which i the bak of coefficiet for the primary atea tap. The Ca Matrix gai wa plotted for each colum of Ca. Thi plot i how below:

19 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 19 The previou two graph were geerated by the M-File: b_1.m

20 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 0 PART B.

21 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 1 PART B.3

22 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC PART B.4

23 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 3 PART B.5 The igal appear at a differet directio from where it i expected. The Filter treat the igal at four degree a iterferece ad geerate a ull i that directio.

24 Erick L. Obertar Fall 001 Project: Sidelobe Caceller & GSC 4 PARTB.6

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