ELG513 Smart Antennas S.Loyka he Concept of Beamformng Generc representaton of the array output sgnal, 1 where w y N 1 * = 1 = w x = w x (4.1) complex weghts, control the array pattern; y and x - narrowband sgnals; "" denotes Hermtan conjugate: w he array pattern s N 1 * = = ( w ) F( k) = w v( k) = w v ( k ) (4.) For narrowband sgnals, the frequency response of array elements s flat and can be presented by a complex number ( w ); x are complex ampltudes, x = a exp( jϕ ) (4.3) Beamformng s to choose w n such a way as to obtan a desred array pattern Fk ( ), ( Fθ ( ) n -D case),.e. fnd such w that provde desred Fk ( ). * 1 Notatons: bold captal (K) matrces; bold lower case (k) vectors; lower case regular (k) scalars; k - -th column of K. Lecture 4 1-Sep-15 1 (17)
ELG513 Smart Antennas S.Loyka Beamformng Usng Pattern Samplng For smplcty, consder -D case: F( θ) and v ( θ) Sample Fθ ( ) at M=N dstnct ponts, F = F( θ ), j = 1,..., M j F j j Defne array manfold matrx he soluton s v j = w v ( θ ) (4.4) = v ( θ ) f = w V j j 1 = w provdes a pattern, whch s the same as the desred pattern at M ponts θ j (but not necessarly n between). w f V (4.5) Important example: formng (N-1) zeros n the pattern (n gven drectons). Assume that the man beam s at broadsde, π θ 1 = and F( θ 1) = 1, F j becomes Usng (4.5), we fnd provdes the desred pattern. [ 1... ] 1 f = = e (4.6) = 1 1 w e V (4.7) Concluson: N-element array s able to form (N-1) nulls. Lecture 4 1-Sep-15 (17)
ELG513 Smart Antennas S.Loyka Null Steerng: Example 1 1 ULA Pattern.8.6.4. 9 6 3 3 6 9 F Complementary angle, deg. F N θ f w = 5; d = λ / ; [, 4, 4,7, 7] = = [ 1,,,, ] = [.9,.5,.33,.5,.9 ] Q1: explan why only 4 zeros can be formed (and not more). Q: do ths example yourself. Lecture 4 1-Sep-15 3 (17)
ELG513 Smart Antennas S.Loyka Null Steerng: Example Be careful when postonng zeros! 3 ULA Pattern.4 1.8 1..6 N θ f w = 5; d = λ / ; [,1, 1,7, 7] = = 9 6 3 3 6 9 F F Complementary angle, deg. [ 1,,,, ] = [.9,., 1.,.,.9] Q1: explan why there s a problem n ths case and suggest a way to avod t. Q: do ths example yourself. Lecture 4 1-Sep-15 4 (17)
ELG513 Smart Antennas S.Loyka Another example: beam steerng n a gven drecton Basc dea - compensate the phase shfts due to the wave propagaton. If the beam s to be steered at k then 1 1 ( 1 1 ) jk p jk p jk p w = v k = e e... e N N N (4.8) he normalzed pattern s N 1 F 1 1 ( k) = w v( k ) = r v( k k ) = N N v k k and F ( k ) = 1. Another form of the pattern, N ( ψ ψ) sn F( θ ) = ψ ψ N sn π π where ψ = d cos θ, ψ = d cosθ. λ λ { } ( ) = (4.9) (4.1) Lecture 4 1-Sep-15 5 (17)
ELG513 Smart Antennas S.Loyka Gratng Lobes Gratng lobe: there may be addtonal man lobe(s) steered n a dfferent (other than θ ) drecton. hs s a parastc lobe and s called gratng lobe. Introduce new angle (measured from broadsde) π π θ = θ ψ = d sn θ λ Gratng lobe condton: (4.11) ψ ψ = ±π sn θ = sn θ ± λ / d (4.1) Doesn t exst when λ d < (4.13) 1 sn θ It s very mportant; f the condton s not respected, there s a spurous response that can not be dstngushed from the man beam of the array. Lecture 4 1-Sep-15 6 (17)
ELG513 Smart Antennas S.Loyka wo specal cases. No steerng: θ = d < λ (4.14) Steerng n the entre half-plane: π λ θ = ± d < (4.15) If ths not feasble to mplement, some other measures must be taken: Element pattern: zero at the gratng lobe drectons. Irregular element locaton (random arrays). Lecture 4 1-Sep-15 7 (17)
ELG513 Smart Antennas S.Loyka Gratng Lobes: Example 1 ULA Pattern: gratng lobes d =.8 λ; N = 1.5 1 9 6 3 3 6 9 tet= tet=6 ULA Pattern: gratng lobes d = λ; N = 1.5 9 6 3 3 6 9 tet= tet=6 Lecture 4 1-Sep-15 8 (17)
ELG513 Smart Antennas S.Loyka Array Performance Parameters Unform lnear array (ULA) gan: N 1 N 1 πd( n m) G = wmwnsnc( ) n= m= λ where snc( x)=sn( x)/ x (see Van rees for a dervaton), and the weghts are normalzed: 1 (4.16) N 1 w 1 1 w = max F ( θ ) = 1 w = = l ( ) = 1 A more compact form of (4.16): where G ( ) 1 = w Sw (4.17) Snm =snc( πd ( n m ) / λ ) (4.18) Note that w ncludes steerng as well. Consder the specal case of d = nλ / and an ULA wth nonunform weghtng: N 1 1 ( ) 1 nm = δ nm, = = = = 1 S G w w w w (4.19) where w = w w s the norm (L ) of a vector (length). If the weghts are of the same magntude, w = 1/ N G = N Lecture 4 1-Sep-15 9 (17)
ELG513 Smart Antennas S.Loyka Note that beam steerng does not change the gan (sotropc elements). he array gan (n ths case) s recprocal of the magntude squared of the weght vector, G = w. If d nλ /, the gan wll depend on the steerng drecton. Unform weghtng maxmzes the gan of the ULA wth d = nλ /. Q.: prove t! Hnt : use Lagrange multplers or, better, the Schwarz nequalty. Lecture 4 1-Sep-15 1 (17)
ELG513 Smart Antennas S.Loyka SNR Gan Important functon of a receve array s to ncrease sgnal to nose rato (SNR), whch s characterzed the SNR gan assumng that the nose s spatally whte. he SNR gan s also known as nose gan. Consder the ncomng wave consstng of the requred sgnal and nose. he array element outputs are where s element. j ae ϕ x = and he array output s = s ξ (4.) ξ are the sgnal and nose at -th (4.1) jϕ y = a w e w ξ = a w w ξ where a s the requred sgnal ampltude. Lecture 4 1-Sep-15 11 (17)
ELG513 Smart Antennas S.Loyka SNR Gan he total (sgnalnose) output power j, j P = y = a w w σ w sgnal power nose power ; (4.) σ = ξ (4.3) R ξ = ξξ = σ I (spatally whte nose) (4.4) Normalze the weghts, w = 1. ( F max =?). he output SNR, SNR he SNR gan out = Ps a SNR P = = n σ w w SNRn n n / (4.5) = a σ (4.6) SNRout A = = w (4.7) SNR Lecture 4 1-Sep-15 1 (17)
ELG513 Smart Antennas S.Loyka SNR Gan For ULA wth d = nλ / A = G In general, d nλ / A G For a unformly weghted array, A = N. Q.: prove the statements above. Q.: can you make a more defnte statement about the relatonshp between A and G, e.g. A G or A G, when d λ /? Justfy your answer. Lecture 4 1-Sep-15 13 (17)
ELG513 Smart Antennas S.Loyka Summary Concept of beamformng. Array pattern synthess for gven levels. Null formng. Gratng lobes. Crtera of no gratng lobes. Array performance parameters: array gan and SNR gan. Effect of random perturbatons. Senstvty functon. References: H.L. Van rees, Optmum Array Processng, Wley, New York,. Homework: fll n the detals n the dervatons above. Do the examples yourself. Lecture 4 1-Sep-15 14 (17)
ELG513 Smart Antennas S.Loyka Appendx: Senstvty and olerance Factor Assume there are random errors (perturbatons) n the array weghts and the element locatons, jϕ w = ge ; g = g(1 ), ϕ = ϕ ϕ p = p p (4.8), ϕ - ampltude and phase errors, p - locaton errors For each array element, g, ϕ, p are the nomnal values. he expected power pattern (conventonal pattern s random), N 1 N 1 j( ϕ kp ( ) ) j( ϕl kpl F g e g e ) k (4.9) = = l= 1 Assume that all varatons are ndependent of each other, and that and ϕ are..d Gaussan. After some manpulatons, N 1 l j( ϕ ϕ ) jk( p p ) l F = F( k) = g g e l e l e N 1 = g g (1 σ ) σϕ - varance of ϕ πp / λ,, σ g - varance of l g, σλ - varance of ϕ λ ( σ σ ) (4.3) Lecture 4 1-Sep-15 15 (17)
ELG513 Smart Antennas S.Loyka F can be presented as ϕ ( ) σ = ϕ ϕ (4.31) ( ) σ g = g g (4.3) π σ λ = p p (4.33) λ = k N 1 ( σ ϕ σλ ) ( σ ϕ σλ ) σ g = F F( ) e g (1 e ) ϕ λ ϕ λ ( σ σ ) ( σ σ ) g (4.34) = F( k) e w (1 σ e ) (4.35) Random varatons has effects: 1st term s an attenuated pattern wthout errors. nd term rases the pattern unformly - sde lobes are not low anymore. nd term s crtcal snce t lmts array ablty to cancel nterference. Lecture 4 1-Sep-15 16 (17)
ELG513 Smart Antennas S.Loyka Introduce the senstvty functon s = w = w (4.36) For small varance, the nd term Note that ϕ λ ( σ σ ) s g e s g ϕ λ ε = [1 σ ] [ σ σ σ ] s 1 A = and, hence, ε depends on A. he larger the nose gan, the smaller the senstvty and ( nose sde lobe level ). Due to ε, we cannot put a perfect null n the drecton of nterference > lmt on null depth. ε (4.37) Example: [ ϕ λ ] 1 1 σ t = σ g σ σ = and A = What s the maxmum null depth? σt 4 L = ε = = 1 = 4dB A If σ t ncreases to.1, null depth decreases to -3 db. he senstvty functon above holds for any array geometry. Lecture 4 1-Sep-15 17 (17)