l

dmin dmin p

k δ

i = m p (p l ) p l µ p BCH µ WB t (q+)

l l i m h(x) A B C = A B k SNR rec. db k SNR rec. db SNR rec. db p = p = p = SNR rec. db

p = k = q = t k σ p(k;{a i} n i= ) n σ p(n;{a i} n i= ) n = p = q = t n k σ p(k;{a i} n i= ) n σ p(n;{a i} n i= ) p = p = T B(x) x T B(x) SNR = db T B(x) x T B(x) SNR = db b a κ (a) b

l l

k f α αe jπft t = f

l x n x n k k x n = Ψ n n s n k s n Ψ n n Ψ n n x n s n s = Ψ x x n k s n x n Ψ s k k x n

k x n y m y m = Φ m n x n = Φ m n Ψ n n s n s n Φ m n x m n x s y m = Φ m n Ψ n n s n = A m n s n s n y m m<n s n s n y m l A

arg min s s n l s.t. Φ m n Ψ n n s n = y m. lp. l p<. l arg min s s n l s.t. Φ m n Ψ n n s n = y m n. l n

l l A k A k k k A k k n δ k < δ k k A m n

s n k δ k A m n s n l s n l + δ k A k δ k δ k k m O ( k log n k ) Am n A µ A max i j a i, a j a i a j, A a j a i n O(n ) m<n m n n µ A m(n m) m µ A m n m O(k ) m = O ( k log n k )

l

l

S m R n S p : d m (S) lp = inf Y sup{ x lp x S Y } n m Y R n S

S S R n l p S s S : s S S S c R : S + S c S S c = S R n Φ m n D S y m = Φ m n x n ˆx n = D ( y m ) D l p E ( x n, Φ m n,d ) = x n ˆx n lp = x n D ( Φ m n x n ) lp D Φ m n E ( S, Φ m n,d ) l p = sup E ( x n, Φ m n,d ) l x n S p Φ m n D S E ( x n, Φ m n,d ) l p m Φ m n D l p m S S E m (S) lp = inf Φ m n,d E( S, Φ m n,d ) l p

Φ m n d m (S) lp E m (S) lp n m dim N Φ m n Y d m (S) lp = inf Y sup x S Y x lp sup x S N Φ x lp D(Φ, x ) = D() x Φ m n x n = m x S N Φ x S N Φ x S N Φ D(Φ x) =D( Φ x) =D() x D() lp + x D() lp x lp E(x n, Φ m n,d) lp x lp or E( x n, Φ m n,d) lp x lp x S N Φ x lp E(S, Φ m n,d) lp = sup E(x n, Φ m n,d) lp x n S sup E(x n, Φ m n,d) lp sup x lp x S N Φ x S N Φ D, Φ m n E m (S) lp d m (S) lp d m (S) lp E m (S) lp m Y Y R n n m Y Φ Y v,...,v m

Y Φ m n = v T v T m u m Φ S u m R m D u m = Φ a a S D(u m ) E m (S) lp S D(u m ) x n S Φ m n ( xn D(Φ m n x n ) ) = Φ m n x n Φ m n D(Φ m n x n ) = Φ m n x n Φ m n x n = m x n D(Φ m n x n ) N Φm n x n D(Φ m n x n ) Y S S c R : x n D(Φ m n x n ) c S x n D(Φ m n x n ) c S Y E(S, Φ m n,d) lp = c sup x n S inf E(S, Φ m n,d) lp c inf Φ m n,d Y x n D(Φ m n x n ) lp c sup c sup x n lp x n S Y x n S Y x n lp E m (S) lp c d m (S) lp d m (S) lp E m (S) lp c d m (S) lp S

l p n U(l n p ) U(l n p )= { x R n x lp = } d m( U(l n p ) ) l q d m( U(l n p ) ) l q D D = E {(x ˆx) } R c ( c ) R = log x ˆx [, ] D = E {(x ˆx) } =

D = c R R h πe R D σ R σ h R R Ψ n n ( n k) k k sn x n = Ψ n n s n y i y m = Φ m n x n Φ m n ŷ i = Q(y i ) β = E { y i ŷ i } E { y i }, ρ = β ŷ i = ρy i + ν i y i ν i β( β)e { y i } ν i y i y i ŷ i ŷ = ρφ Ψ s + ν = A s + ν }{{} A s R Rk x kr n s x D direct kr n

k s R = log ( n k) k D adaptive (R R) kr m m m m m R m kr ln R ln m D CS (R R ) R = log (er ln ) D adaptive D CS D adaptive s m

k n k k k n n k d min n c n u k

c n = G n k u k GF () w n r n = c n + w n H (n k) n k G n k GF () H (n k) n G n k = (n k) k H (n k) n s (n k) = H (n k) n r n = H (n k) n w n s (n k) s (n k) = H (n k) n w n w n H (n k) G n n n n

k δ k k k l l l l δ k < k A m n l x n y m = A m n x n x n x n k x (k) n y m = A m n x n x x l C k x x (k) l x A C A k x n l δ k

δ k < k A m n n m l ϵ y m = A m n x m + n m x n k x (k) n Ax y l ϵ l x n x n x x l C k x x (k) l + C ϵ C,C k l Ax y l ϵ l l x n l y Ax l + λ x l σ n λ λ = σ n log n n

arg min x Ax b l subjectto x l τ λ λ τ λ x l n n Ax b l y m k x y m = A m n x n k A m n k l A m n x n y m A m n

A m n y m ˆx () n r m = y m A m nˆx () n ˆx () n ˆx () n x A m n A i i x A A A k l

arg min x y Ax l + λ x l l x (i+) = H ( x (i) + A H (y Ax (i) ) ) H(x i )= x i λ H x i λ x i > λ l l l l l l

A m n m σ y m = A m n x n x n n i m : y i = a ij x j j= y i µ y = E {y i } = n j= E {a ij}x j = σ y = E {y i } = n j,j = E {a ij a ij }x j x j = x n l m m x l y i y i y i l i = my i x l l = m i= l i = m y l x l µ l = E {l} = m σ x l y = m ( Ax P l P x ( l Ax l x l ) + δ ) δ P ( δ Ax l x l = P ( l m( + δ) ) = P ( l>m( + δ) ) = P ( l m( δ) ) = P ( l<m( δ) ) ) + δ P ( l>m( + δ) ) P ( l<m( δ) ) ν > P ( l>m( + δ) ) < E l { e ν(l m(+δ))} = E {li} i { e ν m i= li (+δ)} = (E l {e ν(l (+δ))}) m = e νm(+δ) ( E l e νl ) m ν < } E l {e νl =( ν) = e ln ν

P ( l>m( + δ) ) <e m(ν(!+δ)+ ln( ν)) ν ν = δ (+δ) P ( l>m( + δ) ) <e m (δ+ln(+δ)) P ( l<m( δ) ) <e m (δ+ln( δ)) δ + ln( + δ) δ δ δ + ln( δ) δ m x n N Q ( P x Q : δ Ax l x l ) + δ N (e m ( δ δ ) + e mδ ) Ne m ( δ δ ) A Q n x n k k n X T {,,...,n} k T k T

X T x n k δ ( ) k δ δ x n X T X T δ X T X T δ u q n δ u + δu = A m n x n l A m n q n l + A m n (x n q n ) l + δ u + δ ( δ ) δ u δ δ x n + δ + + δ u δ ( ) < δ q n x n X T A m n x n l A m n q n l A m n (x n x n ) l δ + δ δ δ ( + δ) δ δ k X T δ ( ) k δ X T {,...,n} T k δ ( ) k ( n ) δ k δ {,...,n} δ ( ) ( k n m( δ δ k)e δ ) ( ) n = O k ( ( n k k n ) ) k ( n ) n k n e O(k ln n k +k) n k k(n k)

δ e O ( k(ln n k ) ++ln δ ) m( δ δ ) m >O ( ) k ln n k k m >O ( ) k ln n k δ A m n ( Am n P x n l E { A m n x n } { l ϵsd Am n x n } ) l e mf(ϵ) SD{.} f R n N Q < δ < u n, v n Q f : R n R m m>m = O ( ) ln N δ ( δ) u v l f(u) f(v) l ( + δ) u v l Q v u m k ln n k

k δ k < k A m n k Ax > x n A k k Ax > x k {x i } n i= A {a i } n i= {x i,...,x ik } x k A m n x n = x ij a ij j= A A (sub) m k =[a i... a ik ] x k R k : δ k A(sub) m k x k l x k l + δ k A(sub) m k x k l x k l A k δ k A k m δ k m δ k ± l l(l+

[ A RM = U U... U l(l ) ] l l U i δ l = A RM U i A RM β m α m αm + β m m a ik A chirp m m i m k = αm + β m a ik = e j π m (αi +βi) m row i, row i = m α= β= m e j π m (αi +βi αi βi) = m m e j π m α(i i ) e j π m β(i i ) α= β= } {{ } = m m m k m+ k k A

F m m I m m a m, b m A m m = [ I m m F m m ] F m m I m m a m F m m I m m a m m b m a m, b m = m m A k = m A µ A = m m D C {d i } m i= {c i } m i= D m m C m m = d i l = Cd i l = C m c j, d i j= m c j, d i m c j D m m C m m m m m

< α β < {a i } R m x R m α x l i a i, x β x l α = β m m m m n m n n m m(n ) n m(m+) n m m n k< p r + p p r+ n = p r+ m = p

Q(x) p GF (p) p ( x, Q(x) ) G(Q) GF (p) B p G(Q) x GF (p) B = {b,...,b p } p GF (p) GF (p) Q b i v Q p v Q =[v,...,v p ] T, v i = b i / G(Q) b i G(Q) P r p(p ) p v Q r GF (p)[x] P r = { a + a x + + a r x r a i GF (p) } p r+ P r P r = { } Q,Q,...,Q p r+ x r Q i (x) =Q j (x) r p v Qi v Qi, v Qj r k< p r + r p m O(k ) m = O(k) k m = k O(log log n) B = k A = n B A G Γ(X) X G d A = O(log log n) A

A k ϵ G X Γ(b) X ϵ d A X k b B A k ϵ > B k m = O(k log n k ) l l δ k k [ δ k, + δ k ] k k (k, δ, ϵ) A m n ( δ) x l A m n x n m l ( + δ) x l

k k ϵ A A A A η > m η ( A m n m c k log n δ ) η k< +(n )η c (k, δ, ϵ) ϵ = e (δ k n ) n η k

±

O(n ) µ A A m n δ k =(k )µ A k< µ A + A k k x n Ax = n i= x i a i = n i,j= x i x j a i, a j = x + i j x i x j a i, a j x i A i x i a i i j x i x j a i, a j ( n µ A x i x j = µ A i j i= x i ) µa x ( n i= x i ) k x k x x i x j a i, a j (k )µ A x i j (k )µ A Ax x +(k )µ A

R(m, w, λ) λ Z w m R(m, w, λ) R(m, w, λ) m w m w m λ...... w λ x x λ c λ a λ λ a λ c λ c λ a (m, w, λ a, λ c )

(m, w, λ) λ a = λ c = λ w m A A A m n λ A w n A λ w A A F α F = GF (q) a N q = a q = d + q D i = {α d+i, α d+i,...,α d+i }, i d q d C i i C i = log α (D i ), i d D D ( a,, ) a ( a ) n n (a )( a ) w wn δ =

p p r+ r p p GF (p) p r lim pr+ p r R(p, p, r) lim p r r i= = lim p r p(p i) p i ( ( r p ) p r ( ) r+ ( p(p r) lim p r p lim r ) r+ p r r p ) r(r+) p lim e r(r+) p = e = p r n k ñ. n C ñ C(ñ, k; ) añ ñ C añ ñ Ãñ k dmin C(ñ, k; ) Añ k ñ( Ãñ k ( ) ) ñ k ñ d min ñ A Ãñ k ( ) ñ k A

A ñ A Ã ãñ, bñ A añ, bñ l b ã a, b = ñ ( (ñ l)+( ) l ) = ñ l ñ ñ l ã ã ñ b l {ã, ã ñ, b} l d min d min ñ l ñ l d min d min l ñ d min ñ l ñ d min d(ñ, ã) d(ñ, ã) ã ñ, ñ C ñ d min ñ a, b ñ d min ñ A ñ k dmin d min d min ñ k + ñ dmin m N ñ = m g(x) x m + g(x) GF ()[x]

x m + = r GF ( m ) r (x r) GF ( m )[x] g(x) α i α GF ( m ) g(x) GF () α i g(x) GF ()[x] {α ij } m j= α i = α i i i (mod m ) i,i α m = d min i,...,i d g(x) α i,...,α i d d min d + dmin h(x) x m + g(x) h(x) h(x) g(x) l< m G (l) m = {α, α,...,α m + l } G (l) m H (l) m α G (l) m H (l) m {r G(l) m j N : r j G (l) m } H (l) m r j r H (l) m

h(x) = r H (l) m (x r) h(x) h(x) r g(x) h(x) GF ()[x] = α G (l) m H(l) m ( + x) h(x) c =[,..., }{{} m ] T c(x) = + x + + x m = x m + x + x m + (x m + ) h(x) + x = c(x)h(x) g(x) = xñ+ h(x) G (l) m h(x) g(x) GF ( m )\G (l) m m + l j m : g(α j )= g(x) α m l d min ( m l )+ = m l d min ñ, k k ñ d min ñ = k + deg ( g(x) ) k = ñ deg ( g(x) ) = ( deg ( g(x) ) + deg ( h(x) )) deg ( g(x) ) = deg ( h(x) ) = H (l) m H (l) m

m H (l) m H (l) m m l β (b m,...,b ) {, } m β =(b m...b ) = m i= b i i β β j α β H (l) m j β = (b m...b ) β = (b m...b b m ) β = (b m...b b m b m ) β m = (b b m...b ) β j = (b m j...b b m b m j ) = m b m j +(b m j...b b m b m j ) β j+ ( mod m ) β j j β ( mod m ) α βj = α j β α β H (l) m α β {α βj } j β j m + l G (l) m < β j β j b m j = β j < m < m + l

m l β j b m j = b m j = = b l j = β j m + l j j= β j m + l h(x) h(x) β m G (l) m α β m + l β j k ( H (l) m O (l+) ln m l ) m l k k ñ = m

m i i = log (k) k m = m i m H seq H dec α GF ( m ) H = {α r r H dec } h(x) = r H(x r) g(x) = x m + h(x) Ã( m ) ( deg(h) ) m à A ( m ) degh m = i H dec = {,,,,..., i } α x + h(x) α h(x) g(x) (x + )g(x) ( i ) ( i ) à i

h(x) x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x +x + x + x + x + x + x + x + m i = m deg h(x) 10 2 i=5 i=4 i=3 i=2 10 1 5 6 7 8 ~ 9 10 11 12 13 m i m h(x) k < i = i m h(x) m d min p p p

p p p GF (p) p C(ñ, k; p) ñ dmin ñ, ñ,...,(p )ñ p cñ a b bñ añ c =(p )ñ... c = ñ c = ñ d(cñ, ñ ) d min d(cñ, ñ ) d min d(cñ, (p )ñ ) d min c / {ñ, ñ,...,(p )ñ } {,,...,p } ñ d min cñ cñ i N i i p p i= N i = Ñ N i ñ d min N i =ñ j i N j ñ (p )(ñ d min ) ñ (p )(ñ d min ) N i ñ }{{} d min, }{{} N min N max N i N min + N max N max N min

{a, a ñ,...,a (p )ñ } {ñ,...,(p )ñ } dmin GF (p) p C(ñ, k; p) Ãñ p k {a, a ñ,...,a (p )ñ } à Añ p k à =[ã αβ ] α,β A = ñ[ e j π p ãαβ ] α,β p(p )ñ p dmin ñ A a β = A ñ[e j π p ã,β... e j π p ãñ,β ] T = à ã α, ã β A a α, a β c i N i c = ã α ã β i p a α, a β = ã H β.ã α = ñ e ñ π j π p (ãi,α ã i,β) = i= ej p ci p = i= N ie j π i p ñ ñ ñ i= γ + x + + x p e j π p p N i e j π i p = p (N i γ)e j π i p i= i= p i= N i γ γ = Nmin+Nmax p N i e j π i p p N max N min i= = p(p )ñ p d min

A a α, a β p(p )ñ p d min ñ N i γ γ k d min ñ > p p p (k ) p p ( kp ) ñ dmin p p ñ dmin p dmin p ñ = p m GF (p) p GF (p)[x] g(x) GF (p m ) h(x) g(x) r GF (p m ) r (x r) =x p m p h(x) = xp m g(x) g(x) {α i,...,α i d } GF (p m ) α

[c,...,cñ] T d min d + i,...,i d α i α i... α (ñ ) i c α i α i... α (ñ ) i c α i d α i d... α (ñ ) i d cñ }{{} H d ñ g(x) ñ j= c jx j = d H d d {i,...,i d } d + H d [c,...,cñ] T g(x) g(x) l< m {α p m + pl p +, α p m + pl p +,...,α p m } α g(x) d min p m p m pl ( p = ñ p d min ñ p p p =(p m ) ( p m p l+ ) p(p )(p m ) ( p l+ (p ) (p m ) ) p m p l ) (p m )(p ) g(x) m = {α, α,...,α p m + p l G (l) p } H (l) m = {r G(l) m j N : r pj G (l) m } GF (p) G (l) m H (l) m x y x y

h(x) h H (l) m (x α h ) GF (p)[x] h(x) g(x) g(x),h(x) h(x) = α H (l) m g(x) xñ =(x )( + x + + xñ ) gcd ( g(x),x ) = g(x) + x + + xñ g(x) d min ñ ñ h(x) H (l) m k m H (l) m l m p x m l x γ H (l) m = O( γ l+) k m n m = p m log p n = H (l) m = O( γ l+) k max p p p m l p m l k max ln γ ln( m l ) m l γ log p kmax log p log p kmax p ( m O k max (log p n) log p kmax log p log p kmax )

p (p l ) p l p µ BCH µ WB p = p = p = p = l = l = l = m O ( k max log p n ) m m = l µbch µ WB p (p l ) p l p H (l) l = {} {p i } l i= {p l+i + p i } l i= {a, a ñ,...,a (p )ñ } p = x = p ñ p {a, a ñ,...,a (p )ñ }

g(x) x = p p m = p m m k< p p p m p l+ + l m m H seq ( m,l) H (l) m m l H seq ( m,l) p GF (p m ) α h(x) = r H (l) m {} ( x α r ) n = p H(l) m h(x) Ãm n g(x) = xp m h(x) A m n = [ e m ãi,j jπ p ] Ã ã i,j

p p p w m µ A k A B wm n C m (n n ) µ B B A µ C α {,,...,n } αn + β l C l n l β α β {,,...,n } k k l A α + i,...,i wm c i,l = b,β+ c i,l = b,β+ c iwm,l = b wm,β+ c s,l =, s / {i,...,i wm },

A B B β + [b,β+,...,b wm,β+] T u wm, v wm C l = α n + β l = α n + β C l l B v u A v u α α w m k k A w k k v u wm v u B u, v = n u i v i i= n i= u i v i = w m ( k wm ) k A v u α = α β + β + B u, v = b β +, b β + k k C

C = A B B mb n b A ma n a C mam b n an b A ma n a B mb n b c η,θ = a γ,τ b ρ,ν, m a,n a,m b,n b γ, τ, ρ, ν θ =(τ )n b + ν η =(γ )m b + ρ C = A B C B A µ C = max { µ A,µ B } k C δ k,a, δ k,b k B A δ k,c δ k,a δ k,b + δ k,a + δ k,b B A k A,k B µ B < k B µ A < k A k C = min{k A,k B } C

B A p p k k x B X B C = min{k a,k b } ( log n a + log n b ) m a m b m b k a log n a m a kx log nx m x X mx n x + m a k b log n b m b = m b B A + m a B B B C B B B A k A m n k s n A S = {i,...,i k } {,...,n} k y m = A s = s ij a ij j= A i a i ŝ n y r m = A(s ŝ) =y Aŝ

A s i max s s k t δ = s ŝ ŝ t s δ i δ i δ i δ ik S r, a i = k δ ij a ij, a i δ i a i, a i k j= j= δ ij a ij, a i A r, a i > δ i k k j= δ ij δ i k k δ i = k k δ i l / S r, a l = k δ ij a ij, a l < j= k k k δ ij k δ i j= r, a l < k k δ i < r, a i a ij a i s n A r m m m

a A a (j) A j a (j) A {a (), a (),...,a (µ) } j a (j), r A µ m a µ(m ) µm r a r r, a (j) = ( r m a ) j m m a r a f r f IDFT{r f a f } = [( r m a ),..., ( r m a ) m ] r f v m u m [v u,...,v m u m ] T m µ µ µ a µ r m a µ µ log µ µ m µ + µ log µ A

Recovery Percentage 1 0.8 0.6 0.4 0.2 OOC Devore brnd63 brnd64 0 0 5 10 15 20 25 30 k k SNR rec. db λ (,, )

Recovery Percentage 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 BCH Rnd64 Devore Ternary Rnd49 0.1 0 0 5 10 15 20 25 30 k k SNR rec. db SNR rec db k SNR rec. db

Output SNR (db) 100 90 80 70 60 50 40 30 20 10 BCH Rnd64 Devore Ternary Rnd49 0 20 30 40 50 60 70 80 90 100 Input SNR (db) p =, (p ) p p = p =

1 Recovery Percentage 0.8 0.6 0.4 0.2 BCH Rnd DFT Complex Rnd Chirp 0 5 10 15 20 25 30 35 40 45 50 k (Sparsity Order) SNR rec. db p = k =

100 Reconstruction SNR (db) 80 60 40 20 0 BCH Rnd DFT Complex Rnd Chirp 0 20 40 60 80 100 Input SNR (db) p = 34 Reconstruction SNR (db) 32 30 28 26 24 22 20 BCH Rnd DFT Complex Rnd Chirp 18 150 160 170 180 190 200 k (Sparsity Order) p =

Recovery Percentage 1 0.8 0.6 0.4 0.2 Rnd DFT Complex Rnd Binary-Mixed Chirp Kronecker-Mixed 0 5 10 15 20 25 30 k (Sparsity Order) SNR rec. db Elapsed Time (seconds) 2.5 2 1.5 1 0.5 Rnd BCH 0 5 10 15 20 25 30 35 40 45 k (Sparsity Order) p =

Ψ Ψ Φ Φ m n Φ m n

Φ m<n n m> v R n, v l = : Φ v = n V R n v V V {u,...,u n } R n {v, u,...,u n } Ψ =[vu... u n ] v = Ψ Φ v = Ψ v Φ f,...,f n y = f (x) = f (x,...,x n ) y m = f m (x) = f m (x,...,x n ) R R n n g : R n R n n y m R n x

k k x n (y,...,y m ) k k = (f,...,f m ) S = { z R n i m : f i (z) =y i } S x n S S x n S x n S x n v S, v x n v = x x, v v, v v v v ṽ ṽ ba a, b {u,...,u n } ṽ ṽ V v, v = x, v x, v v, v v, v = v v ṽ ṽ Ψ R n {ṽ, ṽ, u,...,u n } Ψ Ψ =[ṽ ṽ u... u n ]

v = v l ṽ Ψ v x v l = v x, v v x = v + x, v l v, v v = v l ṽ + x, v ṽ Ψ = x k v x k = k = k = S R n a b S : < a, b < a l b l S a b R n : a, b a l b l b a a, b = a l b l c R : a = cb Ψ S S

a, b a, b S Ψ S Ψ ba Ψ Ψ Ψ a, b ba a, b Ψ Ψ Ψ a, b S f (x) = n i= i sign(x i ) f (x) = x n l = n i= x i sign(a) = a> a = a< x n f f n f n log n, n f (x) = f x n = a.b f ba f (a) =f (b) a = [a... a n ] T b = [b... b n ] T f (a) =f (b) i n : sign(a i ) = sign(b i ) a i b i n i= a i b i a, b

a i b i i a, b > ba f (a) =f (b) Ψ f f ba f f c R : a = cb f (a) = a l = c b l = c f (b) c = c = f (a) =f (b) c = c = a = b a, b = b l f (a) =f (b) Ψ f f f f f C n R n f (x) = n i= (i ) sign ( Rx i ) + i sign ( Ixi ) f (x) = n i= x i f (x) = x i msign(rxi) Ixi Rx i + Ix i msign(a) = a a<

C n k x n k x n Ψ k a,...,a n y = a x + a x + + a n x n y = a x + a x + + a nx n y k = a x k + a x k + + a n x k n k {y,...,y k } x n {y i } k i= {x i,...,x ik } x i k

k c,...,c k R : y i + k j= c j y i j = (k i k ) y i Z x i k k c i y i x ij y k y k... y y k y k... y y k y k... y k y = n i= a i c c = y k y k+ c k y k c i x ij x n c i q(z) =z k + c z k + c z k + + c k q(x i )=q(x i )= = q(x ik )= x n c i q(z) x n x n y,...,y k x ij y i Z d j y l = k d j x l i j, l k j=

a i d j a i d j a i x n x j x i a i a j k y k y k... y y k y det k... y y k y k... y k = k {y i } k x n k y i y i... y y A i = i+ y i... y y i y i... y i, i k i det A i i x n i x n d j d j y,...,y i a j x n a j a j d j a j

d j a j j n : a j = j k a j f() = f f f(x i ) x i f(x i ) x i f() = x i [f(x ),...,f(x n )] [x,...,x n ] x i f k [x,...,x n ] [, ] [ k, k ] k i f i (x) f(x) =e jπ x f x i x i y = a sin ( πx M y = a sin ( πx M y k = a sin ( (k ) πx M ) + a sin ( πx ) + + an sin ( ) πx n M ) + a sin ( πx M M ) + + an sin ( πxn M ) + a sin ( (k ) πx M ) ) + + an sin ( ) (k ) πxn M sin x = ejx e jx j [ M,M] k {y l } l= k k l k : q(z) = k c i z i = i= q(z) k k l= ( z e j πx i l )( M z e j πx i l M ) = k l= ( z z cos πx il M + )

q(z) c = c k = c i = c k l : = k i= k c i y l i = c k y l k + c i (y l i + y l k+i ) i= c i y k + y k y k + y k... y + y y c (y k+ + y k ) y k+ + y k y k + y k... y + y y c (y = k+ + y k ) y k + y y k + y... y k+ + y k y k c k (y k + y ) y l = y l y = k y,...,y k y,...,y k q(z) c,...,c k y i a i [x,...,x n ] {y l } l k x k k x k {y l } l a i k k

SNR rec (db) 250 200 150 100 k=1 k=2 k=3 k=4 k=5 k=6 50 50 100 150 200 SNR input (db) n = k [, ] ( k ) a i = i a i k =,..., k k k y i k

SNR rec (db) 160 140 120 100 80 OSR= 4 OSR= 3 OSR= 2 OSR=1.5 OSR= 1 60 40 50 100 150 200 SNR (db) input k = k = = k k

k l p c k k c R> l p {x i } i= l p x i Ri p

f x {x i } {x i } l t {a i } n i=

k n p> a n a nn {a ni } n i= ( σ p k; {ai } n ) i= = ( a n p + + a nk p) p k l p σ p (k; {a i } n i= ) <p r {a i } n i= { ( κ p r; {ai } n ) i= = min k σ ( p k; {ai } i=) n } ( ) σ p n; {ai } n r i= r κ p (.) l p {a i } i N r< : ( ) κ p r; {ai } n lim i= = n n i N, a i = α i k = ln( rp ) p ln α r< α = ( ) σ p k; {ai } n ( i=) σ p n; {ai } n = i= = ( k ln α ) ( i= e ip p k ln α ) n ln α i= e ip p ln α i= e ip i= e ip ( e ln α kp ) p r n r k κ p ( r; {ai } n i=) k ( ) κ p r; {ai } n lim i= = n n l p p> {a i } i N l p l p {a i } α >

l q q p l p p ( σ p(k;{a i} n i= ) σ p(n;{a i} n i= ) ) p k<n {a i } q p> i j n a ni a nj a ni q p a nj q p a ni q a nj p a ni p a nj q k + j n i k k n a ni q a nj p i= j=k+ k i= a ni q n i=k+ a ni q k n a ni p a nj q i= j=k+ k i= a ni p n i=k+ a ni p k i= a ni q n i= a ni q k i= a ni p n i= a ni p p κ p (r p, {ai } n i= ) q p p l p p p l p l p p

f x {x i } i N p r, δ ( ) ( ( ) σp k; {xi } κ p r, δ,n; fx = min {k n ) P ( i=) σ p n; {xi } n r i= } δ κ p (.) κ p (.) κ p δ δ r κ p l p f x r, δ <, ( ) κ p r, δ,n; fx lim = n n δ =,r = k l p l p f x p, δ <, ( ) κ p, δ,n; fx lim < n n r = x = f x P (x = ) =π > n( π ) n P (x = ) =π >

f x π f x l q l q q p l p l p l l p l p x l y = x p {y i } l {x i } l p l l E {e γ x } γ > f x l y i = x i f x {x i } σ y µ y f y (y) = ( f x (y)+f x ( y) ) u(y) {y i } σ y µ y E x {e γ x } f y γ E y {e γy } <e mγrµy (m) m <r< E {e γy }

EV : σ (k; {y i } mk i= ) r mk i= y i EV : mk i= y i mkµ y (mk) σ y EV : mk i= y i < mkµ y (mk) σ y P (EV ) = P ( EV EV ) P (EV )+P ( EV EV ) P (EV ) P ( EV EV ) + P (EV ) ( P σ (k; {y i} mk i=) r ( mkµ y (mk) ) ) ( mk σ y + P i= y i < mkµ y (mk) σ y ) i= { mk E y i= y i} = mkµy Var { mk i= y i ( mk P y i < mkµ y (mk) ) σ y mk mk k ( mk k ( P σ (k; {y i } mk i= ) r( mkµ y (mk) ) ) σ y y i } = mkσ y ( ) mk P k ) σ (k; {y i } mk ) ( k i= i= y i r ( mkµ y (mk) σ y ) ) k i= y i r ( mkµ y (mk) σ y ) E { k i= y i} = kµy P ( k ( mk k i= k y i T ) = P ( e γ k i= yi e γt ) E { e γ k } i= yi e γt = ( E { e γy} e γ T k ) k < ( e γ(rmµ y T k ) m ) k ) < (em)k ( P σ (k; {y i } mk i= ) r( mkµ y (mk) ) ) ( ) k e σ y < eγrσym k

( σ (k; {x i } mk P (EV )=P i= ) ) σ (mk; {x i } mk i= ) r < ( ) k e + mk eγrσym k k m δ P (EV ) P (EV ) mk k ( ) κ p r, δ,km; fx lim k km m l f x k r m l µ x {x i } i N n n i= (x i µ x ) σ x σ x < α α α = α = α t (α+) n α α α

x < α < x G(t) =P (y >t) y = x α t h(ct) h(t) c> t h(t) =t α G(t) lim t P (x>t) G(t) α {η i } {x i } Γ i = i j= η j α lim a n n ( yn,...,y nn,,,... ) ( = d Γ α, Γ α,... ) { x i } n i= {y ni } n i= = d G ( a n = lim t t nt α ) α f x x l f x α <

( ( ) y i = x i σ k; {xi } i=) n = σ k; {yi } n i= ( ( σ k; {xi } n ) P ( i=) ) σ n; {xi } n r = P i= ( a ( ) n yn(k+) + + y nn a n ) ( ) r yn + + y nn Γ α i a n y ni ( ( σ k; {xi } n ) P ( i=) ) σ n; {xi } n r i= ( Γ α k+ P + + Γ α n Γ α + + Γ α n ) r Γ α k+ + + Γ α n Γ α + + Γ α n = k α( Γ ) α Γ k+ + ( Γ ) α Γ + + ( Γ ) α Γ n k α( Γ ) α Γ k+ }{{} A k + ( Γ k+ ) α Γ k+ + ( Γ k+ ) α Γ k+ + + ( Γ k+ ) α Γ n k α + + ( Γ k+ ) α Γ n k α } {{ } B k,n ( Γ α k+ P + + Γ α n Γ α + + Γ α n ) r P ( A k B k,n r ) P ( A k > ) P ( B k,n > r ) E {A k } r E {B k,n} E {A k } n>k+ k E {B k,n } k E {A k } θ, β {η i } i N E {ηi} i { ( + θ k i= η i ) β } < k + ( θ ) β k

θ η i E {ηi} { ( + Γ k θ ) β } = k ( + γ θ f Γk (γ)dγ + ) β f Γk (γ)dγ = k ( θ k k ( + γ θ Γ k = k i= η i ) β f Γk (γ)dγ + k ) βfγk (γ)dγ P ( Γ k k ) + ( θ k ( + γ θ ) β f Γk (γ)dγ ) β k Γ k P ( Γ k k ) E {Γk } = k ( P Γ k k ) ( = P k i= = e k ( E η {e η } ) k = ( e η i k ) = P (e ) k i= ηi e k E { {η i} i e k } i= ηi e k ) k < k E {A k } { ( E {A k } = k α Γ ) } { { α ( k+ E {ηi} i = k α E η E {ηi} Γ i> + k+ η { ( k+ E {ηi} i> + η i= i= }} ) α η i ) } α η i < k + ( η ) α k E {A k } k α k + k α α Eη {(η) α E {B k,n } lim k E {A k } = < α + α } k {η i } i N E {ηi} i { + k= } ( + k θ i= η ) β i < θ, < β = + θ β

Γ k Γ k = k i= η i k f Γk (γ) = ( ) { f η f η (γ) =F ω ( + jω) k} (γ) }{{} k times E {ηi} { ( + Γ k θ ) β } = R e jωγ dω ( ) + γ β ( + jω) k π dγ θ E {ηi} i { + k= } ( + Γ k ) β θ = = = R R R e jωγ ) β k= e jωγ ( + γ θ ( + γ θ ) β ( + jω ( + jω) k dω π dγ )dω π dγ δ(γ)+ ( ) + γ β dγ = + θ β < θ θ = {Λ k } k + k= Λ β k E {Λ k } = k + Λ k Λ k+ + k= k β β β = E {B k,n } E {B k,n } = k α { E n + t=k+ (Γ k+ Γ t ) } α k α { E + t=k+ (Γ k+ Γ t ) } α E {ηi} i>k+ { + t=k+ (Γ k+ Γ t ) } α = E {ηi} i>k+ { + = + α α Γ k+ t=k+ ( + t i=k+ ηi Γ k+ ) α }

t (q+) f x (t) q λ Γ((q+)/) Γ(q/) πλ t q λe (+ ( + t ) (q+) λ ( + t (q+)/ λ) t ( λ ) q + t ) (q+) λ (q/λ)(t/λ) q ( +(t/λ) q) E {B k,n } k α { α } E Γk+ + α Γ k+ = k α ( + α(k + ) ) α k α < E {B k,n } = E {A k } = ( Γ α lim P k+ + + Γ α n k Γ α + + Γ α n ) r = k E {B k,n } E {A k } ϵ < ϵ < n k >k k ϵ n ϵ k n k ( ( σ k ; {x i } n ) P ( i=) ) σ n; {xi } n r ϵ i= k n n α < t (α+) f x (t) q t (q+) l t (q/p+) f y (t) = fx(t/p )+f x( t /p ) pt /p y = x p l p p>q f x

2000 0 2000 4000 0 20 40 60 80 100 5 x 10 4 0 5 0 200 400 600 800 1000 index q = t t (q+) l p p>α < α < α q = t σ (k;{a i} n i= ) σ (n;{a i} n i= ) k

1 0.9 n σ 1 (k,{a i } i=1 ) / n σ1 (n,{a i } i=1 ) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 Student t Cauchy Laplace Gaussian 0.1 0 0 0.2 0.4 0.6 0.8 1 k / n k σ p(k;{a i} n i= ) n σ p(n;{a i} n i= ) n = p = q = t q = t n = l n k n n l p> t l p = l p p l l l l n

1 p = 1.1 ) ) / σ p (n,{a i } i=1 n σ p (k,{a i } i=1 n 0.95 0.9 0.85 0.8 0.75 p = 0.9 n = 1e7 n = 1e6 n = 1e5 n = 1e4 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 k / n n k n σ p(k;{a i} n i= ) σ p(n;{a i} n i= ) p = p =

B B A k B k

b k k M B B k F M B F F B A B B A B A

T =[t ij ] t ij T T =[t i,j ] n n

r α r α d n D =[d i,j ] j i d i,j i p i,j P =[p i,j ] n n D j P P D d P =[p α i,j ] n n P α d f(a) = [ f(a i,j ) ] m n A = [ a ] i,j m n A =[a i,j] m n

B A f(.) B = f(a) k A B f B T B (x) = det ( B x) T B (q,...,q n ) = det ( [(ln bi,j ) qi]) p> f(x) =x p f B = f(a) A B B A p k A n n rank ( A p) { ( )} k + p min n, p A {v i } k i= k rank(a) =k A c,... c,k v [ k ] A n n = = l= c i,lv l,j c n, }... {{ c n,k v k }}{{} C n k V k n A p = = [ ( k ] [ ) p ( ) p k ] c i,l v l,j = (c i,l v l,j ) p l p l= p + +p k =p,...,p k l= ( )[ p k ] (c i,l v l,j ) p l p p + +p k =p,...,p k l=

k l= (c i,lv l,j ) p l i, j n n [ k l= (c i,lv l,j ) l] p rank(a p ) [ k (c i,l v l,j ) p l] = l= rank(a p ) p + +p k =p k l= cp l,l k l= cp l n,l ([ k ]) rank (c i,l v l,j ) p l l= [ k l= vp l l,... ] k l= vp l l,n ([ k ]) rank (c i,l v l,j ) p l = l= p + +p k =p ( ) k + p = p k A rank(a p ) p(k )+ p p f(x) f(x) = p i= f ix i B = f(a) p f(a) = f i A i i=

rank ( f(a) ) p rank ( A i) i= p i= ( k+i i ) = ( k+p p ) k A A n n p f(x) k B ( k A+p) p B = f(a) n ka ( k B+i ) i i N B i B pi B i A B ( k A+pi) pi A B n k A i rank ( B i) B B i. p B = A p A A A A.. B n n B = A. p B A B x p x B p p B T B (x) B = A p T B () = n> B = n n

x T B (x) T B ( p )=T A() = A T B (x) T B (x) p x B x = T B (x) T B (x) x = q T B (x) T B (x) = q= t qx q t q = π S n sgn(π) ( n i= ln b ) q i,π(i) q! n! {,...,n} S n N(π) ( ) N(π) sgn(π) π π S n π T B (x) = det [ b x ] n i,j = sgn(π) b x i,π(i) = sgn(π)e x n i= ln b i,π(i) π S n i= π S n = ( n sgn(π) i= ln b q i,π(i)) x q x q ( n ) q = sgn(π) ln b i,π(i) q! q! π S n q= q= π S n i= det(m + M ) det ( q x q B x) q x q T B (x) det(m ) + det(m ) T B (x) T B (x) T B (x)

T B (x) n T B (x) = q=n x q q! q,...,q n=q q i Z + ( q q,...,q n ) T B (q,...,q n ) TB ( n ) q ln b i,π(i) = i= q,...,q n=q q i Z + ( π S n q q,...,q n ) n i= ( ln bi,π(i) ) qi T B (x) = q= x q q! q,...,q n=q q i Z + ( q q,...,q n ) T B (q,...,q n ) [( ln bi,j ) qi ] n i= q i = q q i (q,...,q n ) n n T B (q,...,q n )= T B (x) n T B x = q<n x q T B T B (x) n n n + T B (x) E N (x) = q=n+ t qx q x ln B M B

E N (x) ( n n emb x ) N+ ( π(n + ) N+ n embx N+ N em B x ) TB (q,...,q n ) = ( [(ln det bi,j ) qi]) n n nm q i B = n n M i= i= n i= qi B ( n ) ln b i,j qi j= E N (x) = q=n+ q=n+ x q q! x q q! q,...,q n=q q i Z + q,...,q n=q q i Z + ( ) q q,...,q T B (q,...,q n ) n ( ) q q,...,q T B (q,...,q n ) n. E N (x) = n n q=n+ x q q! q=n+ q,...,q n=q q i Z + ( MB x ) q q n q! ( ) q n n M q B q,...,q n E N (x) n n (em B x) n π n n (em B x) n π q=n+ q=n+ n! > ( n e ) n πn ( emb x ) q+ n q ( emb x ) q+ n N + embx N+ < N em B x q=n+ ( emb x N + ) q+ n ( emb x ) N+ n = N + embx N+ B M B T B (x)

B n n A n n m max ( m+k ) m <n m p B = A p A m m T B T B (x) { i p }mmax i= p m max + p k A l ( ) l + k <n l ( ) l + k l B = A A = A i =,,, rank(a i )=i + B p = = p = db B B T B(x) x x

value 10 6 10 8 10 10 10 12 10 14 10 16 deg=9 deg=11 deg=13 deg=17 deg=25 deg=28 deg=30 value 8 x 10 13 6 4 2 0 2 4 10 18 1 0 1 2 3 4 5 6 7 8 x 6 0 5 10 15 20 25 30 35 # coefficient T B(x) SNR = db x T B(x) value 10 5 10 0 10 5 10 10 10 15 deg=9 deg=11 deg=13 deg=17 deg=25 deg=28 deg=30 1 0 1 2 3 4 5 6 7 8 x value 1.5 1 0.5 0 0.5 1 1.5 2 x 10 7 2 0 5 10 15 20 25 30 35 # coefficient T B(x) SNR = db x T B(x) x SNR = db

B p SNR = db {,, e, e, e } A p {,,,, } A

( l >j j ( l ) O ln j ) (l j) j k j + ( p (p l ) p O ln j ) (l j) p j ( ln(log p p p r+ p log r) ) O r log p log r

k m k τ (a) b m l a b a b κ (a) b τ (a) b κ (a) b κ (a) b κ (a) b b b b,..., }{{} a κ (a) b, a + b a b,..., }{{} a, κ (a) b a

k κ (a) b = κ (a) b + κ(a) b a κ (a) b a + b a + : κ (a) b = b + = κ (a) a+ = a + κ (a) (z) = b= κ (a) b Z κ (a) b z b, κ (a) (z) = z z (a+) z z (a+) γ γ b b κ (a) b (, ) f() f() < f(z) =z a+ z a f(z) γ f(z) γ < γ <, f(γ) =γ a+ γ a = δ > γ = + δ < γ < γ a+ γ a = ( + δ ) a = δ δ >a κ (a) b τ (a) b γ a ln a ( ) a ln a = + = δ ln a >a ln a = e > δ κ (a) b τ (a) b a κ (a) b a τ (a) b κ (a) b

k κ b (a) 10 15 10 10 a= 2 a= 3 a= 5 a=10 10 5 10 0 0 20 40 60 80 100 b b a κ (a) b κ (a) b a τ (a) b τ (a) b O(γ b a ) τ (a) b O(γ b ) ( (a)) a τ ln a b O ( b a) k k = τ ( m l ) m ( O ln (l+) m l m l ) k m l log k m = m < m l l+ O (k ( log n ) ) log k ln log k b a κ (a) b κ () b O( b ) κ () b b

δ δ >a a N ( + δ ) a = δ x> f(x) =x x ln x [, ) f f (x) lim x + f(x) =+ f f (x) = x + x x = frac(x ) (x ) + x f y y y + x f( ) > x <f(x) =x ( x x ) ln x x ln ( + x ) ln x e ln x <e x ln ( +x ) ( x ) x+ ( + x ) x <

δ x ψ(x) = ( xa+ ) a = x ( a +x +x) ψ(δ) = ( + δ ) a = δ +x δ >a ψ ( a ) < x = a

p

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k ` ` m w l

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