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Signals and Systems
MT study
80
Mathematics
03/05/2012

Term
 System
Definition
 A transformation from an input space to an output space.
Term
 Discrete System
Definition
 System where both input and output spaces are discrete-time domains.
Term
 Continuous System
Definition
 A system where both the input and output spaces are continuous-time domains.
Term
 Hybrid System
Definition
 System where input and output spaces are not both continuous or both discrete time domains.
Term
 Linear Space
Definition
 Space upon which addition and scalar multiplication are defined such that:for every x,y,z in X: (x+y) + z = x + (y+z) there exists a 0x  s.t. for every x there exists -x s.t. x+ (-x) = 0   for every k in R or C, k(x+y) = kx+ky
Term
 Norm
Definition
 || . ||   i) || x || ≥ 0 , || x || = 0 <=> x=0 ii) ||kx|| = |k| ||x|| iii) ||x-y|| ≤ ||x-z|| + ||z-y||    defined only on linear spaces
Term
 a.Metricb.Metric Space
Definition
 a. d: X x X -> R i) d(x,y) ≥ 0 d(x,y)=0 <=> x=y ii) d(x,y) = d(y,x) iii) d(x,y)≤d(x,z) + d(z,y)   b. collection of points on which a metric is defined
Term
 a. Continuous Transformation b. sequential continuity
Definition
 a. T is continuous at x0 if for every ε>0, there exists a δ>0 such that:   ||x-x0|| ≤ δ => ||T(x)-T(x0)|| ≤ ε   b. T is sequentially continuous at x0 if x->x0 => T(x)->T(x0)   seq. cont. <=> cont.
Term
 Hilbert Space
Definition
 a) inner-product defined <.,.> : H x H -> C   i) ≥ 0, =0 <=> x = 0 ii) = conj() iii) + iv) = k   b) complete ( all Cauchy sequences converge in H)
Term
 Cauchy-Schwarz Inequality
Definition
 ||2 ≤ ||x||2||y||2
Term
 Cauchy
Definition
 {xn} Cauchy if   for every e>0, there exists N st for every n,m≥N   ||xn-xm||
Term
 Complete space
Definition
 all Cauchy sequences converge
Term
 Banach Space
Definition
 complete normed linear space
Term
 T or F: Countable unions of countable sets are countable yet the Cartesian product of a finite # of countable sets is not.
Definition
 false: Both are countable
Term
 Give an example of a countable subspace
Definition
 Q
Term
 Dense subset a. English b. math
Definition
 a. subset in which all elements of another subspace can be arbirtarily closely approximated    b. D C X is a dense subset of a normed linear space X if for all x in X, e>0:   there is a y in D st ||x-y||≤e
Term
 Separable (two definitions)
Definition
 a. X is separable iff it contains a complete orthonormal sequence b. X is separable iff it contains a dense countable subset
Term
 Orthonormal Sequence
Definition
 Ortho => perpendicular  all elements of the sequence are perpendicular to   each other ( = 0)   normal => ||.|| = 1   sequence => bijection between set and N
Term
 Complete sequence
Definition
 only vector orthogonal to all elements of the sequence is the zero vector
Term
 If {en} is c.o.n. seq. in Hilbert space H, then:
Definition
 x = sum(1,inf) {ek}
Term
 T or F: l2 is not separable
Definition
 false
Term
 T or F: L2 is separable
Definition
 true
Term
 Wieirstrass' Approximation Thm
Definition
 let the norm on C([a,b]) be   ||f|| = sup (t in [a,b]) of |f(t)|   the set of polynomials: sum(1,n){aiti}, ai in R, n in Z   is dense in C([a,b])   (i.e. polynomials without constant terms can be used to closely approximate continuous functions)
Term
 Name the discussed c.o.n. seq. in L2([a,b];R)
Definition
 fk(t) = 1/sqrt(2pi) * e^(ikt)
Term
 How can a computer approximate a non-separable space of functions in L2(R+;R) with error less than e?
Definition
 sacrifice e/2 for removal of later terms (L2 f'ns go to zero)   sacrfice e/2 to approximate f with a polynomial (Wieirstrass' approx. thm)   => less than 'e' error
Term
 Lebesgue Measure
Definition
 λ ([a,b]) = b-a
Term
 Lebesgue Integral
Definition
 a) λ(A) = int [ 1{x in A}λ(dx) ] with indicator f'n   b) define simple f'n: fn(x) = sum(1,n){ak 1{x in Ak}}     int [ fn(x)λ(dx) ] = sum(1,n){ak.λ(Ak)}   c) for every f, there is a seq. of simp. f'ns st lim fn(x) = f(x) for every x   int [ f(x)λ(dx) ] = lim int [ fn(x)λ(dx) ]    (indicator f'ns are dense in L2([a,b];R)
Term
 Lebesgue Monotone Convergence thm
Definition
 0≤fn(x)≤f(n+1)(x) and lim fn(x) = f(x)   => lim int [ fn(x)λ(dx) ] =int [ lim fn(x)λ(dx) ]  = int [ f(x)λ(dx) ]
Term
 Lebesgue Dominated convergence thm
Definition
 lim fn(x) = f(x) and there is g(x) st   |f(x)|≤g(x) for every x  and  int [ g(x)λ(dx) ] < inf   => lim int [ fn(x)λ(dx) ] =int [ f(x)λ(dx) ]
Term
 Name a space in which C([a,b]) is dense
Definition
 L2
Term
 Name a dense subset of L2
Definition
 C([a,b])
Term
 Urysohn's Lemma
Definition
 any indicator f'n can be closely approximated by a continuous f'n
Term
 Haar Wavelet f'ns
Definition
 ψ 0,0 = 1 for t in [0,1] φ n,k = {2n/2 for t in [k2-n,(k+1/2)2-n] {-2n/2 for t in [(k+1)2-n,(k+1)2-n] {0 o.w.
Term
 Dual Space of Normed linear space X
Definition
 all linear functionals on X, f such that f is bounded (and hence continuous)
Term
 Bounded
Definition
 sup for ||x||=1 : {|f(x)|/||x||}
Term
 Riesz Representation thm
Definition
 for x in X*:   there is an n in lq st   f(x) = sum(1,n){xi.ni} = for every x in X = lp   where 1/p + 1/q = 1
Term
 Hölder's inequality
Definition
 x in lp, y in lq, 1/p + 1/q = 1   =>sum (0,inf) { xi yi} = ≤||x||p||y||q     hence:   if n in lq   => sup for ||x||=1 of (/||x||p) ≤ ||n||q
Term
 Strong Convergence
Definition
 if xn->x in n.l. X lim ||xn - x|| = 0 in a
Term
 Weak convergence
Definition
 f(xn)->f(x) for every f in X*
Term
 dual space of l1
Definition
 l(inf)
Term
 dual space of lp
Definition
 lq  st 1/p + 1/q = 1 and p,q < inf
Term
 Which subspace are the Haar Wavelets a c.o.n in?
Definition
 L2
Term
 Which subspace are the Haar Wavelets a c.o.n in?
Definition
 L2 ([0,1];R)
Term
 Delta Dirac f'n
Definition
 ∂ (t) = { inf if t = 0, 0 o.w.
Term
 Test f'n space: D
Definition
 smooth, cmpt support no metric defined
Term
 Test f'n space: S
Definition
 Schwarz Space S = {v in T(R;R) st sup for t in R (|ta(db/dtb)v|)
Term
 Convergence in S
Definition
 xn -> x in S if   lim pa,b(xn-x)=0 for every a,b in Z where pa,b (v) = sup for t in R (|ta(db/dtb)v|)
Term
 Distribution
Definition
 a function in the dual space of S* or D (linear, continuous & bounded)
Term
 delta dist'n
Definition
 ∂ (bar) (Φ) = Φ(0) = int (-inf, inf){∂(t)Φ(t)dt}
Term
 Regular Dist'n
Definition
 there exists an f such that:   ƒ = int(-inf, inf){f(t)Φ(t)} < inf for all Φ in S
Term
 Singular Dist'n
Definition
 there does not exist an f such that:   ƒ = int(-inf, inf){f(t)Φ(t)} < inf for all Φ in S
Term
 When does equality exist between two dist'ns f and g?
Definition
 if f(Φ) = g(Φ) for all Φ in S
Term
 When does a seq. of dist'ns fn converge to a dist'n f in S*?
Definition
 when lim fn(Φ) = f(Φ) for every Φ in S
Term
 What is the derivative of a distribution f(Φ)?
Definition
 - f(DΦ)
Term
 Convolution f*g = ?
Definition
 (f*g)(t) = int (-inf, inf) { f(u)g(t-u)du}
Term
 if {Φn} is a seq. of dist'ns:   int(-inf,inf){Φn(t)dt} = ?
Definition
 1
Term
 if {Φn} is a seq. of dist'ns:   lim int(|t|>∂){Φn(t)dt} = ?
Definition
 0 for all ∂>0
Term
 T or F: if {Φn} is a seq. of dist'ns:   Φn≤0 for all t
Definition
 false:    Φn≥0 for all t
Term
 for f in C([a,b]), Φn a seq. of dist'ns   what can we say about (Φn*f)(t)?
Definition
 it converges uniformly to f in t
Term
 Input-output system
Definition
 each input has a unique output
Term
 Time index, T
Definition
 if: T is countable => DT system else: CT system if: input/output T
Term
 Memoryless System
Definition
 output only depends on current and future input values
Term
 Causal system
Definition
 output only depends on past and present inputs
Term
 Time-Invariant System
Definition
 shift in input (u(t-a))   results in:    a shift in output (y(t-a)) i.e. if (u(t),y(t)) is in R (relation) =>  (u(t-a),y(t-a)) is in R
Term
 Linear System
Definition
 i) U, Y linear spaces ii) Relation R is a linear subspace of UxY   s.t. ((u1,y1), (u2,y2)) is in R   => (a*u1+b*u2, a*y1+b*y2) for every a,b in Reals
Term
 Impulse signal
Definition
 ∂(n-n0) = 1{n=n0}
Term
 Impulse Response
Definition
 DT: x(n) = sum(Z){xm.∂(n-m)}   for linear system:   y(n) = sum(Z){h(n,m)u(m)}   if u = ∂(n-n0) =>y(n) = h(n,n0)
Term
 Kernel of system
Definition
 output @ t = n when input is DT ∂ @ t = n0  h(n,m)
Term
 if system is discrete and linear, then...
Definition
 it can be described as:   y(n) = sum(Z){h(n,m)u(m)}
Term
 if a system is continuous and linear, then ...
Definition
 it can be described as:   y(t) = int(u){h(t,u)u(u)du}   with h cont. in u
Term
 if lim T(xn) = T(x)   then:   (T(x))(n) = ?
Definition
 sum(-inf,inf){x(m)h(n,m)}   h(.,.): ZxZ ->C
Term
 Time invariance => ?
Definition
 h(n,n0+T) = h(n-T,n0)   and the system is a convolution system
Term
 h(n-m) = h(n,m) => ?
Definition
 time invariance   and hence system is a convolution system
Term
 BIBO stable
Definition
 u in l(inf) -> y in l(inf) or  u in L(inf) -> y in L(inf)   i.e.      BIBO stable <=> ||h||1
Term
 Eigenfunction property of harmonic signals for convolution systems
Definition
 harmonic input => harmonic scaling of Fcc of impulse response
Term
 Pythagorean Thm
Definition
 ||x+y||2=||x||2+||y||2
Term
 GS process
Definition
 turns x1,x2,...,xn into o.n. seq. e1,e2,...,en   e1 = x1/||x1||   e2 = (x2-e1)/||(x2-e1)|| ... en = (xn-sum(i=1,n){ei)})/ ||(xn-sum(i=1,n){ei)})||
Term
 sum(1,inf){|rk|2}
Definition
 sum (1,n){rk.ek}->x in Hilbert H
Term
 sum (1,n){rk.ek}->x in Hilbert H   iff ....
Definition
 sum(1,inf){|rk|2}
Term
 ˆh(f) = ?
Definition
 ˆh(f) = int(-inf,inf){h(t)e^(-i2pift)dt}
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