[PDF] fiche pédagogique fourmi
[PDF] nouvelle ? chute cm2
[PDF] documentaire fourmi ce1
[PDF] fourmis maternelle activités
[PDF] texte documentaire sur les fourmis ce2
[PDF] leçon sur les fourmis
[PDF] nouvelles ? chute
[PDF] exposé sur les fourmis cm2
[PDF] fiche documentaire sur les fourmis
[PDF] cycle de vie d'une fourmi
[PDF] tpe fourmis experience
[PDF] algorithme colonie de fourmis pdf
[PDF] organisation sociale des fourmis
[PDF] fourmi reine
[PDF] structures des sociétés animales
The Fourier Transform
and its Applications
The Fourier Transform:
F(s) =?
f(x)e-i2πsxdx
The Inverse Fourier Transform:
f(x) =?
F(s)ei2πsxds
Symmetry Properties:
Ifg(x) is real valued, thenG(s) is Hermitian:
G(-s) =G?(s)
Ifg(x) is imaginary valued, thenG(s) is Anti-Hermitian:
G(-s) =-G?(s)
In general:
g(x) =e(x) +o(x) =eR(x) +ieI(x) +oR(x) +ioI(x)
G(s) =E(s) +O(s) =ER(s) +iEI(s) +iOI(s) +OR(s)
Convolution:
(g?h)(x)?=? g(ξ)h(x-ξ)dξ Autocorrelation: Letg(x) be a function satisfying?∞ -∞|g(x)|2dx <∞(finite energy) then g(x)?= (g?? g)(x)?=? g(ξ)g?(ξ-x)dξ =g(x)?g?(-x) Cross correlation: Letg(x) andh(x) be functions with finite energy. Then (g?? h)(x)?=? g?(ξ)h(ξ+x)dξ g?(ξ-x)h(ξ)dξ = (h?? g)?(-x)
The Delta Function:δ(x)
•Scaling:δ(ax) =1|a|δ(x) •Sifting:?∞ -∞δ(x-a)f(x)dx=f(a) -∞δ(x)f(x+a)dx=f(a) •Convolution:δ(x)?f(x) =f(x)•Product:h(x)δ(x) =h(0)δ(x) •δ2(x) - no meaning •δ(x)?δ(x) =δ(x) •Fourier Transform ofδ(x):F{δ(x)}= 1 •Derivatives: -∞δ(n)(x)f(x)dx= (-1)nf(n)(0) -δ?(x)?f(x) =f?(x) -xδ(x) = 0 -xδ?(x) =-δ(x) •Meaning ofδ[h(x)]:
δ[h(x)] =?
iδ(x-xi)|h?(xi)|
The Shah Function: III(x)
•Sampling: III(x)g(x) =?∞ n=-∞g(n)δ(x-n) •Replication: III(x)?g(x) =?∞ n=-∞g(x-n) •Fourier Transform:F{III(x)}= III(s) •Scaling: III(ax) =?δ(ax-n) =1|a|?δ(x-na
Even and Odd Impulse Pairs
Even: II(x) =12
δ(x+12
) +12
δ(x-12
Odd: I
I(x) =12
δ(x+12
)-12
δ(x-12
Fourier Transforms:F{II(x)}= cosπs
F{II(x)}=isinπs
Fourier Transform Theorems
•Linearity:F{αf(x) +βg(x)}=αF(s) +βG(s) •Similarity:F{g(ax)}=1|a|G(sa •Shift:F{g(x-a)}=e-i2πasG(s)
F{g(ax-b)}=1|a|e-i2πsba
G(sa •Rayleigh"s:?∞ -∞|g(x)|2dx=?∞ -∞|G(s)|2ds •Power:?∞ -∞f(x)g?(x)dx=?∞ -∞F(s)G?(s)ds •Modulation:
F{g(x)cos(2πs0x)}=12
[G(s-s0) +G(s+s0)] •Convolution:F{f?g}=F(s)G(s) 1 •Autocorrelation:F{g?? g}=|G(s)|2 •Cross Correlation:F{g?? f}=G?(s)F(s) •Derivative: -F{g?(x)}=i2πsG(s) -F{g(n)(x)}= (i2πs)nG(s) -F{xng(x)}= (i2π)nG(n)(s) •Fourier Integral: Ifg(x) is of bounded variation and is absolutely integrable, then F -1{F{g(x)}}=12 [g(x+) +g(x-)] •Moments:?∞ f(x)dx=F(0) xf(x)dx=i2πF?(0) xnf(x)dx= (i2π)nF(n)(0) •Miscellaneous:
IfF{g(x)}=G(s) thenF{G(x)}=g(-s)
andF{g?(x)}=G?(-s) F ?x g(ξ)dξ? 12
G(0)δ(s) +G(s)i2πs
Function Widths
•Equivalent Width W f?=? -∞f(x)dxf(0)=F(0)f(0) F(0)? -∞F(s)ds=1W F •Autocorrelation Width W f??f?=? -∞f?? f dxf ?? f|x=0 |F(0)|2? -∞|F(s)|2ds=1W |F|2•Standard Deviation of Instantaneous Power: Δx (Δx)2?=? -∞x2|f(x)|2dx? -∞|f(x)|2dx-? -∞x|f(x)|2dx? -∞|f(x)|2dx? 2 (Δs)2?=? -∞s2|F(s)|2ds? -∞|F(s)|2ds-? -∞s|F(s)|2ds? -∞|F(s)|2ds? 2 -Uncertainty Relation: (Δx)(Δs)≥14π
Central Limit Theorem
Given a functionf(x), ifF(s) has a single absolute maximum ats= 0; and, for sufficiently small|s|, F(s)≈a-cs2where 0< a <∞and 0< c <∞, then: lim n→∞[⎷nf(⎷nx)]?na n=?πa 2 e-πac x2 and [f(x)]?n≈an+12 n 12 c e-πacn x2
Linear Systems
For a linear systemw(t) =S[v(t)] with responseh(t,τ) to a unit impulse at timeτ: S[αv1(t) +βv2(t)] =αS[v1(t)] +βS[v2(t)] w(t) =? v(τ)h(t,τ)dτ
If such a system is time-invariant, then:
w(t-τ) =S[v(t-τ)] and w(t) =? v(τ)h(t-τ)dτ = (v?h)(t) The eigenfunctions of any linear time-invariant system areei2πf0t, since for a system with transfer function H(s), the response to an input ofv(t) =ei2πf0tis given by: w(t) =H(f0)ei2πf0t.
Sampling Theory
ˆg(x) = III(xX
)g(x) =X∞? n=-∞g(nX)δ(x-nX) 2
G(s) =XIII(Xs)?G(s)
n=-∞G(s-nX Whittaker-Shannon-Kotelnikov Theorem: For a bandlim- ited functiong(x) with cutoff frequencies±sc, and with no discrete sinusoidal components at frequencysc, g(x) =∞? n=-∞g(n2sc)sinc[2sc(x-n2sc)]
Fourier Tranforms for Periodic Functions
For a functionp(x) with periodL, letf(x) =p(x)?(xL Then p(x) =f(x)?∞? n=-∞δ(x-nL)
P(s) =1L
n=-∞F(nL )δ(s-nL
The complex fourier series representation:
p(x) =∞? n=-∞c nei2πnL x where c n=1L F(nL 1L L/2 -L/2p(x)e-i2πnL xdx
The Discrete Fourier Transform
bandlimited to±BHz, so we samplef(x) every 1/2B seconds, obtainingN=?2BL?samples.
The Discrete Fourier Transform:
F m=N-1? n=0f ne-i2πmnN form= 0,...,N-1
The Inverse Discrete Fourier Transform:
f n=1N N-1? m=0F mei2πmnN forn= 0,...,N-1
Convolution:
h n=?N-1 k=0fkgn-kforn= 0,...,N-1 wheref,gare periodic
Serial Product:
h n=?N-1 k=0fkgn-kforn= 0,...,2N-2 wheref,gare not periodicDFT Theorems •Linearity:DFT {αfn+βgn}=αFm+βGm •Shift:DFT {fn-k}=Fme-i2πN km(fperiodic) •Parseval"s:?N-1 n=0fng?n=1N N-1 m=0FmG?m •Convolution:FmGm=DFT {?N-1 k=0fkgn-k}
The Hilbert Transform
The Hilbert Transform of f(x):
F
Hi(x)?=1π
-∞f(ξ)ξ-xdξ(CPV)
The Inverse Hilbert Transform:
f(x) =-1π -∞F
Hi(ξ)ξ-xdξ+fDC(CPV)
•Impulse response:-1πx •Transfer function:i sgn(s) •Causal functions: A causal functiong(x) has Fourier
TransformG(s) =R(s) +iI(s), whereI(s) =
H{R(s)}.
•Analytic signals: The analytic signal representation of a real-valued functionv(t) is given by: z(t)?=F-1{2H(s)V(s)} =v(t)-ivHi(t) •Narrow Band Signals:g(t) =A(t)cos[2πf0t+φ(t)] -Analytic approx:z(t)≈A(t)ei[2πf0t+φ(t)] -Envelope:A(t) =|z(t)| -Phase: arg[g(t)] = 2πf0t+φ(t) -Instantaneous freq:fi=f0+12πddt
φ(t)
The Two-Dimensional Fourier Transform
F(sx,sy) =?
f(x,y)e-i2π(sxx+syy)dxdy
The Inverse Two-Dimensional Fourier Transform:
f(x,y) =?
F(sx,sy)ei2π(sxx+syy)dsxdsy
The Hankel Transform (zero order):
F(q) = 2π?
0 f(r)J0(2πrq)rdr 3
The Inverse Hankel Transform (zero order):
f(r) = 2π? 0
F(q)J0(2πrq)qdq
Projection-Slice Theorem: The 1-D Fourier transform P θ(s) of any projectionpθ(x?) throughg(x,y) is identi- cal with the 2-D transformG(sx,sy) ofg(x,y), evaluated along a slice through the origin in the 2-D frequency do- main, the slice being at angleθto the x-axis. i.e.: P
θ(s) =G(scosθ,ssinθ)
Reconstruction by Convolution and Backprojection:
g(x,y) =? 0
F-1{|s|Pθ(s)}dθ
0 f
θ(xcosθ+ysinθ)dθ
wherefθ(x?) = (2s2csinc2scx?-s2csinc2scx?)?pθ(x?) compiled by John Jackson 4quotesdbs_dbs5.pdfusesText_9
[PDF] The Fourier Transform and its Applications