The Fourier Transform and its Applications

The Fourier Transform and its Applications The Fourier Transform: Z F (s) =

• Product:

h(x)δ(x) = h(0)δ(x)

• δ 2 (x) - no meaning •

∞

δ(x) ∗ δ(x) = δ(x)

• Fourier Transform of δ(x):

f (x)e−i2πsx dx

−∞

F{δ(x)} = 1

• Derivatives:

The Inverse Fourier Transform: Z ∞ F (s)ei2πsx ds f (x) =

R∞

–

−∞

δ (n) (x)f (x)dx = (−1)n f (n) (0) δ 0 (x) ∗ f (x) = f 0 (x)

–

−∞

–

Symmetry Properties: If g(x) is real valued, then G(s) is Hermitian:

xδ(x) = 0 0

xδ (x) = −δ(x)

– • Meaning of δ[h(x)]:

G(−s) = G∗ (s)

δ[h(x)] =

If g(x) is imaginary valued, then G(s) is Anti-Hermitian:

X δ(x − xi ) i

|h0 (xi )|

G(−s) = −G∗ (s) The Shah Function: III(x) P∞ • Sampling: III(x)g(x) = n=−∞ g(n)δ(x − n) P∞ • Replication: III(x) ∗ g(x) = n=−∞ g(x − n)

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)

• Fourier Transform: Convolution: 4

Z

• Scaling:

∞

(g ∗ h)(x) =

g(ξ)h(x − ξ)dξ

III(ax) =

P

F{III(x)} = III(s) P 1 δ(x − na ) δ(ax − n) = |a|

Even and Odd Impulse Pairs

−∞

Autocorrelation: Let g(x) be a function satisfying Even: II(x) = 1 δ(x + 1 ) + 1 δ(x − 1 ) R∞ 2 2 2 2 2 |g(x)| dx < ∞ (finite energy) then −∞ 1 1 1 1 Odd: II (x) = 2 δ(x + 2 ) − 2 δ(x − 2 ) Z ∞ 4 4 Γg (x) = (g ∗ ? g)(x) = g(ξ)g ∗ (ξ − x)dξ Fourier Transforms:

F{II(x)} = cos πs

−∞

= g(x) ∗ g ∗ (−x)

F{II (x)} = i sin πs Fourier Transform Theorems

Cross correlation: Let g(x) and h(x) be functions with finite energy. Then Z ∞ 4 ∗ (g ? h)(x) = g ∗ (ξ)h(ξ + x)dξ −∞ Z ∞ = g ∗ (ξ − x)h(ξ)dξ

• Linearity:

F{αf (x) + βg(x)} = αF (s) + βG(s)

• Similarity:

F{g(ax)} =

1 s |a| G( a )

F{g(x − a)} = e−i2πas G(s)

• Shift:

−∞

=

(h∗ ? g)∗ (−x)

F{g(ax − b)} =

The Delta Function: δ(x) • Scaling: • Sifting:

δ(ax) = R∞ −∞

R∞ −∞

• Convolution:

• Rayleigh’s: 1 |a| δ(x)

• Power:

δ(x − a)f (x)dx = f (a)

b 1 −i2πs a G( as ) |a| e

R∞ |g(x)|2 dx = −∞ |G(s)|2 ds R∞ R∞ f (x)g ∗ (x)dx = −∞ F (s)G∗ (s)ds −∞ R∞

−∞

• Modulation: F{g(x)cos(2πs0 x)} = 21 [G(s − s0 ) + G(s + s0 )]

δ(x)f (x + a)dx = f (a) δ(x) ∗ f (x) = f (x)

• Convolution: 1

F{f ∗ g} = F (s)G(s)

F{g ∗ ? g} = |G(s)|2

• Autocorrelation:

• Standard Deviation of Instantaneous Power: ∆x "R ∞ #2 R∞ 2 x |f (x)|2 dx x|f (x)|2 dx 4 −∞ −∞ 2 R∞ (∆x) = − R∞ |f (x)|2 dx |f (x)|2 dx −∞ −∞

F{g ∗ ? f } = G∗ (s)F (s)

• Cross Correlation: • Derivative:

0

F{g (x)} = i2πsG(s)

– F{g

–

(n)

2

(∆s)

• Moments: Z

(x)} = (i2πs) G(s)

∞

and

xf (x)dx = −∞ ∞

xn f (x)dx = (

−∞

Z

x

 g(ξ)dξ

−∞

i 0 F (0) 2π

1

∗n

[f (x)]

i n (n) ) F (0) 2π

F{G(x)} = g(−s) F{g ∗ (x)} = G∗ (−s)

=

=

4

=

=

an+ 2

≈

r

1 n2

π − πa x2 e cn c

= 21 G(0)δ(s) +

G(s) i2πs

−∞

If such a system is time-invariant, then: w(t − τ ) = S[v(t − τ )] and

R∞

f (x)dx −∞ f (0)

Z

F (0) = f (0)

w(t)

v(τ )h(t − τ )dτ (v ∗ h)(t)

=

F (0) 1 = W F (s)ds F −∞

The eigenfunctions of any linear time-invariant system are ei2πf0 t , since for a system with transfer function H(s), the response to an input of v(t) = ei2πf0 t is given by: w(t) = H(f0 )ei2πf0 t .

f ∗ ? f dx

−∞ f∗ ?

∞

= −∞

R∞

• Autocorrelation Width R∞ Wf ∗ ?f

1 4π

S[αv1 (t) + βv2 (t)] = αS[v1 (t)] + βS[v2 (t)] Z ∞ w(t) = v(τ )h(t, τ )dτ

• Equivalent Width Wf

(∆x)(∆s) ≥

Linear Systems

Function Widths

4

|F (s)|2 ds

For a linear system w(t) = S[v(t)] with response h(t, τ ) to a unit impulse at time τ :

• Miscellaneous: If F{g(x)} = G(s) then and

F

#2

Given a function f (x), if F (s) has a single absolute maximum at s = 0; and, for sufficiently small |s|, F (s) ≈ a − cs2 where 0 < a < ∞ and 0 < c < ∞, then: r √ √ πa − πa x2 [ nf ( nx)]∗n lim = e c n→∞ an 2

1 [g(x+ ) + g(x− )] 2

∞

−

s|F (s)|2 ds

R−∞ ∞ −∞

Central Limit Theorem

f (x)dx = F (0)

Z

|F (s)|2 ds

"R ∞

– Uncertainty Relation:

−∞

Z

s2 |F (s)|2 ds

R−∞ ∞ −∞

=

• Fourier Integral: If g(x) is of bounded variation and is absolutely integrable, then F −1 {F{g(x)}} =

R∞

n

i n (n) F{xn g(x)} = ( 2π ) G (s)

–

4

Sampling Theory

f |x=0 gˆ(x)

|F (0)|2 1 R∞ = 2 W|F |2 |F (s)| ds −∞

x )g(x) X ∞ X = X g(nX)δ(x − nX) =

III(

n=−∞

2

ˆ G(s)

= XIII(Xs) ∗ G(s) ∞ X n = G(s − ) X n=−∞

DFT Theorems • Linearity:

∞ X

g(

n=−∞

n n )sinc[2sc (x − )] 2sc 2sc

The Hilbert Transform

Fourier Tranforms for Periodic Functions For a function p(x) with period L, let f (x) = Then ∞ X p(x) = f (x) ∗ δ(x − nL)

The Hilbert Transform of f(x): Z ∞ f (ξ) 4 1 FHi (x) = dξ (CPV) π −∞ ξ − x

x ). p(x) u ( L

The Inverse Hilbert Transform: Z 1 ∞ FHi (ξ) f (x) = − dξ + fDC (CPV) π −∞ ξ − x

n=−∞

P (s)

∞ n n 1 X F ( )δ(s − ) L n=−∞ L L

=

The complex fourier series representation: ∞ X

p(x) =

n

cn ei2π L x

where =

1 n F( ) L L

=

1 L

Z

• Impulse response:

1 − πx

• Transfer function:

i sgn(s)

• Causal functions: A causal function g(x) has Fourier Transform G(s) = R(s) + iI(s), where I(s) = H{R(s)}.

n=−∞

cn

2π

DFT {fn−k } = Fm e−i N km (f periodic) PN −1 PN −1 1 ∗ ∗ • Parseval’s: n=0 fn gn = N m=0 Fm Gm PN −1 • Convolution: Fm Gm = DFT { k=0 fk gn−k } • Shift:

Whittaker-Shannon-Kotelnikov Theorem: For a bandlimited function g(x) with cutoff frequencies ±sc , and with no discrete sinusoidal components at frequency sc , g(x) =

DFT {αfn + βgn } = αFm + βGm

• Analytic signals: The analytic signal representation of a real-valued function v(t) is given by:

L/2

n

p(x)e−i2π L x dx

4

z(t) = F −1 {2H(s)V (s)} = v(t) − ivHi (t)

−L/2

The Discrete Fourier Transform Let g(x) be a physical process, and let f (x) = g(x) for 0 ≤ x ≤ L, f (x) = 0 otherwise. Suppose f(x) is approximately bandlimited to ±B Hz, so we sample f (x) every 1/2B seconds, obtaining N = b2BLc samples. The Discrete Fourier Transform: Fm =

N −1 X

−i 2πmn N

• Narrow Band Signals: g(t) = A(t) cos[2πf0 t + φ(t)]

– Phase:

Convolution: PN −1 hn = k=0 fk gn−k Serial Product: PN −1 hn = k=0 fk gn−k

fi = f0 +

1 d 2π dt φ(t)

The Two-Dimensional Fourier Transform

The Inverse Discrete Fourier Transform: N −1 2πmn 1 X Fm ei N N m=0

arg[g(t)] = 2πf0 t + φ(t)

– Instantaneous freq:

n=0

fn =

A(t) =| z(t) |

– Envelope:

for m = 0, . . . , N − 1

fn e

z(t) ≈ A(t)ei[2πf0 t+φ(t)]

– Analytic approx:

Z

for n = 0, . . . , N − 1

∞

Z

∞

f (x, y)e−i2π(sx x+sy y) dxdy

F (sx , sy ) = −∞

−∞

The Inverse Two-Dimensional Fourier Transform: Z ∞Z ∞ f (x, y) = F (sx , sy )ei2π(sx x+sy y) dsx dsy

for n = 0, . . . , N − 1 where f , g are periodic

−∞

−∞

The Hankel Transform (zero order): Z ∞ F (q) = 2π f (r)J0 (2πrq)rdr

for n = 0, . . . , 2N − 2 where f , g are not periodic

0

3

The Inverse Hankel Transform (zero order): Z ∞ f (r) = 2π F (q)J0 (2πrq)qdq 0

Projection-Slice Theorem: The 1-D Fourier transform Pθ (s) of any projection pθ (x0 ) through g(x, y) is identical with the 2-D transform G(sx , sy ) of g(x, y), evaluated along a slice through the origin in the 2-D frequency domain, the slice being at angle θ to the x-axis. i.e.: Pθ (s) = G(s cos θ, s sin θ) Reconstruction by Convolution and Backprojection: Z π g(x, y) = F −1 {| s | Pθ (s)}dθ Z0 π = fθ (x cos θ + y sin θ)dθ 0

where fθ (x0 )

=

(2s2c sinc2sc x0 − s2c sinc2 sc x0 ) ∗ pθ (x0 )

compiled by John Jackson

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