The Fourier Transform • Product: h(x)δ(x) = h(0)δ(x)

and its Applications • δ 2 (x) - no meaning

The Fourier Transform: • δ(x) ∗ δ(x) = δ(x)
Z ∞
F (s) = f (x)e−i2πsx dx • Fourier Transform of δ(x): F{δ(x)} = 1
−∞
• Derivatives:
The Inverse Fourier Transform: R∞
Z ∞ – −∞
δ (n) (x)f (x)dx = (−1)n f (n) (0)
f (x) = F (s)ei2πsx ds – δ 0 (x) ∗ f (x) = f 0 (x)
−∞
– xδ(x) = 0
Symmetry Properties: – 0
xδ (x) = −δ(x)
If g(x) is real valued, then G(s) is Hermitian:
• Meaning of δ[h(x)]:
G(−s) = G∗ (s)
X δ(x − xi )
If g(x) is imaginary valued, then G(s) is Anti-Hermitian: δ[h(x)] =
i
|h0 (xi )|
G(−s) = −G∗ (s)
The Shah Function: III(x)
In general: P∞
• Sampling: III(x)g(x) = n=−∞ g(n)δ(x − n)
g(x) = e(x) + o(x) = eR (x) + ieI (x) + oR (x) + ioI (x) P∞
• Replication: III(x) ∗ g(x) = n=−∞ g(x − n)
G(s) = E(s) + O(s) = ER (s) + iEI (s) + iOI (s) + OR (s)
• Fourier Transform: F{III(x)} = III(s)
Convolution:
1
δ(x − na )
P P
Z ∞ • Scaling: III(ax) = δ(ax − n) = |a|
4
(g ∗ h)(x) = g(ξ)h(x − ξ)dξ
−∞ 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

Cross correlation: Let g(x) and h(x) be functions with Fourier Transform Theorems
finite energy. Then
Z ∞ • Linearity: F{αf (x) + βg(x)} = αF (s) + βG(s)
∗ 4
(g ? h)(x) = g ∗ (ξ)h(ξ + x)dξ • Similarity: F{g(ax)} = 1 s
−∞ |a| G( a )
Z ∞
= g ∗ (ξ − x)h(ξ)dξ • Shift: F{g(x − a)} = e−i2πas G(s)
−∞
b
= (h∗ ? g)∗ (−x) F{g(ax − b)} = 1 −i2πs a
|a| e G( as )

The Delta Function: δ(x) • Rayleigh’s:

R∞ R∞
|g(x)|2 dx = −∞ |G(s)|2 ds
−∞
1
• Scaling: δ(ax) = |a| δ(x) • Power:
R∞ R∞
f (x)g ∗ (x)dx = −∞ F (s)G∗ (s)ds
−∞
R∞
• Sifting: −∞
δ(x − a)f (x)dx = f (a) • Modulation:
R∞
−∞
δ(x)f (x + a)dx = f (a) F{g(x)cos(2πs0 x)} = 21 [G(s − s0 ) + G(s + s0 )]

• Convolution: δ(x) ∗ f (x) = f (x) • Convolution: F{f ∗ g} = F (s)G(s)

1
• Autocorrelation: F{g ∗ ? g} = |G(s)|2 • Standard Deviation of Instantaneous Power: ∆x
R∞ 2 "R ∞ #2
• Cross Correlation: F{g ∗ ? f } = G∗ (s)F (s) 2 4 −∞
x |f (x)|2 dx −∞
x|f (x)|2 dx
(∆x) = R∞ − R∞
−∞
|f (x)|2 dx −∞
|f (x)|2 dx
• Derivative:
R∞ "R ∞ #2
– 0
F{g (x)} = i2πsG(s) 4 s2 |F (s)|2 ds s|F (s)|2 ds
(∆s)2
= R−∞
∞ − R−∞
∞
(n) n −∞
|F (s)|2 ds −∞
|F (s)|2 ds
– F{g (x)} = (i2πs) G(s)
i n (n)
– F{xn g(x)} = ( 2π ) G (s) 1
– Uncertainty Relation: (∆x)(∆s) ≥ 4π
• Fourier Integral: If g(x) is of bounded variation and
Central Limit Theorem
is absolutely integrable, then

1 Given a function f (x), if F (s) has a single absolute

F −1 {F{g(x)}} = [g(x+ ) + g(x− )] maximum at s = 0; and, for sufficiently small |s|,
2
F (s) ≈ a − cs2 where 0 < a < ∞ and 0 < c < ∞,
then: √ √
• Moments: [ nf ( nx)]∗n
r
πa − πa x2
Z ∞
lim = e c
f (x)dx = F (0) n→∞ an 2
−∞
and
∞ 1
an+ 2
Z
i 0
r
∗n π − πa x2
xf (x)dx = F (0) [f (x)] ≈ e cn
−∞ 2π 1 c
n2
Z ∞
i n (n) Linear Systems
xn f (x)dx = ( ) F (0)
−∞ 2π
For a linear system w(t) = S[v(t)] with response h(t, τ )
• Miscellaneous: to a unit impulse at time τ :
If F{g(x)} = G(s) then F{G(x)} = g(−s)
and F{g ∗ (x)} = G∗ (−s) S[αv1 (t) + βv2 (t)] = αS[v1 (t)] + βS[v2 (t)]
Z ∞
Z x 
G(s) w(t) = v(τ )h(t, τ )dτ
F g(ξ)dξ = 21 G(0)δ(s) + −∞
−∞ i2πs
If such a system is time-invariant, then:

Function Widths w(t − τ ) = S[v(t − τ )]

• Equivalent Width and

R∞ Z ∞
4 −∞
f (x)dx F (0) w(t) = v(τ )h(t − τ )dτ
Wf = = −∞
f (0) f (0)
= (v ∗ h)(t)
F (0) 1
= R∞ = The eigenfunctions of any linear time-invariant system
−∞
F (s)ds W F 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:
• Autocorrelation Width w(t) = H(f0 )ei2πf0 t .
R∞
4 −∞
f ∗ ? f dx Sampling Theory
Wf ∗ ?f =
f∗ ? f |x=0
x
ĝ(x) = III(
)g(x)
|F (0)|2 1 X
= R∞
2
= ∞
−∞
|F (s)| ds W|F |2 X
= X g(nX)δ(x − nX)
n=−∞

2
 Ĝ(s) = XIII(Xs) ∗ G(s) DFT Theorems
∞
X n
= G(s − ) • Linearity: DFT {αfn + βgn } = αFm + βGm
n=−∞
X
2π
• Shift: DFT {fn−k } = Fm e−i N km (f periodic)
Whittaker-Shannon-Kotelnikov Theorem: For a bandlim-
PN −1 ∗
PN −1 ∗
ited function g(x) with cutoff frequencies ±sc , and with • Parseval’s: n=0 fn gn = N
1
m=0 Fm Gm
no discrete sinusoidal components at frequency sc , PN −1
∞ • Convolution: Fm Gm = DFT { k=0 fk gn−k }
X n n
g(x) = g( )sinc[2sc (x − )]
n=−∞
2sc 2sc The Hilbert Transform

Fourier Tranforms for Periodic Functions The Hilbert Transform of f(x):

x
For a function p(x) with period L, let f (x) = p(x) u ( L ). 4 1
Z ∞
f (ξ)
Then FHi (x) = dξ (CPV)
π −∞ ξ − x
X∞
p(x) = f (x) ∗ δ(x − nL) The Inverse Hilbert Transform:
n=−∞
1 ∞ FHi (ξ)
Z
∞
1 X n n f (x) = − dξ + fDC (CPV)
P (s) = F ( )δ(s − ) π −∞ ξ − x
L n=−∞ L L
1
The complex fourier series representation: • Impulse response: − πx
∞
X n
• Transfer function: i sgn(s)
p(x) = cn ei2π L x
n=−∞ • Causal functions: A causal function g(x) has Fourier
Transform G(s) = R(s) + iI(s), where I(s) =
where
H{R(s)}.
1 n
cn = F( ) • Analytic signals: The analytic signal representation
L L
of a real-valued function v(t) is given by:
Z L/2
1 n
= p(x)e−i2π L x dx 4
L −L/2
z(t) = F −1 {2H(s)V (s)}
The Discrete Fourier Transform = v(t) − ivHi (t)

Let g(x) be a physical process, and let f (x) = g(x) for 0 ≤ • Narrow Band Signals: g(t) = A(t) cos[2πf0 t + φ(t)]
x ≤ L, f (x) = 0 otherwise. Suppose f(x) is approximately
bandlimited to ±B Hz, so we sample f (x) every 1/2B – Analytic approx: z(t) ≈ A(t)ei[2πf0 t+φ(t)]
seconds, obtaining N = b2BLc samples. – Envelope: A(t) =| z(t) |
The Discrete Fourier Transform:
– Phase: arg[g(t)] = 2πf0 t + φ(t)
N −1
−i 2πmn 1 d
– Instantaneous freq: fi = f0 + 2π dt φ(t)
X
Fm = fn e N for m = 0, . . . , N − 1
n=0
The Two-Dimensional Fourier Transform
The Inverse Discrete Fourier Transform:
N −1 ∞ ∞
1 X
Z Z
2πmn
fn = Fm ei N for n = 0, . . . , N − 1 F (sx , sy ) = f (x, y)e−i2π(sx x+sy y) dxdy
N m=0 −∞ −∞

Convolution: The Inverse Two-Dimensional Fourier Transform:

PN −1 Z ∞Z ∞
hn = k=0 fk gn−k for n = 0, . . . , N − 1
f (x, y) = F (sx , sy )ei2π(sx x+sy y) dsx dsy
where f , g are periodic −∞ −∞

Serial Product: The Hankel Transform (zero order):

PN −1
hn = k=0 fk gn−k for n = 0, . . . , 2N − 2 Z ∞
where f , g are not periodic F (q) = 2π f (r)J0 (2πrq)rdr
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 identi-
cal with the 2-D transform G(sx , sy ) of g(x, y), evaluated
along a slice through the origin in the 2-D frequency do-
main, 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