EE 261 The Fourier Transform
and its Applications
1
cn =
T
Integration by parts:
t=b Z
b
u(t)v0 (t) dt = u(t)v(t)
−
a
t=a
T
−2πint/T
e
0
1
f(t) dt =
T
Z
T /2
e−2πint/T f(t) dt
−T /2
Orthogonality of the complex exponentials:
(
Z T
0,
n 6= m
e2πint/T e−2πimt/T dt =
T,
n=m
0
This Being an Ancient Formula Sheet
Handed Down
To All EE 261 Students
Z
Z
√
The normalized exponentials (1/ T )e2πint/T , n =
0, ±1, ±2, . . . form an orthonormal basis for L2 ([0, T ])
b
u0(t)v(t) dt
Rayleigh (Parseval): If f(t) is periodic of period T
then
Z
∞
X
1 T
|f(t)|2 dt =
|ck |2
T 0
a
Even and odd parts of a function: Any function f(x)
can be written as
k=−∞
f(x) + f(−x)
f(x) =
+
2
(even part)
f(x) − f(−x)
2
(odd part)
The Fourier Transform:
Z ∞
Ff(s) =
f(x)e−2πisx dx
−∞
Geometric series:
N
X
rn =
n=0
N
X
n=M
The Inverse Fourier Transform:
Z ∞
−1
F f(x) =
f(s)e2πisx ds
1 − rN +1
1−r
−∞
rM (1 − rN −M +1 )
rn =
(1 − r)
Symmetry & Duality Properties:
Let f − (x) = f(−x).
2
Complex numbers: z = x + iy, z̄ = x − iy, |z| = zz̄ =
x2 + y 2
1
= −i
i
z + z̄
z − z̄
x = Re z =
, y = Im z =
2
2i
Complex exponentials:
cos 2πt =
e
−2πit
+e
2
2πit
,
sin 2πt =
e
z = reiθ ,
r=
−e
2i
e2πint =
n=−N
p
x2 + y2 , θ = tan−1 (y/x)
sin(2N + 1)πt
sin πt
∞
X
–
Ff − = (Ff)−
–
If f is even (odd) then Ff is even (odd)
–
If f is real valued, then Ff = (Ff)−
Z
∞
f(x − y)g(y) dy
–
f ∗g = g∗f
–
(f ∗ g) ∗ h = (f ∗ g) ∗ h
–
f ∗ (g + h) = f ∗ g + f ∗ h
Smoothing: If f (or g) is p-times continuously differentiable, p ≥ 0, then so is f ∗ g and
dk
dk
(f ∗ g) = ( k f) ∗ g
k
dx
dx
Fourier series If f(t) is periodic with period T its
Fourier series is
f(t) =
F −1f = Ff −
−∞
Symmetric sum of complex exponentials (special
case of geometric series):
N
X
–
(f ∗ g)(x) =
−2πit
Polar form:
z = x + iy
FFf = f −
Convolution:
e2πit = cos 2πt + i sin 2πt
2πit
–
Convolution Theorem:
F(f ∗ g) = (Ff)(Fg)
cne2πint/T
F(fg) = Ff ∗ Fg
n=−∞
1
Autocorrelation: Let g(x) be a function satisfying
R∞
2
|g(x)| dx < ∞ (finite energy) then
−∞
(g ? g)(x)
=
Z
Linearity:
F{αf(x) + βg(x)} = αF (s) + βG(s)
Stretch:
F{g(ax)} =
1
G( as )
|a|
∞
F{g(x − a)} = e−i2πas G(s)
Shift:
g(y)g(y − x) dy
−∞
=
Shift & stretch:
g(x) ∗ g(−x)
−∞
−∞
−∞
Modulation:
(h ? g)(−x)
F{g(x) cos(2πs0 x)} = 12 [G(s − s0 ) + G(s + s0 )]
Rectangle and triangle functions
(
(
1, |x| < 12
1 − |x|,
Π(x) =
Λ(x) =
1
0,
0, |x| ≥ 2
sin πs
,
πs
Scaled rectangle function
(
Πp (x) = Π(x/p) =
Cross Correlation:
Λp (x) = Λ(x/p) =
F{g ? f} = G(s)F (s)
Derivative:
1,
0,
|x| <
|x| ≥
p
2
p
2
–
F{g0 (x)} = 2πisG(s)
–
F{g(n) (x)} = (2πis)n G(s)
–
i n (n)
F{xng(x)} = ( 2π
) G (s)
,
Moments:
Z
FΠp (s) = p sinc ps
Scaled triangle function
(
F{g ? g} = |G(s)|2
Autocorrelation:
|x| ≤ 1
|x| ≥ 1
FΛ(s) = sinc2 s
FΠ(s) = sinc s =
1 − |x/p|,
0,
Z
|x| ≤ p
,
|x| ≥ p
∞
f(x)dx = F (0)
−∞
Z
∞
xf(x)dx =
−∞
∞
xnf(x)dx = (
−∞
2
FΛp (s) = p sinc ps
Gaussian
2
2
Gaussian with variance σ2
2
2
1
√ e−x /2σ
σ 2π
i n (n)
) F (0)
2π
The Delta Function: δ(x)
Fg(s) = e−2π
2
σ 2 s2
Scaling:
One-sided exponential decay
(
0,
t < 0,
1
f(t) =
Ff(s) =
a + 2πis
e−at ,
t ≥ 0.
Sifting:
δ(ax) =
Convolution:
1
δ(x)
|a|
R∞
δ(x − a)f(x)dx = f(a)
R∞
δ(x)f(x + a)dx = f(a)
−∞
−∞
Two-sided exponential decay
F(e−a|t| ) =
i 0
F (0)
2π
Miscellaneous:
Z x
G(s)
F
g(ξ)dξ = 12 G(0)δ(s) +
i2πs
−∞
F(e−πt ) = e−πs
g(t) =
1 −i2πsb/a
G( as )
|a| e
Rayleigh (Parseval):
Z ∞
Z ∞
2
|g(x)| dx =
|G(s)|2 ds
−∞
−∞
Z ∞
Z ∞
f(x)g(x) dx =
F (s)G(s) ds
Cross correlation: Let g(x) and h(x) be functions with
finite energy. Then
Z ∞
(g ? h)(x) =
g(y)h(y + x) dy
−∞
Z ∞
=
g(y − x)h(y) dy
=
F{g(ax − b)} =
δ(x) ∗ f(x) = f(x)
δ(x − a) ∗ f(x) = f(x − a)
2a
a2 + 4π2s2
δ(x − a) ∗ δ(x − b) = δ(x − (a + b))
Fourier Transform Theorems
Product:
2
h(x)δ(x) = h(0)δ(x)
Fourier Transform:
Fδ = 1
The complex Fourier series representation:
F(δ(x − a)) = e−2πisa
R∞
–
−∞
where
δ (n)(x)f(x)dx = (−1)n f (n) (0)
δ 0(x) ∗ f(x) = f 0 (x)
–
–
αn
=
1
n
F( )
L L
=
1
L
xδ(x) = 0
0
–
xδ (x) = −δ(x)
Fourier transform of cosine and sine
(
−1,
sgn t =
1,
t<0
t>0
∞
X
n=−∞
III(x)g(x) =
Periodization:
P∞
n=−∞
III(x) ∗ g(x) =
w(t − τ ) = L[v(t − τ )]
In this case L(δ(t − τ ) = h(t − τ ) and L acts by
convolution:
Z ∞
w(t) = Lv(t) =
v(τ )h(t − τ )dτ
−∞
δ(x − np)
=
P∞
n=−∞
Fourier Transform:
FIII = III ,
g(x − n)
a>0
Le2πiνt = H(ν)e2πiνt
FIIIp = 1p III1/p
The Discrete Fourier Transform
Sampling Theory For a bandlimited function g(x) with
Fg(s) = 0 for |s| ≥ p/2
N th root of unity:
Let ω = e2πi/N . Then ωN = 1 and the N powers
1 = ω0 , ω, ω2 , . . . ωN −1 are distinct and evenly
spaced along the unit circle.
Fg = Πp (Fg ∗ IIIp )
g(t) =
∞
X
g(tk ) sinc(p(x − tk ))
(v ∗ h)(t)
The transfer function is the Fourier transform of the
impulse response, H = Fh The eigenfunctions of any
linear time-invariant system are e2πiνt, with eigenvalue H(ν):
g(n)δ(x − n)
III(ax) = a1 III1/a (x),
Scaling:
−L/2
A system is time-invariant if:
n=−∞
Sampling:
n
p(x)e−2πi L xdx
−∞
1
Fsgn (s) =
πis
IIIp (x) =
L/2
Superposition integral:
Z ∞
w(t) =
v(τ )h(t, τ )dτ
The Shah Function: III(x)
δ(x − n) ,
Z
Linear Systems Let L be a linear system, w(t) = Lv(t),
with impulse response h(t, τ ) = Lδ(t − τ ).
1
(δ(s − a) + δ(s + a))
2
1
F sin 2πat = (δ(s − a) − δ(s + a))
2i
Unit step and sgn
(
0, t ≤ 0
1
1
H(t) =
FH(s) =
δ(s) +
2
πis
1, t > 0
F cos 2πat =
∞
X
n
αne2πi L x
n=−∞
Derivatives:
III(x) =
∞
X
p(x) =
tk = k/p
k=−∞
Vector complex exponentials:
Fourier Transforms for Periodic Functions
x
For a function p(x) with period L, let f(x) = p(x)Π( L
).
Then
∞
X
p(x) =
f(x) ∗
P (s) =
∞
n
n
1 X
F ( )δ(s − )
L n=−∞ L
L
1 = (1, 1, . . ., 1)
ω = (1, ω, ω2, . . . , ωN −1)
ω k = (1, ωk , ω2k, . . . , ω(N −1)k )
δ(x − nL)
n=−∞
Cyclic property
ωN = 1 and
3
1, ω, ω2 , . . .ω N −1
are distinct
Ff=
Impulse response:
1
− πx
Transfer function:
i sgn (s)
Causal functions: g(x) is causal if g(x) = 0 for x < 0.
A casual signal Fourier Transform G(s) = R(s) +
iI(s), where I(s) = H{R(s)}.
The DFT of order N accepts an N -tuple as input
and returns an N -tuple as output. Write an N -tuple
as f = (f[0], f[1], . . ., f[N − 1]).
N
−1
X
Analytic signals: The analytic signal representation
of a real-valued function v(t) is given by:
f[k]ω−k
Z(t)
k=0
Inverse DFT:
F −1 f =
N −1
1 X
f[k]ωk
N
=
=
F −1 {2H(s)V (s)}
v(t) − iHv(t)
Narrow Band Signals: g(t) = A(t) cos[2πf0 t + φ(t)]
Analytic approx:
k=0
Envelope:
Periodicity of inputs and outputs: If F = F f then
both f and F are periodic of period N .
arg[z(t)] = 2πf0 t + φ(t)
Instantaneous freq:
(f ∗ g)[n] =
f[k] g[n − k]
Discrete δ:
m≡k
m 6≡ k
mod N
mod N
F δ k = ω−k
DFT of vector complex exponential
F ωk = N δ k
In 2-dimensions (in coordinates):
Z ∞ Z ∞
Ff(ξ1 , ξ2) =
e−2πi(x1 ξ1 +x2 ξ2 ) f(x1 , x2) dx1dx2
−∞
F f − = (F f)−
Parseval:
F {αf + βg) = αF f + βF g
0
The Inverse Hankel Transform (zero order):
Z ∞
f(r) = 2π
F (ρ)J0 (2πrρ)ρdρ
F f · F g = N (f · g)
Shift: Let τp f[m] = f[m − p]. Then F (τp f) = ω−p F f
Modulation:
F (ωp f) = τp (F f)
Convolution:
F (f ∗ g) = (F f)(F g)
F (f g) =
1
N
−∞
The Hankel Transform (zero order):
Z ∞
F (ρ) = 2π
f(r)J0 (2πrρ)rdr
DFT Theorems
Linearity:
1 d
φ(t)
2π dt
Inverse Fourier Transform:
Z
−1
F f(x) =
e2πix·ξ f(ξ) dξ
Rn
DFT of the discrete δ
Reversed signal: f − [m] = f[−m]
fi = f0 +
Higher Dimensional Fourier Transform In ndimensions:
Z
Ff(ξ) =
e−2πix·ξ f(x) dx
Rn
k=0
(
1,
δ k [m] =
0,
z(t) ≈ A(t)ei[2πf0 t+φ(t)]
| A(t) |=| z(t) |
Phase:
Convolution
N
−1
X
H−1 f = −Hf
Inverse Hilbert Transform
The vector complex exponentials are orthogonal:
(
0,
k 6≡ ` mod N
ωk · ω` =
N,
k ≡ ` mod N
0
Separable functions: If f(x1 , x2) = f(x1 )f(x2 ) then
Ff(ξ1 , ξ2) = Ff(ξ1 )Ff(ξ2 )
(F f ∗ F g)
Two-dimensional rect:
The Hilbert Transform The Hilbert Transform of
Π(x1, x2) = Π(x1)Π(x2 ), FΠ(ξ1 , ξ2) = sinc ξ1 sinc ξ2
f(x):
Z
Two dimensional Gaussian:
1 ∞ f(ξ)
1
∗ f(x) =
dξ
Hf(x) = −
2
2
πx
π −∞ ξ − x
g(x1 , x2) = e−π(x1 +x2 ) , Fg = g
Fourier transform theorems
(Cauchy principal value)
4
Shift: Let (τb f)(x) = f(x − b). Then
F(τb f)(ξ) = e−2πiξ·b Ff(ξ)
Stretch theorem (special):
F(f(a1 x1 , a2, x2)) =
1
ξ1 ξ2
Ff( , )
|a1||a2|
a1 a2
Stretch theorem (general): If A is an n × n invertible
matrix then
1
F(f(Ax)) =
Ff(A−T ξ)
| det A|
Stretch and shift:
F(f(Ax + b)) = exp(2πib · A−T ξ)
1
Ff(A−T ξ)
| det A|
III’s and lattices III for integer lattice
X
IIIZ 2 (x) =
δ(x − n)
2
n∈Z
∞
X
=
δ(x1 − n1 , x2 − n2 )
n1 ,n2 =−∞
FIIIZ = IIIZ 2
A general lattice L can be obtained from the integer lattice by L = A(Z2 ) where A is an invertible matrix.
X
1
IIIL (x) =
δ(x − p) =
III 2 (A−1 x)
| det A| Z
2
p∈L
2
If L = A(Z ) then the reciprocal lattice is L∗ = A−T Z2
Fourier transform of IIIL :
1
FIIIL =
IIIL∗
| det A|
Radon transform and Projection-Slice Theorem:
Let µ(x1 , x2) be the density of a two-dimensional region.
A line through the region is specified by the angle φ of
its normal vector to the x1-axis, and its directed distance
ρ from the origin. The integral along a line through the
region is given by the Radon transform of µ:
Z ∞ Z ∞
Rµ(ρ, φ) =
µ(x1, x2)δ(ρ−x1 cos φ−x2 sin φ) dx1dx2
−∞
−∞
The one-dimensional Fourier transform of Rµ with respect to ρ is the two-dimensional Fourier transform of µ:
Fρ R(µ)(r, φ) = Fµ(ξ1 , ξ2),
ξ1 = r cos φ, ξ2 = r sin φ
The list being compiled originally
by John Jackson
a person not known to me,
and then revised here
by your humble instructor
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