Fourier Transform Cheat Sheet

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, z¯ = x − iy, |z| = z¯ z= x2 + y 2 1 = −i i z + z¯ z − z¯ 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,

t0

∞ 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

5