Collection of Formulae Basics EIT140 (OFDM) 1

Collection of Formulae Basics EIT140 (OFDM) 1
EIT140 (OFDM)
Collection of Formulae
1
Collection of Formulae
Basics
Euler relations
e±jα = cos α ± j sin α
Goniometric relations
sin(α ± β) =
cos(α ± β) =
sin 2α
cos 2α
=
=
sin α + sin β
=
sin α − sin β
=
cos α + cos β
=
cos α − cos β
=
sin α sin β
=
cos α cos β
=
sin α cos β
=
sin2 α
=
cos2 α
=
sin α cos β ± cos α sin β
cos α cos β ∓ sin α sin β
2 sin α cos α
cos2 α − sin2 α
α+β
α−β
2 sin
cos
2
2
α−β
α+β
sin
2 cos
2
2
α−β
α+β
cos
2 cos
2
2
α+β
α−β
−2 sin
sin
2
2
1
(cos(α − β) − cos(α + β))
2
1
(cos(α − β) + cos(α + β))
2
1
(sin(α − β) + sin(α + β))
2
1
(1 − cos 2α)
2
1
(1 + cos 2α)
2
Geometric sum
infinite length:
∞
X
qn =
1
,
1−q
qn =
(
N,
n=0
finite length:
N
−1
X
n=0
for |q| < 1
if q = 1
1−q N
1−q
, otherwise
Correlation
Correlation of two sequences a(n) ∈ C, b(n)
X ∈ C, n ∈ Z:
x(m) =
a(n)b∗ (n − m),
n
x(0) = ha, bi
Autocorrelation sequence rxx (l) and crosscorrelation sequence rxy (l) of deterministic signals:
P
- Definition for energy signal: rx{x,y} (l) =
x(n){x, y}(n − l)
n
K
P
1
2K+1
K→∞
n=−K
- Definition for power signal: rx{x,y} (l) = lim
- Definition for periodic signal: rx{x,y} (l) =
1
N
NP
−1
n=0
x(n){x, y}(n − l)
x(n){x, y}(n − l)
Autocorrelation sequence φxx (l) and crosscorrelation sequence φxy (l) of wide-sense stationary random
signals:
φx{x,y} (l) = E {x(n){x, y}(n − l)}
Frequently updated. Version: 0801. Comments, corrections, suggestions, questions are welcome (tom<AT>eit.lth.se).
EIT140 (OFDM)
Collection of Formulae
2
Convolution
Linear convolution of two sequences x(n) ∈ R, h(n) ∈ R, n ∈ Z:
X
y(n) =
x(k)h(n − k) = x(n) ∗ h(n)
k
N -circular convolution of two length-N sequences x(n) ∈ R, h(n) ∈ R, n = 0, . . . , N − 1 ([ ]N denotes
modulo-N of the argument):
y(n) =
N
−1
X
k=0
x(k)h([n − k]N ) = x(n) N h(n)
Fourier transform
Z∞
X(F ) =
−j2πF t
x(t)e
t=−∞
dt, F ∈ R, F in Hertz
x(t) =
Z∞
X(F )ej2πF t dF, t ∈ R, t in seconds
F =−∞
Z∞
X(jΩ) =
t=−∞
x(t)e−jΩt dt, Ω ∈ R, Ω = 2πF
Z∞
1
2π
x(t) =
X(jΩ)ejΩt dΩ, t ∈ R
Ω=−∞
z-transform
X(z) =
X
x(n)z
−n
n
1
x(n) =
2πj
,z ∈ C
I
X(z)z n−1 dz
C
Discrete-time Fourier transform (DTFT)
1
X(f ) =
X
n
jω
X(e ) =
X
n
x(n)e−j2πf n , −
1
1
≤f ≤
2
2
x(n) =
Z2
X(f )ej2πf n df, n ∈ Z
f =− 12
−jωn
x(n)e
1
x(n) =
2π
, −π ≤ ω ≤ π, ω = 2πf
Zπ
ω=−π
X(ejω )ejωn dω, n ∈ Z
DTFT Φxx (f ) of the autocorrelation sequence φxx (l) = power spectral density (PSD) Pxx (ω):
X
Pxx (ω) = Φxx (f ) =
φxx (l)e−j2πf l
l
Discrete Fourier transform (DFT)
X[k] =
N
−1
X
n=0
x(n)e−j2πkn/N , k = 0, . . . , N − 1
Sampling the DTFT:
Y [k] = X(ej2πk/N ), k = 0, . . . , N − 1
x(n) =
N −1
1 X
X[k]ej2πkn/N , n = 0, . . . , N − 1
N
k=0
y(n) =
X
m
x(n + mN ), n = 0, . . . , N − 1
EIT140 (OFDM)
Collection of Formulae
3
Properties of transforms
Fourier-transform
z-transform
DTFT
DFT
x(t) ←→ X(jΩ)
y(t) ←→ Y (jΩ)
x(n) ←→ X(z), Rx
y(n) ←→ Y (z), Ry
x(n) ←→ X(ejω )
y(n) ←→ Y (ejω )
x(n) ←→ X[k]
y(n) ←→ Y [k]
Linearity:
αx(t) + βy(t) ←→ αX(jΩ) + βY (jΩ)
αx(n) + βy(n) ←→ αX(z) + βY (z), incl. Rx ∩ Ry
αx(n) + βy(n) ←→ αX(ejω ) + βY (ejω )
Time shift:
x(t ± t0 ) ←→ e±jΩt0 X(jΩ)
x(n ± n0 ) ←→ z
x(n ± n0 ) ←→ e
Time reversal:
x(−t) ←→ X(−jΩ)
o
n ˛
˛ 1
1
x(−n) ←→ X(z −1 ), R−1
x = z ˛ r < |z| < r
Frequency shift:
e±jΩ0 t x(t) ←→ X(j(Ω ∓ Ω0 ))
n
z0
x(n) ←→ X(z/z0 ), |z0 |Rx
e
Convolution:
x(t) ∗ y(t) ←→ X(jΩ)Y (jΩ)
x(n) ∗ y(n) ←→ X(z)Y (z), incl. Rx ∩ Ry
x(n) ∗ y(n) ←→ X(ejω )Y (ejω )
property
Modulation:
1
x(t)y(t) ←→ 2π
∞
R
x∗ (t) ←→ X ∗ (−jΩ)
Diff. in trans.-domain:
tx(t) ←→ j
Parseval:
∞
R
X(jΦ)Y (j(Ω − Φ))dΦ
1
x(n)y(n) ←→ j2π
−∞
H
1
X(v)Y (z/v)v −1 dv, incl. Rx Ry
C
nx(n) ←→ −z
∞
R
X(jΩ)Y ∗ (jΩ)dΩ
−∞
∞
P
n=−∞
jω
X(e
)
x(−n) ←→ X(e−jω )
±jω0 n
x(n) ←→ X(e
1
x(n)y(n) ←→ 2π
Rπ
αx(n) + βy(n) ←→ αX[k] + βY [k]
x(hn ± n0 iN ) ←→ e±j2πkn0 /N X[k]
x(h−niN ) ←→ X[h−kiN ]
j(ω∓ω0 )
)
X(ejϕ )Y (ej(ω−ϕ) )dϕ
x∗ (n) ←→ X ∗ (e−jω )
dX(z)
, Rx except possibly z = 0, z = ∞
dz
1
x(n)y ∗ (n) = j2π
±jωn0
e±j2πk0 n/N x(n) ←→ X[hk ∓ k0 iN ]
x(n) N y(n) ←→ X[k]Y [k]
1
x(n)y(n) ←→ N
−π
x∗ (n) ←→ X ∗ (z ∗ ), Rx
dX(jΩ)
dΩ
1
x(t)y ∗ (t)dt = 2π
X(z), Rx \ {0(n0 > 0), ∞(n0 < 0)}
2
−∞
Conjugation:
±n0
H
C
X(v)Y ∗ (1/v ∗ )v −1 dv
nx(n) ←→ j
∞
P
n=−∞
NP
−1
m=0
X[m]Y [hk − miN ]
x∗ (n) ←→ X ∗ [h−kiN ]
dX(ejω )
dω
1
x(n)y ∗ (n) = 2π
Rπ
−π
X(ejω )Y ∗ (ejω )dω
NP
−1
n=0
1
x(n)y ∗ (n) = N
NP
−1
k=0
X[k]Y ∗ [k]
EIT140 (OFDM)
Collection of Formulae
4
Transforms of some elementary signals
unit sample
time-domain
(
1, n = 0
δ(n) =
0, n 6= 0
unit step
(
1, n ≥ 0
µ(n) =
0, n < 0
1
1−z −1 , R
αn µ(n)
1
1−αz −1 , R
1
πn
z-transform
DTFT
1, R = {z}
1
1
1−e−jω
= {z | |z| > 1 }
= {z | |z| > |α| }
(
1,
0,
n = −M, . . . , M
otherwise
2πδ(ω + 2πk)
k
1
1−αe−jω , |α|
(
1,
0,
sin(ωc n), n ∈ Z
P
<1
0 ≤ |ω| ≤ ωc
otherwise
sin((M + 21 )ω)
sin(ω/2)
Some probability distributions
pdf
cdf
mean
variance
Exponential(λ)
λ exp(−λx), x ∈ [0, ∞)
1 − exp(−λx)
λ−1
λ−2
Nakagami(µ, ω)
2µµ
2µ−1
exp
Γ(µ)ω µ x
γ(µ, xωµ )
Γ(µ)
Γ(µ+ 21 ) ω 1
2
Γ(µ) ( µ )
ω(1 − µ1 (
Normal N (µ, σ 2 )
√ 1
exp
2πσ 2
µ
σ2
Chi square χ2k
(1/2)k/2 k/2−1
exp(−x/2),
Γ(k/2) x
k
2k
Rayleigh R(σ)
x
σ2
x2
, x ∈ [0, ∞)
exp − 2σ
2
Rice RICE(σ, v)
x
σ2
exp
Uniform U(a, b)
1/(b − a), x ∈ [a, b]
erf(z)
erfc(z)
Γ(z)
...
...
...
− ωµ x2 , x ∈ (0, ∞)
− (x−µ)
2σ 2
−(x2 +v 2 )
2σ 2
2
, x ∈ (−∞, ∞)
I0 ( xv
σ 2 ), x ∈ [0, ∞)
error function. erf(z) =
√2
π
...
Rz
0
1
2
√ )
1 + erf( x−µ
σ 2
γ(k/2,x/2)
Γ(k/2)
x2
1 − exp − 2σ
2
σ
1 − Q1 ( σv , σx )
σ
x−a
b−x
a+b
2
exp(−t2 )dt, z ∈ R+
complementary error function. erfc(z) = 1 − erf(z) =
Gamma function. Γ(z) =
R∞
0
γ(a, z)
x ∈ [0, ∞)
2
tz−1 exp(−t)dt, z ∈ R
lower incomplete Gamma function. γ(a, z) =
Rz
0
In (z)
Qn (a, b)
L 12 (z)
...
...
...
modified Bessel function of first kind. In (z) =
Marcum Q-function. Qn (a, b) =
1
an−1
R∞
b
√2
π
R∞
z
pπ
2
4−π 2
2 σ
−v 2
1
2 L 2 ( 2σ 2 )
2σ 2 +v 2 − πσ2 L21 ( −v
2σ 2 )
pπ
Rπ
0
2
2
2
(b−a)2
12
exp(−t2 )dt, z ∈ R+
ta−1 exp(−t)dt, z, a ∈ R
1
π
Γ(µ+ 21 ) 2
Γ(µ) ) )
exp(z cos(t)) cos(nt)dt, n ∈ Z, z ∈ R
tn exp(−(t2 + a2 )/2)In−1 (at)dt, n ∈ Z, a, b ∈ R
Laguerre polynomial. L 12 (z) = exp(z/2) ((1 − z)I0 (−z/2) − zI1 (−z/2)) , z ∈ R
EIT140 (OFDM)
Collection of Formulae
5
Note:
X, Y ∼ N (0, σ 2 ), X and Y independent
−→
√
X 2 + Y 2 ∼ R(σ)
X ∼ N (v cos θ, σ 2 ), Y ∼ N (v sin θ, σ 2 ), X and Y independent, θ ∈ R −→
√
X 2 + Y 2 ∼ RICE(σ, v)
Xi ∼ N (0, 1), i = 1, . . . , k, Xi independent
Pk
−→
i=1
Xi2 ∼ χ2k
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