collection of formulas in signal processing
TRANSCRIPT
-
8/14/2019 Collection of Formulas in Signal Processing
1/21
Collection of Formulas in Signal Processing
Stockholm October 6, 2006
Kungliga Tekniska Hogskolan
Signal Processing LabElectrical Engineering
SE-100 44 Stockholm
-
8/14/2019 Collection of Formulas in Signal Processing
2/21
-
8/14/2019 Collection of Formulas in Signal Processing
3/21
1
Contents
1 Deterministic Signals 2
2 Wide-sense stationary processes 4
3 Some common distributions 6
4 Continuous Fourier transform 8
5 The Laplace transform 10
6 z-Transform 14
7 Discrete Time Fourier Transform 17
8 Discrete Fourier Transform (DFT) 19
-
8/14/2019 Collection of Formulas in Signal Processing
4/21
2 1 Deterministic Signals
1 Definitions and Relations for DeterministicSignals
Time continuous Time discreteFundamentals
Spectrum X(f) =
x(t)ej2f tdt Xd() =
n=
x[n]ej2n
Inverse x(t) =
X(f)ej2ftdf x[n] =
1/21/2
Xd()ej2nd
EnergySpectrum
SX
(f) =|X(f)
|2 S
X() =
|X
d()
|2
Total Energy
|x(t)|2dt =
|X(f)|2df
n=
|x[n]|2 =1/21/2
|Xd()|2d
Linear FilteringOutputSignal
y(t) = h(t) x(t) y[n] = h[n] x[n]
=
h(u)x(t u)du =
m=
h[m]x[n m]
Output
Spectrum
Y(f) = H(f)X(f) Yd() = Hd()Xd()
Frequencyresponse
H(f) =
h(t)ej2f tdt Hd() =
n=
h[n]ej2n
Transfer function (causal systems)
H(s) =
0
h(t)estdt H(z) =n=0
h[n]zn
Transfer function (non-causal systems)
H(s) =
h(t)estdt H(z) =
n=
h[n]zn
-
8/14/2019 Collection of Formulas in Signal Processing
5/21
1 Deterministic Signals 3
Sampling
x(t)
X(f)
y[n]
Yd()nT
y[n] = x(nT) = Yd() = 1T
m
X
m
T
Pulse amplitude modulation
z(t)
Z(f)
y[n]
Yd()
PAM
p(t)
z(t) =n
y[n]p(t nT) = Z(f) = P(f)Yd(f T)
Reconstruction continuous time
x(t) =n
x(nT)h(t nT) E =
|x(t) x(t)|2 dt
E =
H(f)
T 1
X(f) +
m=0
H(f)
TX(f m/T)
2
df
The sampling theorem
If X(f) = 0 for
|f
| 12T
, then
x(t) =
n=
x(nT)sin((t nT)/T)
(t nT)/T
-
8/14/2019 Collection of Formulas in Signal Processing
6/21
4 2 Wide-sense stationary processes
2 Definitions and relations for wide-sense sta-tionary processes
Continuous time Discrete timeFundamentals
acf rX() = E[X(t + )X(t)] rX(k) = E[X(n + k)X(n)]
PSD RX(f) =
rX()ej2fd RX() =
k=
rX(k)ej2k
Inverse rX() =
RX(f)ej2ftdf rX(k) =
1/2
1/2
RX()ej2kd
Total power
RX(f)df = E[X(t)2]
1/21/2
RX()d = E[X(n)2]
Linear filteringFiltered signal Y(t) = h(t) X(t) Y(n) = h(n) X(n)
=
h(u)X(t u)du =
m=
h(m)X(n m)
Expected value mY = mX
h(u)du mY = mX
m=
h(m)
= mXH(0) = mXH(0)
acf rY() =
h(u)h(v) rY(k) =
=
m=
h()h(m)
rX( u + v)dudv rX(k + m)PSD RY(f) = |H(f)|2RX(f) RY() = |H()|2RX()
-
8/14/2019 Collection of Formulas in Signal Processing
7/21
2 Wide-sense stationary processes 5
Sampling
X(t)
RX(f)
Y[n]
RY()nT
Y[n] = X(nT) = RY() = 1T
m
RX
m
T
Pulse amplitude modulation
Z(t)
RZ(f)
Y[n]
RY()
PAM
p(t )
Z(t) =n
Y[n]p(t nT ) =
RZ(f) =1T|P(f)|2RY(f T)
E[Z(t)] = 1T
P(0)E[Y(n)]
Reconstruction contionuous time
X(t) =n
X(nT + )h(t nT ) P = E[(X(t) X(t))2]
P =
H(f)
T 1
2
RX(f) +m=0
H(f)T2
RX(f m/T) df
-
8/14/2019 Collection of Formulas in Signal Processing
8/21
6 3 Some common distributions
3 Some common distributions
Uniform distribution Re(a, b)
fX(x) =
1b a a < x < b0 otherwise
mX = E[X] =a + b
2
2 = E[(X mX)2] = (b a)2
12
Rayleigh distribution
fX(x) =
xa2
ex2
2a2 x 00 x < 0
mX = E[X] = a
2
2 = E[(X mX)2] = a2 (2 /2)
One-sided exponential distribution
fX(x) =
aeax x 00 x < 0
mX = =1
a
Two-sided exponential distribution
fX(x) =a
2ea|x|
mX = 0, 2 =
2
a2
Poisson distribution Discrete integer distribution
P(X = k)= pk = e
a ak
k!for k = 0, 1, 2, . . .
mX = 2 = a
-
8/14/2019 Collection of Formulas in Signal Processing
9/21
3 Some common distributions 7
Normal distribution One-dimensional, N(mX , )
fX(x) =1
2e
(xmX)2
22
Two-dimensional, N(mX , mY, X , Y, )
fX,Y(x, y) =1
2XY
1 2e
g(x,y)
2(12)
where
g(x, y) =
x mX
X
2 2x mX
X
y mYY
+
y mY
Y
2
and
= (X, Y) =E[XY] mXmY
XY
If A,B,C and D are all Normal distributed, then
E[ABCD] = E[AB] E[CD]+E[AC] E[BD]+E[AD] E[BC]2 E[A] E[B] E[C] E[D]
The so-called Q-function can also be convenient to use. If X is N(mX , ),then Pr[X > a] = Q
amX
where
Q(x) =12
x
eu2/2du
For x > 1,
1 x2x
2ex
2/2 < Q(x) 0 X(f), ( = 2f)
(t) 1 (4.15)
1 (f) (4.16)
rectP(t) =
1 for |t| P
2
0 for |t| > P2
Psinc(f P) =sin(f P)
f(4.17)
Bsinc(Bt) = sin(Bt)t
rectB(f) =
1 for |f| B20 for |f| > B
2
(4.18)
ej2f0t (f f0) (4.19)sin(2f0t)
1
2j((f f0) (f + f0)) (4.20)
cos(2f0t)1
2((f f0) + (f + f0)) (4.21)
u(t) =
1, t > 0
1/2, t = 0
0, t < 0
1
j2f
+1
2
(f) (4.22)
eatu(t)1
a + j2f(4.23)
eatu(t) 1a j2f (4.24)
ea|t|2a
a2 + (2f)2(4.25)
eat sin(2f0t)u(t)0
(j + a)2 + 20(0 = 2f0) (4.26)
eat cos(2f0t)u(t)j + a
(j + a)2 + 20(4.27)
ea|t| sin(2f0|t|) 20(a2 + 20 2)
(a2 + 20 2)2 + (2a)2(4.28)
ea|t| cos(2f0t)2a(a2 + 20 +
2)
(a2 + 20 2)2 + (2a)2(4.29)
-
8/14/2019 Collection of Formulas in Signal Processing
12/21
10 5 The Laplace transform
5 The Laplace transform
Properties of the two-sided Laplace transform
x(t) X(s)
x(t)
x(t)est dt (5.1)
12j
+jj
X(s)est ds X(s) (5.2)
cx(t) + dy(t) cX(s) + dY(s) (5.3)
x(ct)1
cX(
s
c), c > 0 (5.4)
x(t P) esPX(s) (5.5)
eatx(t) X(s + a) (5.6)
tnx(t) (1)n nX(s)
sn(5.7)
nx(t)
tnsnX(s) (5.8)
t
x()d1
sX(s) (5.9)
x(t) y(t) X(s)Y(s) (5.10)
-
8/14/2019 Collection of Formulas in Signal Processing
13/21
5 The Laplace transform 11
Properties of the one-sided Laplace transform
x(t) X(s)
x(t)
0
x(t)est dt (5.11)
1
2j
+jj
X(s)est ds X(s) (5.12)
cx(t) + dy(t) cX(s) + dY(s) (5.13)
x(ct)1
cX(
s
c), c > 0 (5.14)
x(t P) esPX(s) (5.15)
eatx(t) X(s + a) (5.16)
tnx(t) (1)n nX(s)
sn(5.17)
nx(t)
tnsnX(s)
n1i=0
sn1iix(t)
ti
t=0
(5.18)
t0
x()d1
sX(s) (5.19)
x(t) y(t) X(s)Y(s) (5.20)
Initial-value theoremIf X(s) is rational, i.e., X(s) = P(s)
Q(s), where order P(s) < order Q(s), then
limt0+
x(t) = lims
sX(s) (5.21)
Final-value theoremIf sX(s) has all poles in the left halfplane, then
limt+
x(t) = lims0
sX(s) (5.22)
-
8/14/2019 Collection of Formulas in Signal Processing
14/21
12 5 The Laplace transform
Common transform pairs (one-/two-sided Laplace transfrom)x(t) X(s) ROC
(t) 1 s (5.23)
u(t) =
1, t > 0
1/2, t = 0
0, t < 0
1
sRe{s} 0 (5.24)
tu(t) 1s2
Re{s} > 0 (5.25)tn
n!u(t)
1
sn+1Re{s} > 0 (5.26)
eatu(t)1
s + aRe{s} > a (5.27)
tneat
n!u(t)
1
(s + a)n+1Re{s} > a (5.28)
sin(0t)u(t)0
s2 + 20Re{s} > 0 (5.29)
cos(0t)u(t)
s
s2 + 20 Re{s} > 0 (5.30)eat sin(0t)u(t)
0(s + a)2 + 20
Re{s} > a (5.31)
eat cos(0t)u(t)s + a
(s + a)2 + 20Re{s} > a (5.32)
eat
cos 0t a0
sin 0t
u(t)
s
(s + a)2 + 20Re{s} > a (5.33)
1 eat
cos 0t +a
0sin 0t
u(t)
a2 + 20s [(s + a)2 + 20 ]
Re{s} > a (5.34)
-
8/14/2019 Collection of Formulas in Signal Processing
15/21
5 The Laplace transform 13
Common transform pairs (two-sided Laplace transform)x(t) X(s) ROC
u(t) 1s
Re{s} 0 (5.35)
tu(t) 1s2
Re{s} < 0 (5.36)
tn
n!u(t) 1
sn+1Re{s} < 0 (5.37)
eatu(
t)
1
s + a
Re
{s
} 0 zkX(z) +k
m=1
x[m]zmk (6.14)
x[n + k], k > 0 zkX(z) k1m=0
x[m]zkm (6.15)
anx[n] X(z/a) (6.16)
x[n] y[n] X(z)Y(z) (6.17)nx[n] z X(z)
z(6.18)
Initial-value theoremIf X(z) is rational, i.e., X(z) = P(z)
Q(z)where order P(z) order Q(z), then
x[0] = limz
X(z) (6.19)
Final-value theoremIf (z 1)X(z) has all poles strictly inside the unit circle, then
limn
x[n] = limz1
(z 1)X(z) (6.20)
-
8/14/2019 Collection of Formulas in Signal Processing
18/21
16 6 z-Transform
Common transform pairs (one-/two-sided z-transform)x[n] X(z) ROC
[n] 1 (6.21)
u[n] =
1, n 00, n < 0
z
z 1 |z| > 1 (6.22)
nu[n]z
(z
1)2|z| > 1 (6.23)
n2u[n]z(z + 1)
(z 1)3 |z| > 1 (6.24)
anu[n]z
z a |z| > |a| (6.25)
nanu[n]za
(z a)2 |z| > |a| (6.26)
n2anu[n]z(z + a)a
(z a)3 |z| > |a| (6.27)
an sin(n)u[n]za sin()
z2
2za cos() + a2
|z
|>
|a
|(6.28)
an cos(n)u[n]z(z a cos())
z2 2za cos() + a2 |z| > |a| (6.29)
Common transform pairs (two-sided z-transform)
x[n] X(z) ROC
anu[n] 11 az |z| < 1|a| (6.30)
a|n|(a2 1)z
az2 (1 + a2)z + a |a| < |z| K
1,(2K+1)() =sin((2K+ 1))
sin()(7.15)
u[n] =
1, n 00, n < 0
1
1 ej2 +1
2() (7.16)
ej20n ( 0) (7.17)
anu[n]1
1 aej2 (7.18)
anu[n] 11 aej2 (7.19)
a|n|1 a2
1 + a2
2a cos(2)(7.20)
an sin(20n)u[n]a sin(20)ej2
1 2a cos(20)ej2 + a2ej4 (7.21)
an cos(20n)u[n]1 a cos(20)ej2
1 2a cos(20)ej2 + a2ej4 (7.22)
-
8/14/2019 Collection of Formulas in Signal Processing
21/21
8 Discrete Fourier Transform (DFT) 19
8 Discrete Fourier Transform (DFT)Properties N = circular convolution with N values
x[n] X[m]
x[n]N1n=0
x[n]ej2mn/N (8.1)
1N
N1m=0
X[m]ej2mn/N X[m] (8.2)
cx[n] + dy[n] cX[m] + dY[m] (8.3)
x[N n] X[m] (8.4)x[n] X[N m] (8.5)x[n k] ej2km/NX[m] (8.6)ej2nk/Nx[n] X[m k] (8.7)N1X[n] x[m] (8.8)x[n] N y[n] X[m]Y[m] (8.9)x[n]y[n] N1X[m] N Y[m] (8.10)
Parsevals theorem
N1n=0
|x[n]|2 = 1N
N1m=0
|X[m]|2 (8.11)