Collection of Formulas in Signal Processing

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Collection of Formulas in Signal Processing

Stockholm October 6, 2006

Kungliga Tekniska Hogskolan

Signal Processing LabElectrical Engineering

SE-100 44 Stockholm


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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


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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


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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


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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()


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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


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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


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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)


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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)


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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)


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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)


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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)


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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)


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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)

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Collection of Formulas in Signal Processing

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