formula table digital signal processing

8/13/2019 Formula Table Digital Signal Processing

1/43


2/43

X(f)


3/43


4/43

sin = cos( /2) sin(+ ) = sin cos + cos sin cos = sin(+ /2) cos(+ ) = cos cos sin sin cos2 + sin2 = 1 2 sin sin = cos( ) cos(+ )cos2 sin2 = cos 2 2sin cos = sin(+ ) + sin( )2sin cos = sin 2 2cos cos = cos(+ ) + cos( )sin() = sin sin + sin = 2 sin +

2 cos

2

cos() = cos cos + cos = 2 cos +2

cos 2

cos2 = 12

(1 + cos 2)

cos = 12

(ej + ej), sin = 12j

(ej ej), ej = cos +jsin

A cos + B sin =

A2 + B2 cos( )

cos = A

A2+B2, sin = B

A2+B2

=

arctan BA

A 0

arctan BA + A


5/43

x

mxn x n

A=

a11 a12 . . . a1na21 a22 . . . a2n

am1 am2 . . . amn

x=

x1x2

xn

aij =aji ij

B= AT bij =aji

(AB)T =BTAT

detA=

|A

|=

a11 a12 . . . a1na21 a22 . . . a2n

am1 am2 . . . amn

=n

i=1 aij(1)i+jdetMij

Mij i j

detAB= detA detB

detA= a11a22 a12a21

A1A= AA1 =I ( detA#0)

A1 = 1

detACT

cij = (1)i+j detMij(AB)1 =B1A1

A1 = 1

detA

a22 a12a21 a11


6/43

(i, i = 1, 2, . . . , n) (qi, i = 1, 2, . . . , n)

Aq= q (A I)q= 0

det(IA) =n + n1n1 + + 0 = 0det(IA)

u(t) = 1 t 0

0 t 1

2

sinc x= sin xx

diric(x, N) =

sin

Nx2

Nsin

x2

est =etejt

ejt = cos t +j sint

N1n=0

an =

N

a= 1

1aN1a a = 1

N1n=0

ej2 kn/N =

N k= 0, N , . . .0


7/43

y(t) = h(t) x(t) y(n) = h(n) x(n)Y(F) = H(F) X(F) Y(f) = H(f) X(f)ryy() = rhh() rxx() ryy (n) = rhh(n) rxx(n)Ryy (F) = |H(F)|2Rxx(F) Ryy(f) = |H(f)|2 Rxx(f)ryx() = h() rxx() ryx(n) = h(n) rxx(n)Ryx(F) = H(F) Rxx(F) Ryx(f) = H(f) Rxx(f)rxx() =

t x(t)x(t )dt rxx(n) =

x()x( n)

ryx() =

t y(t)x(t )dt ryx(n) =

y()x( n)xx() = E{x(t)x(t )} xx(n) = E{x()x( n)}yx() = E

{y(t)x(t

)

} yx(n) = E

{y()x(

n)

} Xi N(mi, i)

E{X1X2X3X4} = E{X1X2} E{X3X4} + E{X1X3} E{X2X4} ++ E{X1X4} E{X2X3} 2m1m2m3m4

+

z-1

v (n)N-1

z-1

+

+

+

+

1-a

-a2

-a N-1

-aN

b1

b0

b2

bN-1

bN

z-1

v N (n)

v1

(n)

+

+

x(n)

+

y(n)

y(n) = Nk=1

aky(n k) +Mk=0

bkx(n k)


8/43

v(n + 1) = Fv(n) + q x(n)

y(n) = gT

v(n) + d x(n)

F=

0 1 . . . 0 0

0 0 0 1ak ak1 . . . a2 a1

; q=

00

1

gT = (bk, . . . , b2, b1) b0(ak, . . . , a2, a1) ; d= b0

H(z) = b0+ b1z1 + + bMzM

1 + a1z1 + + aNzN

y(n) =h x=

k=h(k)x(n k) =

k= h(n k)x(k)

y(n) =gT Fnv(0) +n1k=0

gT Fn1kqx(k)u(n 1) + dx(n)

h(n) =gT Fn1qu(n 1) + d(n)

H(z) =gT[zI F]1q + d


9/43

x(t) =ej0t = (cos(0t) +j sin(0t)) < t <

y(t) =

=0h()x(t )d=

=0

h()ej0(t)d=H(s)|s=j0 ej0t

x(t) =ej0tu(t) = (cos(0t) +j sin(0t))u(t); X(s) = 1

s j0

Y(s) =H(s)X(s) = T(s)

N(s)

1

s j0 = T1(s)

N(s)

+ H(s)|s=j01

s j0

y(t) = + H(s)|s=j0 ej0t

x(t) =Re{ej0t

} = cos(0t) < t < y(t) =

=0

h()x(t )d=

=0h()

1

2(ej0(t) + ej0(t))d=

= |H(s)|s=j0 cos(0t + arg{H(s)|s=j0})

x(t) =Re

{ej0t

}u(t) = cos(0t)u(t); X(s) =

s

s2

+ 2

0

Y(s) =H(s)X(s) = T(s)

N(s)

s

s2 + 20=

T1(s)

N(s)

+C1s + C0

s2 + 20

H(s)|s=j0 =A ej ; C1=A cos(); C0= A0sin

y(t) = + C1cos(0t) +C00

sin(0t)

=

=

+ |H(s)|s=j0 cos(0t + arg{H(s)|s=j0})


10/43

x(n) =ej0n = (cos(0n) +j sin(0n)) < n <

y(n) =

k=0

h(k)x(n k) =

k=0

h(k)ej0(nk) =H(z)|z=ej0 ej0n

x(n) =ej0nu(n) = (cos(0n) +j sin(0n))u(n); X(z) = 1

1 ej0z1

Y(z) =H(z)X(z) = T(z)N(z)

11 ej0z1 =

T1(z)N(z)

+ H(z)|z=ej0 11 ej0z1

y(n) = + H(z)|z=ej0 ej0n

x(n) =Re{

ej0n

}= cos(0n)

< n 0

32. sinc(at) = sin(at)at

1a

rect( Fa

) a >0

33. repT(w(t)) =

m= w(t mT) 1|T| comb1/T(W(F))

34. |T|combT(w(t)) = rep1/T(W(F))|T|m= w(mT)(t mT)35.

n= cn(t nT)

n=

1T

cn(F nT) =

cnej2nTF


15/43

1. X(z) =Z[x(n)] =n= x(n)zn 2. x(n) =Z1[X(z)] = 1

2j

X(z)zn1dz

3.

ax(n)

aX(z)

4. x(n n0) zn0X(z) n0

5. nx(n)

z d

dz

X(z) n

6. anx(n) X

za

7. x(n) X1z

8.n

= x() z

z1X(z)

9. x y X(z) Y(z)

10. x(n) y(n) 1

2j Y()Xz 1d 11. x(0) = limzX(z)

12. limn x(n) = limz1(z 1)X(z)

13.

= x()y() = 12j

x(z)y

1z

z1dz

14.= x2() = 12j X(z)X(z1)z1dz


16/43

x(n) X(z)15. (n) 1

16. u(n) 11 z1

17. nu(n) z1(1 z1)2

18. nu(n)

1

1

z1

19. (n + 1)nu(n) 1(1 z1)2

20. (n+ 1)(n + 2) . . . (n + r 1)

(r 1)! nu(n) 1

(1 z1)r

21. n cos n u(n) 1 z1 cos 1 z12 cos + 2z2

22. n sin n u(n)

z1 sin

1 z1

2 cos + 2

z2

23. Fnu(n) (I z1F)1

X+(z) =n=0 x(n)zn x(n)

X(z) = X+(z)

x(n)

x(n 1) z1X+(z) + x(1)x(n + 1) zX+(z) x(0) z

n0 (n0 0)

x(n n0) zn0X+(z) + x(1)zn0+1 ++x(

2)zn0+2 + . . .+ x(

n0)

x(n + n0) zn0X+(z) x(0)zn0 x(1)zn01 . . . x(n0 1)z


17/43

1. X(f) = F(x(n)) = == x()ej2f = 2f

2. x(n) =1/21/2 X(f)e

j2f ndf=

= 12

X(f)e

jn d

3.

ax(n) aX(f) 4. x(n n0) X(f) ej2f n0

5. x(n)ej2f0n X(f f0)

6. x(n) cos2f0n 12 [X(f f0) + X(f+ f0)]

7. x(n) sin2f0n 12j [X(f f0) X(f+ f0)]

8. x y X(f) Y(f)

9. x y 1/21/2 X() Y(f )d 10.

= x()y() =

=1/21/2 X(f)Y

(f)df

11. X(f) = X(ej ) x(n) = 0 n < n0 = |x()|2 <

||


18/43

16. 2f1 sinc(2f1 n) = 2f1 sin(2f1n)2f1n

rectp

f2f1

=

1 |f n| < f1


19/43

T0

f(t) = f(t

T0)

f(t) =

k=ck e

j2kF0t

ck = 1

T0

T0

f(t)ej2kF0tdt ; F0 = 1

T0 f(t)

f(t) = c0+ 2

k=1

|ck| cos(2kF0t + k) =

= a0+

k=1

akcos 2kF0t bksin 2kF0t

a0 = c0 = 1

T0

T0

f(t)dt

ak = 2|ck| cos k = 2T0

T0

f(t) cos(2kF0t)dt

bk = 2|ck| sin k =2T0

T0

f(t) sin(2kF0t)dt

P = 1

T0

T0

|f(t)|2dt=

k=|ck|2

P =c20+ 2

k=1

|ck|2 =a20+1

2

k=1

(a2k+ b2k)

N f(n) = f(n N)

f(n) =N1k=0

ck ej2k n/N

ck= 1

N

N1n=0

f(n) ej2k n/N, k = 0, . . . , N 1

f(n)

f(n) = c0+ 2L

k=1

|ck| cos2 knN

+ k

=

= a0+L

k=1

akcos

2

kn

N

bksin

2

kn

N


20/43

a0=c0

ak= 2

|ck

|cos(k)

bk= 2|ck| sin(k)

L=

N2

N

N12

N

P= 1

N

N1n=0

|f(n)|2 =N1k=0

|ck|2

EN=

N

1

n=0|f(n)|2 =N

N

1

k=0|ck|2

Xk =DF T (xn) =N1n=0

xnej2nk/N k= 0, 1, . . . , N 1

xn=IDFT (Xk) = 1N

N

1

k=0Xkej2nk/N n= 0, 1, . . . , N 1

N1n=0

ej2 k k0N n =N (k k0, ( N))

xn N yn =N1

=0xyn,moduloN

DFT XkYk

N x(n) y(n)

x(n) = 0

n = [0, L 1]

y(n) = 0

n = [0, M 1]

x y = 0

n = [0, N 1] N L + M 1

x

y =

x N y= (XkYk) n= 0, 1, . . . , N 1

0

Xk = (x(n))Yk = (y(n))


21/43

X(f)

X(k/N) = Xk =DF T(x(n))

x(n) = 0

n = [0, N 1]X(k/N) = Xk =DF T(xp(n)) x(n) xp(n) =

=

x(n N)

X

k

N

= Xk =DF T(x(n)) =N ck

x(n) =xp(n), 0 n N 1

xp(n) =N1k=0

ckej2 nkN < n <

ck= 1

N

N1n=0

xp(n)ej2 nkN k= 0, 1, . . . , N 1

N1n=0

x(n)y(n) = 1

N

N1k=0

XkY(k)


22/43

x(n), y(n) X(k), Y(k)x(n) =x(n + N) X(k) =X(k+ N)x(N 1) X(N k)x((n 1))N X(k)ej2k1/Nx(n)ej21n/N X((k 1))Nx(n) X(N k)x1(n)N x2(n) X1(k)X2(k)x(n)N y(n) X(k)Y(k)x1(n)x2(n)

1N

X1(k) N X2(k)

N1n=0 x(n)y

(n) 1N

N1k=0 X(k)Y

(k)

M

(M 1)/2 n (M 1)/2

wrect(n) = 1

Wrect(f) = Msin(f M)Msin(f)

whanning(n) = 0.5 + 0.5cos

2n

M 1

Whanning(f) = 0.5 Wrect(f) +

+0.25 Wrect

f 1

M 1

+

+0.25 Wrect

f+

1

M

1

whamming(n) = 0.54 + 0.46cos

2n

M 1

Whamming(n) = 0.54 Wrect(f) +

+0.23 Wrect

f 1

M 1

+

+0.23 Wrect

f+

1

M 1

wblackman(n) = 0.42 + 0.5cos 2n

M 1+ 0.08cos 4n

M 1


23/43

Wblackman(f) = 0.42 Wrect(f) +

+0.25 Wrect

f 1

M 1

+

+0.25 Wrect f+ 1M 1 ++0.04 Wrect

f 2

M 1

+

+0.04 Wrect

f+

2

M 1

wtriangel(n) = 1 |n|(M 1)/2

Wtriangel (f) = M

2

sin f M

2M2

sin(f)

2 2

M W2rect

f

2

f

0 n M 1

M

whanning(n) = 0.5

1 + cos2

n M1

2

M 1

=

= 0.5 1 cos2 nM 1

whamming(n) = 0.54 + 0.46 cos2

n M1

2

M 1 =

= 0.54 0.46 cos 2nM 1

wblackman(n) = 0.42 + 0.5 cos2 n M12

M 1 +

+0.08 cos4

n M1

2

M 1 =

= 0.42 0.5 cos 2nM 1+ 0.08 cos

4n

M 1

wtriangel(n) = 1 n M12

M12


24/43

Xa(F) =

xa(t)ej2F tdt

xa(t) =

Xa(F)ej2F tdF

X(f) =

n= x(n)ej2f n

x(n) =1/21/2 X(f)ej2f ndf

xa(t) Xa(F) = 0 |F| 1/2T

xa(t) =

n=x(n)

sin T

(t nT)T

(t nT)

Fs= 1/T

x(n) = xa(nT) ; T = 1

Fs

X(f) = X

F

Fs

= Fs

k=

Xa(F kFs)

(f) =

F

Fs

= Fs

k=

a(F kFs)

xa(t) =

n=x(n)

sin T

(t nT)T

(t nT)Xa(F) =

1

FsX

F

Fs

|F| Fs

2

a(F) = 1

Fs

F

Fs

|F| Fs

2

Xa(F) =

1

Fs X F

Fs sin(F T)F T ej2F T2 HLP(F)

a(F) = 1

Fs

F

Fs

sin(F T)F T2

|HLP(F)|2


25/43

Lgpass

-Fs/2 Fs/2

( t-n T)n y

a(t)

|Y (F)|a

Lgpass

-Fs/2 Fs/2

( t-n T)n

ya

(t)

ya

(t)

|Y (F)|a

FFs

|X( ) T |

x(n)

Tx(n)

x(1) T

| X(f)|

ya

(t)

x(n)

Tx(n)

x(1) T

| X(f)|

n t t

f F F

-.5 .5 -Fs/2 Fs/2 -Fs/2 Fs/2

ya(t) =

n=

x(n) sin

T (tnT)

T

(tnT)

Ya(F) = 1

FsX

F

Fs

|F|

Fs

2

Lgpass

-Fs/2 Fs/2

1

T

hSH

(t) ya

(t)

ya

(t)

|Y (F)|aSH

|X( ) H (F)|Fs

F

( t-n T)n

x(n)

x(n)

| X(f)|

n t t

f F F

-.5 .5 -Fs/2 Fs/2 -Fs/2 Fs/2

Ya(F) = 1

FsXF

Fs

sin(FT)

FT ej2F

T

2 HLP(F)


26/43

in(F)

DA = 2

FsFpin(F)dF

Ds= 2 Fp0

in(F)dF

0

Fp

Fs/2

SDRA =

DSDA

=

Fp0

in(F)dF

FsFpin(F)dF

SDR0A = min|F|Fpin(F)

in(FsF)

SDR0

A

= in(Fp)

in(Fs Fp)

DP = 2

Fs/2out(F)dF

DS= 2

Fs/2

0out(F)dF

SDRP = DSDP

=

Fs/20

out(F)dF

Fs/2out(F)dF

SDR0P= min|F|


27/43

DQ

2

12

SDRQ=

Dq

r

SDRQ= 1.76 + 6 r[dB]

r

SDRQ= 6 r+ 1.76 1010

log Apeak

ARMS 22

1010

log V

Apeak2

[V, V]

M

M y(n) = {. . . u(0), u(M), u(2M) . . .}

Y(f) = 1M M1i=0 U fiM

L

L w(n) = {. . . x(0), 0, 0, . . .

, x(1), 0, 0, . . .

, x(2) . . .}

W(f) =X(f L)


28/43

|H()| = K1 +gN

p

2 = 2F

gN

p

2

1 p

< 1 1

p

> 1

p p

p= 1

|H()| = K1 +

p

2N

2K

|H()| Kurvan gr alltidgenom dessa punkter

K3dB

K K = 0

p= 1 p

H(s) = K

sp

N+ aN1

sp

N1+ + a1

sp

+ 1

a1, . . . , aN1


29/43

a sN + aN1sN1 + . . . + a1s + 1

N a1 a2 a3 a4 a5 a6 a7

2

p= 1 p= 1 s s/p

N (s + 1)

(s2 +

2s + 1) (s2 + s + 1)(s + 1) (s2 + 0.76536s + 1)(s2 + 1.84776s + 1)

(s + 1)(s2 + 0.6180s + 1)(s2 + 1.6180s + 1)

(s2 + 0.5176s + 1)(s2 +

2s + 1)(s2 + 1.9318s + 1)

(s + 1)(s2

+ 0.4450s + 1)(s2

+ 1.2465s + 1)(s2

+ 1.8022s + 1) (s2 + 0.3896s + 1)(s2 + 1.1110s + 1)(s2 + 1.6630s + 1)(s2 + 1.9622s + 1)

|H()| = K1 + 2T2N

p

|H()|

p

2

K

1+

Kurvan gr alltidgenom denna punkt

N udda

N jmn

RippleK

OBS! E ndvndi tvis 3dB- rns

= 10log(1 +2) K= K= = 0 N


30/43

TN(p

) CNp

p= 1 p= 1 p

H(s) =K a0

1 N

11+2

N

sp

N+ aN1

sp

N1+ + a0

, a0, . . . , aN1 H(s) p = 1 p= 1 p

TN() =

cos(N arccos) || 1

cosh(N arccosh) || 1 = 2F

TN() =

+

2 1

N+

+

2 1N

2 || 1

TN+1() = 2TN() TN1()

N TN()

22 1

43 3

84 82 + 1 165 203 + 5

326 484 + 182 1 647 1125 + 563 7

1288 2566 + 1604 322 + 1 2569 5767 + 4325 1203 + 9

51210 12808 + 11206 4004 + 502 1


31/43

a

(= 0.349, 2 = 0.122) a7 a6 a5 a4 a3 a2 a1 a0

(= 0.509, 2 = 0.259)

a7 a6 a5 a4 a3 a2 a1 a0

(= 0.765, 2 = 0.585)

a7 a6 a5 a4 a3 a2 a1 a0

) (= 0.998, 2 = 0.995)

a7 a6 a5 a4 a3 a2 a1 a0

20 log 2 = 1 a0= 1 N= 1


32/43

(= 0.349, 2 = 0.122) N= 1

(= 0.509, 2 = 0.259)

N= 1

(= 0.765, 2 = 0.585)

N= 1

) (= 0.998, 2 = 0.995)

N= 1


33/43

n a0 a1 a2 a3 a4 a5

n

n

s + 1 s2 + 3s + 3 (s2 + 3.67782s + 6.45944)(s + 2.32219) (s2 + 5.79242s + 9.14013)(s2 + 4.20758s + 11.4878) (s2 + 6.70391s + 14.2725)(s2 + 4.64934s + 18.15631)(s + 3.64674) (s2 + 8.49672s + 18.80113)(s2 + 7.47142s + 20.85282)

(s2 + 5.03186s + 26.51402) 10395


34/43

12 =

lu

p= 1, r

s H(s) H(s)

1 2 A =B

21 l u u r r l 2 u1

() () ()

LP HP BP BS

()

1

S

u

s

(s2 + lu)

s(u l)

s(u l)

(s2 + lu)

r u/r r = u/

r

av= (u l)/2 r =min(|A|, |B|)

1=

2r2av+ lu avr A= (

2

1+ lu)/[1(u l)]

2=

2r2av+ lu+ avr B= (+

2

2 lu)/[2(u l)]

sBP =SLPav (SLPav)2 ul)

av= (u l)/2 r =min(|A|, |B|)

1=

2av/2r+ lu av/r A= 1(u l)/(

2

1+ lu)

2=

2av/2r+ lu+ av/r B= 2(u l)/(

2

2+ lu)

sBP = av/SLP

(av/SLP)2 ul)


35/43

H(z) = b0+ b1z1 + . . . + bMzM

h(n) =

bn 0 n M0

H(z) = b0+ b1z1 + . . . + bMzM

1 +a1z1 + . . .+ aNzN

h(n) = Z1

{H(z)

}

h(n) =hd(n) w(n) hd(n) Hd() 0 w(n)

hd(n) = c

sin cn M1

2

c

n M12

Hd() =

ej (M1)/2 || < c0

hd(n) = 2 cos

0

n M 1

2

c

sin c

n M12

c n

M12

Hd() =

ej (M1)/2 0 c< || < 0+ c0

hd(n) =

n M 12

c

sin c

n M12

c

n M1

2

Hd() =

ej (M1)/2 || > c0

H() =Hd() W() c

M


36/43

4/M 8/M 8/M 8/M 12/M

M M M

f

M

sin(f M)Msin(f)

sin(f M)f M

f

M

H(f) x = (f fc) M M = 99 fc = 0.1


37/43

x=

(f fc) M (f fc) M

1 0.5 0 0.5 1 1.5 2 2.5 3 3.5 490

80

70

60

50

40

30

20

10

0

10

x=(ffc)*M

DmpningidB

1.2 1 0.8 0.6 0.4 0.2 0 0.2 0.49

8

7

6

5

4

3

2

1

0

1

x

|H(f)|/dB

M = 99 fc = 0.1

x= (f fc)M x= (f fc)M


38/43

s

fs

1+p

p1-

f

f

|H(f)|

1

N = D(p, s)

f + 1

f = fsfp

D(p, s) =20 log

ps13

14.6

E=n

[x(n)h(n)d(n)]2

M1n=0

h(n)rxx(n) =rdx() = 0, . . . ,M 1

Emin =rdd(0)M1k=0

h(k)rdx(k)

rxx() x(n) rdx() d(n) x(n)

Rxx h= rdx

h= R1xx rdx

Emin =rdd(0) hT rdx


39/43

h(n) =ha(nT)

ha(t) =e0t Ha(s) = 1

s + 0

H(z) = 11

e0Tz1

ha(t) =e0t cos0t Ha(s) = s + 0

(s + 0)2 + 20

H(z) = 1 z1e0T cos0T

1 2z1e0T cos0T+ z2e20T

ha(t) =e0t sin0t

Ha(s) =

0

(s + 0)2

+ 20

H(z) = z1e0T sin0T

1 2z1e0T cos0T+ z2e20T

Fprewarp = 1

T

tan(f)

prewarp

H(z) = Ha(s) s= 2T

1 z11 +z1

T


40/43

a1, . . . , ak a1, . . . , ak

pi pia1 a1+ + pi

akak

piaj

= pkji

(pi p1)(pi p2) . . . (pi pk) k1

(pi pi)

z -1

+

+g (n)0 g (n)1

z -1

f (n)0 f (n)1

+

+f (n)2

z -1 +

+

K1 K2

K1 K2

f (n) = y(n)M-1

g (n)2 M-1g (n)

. . .

. . .

Kx(n)

Lattice FIR

z -1

+

+ z -1

+

+ z -1

Nf (n)

f (n)N-1 f (n)2 f (n)1

K1

K1

K2

K2

KN

KN

+

+

g (n)N

+ +

g (n)0

0vv1Nv

f (n)0

g (n)2 g (n)1

v2

+

z -1

+

+

+

+ z -1

+

+ z -1

Nf (n)

f (n)N-1 f (n)2f (n)1 f (n) = y(n)0

g (n)Ng (n)2 g (n)1

g (n)0

K1

K1

K2

K2

KN

KN

Lattice all-pole IIR

. . .

. . .

x(n)

y(n)

- --

. . .

Lattice-ladder

. . .

. . .

x(n)

- --


41/43

A0(z) =B0(z) = 1

Am(z) =Am1(z) +Kmz1Bm1(z)Bm(z) =KmAm1(z) +z1Bm1(z)

Am1(z) = 1

1 K2m(Am(z) KmBm(z))

Am(z) = Z{m(n)} Km=m(m)Bm(z) = Z{m(n)}

Am(z) Bm(z)

Bm(z) = zmAm(z1)

m(k) = m(m k)

H(z) =AM1(z)

H(z) = 1

AN(z)

H(z) =CN(z)

AN(z)

=c0+ c1z

1 . . . cNzN

AN(z)

Cm(z) =Cm1(z) + vmBm(z)

cm(m) =vm m= 0, 1, . . . , N


42/43

xx(m) = E{x(n)x(n + m)}

xx(f) =

m=xxe

j2f m

rxx(m) = 1

N

Nm1n=0

x(n)x(n + m) 0 m N 1

Pxx(f) =N1

m=N+1rxx(m)e

j2f m = 1

N

N1m=0

x(m)ej2f m2

E{rxx(m)} = 1 |m|N xx(m) xx(m) N var(rxx(m)) 1

N

n=

[2xx(n) + xx(n m)xx(n + m)] 0 N

E{Pxx(f)} = 1/21/2

xx()WB(f )d

WB(f)

1 |m|N

var(Pxx(f)) = 2xx(f) 1 + sin2f NNsin2f2 2xx(f) N

x(n)

Pxx

k

N

=

1

N

N1n=0

x(n)ej2 nkN

2

k= 0, . . . , N 1


43/43

Q=[E{Pxx(f)}]2

var(Pxx(f))

1Q

Q

f= 0.9M

Q= 1

(N=K M) f= 0.9

M QB =

NM

(N=L M) f= 1.28

M QB = 16

9 N

M

f= 0.6M

QB = 12 N

M

f= 0.9M

QB = 32 N

M

f

formula table digital signal processing

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