formula table digital signal processing
TRANSCRIPT
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X(f)
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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
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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
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(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
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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)
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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
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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})
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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
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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
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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
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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 <
||
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16. 2f1 sinc(2f1 n) = 2f1 sin(2f1n)2f1n
rectp
f2f1
=
1 |f n| < f1
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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
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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))
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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)
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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
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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
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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
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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)
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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|
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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)
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|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
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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
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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
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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
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(= 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
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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
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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)
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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
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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
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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
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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
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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
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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)
- --
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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
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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
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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