数字信号处理chapt2_rev

Y (e jω ) FT [ y (n)] =
[
∞ m =−∞
x(m)h(n − m)]e − jω
k=n-m Y (e jω ) =
=
∞ k =−∞ jω
h(k )e − jω k x(m)e − jωk e− jω x(m)e − jω k
k =−∞
h(k )e − jωk
jω
= H (e ) X (e )
X ( k )e
j
X ( k ) = N ⋅ ak
(2.3.6)
x ( n)
(2.3.7)
DFS N
X (k )
2
2.3.1 8
x(n)=R4(n)
x ( n)
7 n =0
x(n) 2.3.1(a)
N=8
x ( n)
3 n =0 − kn 4
DFS
X (k ) = =
x ( n )e
− j k ⋅4 4 −j k 4
2
(2.2.11) (2.2.12)
* xo (n) = − xo ( − n)
(2.2.13)
2
7
2
2.2.2 x(n) n -n
x(n)=x*(-n)
(2.2.10)
x(n)
x(n)=cos( n)+j sin( n) (END)
2
x(n) = xe (n) + xo (n)
xe(n), xo(n) (2.2.16) (2.2.16) n -n x(n)
h(n)=he(n)+ho(n) h(n)+h(-n) h(n)-h(-n)
h(n) = 0, n < 0
he(n)
ho(n)
h(o),
n=0
(2.2.27)
he ( n ) =
1 h ( n ), n > 0 2 1 h ( − n ), n < 0 2
12
2
0,
n=0
(2.2.28)
ho (n ) =
FT FT xe(n) FT xo(n) =1/2 =1/2 X(ej )+X*(ej ) =Re X(ej ) =XR(ej ) =jXI(ej ) X(ej )-X*(ej ) =jIm X(ej )
(2.2.18)
(2.2.19)
10
2
xe(n) xo(n)
FT FT
XR(ej ) j XI(ej ) ( j)
(2.2.23) (2.2.24)
x(n) = xr (n) + jxi (n)
FT X(e j )=Xe(e j )+Xo(e j )
X e (e jω ) = FT [ xr (n)] =
∞ n =−∞
xr (n)e − jωn =X e* (e- jω )
* xi (n)e − jω n = − X o (e- jω )
X o (e jω ) = FT [ jxi (n)] = j
∞ n =−∞
9
2
• • j
FT FT
2
(b)
xe(n) x(n)=xe(n)+xo(n)
xo(n) (2.2.25)
1 1 xe (n) = [ x(n) + x∗ (−n)], xo (n) = [ x(n) − x∗ (− n)] 2 2
2
h(n) i.e.,
FT
He(ej )
H(ej )
11
2
X (e jω ) = X (e − jω ) =
h[n] ⋅ e − jnω h[n] ⋅ e jnω
h[ n]
X (e jω ) = X * (e − jω )
2
(2.2.18) he(n)=1/2 ho(n)=1/2 h(n)
(2.2.19)
=
1 n a ,n > 0 2 1 − a −n , n < 0 2
13
2
2.2.3
2.2.3
2
5. y(n)=x(n)*h(n), Y(e j )=X(e j )·H(e j )
y (n) =
∞ m =−∞
(2.2.32)
x ( m) h( n − m)
∞ n =−∞ ∞ ∞ m =−∞ ∞ m =−∞
1 h(n), n > 0 2 1 − h(−n), n < 0 2
h(n) he(n) ho(n)
h(n)= he(n)u+(n) h(n)= ho(n)u+(n)+h(o) (n)
(2.2.29) (2.2.30)
2
2.2.3
x(n)=an u(n); 0<a<1; xo(n) (2.2.xx)
15
2
7.
∞ n =−∞
2
(Parseval)
1 2π
∞ n =−∞
x ( n) =
π
−π
X (e jω ) d ω
(2.2.34)
2
∞ n =−∞
2
x ( n) =
x ( n ) x* ( n ) =
π
−π
∞ n =−∞
x* (n)[
1 2π
π
−π
X (e jω )e jω n d ω )]
1 2π 1 = 2π =
n =−∞
e
jω ( m − n )
dω = 2πδ (n − m)
π
−π
DTFT (2.2.3)
x ( m) =
1 2π
X (e jω )e jω m dω
m≠n
m=n
3
2
FT (2.2.2)
(2.2.1) FT
(2.2.4)
Common DTFT pairs
MMTICA DEMO
2
2.2.1
x(n)=RN(n)
(2.2.16)
(2.2.17) (2.2.17) (2.2.18) (2.2.19)
1 xe ( n ) = [ x (n ) + x ∗ ( −n )] 2 1 xo (n ) = [ x ( n ) − x ∗ ( −n )] 2
8
2
xe(n) X(ej ) X(ej )=Xe(ej )+Xo(ej ) Xe(ej ) Xo(ej )
∞ n =−∞ ∞ n =−∞
x(n)h(n)e − jω n x( n)[ 1 2π
π
−π
H (e jθ )e jθ n dθ ]e − jω n
Y (e ) =
jω
∞ 1 π H (e jθ )[ x(n)e − j (ω −θ ) n ]dθ 2π −π n =−∞ π 1 = H (e jθ ) X (e j (ω −θ ) )dθ − π 2π 1 = H (e jω ) * H (e jω ) 2π
(2.2.2)
2
2
DTFT Definition in Mathematica
2
FT - ~
π
−π
ej
n
(2.2.1)
X (e jω )e jω m d ω = =
π
−π
π
−π ∞
[
∞ n =−∞
x(n)e − jω n ]e jω m dω
∞ −∞
x(n)
e jω ( m − n ) d ω
(2.2.3) (2.2.4)
X ( e jω )
∞ n =−∞
x ∗ ( n )e jω n dω 1 2π
π
−π
π
−π
X ( e jω ) X * ( e jω )dω =
X ( e jω ) dω
2
1/(2 )
|X(e j )|2 2.2.1 FT
2 2.2.1
x(n)e jnωo → X (ω − ω0 )
16
2
2.3
2
2
2.1 2.2 2.3 2.4 2.5 2.6 Z Z
2
2.1
t
(
)
1
2
Z
DTFT
Z
Z
2
2.2
2.2.1 DTFT
X (e jω ) =
x(n) DTFT
∞ n =−∞
∞ n =−∞
x ( n )e − jω n
(2.2.1)
,
FT(Fourier Transform) x(n)
x(n) < ∞
2.3.1
x ( n)
(DFS) N
∞ k =−∞ j 2π kn N
x ( n) =
ak
2π − j mn N
ak e
(2.3.1)
ak
e
n
N
2
N −1 n =0 N −1 n =0
x ( n )e
−j
2π mn N
=
N −1 n=0
[
∞ k =−∞
ak e =
−j
2π mn N
=
∞ k =−∞
ak
,
e
j 2π kl N
DFS (Discrete Fourier Series k
j 2π kn N j 2π kl N
X ( k )e
= e
j
[
N −1 n =0
x ( n )e
]e
=
N −1 n =0
x ( n)
N −1 k =0
2π (l −n ) k N
= x(l )
j 2π kn N
x ( n) =
xe(n)
x(n)=xe(n)+xo(n),
x (0),