1. ax [n ]+by [n ]aX (e j ω) +bX (e j ω) Proof:
∞
∑(ax [n ]+by [n ])e
-∞
∞-j ωn ∞
-j ωn =a ∑x [n ]e
-∞+b ∑y [n ]e -∞
j ω-j ωn =aX (e j ω) +bX (e )
2. (1)x [n -n d ]X (e
Proof:
∞j ω) e -j ωn d
∑
n =-∞x [n -n d ]e
∞-j ωn
=∑
n =-∞x [n -n d ]e
j ω-j ω(n -n d ) . e -j ωn =X (e ) e -j ωn d
(2) e
Proof:
∞j ω0n x [n ]X (e j (ω-ω0) ) ∞
∑
n =-∞e j ω0n x [n ]e -j ωn =∑n =-∞x [n ]e -j (ω-ω0) n =X (e j (ω-ω0) )
3. x [-n ]X (e -j ω)
Proof:
∞∞
∑
n =-∞x [-n ]e -j ωn =∑n =-∞x [-n ]e j ω(-n ) =X (e -j ω) if x [n ] is real X (e -j ω) =X (e *j ω) 4. nx [n ]j
Proof:
∞dX (e d ωj ω) X (e
⇒j ω) =j ω∑n =-∞x [n ]e ∞-j ωn dX (e
d ω)
j ω=) ∑n =-∞∞(-jn ) x [n ]e
nx [n ]e -j ωn ⇒j dX (e
d ω=∑
n =-∞-j ωn
∞
5. (1)∑
n =-∞|x [n ]|=212ππ⎰
-π|X (e j ω) |d ω 2
Proof:
1
2π
=π⎰|X -π(e j ω) |d ω21
2π
1
2π
1
2π
∞π⎰-πX (e ∞j ω) X e (-j ωd ) ω∞π= =-πn =-∞⎰∑∞x [n e ]-j ωn ∑n =-∞x n [e ]d ω*j ωn π -πn =-∞
2⎰∑|x [n ]|d ω1
2ππ2=∑
n =-∞
∞|x [n ]|
2⎰-πd ω=∑
n =-∞
∞|x [n ]|π
(2) ∑
n =-∞x [n ]y [n ]=*12π⎰
-πX (e j ω) Y (e *j ω) d ω Proof:
1
2π
=π⎰-πX (e πj ω) Y (e *j ω) d ω1
2π
1
2π
π⎰-πX (e ∞j ω) Y (e -j ω) d ω∞π=1-πn =-∞∞
*⎰∑x [n ]e -j ωn ∑n =-∞y [n ]e *j ωn d ω
2π
=-πn =-∞∞
*⎰∑x [n ]y [n ]d ω1
2ππ∞∑
n =-∞
∞x [n ]y [n ]
*-πn =-∞⎰∑d ω=∑
n =-∞x [n ]y [n ]
6. x [n ]*y [n ]X (e
Proof: j ω) Y (e j ω)
∞
∑
n =-∞{x [n ]*y [n ]}e
∞j ωn
=
=∑n =-∞∞{x [i ]y [n -i ]}e ∞-j ωn
y [n -i ]e -j ω(n -i ) ∑
n =-∞x [i ]e
j ω-j ωi ∑n -i =-∞=X (e ) Y (e j ω)
7. x [n ]y [n ]
Proof: 1
2π
=π12ππ⎰-πX (e j θ) Y (e （j ω-θ）) d θ ⎰-π[π12ππ⎰-πX (e j θ) Y (e 1
2π（j ω-θ）) d θ]d ω12π
1
2π
1
2ππ⎰-πX (e j θ) e j θn ⎰-πY (e （j ω-θ）) d ωd θπ =⎰-πX (e j θ) e j θn y [n ]d θ π=⎰
-πX (e j θ) e j θn d θ. y [n ]
=x [n ].y [n ]
1. ax [n ]+by [n ]aX (e j ω) +bX (e j ω) Proof:
∞
∑(ax [n ]+by [n ])e
-∞
∞-j ωn ∞
-j ωn =a ∑x [n ]e
-∞+b ∑y [n ]e -∞
j ω-j ωn =aX (e j ω) +bX (e )
2. (1)x [n -n d ]X (e
Proof:
∞j ω) e -j ωn d
∑
n =-∞x [n -n d ]e
∞-j ωn
=∑
n =-∞x [n -n d ]e
j ω-j ω(n -n d ) . e -j ωn =X (e ) e -j ωn d
(2) e
Proof:
∞j ω0n x [n ]X (e j (ω-ω0) ) ∞
∑
n =-∞e j ω0n x [n ]e -j ωn =∑n =-∞x [n ]e -j (ω-ω0) n =X (e j (ω-ω0) )
3. x [-n ]X (e -j ω)
Proof:
∞∞
∑
n =-∞x [-n ]e -j ωn =∑n =-∞x [-n ]e j ω(-n ) =X (e -j ω) if x [n ] is real X (e -j ω) =X (e *j ω) 4. nx [n ]j
Proof:
∞dX (e d ωj ω) X (e
⇒j ω) =j ω∑n =-∞x [n ]e ∞-j ωn dX (e
d ω)
j ω=) ∑n =-∞∞(-jn ) x [n ]e
nx [n ]e -j ωn ⇒j dX (e
d ω=∑
n =-∞-j ωn
∞
5. (1)∑
n =-∞|x [n ]|=212ππ⎰
-π|X (e j ω) |d ω 2
Proof:
1
2π
=π⎰|X -π(e j ω) |d ω21
2π
1
2π
1
2π
∞π⎰-πX (e ∞j ω) X e (-j ωd ) ω∞π= =-πn =-∞⎰∑∞x [n e ]-j ωn ∑n =-∞x n [e ]d ω*j ωn π -πn =-∞
2⎰∑|x [n ]|d ω1
2ππ2=∑
n =-∞
∞|x [n ]|
2⎰-πd ω=∑
n =-∞
∞|x [n ]|π
(2) ∑
n =-∞x [n ]y [n ]=*12π⎰
-πX (e j ω) Y (e *j ω) d ω Proof:
1
2π
=π⎰-πX (e πj ω) Y (e *j ω) d ω1
2π
1
2π
π⎰-πX (e ∞j ω) Y (e -j ω) d ω∞π=1-πn =-∞∞
*⎰∑x [n ]e -j ωn ∑n =-∞y [n ]e *j ωn d ω
2π
=-πn =-∞∞
*⎰∑x [n ]y [n ]d ω1
2ππ∞∑
n =-∞
∞x [n ]y [n ]
*-πn =-∞⎰∑d ω=∑
n =-∞x [n ]y [n ]
6. x [n ]*y [n ]X (e
Proof: j ω) Y (e j ω)
∞
∑
n =-∞{x [n ]*y [n ]}e
∞j ωn
=
=∑n =-∞∞{x [i ]y [n -i ]}e ∞-j ωn
y [n -i ]e -j ω(n -i ) ∑
n =-∞x [i ]e
j ω-j ωi ∑n -i =-∞=X (e ) Y (e j ω)
7. x [n ]y [n ]
Proof: 1
2π
=π12ππ⎰-πX (e j θ) Y (e （j ω-θ）) d θ ⎰-π[π12ππ⎰-πX (e j θ) Y (e 1
2π（j ω-θ）) d θ]d ω12π
1
2π
1
2ππ⎰-πX (e j θ) e j θn ⎰-πY (e （j ω-θ）) d ωd θπ =⎰-πX (e j θ) e j θn y [n ]d θ π=⎰
-πX (e j θ) e j θn d θ. y [n ]
=x [n ].y [n ]