《信号与系统》第一章习题解答

y[n] = A 2δ [n]δ [n − 2] = 0
(c) Is this system invertible?
x[n] = Aδ [n − 1]
y[n] = A 2δ [n − 1]δ [n − 3] = 0
No.
Chapter 1 1.17
y (t ) = x (sin (t ))
Problems Solution
Problems Solution Determine and sketch x2 (t ) → y2 (t ) = ?
(a ) x1 (t ) → y1 (t )
(b) Determine and sketch the response of the system considered in part of (a) to the input x3(t).
8
10
12
t
Chapter 1
Problems Solution
1.22 (d) x[3n + 1] (g)
1 2
x[n]
x[n] +
1 2
(− 1)n x[n]
1
L
1 2
-4 -3 -2 -1 0 1 2 3 4
− 1 2
x[3n + 1] =
1 2
0 n ≤ -2 1 n = -1 2 1 n=0 0 n≥1
(d) x (4 − t / 2 ) (e) [x (t ) + x (− t )]u (t ) (f) x (t )[δ (t + 3 / 2 ) − δ (t − 3 / 2 )]
-2
Problems Solution
x (t )
2 1
-1
−1
0
1
2 t
2 1
x (4 − t / 2 )
−1
4
6
3/2
1 / 21
−1 0
x 5 (t )
1
1 2 t
0
1
2 3
t
(c ) y(t ) = ∫−∞ x (τ )dτ
2t
Linear , Time-varying , with memory , not causal , not stable
Chapter 1
Problems Solution
0, (d ) y(t ) = x (t ) + x (t − 2) 0, (e) y(t ) = x (t ) + x (t − 2)
−1
1 2
L
n
x[n] +
1 2
(− 1)n x[n]
x[n] n is even = 0 n is odd
1 2
1 x[3n + 1]
n
1
L L
-2 -1 0 1 2
L
-4 -3 -2 -1 0 1 2 3 4
−1
L
n
Chapter 1
Problems Solution
1.27 A system may or may not be (1) Memoryless (2) Time invariant (3) Linear (4) Causal (5) Stable Determine which of these properties hold and which do not hold for each of the following continuous-time systems.
Linear , Time-varying , with memory , not causal , Stable
(g ) y(t ) =
dx (t ) dt
Linear , Time-invariant , with memory , causal , not stable
Chapter 1 1.31 Consider an LTI system:
x1 (t )
1
0
y1 (t )
1
2
1
2
t
0
1
2
t
x 2 (t )
1
0 −1
x 3 (t ) 2 1
2 3
1
4
t
−1 0
1
2
t
Chapter 1
Problems Solution
y 2 (t )
2
1
0 1 2 3 4 t
1. x 2 (t ) = x1 (t ) − x1 (t − 2 ) y2 (t ) = y1 (t ) − y1 (t − 2 )
t<0 t≥0 x (t ) < 0 x (t ) ≥ 0
Linear , Time-varying , with memory , Causal , Stable
Nonlinear , Time-invariant , with memory , Causal , Stable
(f ) y(t ) = x (t / 3)
Chapter 1 1.14
1 , 0≤t ≤1 x (t ) = T =2 -2, 1<t < 2
1
-2 -1 0 1
Problems Solution
x (t )
g (t ) =
k = −∞
∑
+∞
δ (t − 2k )
L
2
3
−2
(1) -2 (1) 0 (1)
L
t
t
dx (t ) = 3 g (t ) − 3 g (t − 1 ) dt
L
(3)
-2
A1 = 3 t1 = 0
A2 = −3 t1 = 1
L L
(3)
-1 0
dx (t ) dt
1
2
L
(3)
2
3
(-3)
(-3)
(-3)
L L
t
Chapter 1 1.16
y[n] = x[n]x[n − 2]
Problems Solution
(a) No ,it is a system with memory. (b) x[n] = Aδ [n]
(a) Is this system causal? (b) Is this system linear? (a) No.
y (− π ) = x (sin (− π )) = x (0 )
(b) Yes.
Chapter 1 1.21. A continuous-time signal x(t) is shown in Figure P1.21.Sketch and label carefully each of the following signals:
(a ). y(t ) = x (t − 2) + x (2 − t )
Linear , Time-varying , with memory , not causal , stable
(b ) y (t ) = (cos 3 t )x (t )
Linear , Time-varying , memoryless , causal , stable
(a ) x1 (t ) → y1 (t )
(b) Determine and sketch the response of the system considered in part of (a) to the input x5(t).
x1 (t )
10y1 (t源自)1212
t
0
1
2
t
x 4 (t )
2. x 3 (t ) = x1 (t ) + x1 (t + 1)
−2
y3 (t ) = y1 (t ) + y1 (t + 1)
y1(t +1)
2
1 0
y1(t )
1
2
y 3 (t )
2 1
t
−1 0
1
2
t
Chapter 1 Consider an LTI system:
Problems Solution Determine and sketch x 4 (t ) → y 4 (t ) = ?