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8/10/2019 An Interactive Approach to Signals and Systems Lab
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An Interactive Approach to Signals andSystems Laboratory
By:
Nasser Kehtarnavaz
Philipos Loizou
Mohammad Rahman
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An Interactive Approach to Signals andSystems Laboratory
By:
Nasser Kehtarnavaz
Philipos Loizou
Mohammad Rahman
Online:< http://cnx.org/content/col10667/1.14/ >
C O N N E X I O N S
Rice University, Houston, Texas
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x
= 1
N
Nj=1
(xj )2
n
N
N
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a3 +
4c a= 1 2 b= 2 3 c= 4 5 d= 4
= [x1, x2, , xn] = [y1, y2, , yn]
= [x1y1, x2y2, , xnyn]
=x1y1
, x2y3 , ,xnyn
= [xp1, x
p2, , x
pn]
1 3
i xiyi
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[1 4, 3 5, 4 6]
4
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>
>
>
<
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>
|
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t
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x1 (t) x2 (t)
t= 0, 0 , 0 , 0 , , 2
x1 (t)
x1 (t) = {0 0 t < 11 1 t < 20 2 t < 3
x2 (t)
x2 (t) = {2t 0 t < 1
2t + 4 1 t < 20 2 t < 3
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<
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x (t) = { 0 2 t
0 2
02 t < 0 2
t
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y (t) x (t) h (t)
y (t) =
h (t ) x () d
(t) t= 0
(t) = { 1 /2 t /20
x (t) x(t) x (k)
x(t) =
k=
x (k) (t k)
0, x(t) x (t)
x (t) = t2 = 1 h (t)
h(t) =
k=
h (k) (t k)
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y(t) =
h(t ) x() d
y(t)
y(n) =
k=
x (k) h ((n k) )
h(t) x(t) t= n x (t) h (t)
k= x (k) h ((n k) )
y (t) t= n y (t)
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x (t) h (t) = h (t) x (t)
x (t) h1(t) h2(t) = x (t) {h1(t) h2(t)}
x (t)
{h1(t) + h2(t)
}=x (t)
h1(t) + x (t)
h2(t)
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x (t) = (
) u (t)
h (t) = ( ) u (t) u (t) 0 t 8 = 0 5, 01, 0 , 0 , 0 , 0 h (t) x (t)
y (t) = 1
a b
e e u (t)
y (n) y (t)
y (n)
y (n)
y (t) y (t)
= 1
N
Nn=1
y (n) y (n)
2
N = T
T= 8
a b T
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x (t) =
(2t) u (t)
h (t) = t2
2
u (t)
(t) = { 1 05 t < 0 50
= 0
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x (t) h (t) x (t)h (t) () (=)
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h (t)
v (t) i (t)
h (t) = 1
RC
1
t
v (t) i (t)
i (t) = h (t)
v (t)
h (t) =R
L
R
Lt
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v (t) i (t)
v (t) h (t) i (t)
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= 1/
(t ) a
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a N
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x (t)
x (t) =
n=
cnej 2 T
cn
cn= 1
T
T
x (t) ej2 T
| cn|=| cn|
cn= cn | |
x (t)
x (t) = a0+n=1
an
2
T
+ bn
2
T
a0 = 1
T
T
x (t)
an= 2
T
T
x (t)
2
T
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bn= 2
T
T
x (t)
2
T
a0 = c0
an= 2 {cn}
bn= 2 {cn}
cn=1
2(an n)
x (t)
cn
1
T
T
| x (t) |2 =
n=
| cn|2
x (t) t
a0= 1
M
Mm=1
x (mt)
an= 2
M
Mm=1
x (mt)
2
M
bn= 2
M
Mm=1
x (mt)
2
M
x (mt) M x (t) T t t = TM
t
cn= 1
M
Mm=M
x (mt)
j2
M
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x (mt)
t t
{ (2 f t) 0 t < T /20
T /2 t < T f =
1/T
a0, an bn
x (t) x (t)
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(vc) n (vc)
vcn = 1/ (
C)R + 1/ ( C)
v n
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v (t)
i (t)
v (t)
P
v (t) T
V
=
1T
T
v2
v (t) = V0+ V1 (1t 1) + V2 (2t 2) + + VN (Nt N) V
V
=
V02 +
V1
2
2+
V2
2
2+ +
VN
2
2
P
=V0I0+ V1 I1 1+ V2 I2 2+
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y (t) = A1 (1t + y (t))
y (t) = A1 (1t + A2 (2t)) + A3 (3t + A4 (4t + A5 (5t + y6(t))))
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x (t)
X() =
x (t) ejt
x (t) X()
x (t) = 1
2
X() ejtd
x (t t0) X() ejt0
x ( ) 1|a|X
a
a1
x1(t) + a
2x2
(t) a1
X1
() + a2
X2
()
x (t) h (t) X() H()
x (t) h (t) X() H()
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x (t) t
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x (t) t0 t0
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x1(t)
x2(t)
a1
a2
a1x1(t) + a2x2(t)
a1X1() + a2X2()
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x1(t) x2(t) x1(t) x2(t) X1() X2()
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x (t) = xm(t) (2fct)
xm(t) fc
(2fct) =1
2
e2fct + e2fct
=
1
2
ect + ect
X() =1
2(Xm( c) + Xm(+ c))
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x (t) = xm1(t) (2f1t) + xm2(t) (2f2t) + xm3(t) (2f3t)
xm1(t) , xm2(t)
xm3(t)
xm1(t) , xm2(t) xm3(t)
A H O V B I P W C J Q X D K R Y E L S Z F M T
G N U
xm1(t) = d (t) + (t) d (t) (t) xm1(t) = 3 (t)
d (t) (t)
d (t) (t)
f1, f2 f3
T
xm1(t) , xm2(t) , xm3(t)
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dl12/ l1
E0(l0) = Kej(2d/)
jd ej(/d)l02
E1(l1) ej(2/d)l0l1
1
E1 E0 K
(t)
1fl0/d
0(l0) = Kej(2d/)
jd ej(/d)l02
I(l0)
I(l0) = | E0(l0) |2
d
3
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x (t) =A (t + ) t= s Ts
x [n] = A ( s+ ) = A (n + ) , n= 0, 1, 2, ,
= Ts= 2ffs
Ts
f
fs
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{ X(j) =
x (t) ejt
x (t) = 12
X(j) ejtd
{ X
ej
=n= x [n] e
, = Ts
x [n] = 12X
ej
e d
W f
Ts = 1fs
2 fs
[0, fs/2]
fs 2f
fs = 0 8
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x (t) = (2 t)
x (t)
x (t)
x [n]
z (t) =
k=
z [ s] ct s
Ts
f fs
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{ Xk = 1T
T/2T/2 x (t) e
j0
x (t) =
k= Xkej0
{ X[k] =N1
n=0 x [n] ej 2N , k = 0, 1, , N 1
x [n] = 1NN1
k=0 X[k] ej 2N , n= 0, 1, , N 1
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f = 2ffs
fs
fs = fs= fs=
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f=
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(
= 0
)
( 1 = 0 )
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T = 01
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T =
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k
k
k
(k0, k1, , k9)
k
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V
= 1/ (jC)
R + 1/ (jC)V
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| H() |=| V V
|= 11 + 2R2C2
H() =
V
V
= 1 ( )
V
= R
R + 1/ (jC)V
| H() |=| V V
|= 11 + 1/ (2R2C2)
H() =
V
V
= 1
1
y [n] =Nk=0
bkx [n k]
b x [n] x [n k] y [n]
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y [n] =N
k=0
bkx [n k] M
k=1
aky [n k]
b a N M y [n k] y [n]
b a y [n k]
y [n]
H() = b0+ b1e
j + + bNe
1 + a1ej + + aMe
H() =
Nsk=1
b0k+ b1kej + b2ke
j2
1 + a1kej + a2kej2
Ns = N/2
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x [n]
y [n] =i
bix [n i] +i
aiy [n i]
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A
y [n] =1
2
y [n 1] + x [n]
y [n 1]
x [n] y [n]
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x1(t) , x2(t) x3(t)
x1(t)
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