An Interactive Approach to Signals and Systems Lab

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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An Interactive Approach to Signals and Systems Lab

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