Homework Solutions #2

Problem 2.11

• Given:

• Determine y[n] if

• Solution: – Find transform of x[n]– Compute – Compute y[n] =

Problem 2.11

• Compute

1. Use Transform identity:

)2()2()cos( 000 keken jj

24cos)

4sin(][

nnnx

2. Using identity find 24

)( 0

andwhereeX j .

)2

4()2

4()( 2/2/ kekeeX jjj

Problem 2.113. Next Compute )()()( jjj eHeXeY

kandkforexceptallforeY j 24

24

0)(

Let k 24

4/4/2/

2/2/

2/

44

42

2/

24

4

24

2

2/

2222)(

)1(2

2

11

1

2

11

1

2

11

1)(

jjjj

jj

j

j

j

j

kj

kj

jj

eeeeY

jee

e

e

ee

e

eeeY

Let k 24

4/4/2/

2/2/

2/

44

42

2/

24

4

24

2

2/

2222)(

)1(2

2

11

1

2

11

1

2

11

1)(

jjjj

jj

j

j

j

j

kj

kj

jj

eeeeY

jee

e

e

ee

e

eeeY

)2

4()2

4(22)( 4/4/ kekeeY jjj

Problem 2.114. Now using the above identity we convert to y[n]

))1(4

sin(22][

)44

sin(22][

)44

cos(22][

nny

nny

nny

This transfer function can be viewed as an all-pass phase shifting filter.

Problem 2.29

• Given an LTI system:• h[n]=• Plot y[n], when:• x[n]=u[n], x[n]=u[n-4], x[n]=u[n]-u[n-4]

Problem 2.29

• Using Convolution:• (a) • (b) Use shift property on (a)

y[n]=

Problem 2.29

• (c) Subtract (b) from (a)• y[n]={1,2,3,4,1,-2,-3,-4,-2,0,0….}u[n]

Problem 2.42

• Given

• Find • Is it causal?

Problem 2.42

jjwjwj

eeHandeeH

1

1)()( 21

]1[][][ 1 nununh nn ]1[][]1[][

)1)(()1)((

1

1

)(

)(

nxnxnyny

eeXeeY

e

e

eX

eY

jwjjwj

j

j

j

j

causal

Problem 2.55

• Given

• +1+2+2++2

• +1+2+2++2 = 8• +1+2+2++2 = -4

𝑥 [𝑛 ]=−𝛿 [𝑛+3 ]+2𝛿 [𝑛+1 ]+𝛿 [𝑛 ]+2𝛿 [𝑛−1 ]+2𝛿 [𝑛−3 ]+𝛿 [𝑛−4 ]+𝛿 [𝑛−5 ]−𝛿[𝑛−7]

Problem 2.55

• Evaluate • Inverse Fourier transform:

Problem 2.55

• Evaluate

• +1+2+2++2