homework solutions #2
DESCRIPTION
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. Problem 2.11. Problem 2.11. 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]. - PowerPoint PPT PresentationTRANSCRIPT
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