Great Deal! Get Instant $10 FREE in Account on First Order + 10% Cashback on Every Order Order Now

Convolution,LTI system, impulse, transfer function,etc. 8 problems

1 answer below »
Convolution,LTI system, impulse, transfer function,etc. 8 problems
005_5k-wuoy4lw5.docx
Answered Same Day Dec 21, 2021

Solution

David answered on Dec 21 2021
120 Votes
Sol. The convolution g(t) of the two signals is given by,
g(t) = ∫ ( ) ( )


X1(t) = 2 for -2 = 0, everywhere
and, X2(t) = -2t + 2
It can be solved graphically where the amount of overlap of two signal represents g(t).
Flipping X2(t) around t = 0, we get the two signals as
x1 x2(-Ï„)
2
-2 1 -1
2
(Ï„)

for t<-2
x1(Ï„)
2
-2 1-1+t t
x2(t-Ï„)

for t<-2, g(t) = 0 {since there is no overlap between two signals}
for -2≤t<-1
x1(Ï„)
2
-2 1-1+t t
x2(t-Ï„)

( ) ∫ ( ( ) )
= -2t(t + 2)
for -1 ≤ t < 1
x1(Ï„)
2
-2 1-1+t t
x2(t-Ï„)

( ) ∫ ( ( ) )
= 2
for 1 ≤ t < 2
x1(Ï„)
2
-2 1-1+t t
x2(t-Ï„)

( ) ∫ ( ( ) )
= 2(t2+4)
for t > 2
x1(Ï„)
2
-2 1 -1+t t
x2(t-Ï„)

g(t) = 0 {since there is no overlap between two signals}
Therefore, the convolution of given two signals is given by
( )
{





( )

( )
Sol. The expression for this signal can be written as,
x(t) = et for t<0
= e-t for t>0
Now, X(jw) = ∫ ( )



{defining integral equation for the FT}
= ∫


∫


= ∫


∫


= ∫ ( )


∫ ( )


=

( )
[ ( ) ]
( ( ))
[ ( ) ]


=

( )
( )
=

Sol. Given, the system input is
( ) ( )
And the system impulse response is
( ) ( )
The system response ( ) is given by
( ) ( ) ( ) where * denotes convolution
when ( ) = ( ) {impulse}, ( ) ( )
The laplace transform of the above equation is given by
( ) ( ) ( )
( )
( )
( )

( )
( ( ))
( ( ))

Now,
( ( ))
( ( )) ( ( ))
( ( ))




therefore,
( )



Now,
( )


( )

The response of the system to this input is given by
( ) ( ) ( )
(


( )...
SOLUTION.PDF
SOLUTION.PDF

Answer To This Question Is Available To Download

Related Questions & Answers

More Questions »

Submit New Assignment

Copy and Paste Your Assignment Here
Attach File