a=3 b=4 c=8 d=7 Please determine r, p, and k by using MATLAB only! Please show t
ID: 2084373 • Letter: A
Question
a=3
b=4
c=8
d=7
Please determine r, p, and k by using MATLAB only!
Please show the partial-fraction expansion by hand writen.
Thank you!!
Below is an example problem discussing r, p, and k
Consider the Laplace transform s(a s) (b s) X (s) (c s)2 (d s) Determine r, p, and k. Print out the results. Hint: First use conv to get num and denom. Demonstrate that you understand the format by writing down the partial-fraction expan- sion. Also write down the time-domain expression if the ROC is Retst mint C dt.Explanation / Answer
>> %To use convolution theorem and derive the Laplace transform%
>> x=[1 0];
>> y=[1 3];
>> p=conv(x,y);
>> p=conv(x,y)
p =
1 3 0
>> %In the above we have convoluted s and (s+3)%
>> a=[1 4];
>> b=[1 4];
>> q=conv(a,b)
q =
1 8 16
>> %In the above we have convoluted (s+4) and (s+4)%
>> num=conv(p,q)
num =
Columns 1 through 4
1 11 40 48
Column 5
0
>> %In the above, we have convoluted the above two expressions p and q%
>> c=[1 8];
>> d=[1 8];
>> r=conv(c,d)
r =
1 16 64
>> %In the above ,we have convoluted (s+8) and (s+8)%
>> e=[1 7];
>> f=[1 7];
>> s=conv(e,f)
s =
1 14 49
>> %In the above expression we have convoluted (s+7) and (s+7)%
>> den=conv(r,s)
den =
Column 1
1
Column 2
30
Column 3
337
Column 4
1680
Column 5
3136
>> %The denominator is obtained by convoluting r and s%
>> %Now we have to find r,p,k values using residue command%
>> [r,p,k]=residue(num,den)
r =
752.0000
640.0000
-771.0000
252.0000
p =
-8.0000
-8.0000
-7.0000
-7.0000
k =
1
>> syms s
>> f=(s^4+11*s^3+40*s^2+4*s+5)/(s^4+30*s^3+337*s^2+1608+3136);
>> ilaplace(f)
ans =
dirac(t)-1/2688941905488*sum((1617616568080-237573334631*_alpha+94409086967*_alpha^2+4851584456*_alpha^3)*exp(_alpha*t),_alpha = RootOf(_Z^4+30*_Z^3+337*_Z^2+4744))
>> %Thus r,p,k values are found and time domain function is found%
>> %Th partial fraction is (752/(s+8)+640/(s+8)^2-771/(s+7)+252/(s+7)^2%
>>
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