Question 1, Question 2

Question 1, Question 2


Signals and Systems Analysis Assessment on the Laplace Transform March 25, 2024 Question 1 Answer each of the following questions. 1. For each of the following transforms, find the original signal in the time domain. In each case, state which property of the Transform you are using. f̂(s) = 40 (2s+ 1)3 , ŷ(s) = 8 + 3s− 2s2 s3 , x̂(s) = 4 s+ 0.2 ( e−5s−1 − e−15s−3 ) . 2. Find the functions x and y with these transforms: x̂(s) = 50e−4s( 5s+ 1 )2 , ŷ(s) = 3s2 1− e−5s − 5se−5s1− e−15s . Question 2 1. Solve the following differential equation for the given initial conditions by direct use of the transform: dx dt + 0.6x = 2te−0.1t, x(0) = 2. 2. The output to a mechanical system is denoted by y(t). The system is governed by the following ODE with the input denoted as x(t). dy dt + 0.25y = x(t). Find the transfer function and the impulse response for this system and use them to write out a general expression for the output y in terms of the input x. 3. For the system described in the previous question, use the impulse response and the convolution equation to find the output signal y(t) for the following input: x(t) = 4 sin(0.5t). 1 Information General The roots of the quadratic ax2 + bx+ c = 0 are x = −b± √ b2 − 4ac 2a If the roots of the quadratic s2+bs+c = 0 turn out to be the complex conjugates −a±ωj, then the quadratic can be written as follows: s2 + bs+ c = (s+ a)2 + ω2. The Heaviside Step Function: the function H(u) is given by: H(u) = { 0 when u < 0="" 1="" when="" u="" ≥="" 0="" calculus="" common="" derivatives="" y="xn" ⇒="" dy="" dx="nxn−1" y="eax" ⇒="" dy="" dx="aeax" y="loge(x+" a)⇒="" dy="" dx="1" x+="" a="" y="sin(ax)⇒" dy="" dx="a" cos(ax),="" y="cos(ax)⇒" dy="" dx="−a" sin(ax)="" common="" integrals="" ∫="" xndx="1" n+="" 1="" xn+1,="" ∫="" eaxdx="1" a="" eax∫="" sin(ax)dx="−1" a="" cos(ax),="" ∫="" cos(ax)dx="1" a="" sin(ax)∫="" uebudu="1" b="" uebx="" −="" 1="" b2="" ebx∫="" eβx="" cos(ωx)="" dx="1" ω2="" +="" β2="" eβx="" (="" ω="" sin(ωx)="" +="" β="" cos(ωx)="" )="" .∫="" eβx="" sin(ωx)="" dx="eβx" ω2="" +="" β2="" (="" −="" ω="" cos(ωx)="" +="" β="" sin(ωx)="" )="" .="" page="" 2="" differentiation="" rules="" if="" y="uv" then="" dy="" dx="v" du="" dx="" +="" u="" dv="" dx="" .="" if="" y="u" v="" then="" dy="" dx="v" dudx="" −="" u="" dv="" dx="" v2="" integration="" by="" parts="" ∫="" u="" dv="" dx="" dx="uv" −="" ∫="" v="" du="" dx="" dx.="" the="" laplace="" transform="" definition="" of="" the="" laplace="" transform="" let="" f="" be="" a="" function="" defined="" for="" t="" ≥="" 0.="" the="" laplace="" transform="" of="" f="" is="" defined="" by="" the="" integral="" shown,="" where="" it="" exists:="" f̂(s)="∫" ∞="" 0="" e−stf(t)="" dt.="" transforms="" of="" common="" functions="" •="" the="" single="" pulse="" at="" a:="" f(t)="δ(t−" a)⇒="" f̂(s)="e−as" •="" the="" constant="" function:="" f(t)="1⇒" f̂(s)="1" s="" •="" the="" exponential="" function:="" f(t)="e−at" ⇒="" f̂(s)="1" s+="" a="" •="" the="" linear="" function:="" f(t)="t⇒" f̂(s)="1" s2="" •="" the="" power="" function:="" f(t)="tn" ⇒="" f̂(s)="n!" sn+1="" •="" the="" sine="" function:="" f(t)="sin(at)⇒" f̂(s)="a" s2="" +="" a2="" •="" the="" cosine="" function:="" f(t)="cos(at)⇒" f̂(s)="s" s2="" +="" a2="" page="" 3="" •="" the="" step="" function:="" ha(t)="H(t−" a)⇒="" ĥa(s)="1" s="" e−as.="" •="" the="" pulse:="" f(t)="{" m="" when="" a="" ≤="" t="">< b="" 0="" otherwise="" ⇒="" f̂(s)="m" s="" (e−as="" −="" e−bs).="" properties="" of="" the="" laplace="" transform="" the="" shift="" theorem="" let="" f(t)="" be="" a="" function="" of="" t="" ≥="" 0="" with="" laplace="" transform="" f̂(s).="" then="" for="" a="" function="" g(t):="" g(t)="e−atf(t)⇒" ĝ(s)="f̂(s+" a).="" multiplication="" by="" t="" let="" f(t)="" be="" a="" function="" of="" t="" ≥="" 0="" with="" laplace="" transform="" f̂(s).="" then="" for="" a="" function="" g(t):="" g(t)="tf(t)⇒" ĝ(s)="−" d="" ds="" f̂(s).="" periodic="" functions="" let="" f(t)="" be="" a="" periodic="" function="" defined="" for="" t="" ≥="" 0,="" then:="" f(t)="f(t+" t="" ),="" when="" all="" t⇒="" f̂(s)="1" 1−="" e−st="" ∫="" t="" 0="" e−stf(t)="" dt.="" delayed="" functions="" let="" function="" f="" ,="" defined="" for="" t=""> 0, have well defined transform f̂(s). Set fa to be the delayed function of f : fa(t) = f(t− a)H(t− a)⇒ f̂a(s) = e−asf̂(s). The Transform of the Derivative Let x be a function of variable t whose Laplace transform is well defined. Let L[f ] denote the transform of a function f . Then L [ dx dt ] = sL[x]− x(0). Page 4 The Convolution Equation In the Transform domain we have ŷ(s) = H(s)x̂(s). Then x and y are linked in the time domain by: y(t) = ∫ t 0 x(u)h(t− u)du. Page 5