Transforms and its Applications

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Arulmigu Meenakshi Amman college of Engineering
Vadamavandal-604410.
Prepared by,
V.Umapathy,
Prof./Mech.

Important Questions and Answers
MA25C03- Transforms and its Applications
Syllabus
1. Laplace Transforms: Existence conditions, Properties of Laplace transform, Laplace
transform of standard functions, derivatives and integrals, Unit step function and Dirac delta
function, Laplace transform of periodic functions; Inverse Laplace transform: Partial fraction
technique, Convolution theorem. Application: Solution of second order ordinary differential
equations using Laplace transform.
Activities: Compute the Laplace transform of time-domain functions, Inverse Laplace
transform, Solution of ordinary differential equations using Laplace transform.
2. Z-Transform: Z-transform of standard functions, properties; Inverse Z – transform:
Standard functions, Partial fraction technique, Convolution theorem. Application: Solution of
difference equation using Z – transform.
Activities: Compute the Z-transform of a discrete-time signal, Solution of linear constant-
coefficient difference equations using Z-transform.
3. Fourier Series: Dirichlet’s conditions, General Fourier series, Convergence of Fourier
series, Odd and even functions; Half range sine series, Half range cosine series, Root mean
square value, Parseval’s identity. Application: Solution of one-dimensional wave and heat
equation. Activities: Compute Fourier coefficients, Reconstruct signal using Fourier series
(Partial sum), Plot convergence of Fourier series.
4. Fourier Transform: Complex Fourier transform, Properties, Relation between Fourier and
Laplace transform, Fourier sine and cosine transforms, Parseval’s identity, Convolution
theorem. Application: Simple applications to solve partial differential equations using
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Fourier transform.
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Activities: Compute the Fourier and inverse Fourier transform, Parseval’s theorem
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Unit-I
Laplace Transforms
Introduction
Laplace transform of f(t):
F(s) = ∫₀∞ e(-st)f(t)dt

Existence(t) is piecewise continuous, exponential order

Properties_
1. Linearity: L{af+bg} = aL{f}+bL{g}
2. Shift: L{e(at)f(t)} = F(s-a)
3. Derivative: L{f'(t)} = sF(s) - f(0)

_Standard Transforms_




Unit Step & Dirac Delta_
L{u(t-a)} = e(as)/s
L{δ(t-a)} = e(-as)

Inverse Transform
_Partial fractions, convolution
Basic Concepts
Existence Conditions
_f(t) piecewise continuous, exponential order (|f(t)| ≤ Me(at))
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Properties_
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1. Linearity: L{af+bg} = aL{f}+bL{g}
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2. Shift: L{e(at)f(t)} = F(s-a)
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3. Derivative: L{f'(t)} = sF(s) - f(0)
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4. Integral: L{∫f(t)dt} = F(s)/s

Standard Transforms_




Special Functions
1. Unit Step: L{u(t-a)} = e(-as)/s
2. Dirac Delta: L{δ(t-a)} = e(-as)

Inverse Transform_
1. Partial Fractions
2. _Convolution: L⁻¹{F(s)G(s)} = fg
Two Marks QA
1. Define Laplace transform.
L{f(t)} = ∫₀∞ e(-st)f(t)dt

2. State linearity property.
L{af+bg} = aL{f}+bL{g}

3. L{1} = ?
1/s

4. L{e(at)} = ?
1/(s-a)
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5. L{δ(t-a)} = ?
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e(-as)
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6. Inverse L{1/(s-a)} = ?
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e(at)
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7. Convolution theorem.
_L⁻¹{F(s)G(s)} = f_g
8. State the sufficient conditions for the existence of the Laplace transform.
Existence Conditions
L{f(t)} exists if

2. f(t) is of exponential order (∃ M, a: |f(t)| ≤ Me^(at) ∀ t ≥ 0)
1. f(t) is piecewise continuous on [0,∞)


Then L{f(t)} exists for s > a.
9. Find the Laplace transform of f(t) = sin(at) and f(t) = cosh(at).




10.What is the Laplace transform of the derivative f'(t).
L{f'(t)} = sF(s) - f(0)
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11. How do you handle the transform of an integral ∫ f ( u ) du ?
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L{∫₀t f(u)du} = F(s)/s
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12. Define the Unit Step Function (Heaviside Function) and its transform.
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u(t-a) = {0, t<a; 1, t≥a}
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