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1. MECHANICAL
ENGINEERING
Convolution and Fourier
Transforms
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2. Problem 1: (a) Plot the result of convolving a one-dimensional function consisting of
three impulses with a triangular function as shown below:
Answer: Use the convolution definition y(t) = f x h = ʃf (Ʈ )h(t - Ʈ)d Ʈ
and using the sifting property of the impulse function,
y (t) = h(t + 1.5) +h (t- 0.75)
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3. Answer: h(t) is the as same used in part (a)
(b) Plot the result of convolving a one-dimensional function consisting of a rectangular
function with the same triangular function used in part(a), as shown below:
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4. The result of the convolution, y(t), is plotted in the following figure
Four basic cases can be observed while varying t (sliding the triangle waveform).,
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5. (c) Plot the result of convolving a pair of identical even “top-hat” (pulse) functions:
f(t) = 1 for |t| < T/2 and f(t) = 0 otherwise. Use your result show that the Fourier
transform 2 of a triangular pulse (such as used in parts (a) and (b)) is of the form
(sin(x)/x)2 .
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7. Problem 2: Show that when two gaussian functions
f1(x) = e−ax2
f2(x) = e−bx2
are convolved, the result is another gaussian function. Hint: Complete the square in
the exponent, change the variable of integration, and recognize that
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8. Answer:
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