# Fourier transform and convolution

Please see the attach file, it take from "Optimization of electronic measurements module 4 text with experiments" by Howard V.Malmstadt

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#### Solution Preview

The transform integral is

(1.1)

Note that the actual transform occurs when

In our case

(1.2)

To evaluate the integral we use the trigonometric identities:

(1.3)

Therefore:

(1.4)

So the integral is now:

(1.5)

This is a simple integral to evaluate and we get:

(1.6)

For and the amplitude (left) and the Power spectrum (right) look like

When we change T to 1s the amplitude they look like:

As we see that the peak gets narrower and narrower, until it ...

#### Solution Summary

The solution shows how the convolution of two signals on the time domain translate to the frequency domain

$2.19