Fourier transform of a single pulse and pulse sequences

[bfusco]

Homework Statement

What is the Fourier transform of a single short pulse and of a sequence of pulses?

The Attempt at a Solution

In class we haven't dealt with the mathematics of a Fourier transform, however my professor has simple stated that a Fourier transform is simply a equation converter. You can take equations that are a function of frequency and change them to a function of time, and vice versa.

With that said, i think that the Fourier transform of a single short pulse consists of an infinite amount of waves, that destructively interfere everywhere except one place.

From that i want to say a series of pulses is the sum of many waves that have phase relations that there is periodic constuctive interference, but destructive interference in between. I have no clue if this is correct.

 

[barryj]

I think you are basically correct

 

[rude man]

By 'sequence of pulses' do you mean a finite number of pulses or an infinite number?

 

[bfusco]

I believe infinite pulses, although the professor didn't specify

 

[barryj]

A single very short pulse has (in the limit) an infinite frequency spectrum. This is why one can determine characteristics of a filter by exciting the filter with a single pulse and then doing a Fourier transform on the output waveform to determine the filter characteristics.

A repeated pulse stream has discrete frequencies in the transform. The frequencies are multiples of the frequency of the puilse stream. However, the amplitude of the frequencies are determined by the width of the pulses and will form a Sin(x)/x envelope so to speak.

 

[rude man]

Homework Statement

What is the Fourier transform of a single short pulse and of a sequence of pulses?

The Attempt at a Solution

In class we haven't dealt with the mathematics of a Fourier transform, however my professor has simple stated that a Fourier transform is simply a equation converter. You can take equations that are a function of frequency and change them to a function of time, and vice versa.

With that said, i think that the Fourier transform of a single short pulse consists of an infinite amount of waves, that destructively interfere everywhere except one place.

What do you mean " .. except in one place"?

From that i want to say a series of pulses is the sum of many waves that have phase relations that there is periodic constuctive interference, but destructive interference in between. I have no clue if this is correct.

I don't see how you can find the Fourier transform of anything if you haven't dealt with the math.

If the sequence is infinite in duration (past and future) then the Fourier transform represents an infinite number of discrete frequencies. Those frequencies appear in the Fourier series.