# What is the meaning of the difference between the Fourier transforms of a single pulse in these two cases?

In texts the Fourier transform of a single pulse is explained as below:

They always create the pulse from -T to T, and they always show the sinc function symmetric around the origin.

But in reality there is no -T(there is not negative time on a scope screen) and even when I take the FFT of a single pulse in a circuit simulator I get a sinc function only for positive frequencies not anything symmetric with negative frequencies.

Below the example circuit which outputs a single 1V pulse with a total ON time of 1ms:

And when I take the FFT of the above pulse I obtain the following plot in linear scale:

Mathematical description of the pulse and the Fourier transform is different than what I get in the simulation. What is the difference between these two?

• The magnitude is shown but not the phase inversion on each alternate lobe. – Tony Stewart Sunnyskyguy EE75 Sep 23 '18 at 20:07
• "there is not negative time on a scope screen" citation needed. There is indeed, it depends on where you set the origin. – Vladimir Cravero Sep 23 '18 at 20:32

The differences:

1) Your pulse starts at t=0,5 seconds. The pulse in the printed book starts at t=minus half of the pulse length

2) Your spectrum discards negative frequencies and phase angles. It shows only the magnitudes of the components at positive frequencies. The spectrum formula and also the graph in the printed book are exact, they have magnitudes and phase angles and give them at positive and negative frequencies. Just this lucky case - a symmetric pulse around t=0 kills all imaginary numbers in the spectrum, one curve and the imaginary unit j nowhere in the formula.

3) If you zoom in your spectrum, you see it's discrete. It contains only frequencies N x 1/total sampled time period. N is integer and the covered frequency range is 0...half of the sampling frequency. The exact spectrum in the printed book is continuous in frequency domain. The printed spectrum formula covers all frequencies from -infinty to +infinity

You may ask how FFT spectrum would still be errorneous if the phase angles and negative frequencies were added. Answer: It would give the same pulse infinitely repeating. The repeating period is = the total sampled time period. That is not considered harmful because the user of the spectrum should know that only one pulse really existed.

• You wrote a very very broad answer explaining why complex exponentials used ect. which does not focus on this particular question. I need some mathematical or graphical explanation to that particular question. – cm64 Sep 23 '18 at 22:14
• @cm64 It's fixed. No stories, – user287001 Sep 23 '18 at 22:48
• The Fourier transform of a real-valued function is always symmetric. Often the negative frequency part is not displayed, because it's the same as the positive frequency part, only backwards.

• LTSpice is displaying the absolute value of the Fourier transform result. The practical effect of this is that phase differences are not displayed. Often this is what you want. It may have a separate phase graph, which would display 0 degrees for the positive parts, and 180 degrees for the negative parts.