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What is the Fourier transform of a pulse train?
find its area. Therefore, the Fourier transform of a periodic impulse train in time is a periodic impulse train in frequency. Therefore, the Fourier transform of the periodic impulse train has an impulse at the frequency of each Fourier series component and the area of the impulse equals the Fourier series coefficient.
What is pulse train in laser?
Pulse train shaping allows the energy and/or width of each of a group of laser pulses to be delivered sequentially. This allows the energy input from the laser to be tailored to the specific application.
What is periodic pulse train?
A periodic pulse train with period T0 consists of rectangular pulses of duration T. The duty cycle of a periodic pulse train is defined by T/T0. An application of the periodic pulse train is in the practical sampling process.
What is a pulse train with a 50% duty cycle called?
A pulse wave or pulse train is a kind of non-sinusoidal waveform that includes square waves (duty cycle of 50%) and similarly periodic but asymmetrical waves (duty cycles other than 50%). The pulse wave is also known as the rectangular wave, the periodic version of the rectangular function.
How to write the Fourier transform of a pulse train?
You should start by writing the Fourier series of representation of $p(t)$, i.e.: From the delta definition of $p(t)$, it follows that $a_k=1$ for all $k$. The Fourier transform, $P(\\omega)$, follows from the Fourier transform of an exponential, yielding:
How to find the coefficients of a Fourier transform?
Let’s find the Fourier Series coefficients Ck for the periodic impulse train p ( t ): by the sifting property. Therefore Thus, an impulse train in time has a Fourier Transform that is a impulse train in frequency. The spacing between impulses in time is Ts, and the spacing between impulses in frequency is ω0 = 2π / Ts.
How to model the Fourier transform of a sampled signal?
As you just saw, p ( t) is an infinite train of continuous time impulse functions, spaced Ts seconds apart. Now, x ( t) is the continuous time signal we wish to sample. We will model sampling as multiplying a signal by p ( t ). Let xs ( t) = x ( t) p ( t ) be the sampled signal.
How are the FS and the ft of a pulse train related?
Your pulse train will have the Fourier series (FS) in stead of Fourier transform (FT). Fortunately the FT of a single pulse and the FS of the pulse train are related. Let the pulse train be periodic with P Hz. Then the FS of the pulse train will be sampled version of FT with a sampling rate of P Hz. i.e.