What is periodicity in Fourier transform?
If X[n] is, as above, a signal that repeats every samples, the Fourier transform of X[n] also repeats itself every units of frequency, that is, for all real values of . This follows immediately from the definition of the Fourier transform, since the factor.
Why is the period Fourier transform?
Thus, the Fourier transform of a periodic signal having the Fourier series coefficients is a train of impulses, occurring at multiples of the fundamental frequency, the strength of the impulse at being .
Is the discrete time Fourier transform a periodic transform?
In my signals and systems course, we have learned that the discrete-time Fourier transform is 2 π periodic, but the continuous-time Fourier transform is not periodic in general. For reference, we are using the following definitions of each transform: I’m searching for some intuition as for why the DTFT is periodic, but the CTFT is not.
How is the Fourier transform of a periodic impulse train related?
The Fourier transform of a periodic impulse train is also a periodic impulse train with period equal to the sampling frequency [6] [8]. The result is that the Fourier transform of Eq. 2 is then, using notation consistent with O&S, where Ω s = 2π/T is the sampling frequency in radians/second.
How is the periodic spectrum related to the Fourier spectrum?
Many mathematical treatments of periodic discrete-time sampling that address the periodic spectrum make reference to the Poisson Summation Formula or just state that if a time series is periodic over period N then the spectrum will also be periodic over N samples.
Why is the periodicity of the DTFT constant?
That’s why the DTFT is periodic. There cannot be any new spectral information outside the frequency interval [ − f s / 2, f s / 2]. If you haven’t learned it yet, check out the sampling theorem, which will also help you understand the nature of spectra of discrete-time signals.