Why is DTFT of $e^{jn\omega_0}$ an impulse train?

1. What is DTFT of unit impulse?
2. What is an impulse train?
3. What is the Fourier transform of an impulse train?
4. Does the impulse train in the time domain yield an impulse train in the frequency domain?

What is DTFT of unit impulse?

The DTFT is a frequency-domain representation for a wide range of both finite- and infinite-length discrete-time signals x[n]. The DTFT is denoted as X(ej ˆω), which. shows that the frequency dependence always includes the complex exponential function.

What is an impulse train?

Impulse trains are trains of action potentials spaced over time, with varying time intervals between them. The brain thus includes a massively parallel impulse train generator and processor. Simultaneously generated impulse trains can have patterns that are a function of the activity of ensembles of neurons.

What is the Fourier transform of an impulse train?

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. ⇐⇒ X(f) = XT (f) × S(f).

Does the impulse train in the time domain yield an impulse train in the frequency domain?

Because the infinite impulse train is periodic, we will use the Fourier Transform of periodic signals: where C_k are the Fourier Series coefficients of the periodic signal. Thus, an impulse train in time has a Fourier Transform that is a impulse train in frequency.