Q4Signals and Systems

Ideal sampling of a continuous-time signal $x(t)$ is modeled mathematically by multiplying $x(t)$ with a periodic impulse train $p(t)$, also known as a Dirac comb.

Where $T_s$ is the sampling period. The sampled signal $x_p(t)$ is:

In the frequency domain, multiplication in time becomes convolution in frequency. The Fourier transform of the impulse train is another impulse train in frequency. Thus, the spectrum of the sampled signal $X_p(j\omega )$ becomes:

This equation shows that the original spectrum $X(j\omega )$ is scaled by $1/T_s$ and infinitely replicated across the frequency axis at integer multiples of the sampling frequency ${\omega }_s=2\pi /T_s$. If ${\omega }_s>2{\omega }_m$ (Nyquist rate), the replicas do not overlap, and the signal can be reconstructed without aliasing.

(A diagram showing a triangle spectrum at origin being replicated at $\pm {\omega }_s,\pm 2{\omega }_s$ is expected here).