Consider a discrete-time Fourier transform (DTFT) pair, differentiate both sides with respect to Ω, and then multiply by j.

The right-hand side becomes the Fourier transform of nx[n], illustrating the frequency differentiation property.

 Apply discrete-time convolution to the DTFT pairs, change the order of the summations on the right-hand side, and apply the time-shifting property.

The substitution of the equations gives the time convolution property.

Multiply the two-time domain signals and interchange the order of integration and summation on the right-hand side.

This corresponds to periodic convolution in the frequency domain, scaled by the inverse of the period, demonstrating frequency convolution.

The accumulation property focuses on the summation of a discrete-time signal.

The DTFT of the accumulated signal has the original DTFT scaled by an exponential factor plus a delta function term, indicating periodic components at multiples of 2π.

The energy of the signal E_x is the sum of the squared magnitudes in the time domain. This is equal to the integral of the squared magnitudes of its DTFT, demonstrating Parseval's Relation.