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 Ex 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.