Each entry is one standard property of the discrete-time Fourier transform: (A)-(II), (B)-(IV), (C)-(I), (D)-(III) — option 2.

(A) — frequency shifting. Multiplying a sequence by a complex exponential translates its spectrum:

\(e^{j\Omega_{0}n}x[n]\ \longleftrightarrow\ X\left(\Omega-\Omega_{0}\right)\)

It is the dual of time shifting, and it is exactly what modulation does — multiplying by a carrier moves the baseband spectrum up to the carrier frequency.

(B) — differentiation in frequency. Differentiating the transform brings down a factor of n:

\(n\,x[n]\ \longleftrightarrow\ j\dfrac{dX(\Omega)}{d\Omega}\)

This can be seen directly from the definition \(X(\Omega)=\sum x[n]e^{-j\Omega n}\): differentiating with respect to \(\Omega\) brings down \(-jn\) from the exponent, and multiplying by j corrects the sign.

(C) — the first difference. Time shifting by one sample multiplies the transform by \(e^{-j\Omega}\), so by linearity

\(x[n]-x[n-1]\ \longleftrightarrow\ X(\Omega)-e^{-j\Omega}X(\Omega)=\left(1-e^{-j\Omega}\right)X(\Omega)\)

The factor \(\left(1-e^{-j\Omega}\right)\) vanishes at \(\Omega=0\), which is the discrete-time counterpart of differentiation removing a constant — a first difference is a high-pass operation.

(D) — the shifted impulse. The sum collapses to a single term:

\(\sum_{n}\delta\left[n-n_{0}\right]e^{-j\Omega n}=e^{-j\Omega n_{0}}\)

Its magnitude is 1 at every frequency — an impulse contains all frequencies equally — and the shift appears entirely as a linear phase. This is the origin of the rule that a pure delay is a linear-phase system.

A quick way to check the code. Only (D) has a transform that is a bare function of \(\Omega\) with no \(X\) in it, since \(\delta[n-n_{0}]\) is the only entry in List I that is a specific sequence rather than a general one. Pairing (D) with (III) alone eliminates options 1 and 3, and (B) with (IV) settles the rest.

Hence, the correct code is (A)-(II), (B)-(IV), (C)-(I), (D)-(III).