Which statement correctly distinguishes continuous-time (CT) signals from discrete-time (DT) signals and provides valid examples?

• A. CT signals are defined for all real time t; examples: x(t)=cos(2π f0 t), x(t)=e^{−α t} u(t).
• B. CT signals are defined only at integer times; examples: x(t)=cos(2π f0 t), x(t)=e^{−α t} u(t).
• C. DT signals are defined for all real time t; examples: x[n]=cos(ω0 n).
• D. DT signals are defined only at even integers; examples: x[n]=(1/2)^n u[n].

Answer: (A)

The main idea here is the time-domain domain of the signal: continuous-time signals are defined for every real time t, while discrete-time signals are defined only at isolated time instants, usually integer indices n.

Continuous-time signals are defined for all real t, which is why a sinusoid like x(t) = cos(2π f0 t) fits this category—it exists for every real time t. A decaying exponential like x(t) = e^(−αt) u(t) is also continuous-time; even though the unit step u(t) makes it nonzero only for t ≥ 0, the function is still described as a function of the real variable t and is defined for all real times.

Discrete-time signals, on the other hand, are sequences indexed by integers n. An example is x[n] = cos(ω0 n), which is defined only at integer n. The sequence x[n] = (1/2)^n u[n] is defined for n ≥ 0 (integer values), not just at even integers; discrete-time signals are not restricted to even indices unless explicitly specified.

So the correct view is that continuous-time signals are defined for all real time, with examples like the cosine and the exponential-with-step; discrete-time signals are defined at integer times, with examples like the discrete cosine and the exponential sequence.