Questions: Signal Classification: Continuous vs. Discrete Time

Question 1

A discrete-time signal x[n] is defined for all integers n. What is the value of x[1.5]?

• A. The average of x[1] and x[2]
• B. Zero, since there is no sample at that index
• C. Undefined — the question is meaningless for a DT signal
• D. It depends on the interpolation method used

Answer: (C)

A discrete-time signal x[n] is only defined at integer indices. x[1.5] is literally meaningless — not zero, not interpolated, but undefined. This is not a gap in the data; it is a fundamental property of the mathematical object. DT signals are sequences indexed by integers, not functions on the real line. The common confusion is treating a DT signal like a CT signal with 'missing' values, but they are categorically different mathematical objects.

Question 2

Which mathematical operation in discrete-time signal analysis is the direct analog of integration in continuous-time analysis?

• A. Differentiation — differences replace derivatives
• B. Summation — sums replace integrals
• C. Convolution — it works the same in both domains
• D. Sampling — it converts between the two domains

Answer: (B)

In DT, summations (∑) replace the integrals (∫) of CT analysis. The Discrete-Time Fourier Transform is a sum ∑ x[n] e^{−jωn} rather than the integral ∫ x(t) e^{−jωt} dt. Similarly, differences replace derivatives. This parallel structure is why DT signals form their own complete mathematical world — every CT concept has a DT counterpart, but the tools are summation-based rather than calculus-based.

Question 3

Discrete-time signals are simply continuous-time signals sampled at regular intervals — they contain exactly the same information, just with gaps between samples.

• T. True
• F. False

Answer: False

DT signals are their own mathematical objects, not CT signals 'with gaps.' They are sequences indexed by integers, with no meaning between those indices. Furthermore, sampling a CT signal to create a DT signal can result in information loss (aliasing) if the sampling rate is insufficient — so the DT signal does not necessarily contain the same information. The Sampling Theorem governs when recovery is possible. Treating DT signals as 'gapped' CT signals leads to serious errors in analysis.

Question 4

A discrete-time sinusoid at frequency ω₀ and a DT sinusoid at frequency ω₀ + 2π produce identical sequences — they are the same signal.

• T. True
• F. False

Answer: True

This is one of the most fundamental differences between CT and DT frequency. Because e^{j(ω₀ + 2π)n} = e^{jω₀n} · e^{j2πn} = e^{jω₀n} · 1 for all integers n, DT frequency is periodic with period 2π. Two sinusoids that differ by 2π in frequency are indistinguishable as sequences. No such periodicity exists in continuous time, where higher frequencies genuinely oscillate faster. This periodicity is why DT frequency is typically restricted to the interval [−π, π] or [0, 2π).

Question 5

Explain why 'frequency' means something fundamentally different for discrete-time signals compared to continuous-time signals.

Model answer: In continuous time, frequency is unbounded and unique — higher frequency means faster oscillation, and no two distinct frequencies produce the same waveform. In discrete time, frequency is periodic with period 2π: a sinusoid at ω₀ and one at ω₀ + 2π are identical sequences, because e^{j2πn} = 1 for all integers n. This means DT frequency only needs to be specified in a 2π-wide window (typically [−π, π]). The periodicity arises from the discrete nature of the index n, which constrains what frequency differences are detectable.

The deeper reason is that DT signals are only defined at integers, and integer-indexed complex exponentials repeat with period 2π. This is not a limitation — it is a structural property that shapes everything in DT signal processing, from filter design to the definition of the DFT.