Continuous-Time and Discrete-Time Signals

Why it matters

Continuous-time and discrete-time signals form the foundation of signal processing, which is essential in telecommunications, control systems, and electronics. Understanding these signals helps in designing systems that can efficiently process real-world data.

Key ideas

• Continuous-Time Signals: These signals are defined for every instant of time. They are represented as a function defined at every real time; the function can contain jumps, such as x(t). Examples include analog audio signals and temperature readings.
• Discrete-Time Signals: These signals are defined only at discrete intervals of time. They are represented as a sequence, such as x[n]. Examples include digital audio signals and sampled data.
• Signal Representation: Continuous-time signals can be converted to discrete-time signals through sampling, which is covered under the Sampling Theorem.
• Periodic and Aperiodic Signals: Signals can be classified based on their repetition over time. Periodic signals repeat after a fixed interval, while aperiodic signals do not.
• Even and Odd Signals: A signal is even if it is symmetric around the vertical axis, and odd if it is symmetric around the origin.

Discrete time does not by itself imply quantized amplitude; a digital signal has a finite representation. For a discrete sinusoid sin(Ωn + φ), periodicity requires Ω/(2π) rational, with an integer sample period.

Formulas

• Continuous-time signal: x(t)
  • x(t): Signal value at time t (unit: depends on the context, e.g., volts for voltage signals)
• Discrete-time signal: x[n]
  • x[n]: Signal value at discrete time n (unit: depends on the context)
• Periodic signal condition: x(t) = x(t + T)
  • T: Period of the signal (unit: seconds)

Worked example

Given: A continuous-time signal x(t) = 5 sin(2π60t) is sampled at a rate of 200 samples per second.

1. Determine the sampling interval:

   Formula: T_s = 1 / f_s

   Where f_s is the sampling frequency.

   T_s = 1 / 200 = 0.005 seconds

2. Find the discrete-time signal representation:

   Formula: x[n] = x(nT_s)

   x[n] = 5 sin(2π60n(0.005))

   x[n] = 5 sin(0.6πn)

Final Answer: The discrete-time signal is x[n] = 5 sin(0.6πn).

Common mistakes

• Confusing continuous-time signals with discrete-time signals.
• Incorrectly applying the sampling theorem, leading to aliasing.
• Misidentifying even and odd signals.

For GATE EC

Questions often involve converting continuous-time signals to discrete-time signals, analyzing periodicity, and identifying signal properties. Practice problems on sampling, signal transformations, and periodicity analysis.

Quick check

1. What is the main difference between continuous-time and discrete-time signals?
2. How do you determine if a signal is periodic?
3. What is aliasing in the context of signal processing?

Answers: 1. Continuous-time signals are defined for every instant, while discrete-time signals are defined at discrete intervals. 2. A signal is periodic if it repeats after a fixed interval T. 3. Aliasing occurs when a signal is undersampled, causing different signals to become indistinguishable.