Continuous-Time and Discrete-Time Signals

Why it matters

Continuous-Time and Discrete-Time Signals are fundamental in the field of signal processing, which is crucial for designing and analyzing systems in telecommunications, control systems, and electronics. Understanding these signals helps in the transition from analog to digital systems, which is essential for modern technology.

Key ideas

• Continuous-Time Signals: These signals are defined for every instant of time. They are represented as a function defined at every time instant (which can still contain jumps), such as x(t). Examples include analog audio signals and temperature readings over time.
• Discrete-Time Signals: These signals are defined only at discrete intervals of time. They are represented as a sequence of numbers, such as x[n]. Examples include digital audio signals and sampled data from sensors.
• Sampling: The process of converting a continuous-time signal into a discrete-time signal by taking samples at regular intervals.
• Aliasing: A phenomenon that occurs when a signal is undersampled, causing different signals to become indistinguishable.
• Nyquist Rate: The minimum sampling rate required to avoid aliasing, which is twice the highest frequency present in the signal.

Formulas

• x(t) = A·sin(2πft + φ)
  • x(t): Continuous-time signal
  • A: Amplitude (unit: V)
  • f: Frequency (unit: Hz)
  • φ: Phase (unit: radians)
• x[n] = x(nT)
  • x[n]: Discrete-time signal
  • T: Sampling period (unit: s)

Worked example

Given: A continuous-time signal x(t) = 5·sin(100πt) is sampled at a rate of 100 Hz.

1. Determine the Nyquist Rate

   • Formula: Nyquist Rate = 2·f_max
   • f_max = 50 Hz (from 100πt, f = 50 Hz)
   • Nyquist Rate = 2·50 Hz = 100 Hz

2. Check if the sampling rate is adequate

   • Given sampling rate = 100 Hz
   • Here equality is insufficient: every sample falls at a zero crossing, so the nonzero sine wave is lost. Use f_s > 2f_max for the usual arbitrary-phase baseband bandlimited reconstruction guarantee, with practical anti-alias filtering and margin.

3. Find the discrete-time signal

   • Formula: x[n] = x(nT)
   • T = 1/100 s
   • x[n] = 5·sin(100π·n/100)
   • x[n] = 5·sin(πn)

Final Answer: x[n] = 5·sin(πn) = 0 for every integer n. These samples do not preserve the original sinusoid.

Common mistakes

• Confusing continuous-time and discrete-time signals.
• Incorrectly calculating the Nyquist Rate, leading to aliasing.
• Forgetting to convert frequency units when calculating the Nyquist Rate.

For GATE EE

Questions often involve calculating the Nyquist Rate, determining if a given sampling rate is adequate, and converting continuous-time signals to discrete-time signals. Practice problems on aliasing and sampling theorem are crucial.

Quick check

1. What is the Nyquist Rate for a signal with a maximum frequency of 200 Hz?
2. How do you represent a continuous-time signal mathematically?
3. What is aliasing?

Answers: 1. 400 Hz 2. As a function of time, x(t) 3. A phenomenon where different signals become indistinguishable due to undersampling.