Continuous-time vs discrete-time signals in Signal Processing

Introduction

We use signals to represent information. Continuous-time signals change smoothly over time, while discrete-time signals change only at specific moments. Understanding the difference helps us analyze and process data correctly.

When measuring temperature every second (discrete-time) versus tracking temperature continuously.

When recording audio digitally (discrete samples) versus analog audio signals.

When analyzing stock prices recorded at closing time each day (discrete) versus continuous price changes.

When designing digital filters that work on sampled data.

When simulating real-world signals on a computer that only handles discrete data.

Syntax

Signal Processing

Continuous-time signal: x(t) where t is any real number
Discrete-time signal: x[n] where n is an integer index

Continuous-time signals are functions of a continuous variable (usually time).

Discrete-time signals are sequences indexed by integers, representing samples.

Examples

This signal is defined for every time t, smoothly changing.

Signal Processing

x(t) = sin(2 * pi * 5 * t)  # Continuous-time sine wave at 5 Hz

This signal is defined only at integer values n, representing samples.

Signal Processing

x[n] = sin(2 * pi * 5 * n / 100)  # Discrete-time sine wave sampled at 100 Hz

Sample Program

This code plots a smooth continuous sine wave and its discrete samples taken at 20 points per second. You can see how the discrete points represent the continuous signal at specific times.

Signal Processing

import numpy as np
import matplotlib.pyplot as plt

# Continuous-time signal: sine wave from 0 to 1 second
t = np.linspace(0, 1, 1000)  # 1000 points for smooth curve
x_t = np.sin(2 * np.pi * 5 * t)  # 5 Hz sine wave

# Discrete-time signal: sampled version at 20 samples
n = np.arange(20)
x_n = np.sin(2 * np.pi * 5 * n / 20)  # samples at 20 Hz

plt.figure(figsize=(8,4))
plt.plot(t, x_t, label='Continuous-time signal')
plt.stem(n/20, x_n, linefmt='r-', markerfmt='ro', basefmt='k-', label='Discrete-time samples')
plt.xlabel('Time (seconds)')
plt.ylabel('Amplitude')
plt.title('Continuous-time vs Discrete-time Signals')
plt.legend()
plt.grid(True)
plt.show()

Important Notes

Discrete-time signals come from sampling continuous signals at regular intervals.

Sampling rate affects how well the discrete signal represents the continuous one.

Continuous signals cannot be directly processed by computers without sampling.

Summary

Continuous-time signals vary smoothly over all time values.

Discrete-time signals are sequences defined only at specific time points.

Sampling converts continuous signals into discrete signals for digital processing.

Practice

1. Which of the following best describes a continuous-time signal?

easy

A. A signal defined only at specific time points

B. A signal that is always zero

C. A signal that only takes integer values

D. A signal defined for every instant of time

Solution

1. Step 1: Understand continuous-time signal definition

   A continuous-time signal is defined for every instant in time, meaning it varies smoothly without gaps.

2. Step 2: Compare with discrete-time signals

   Discrete-time signals are defined only at specific time points, not continuously.

3. Final Answer:

   A signal defined for every instant of time -> Option D

4. Quick Check:

   Continuous-time = defined at all times [OK]

Hint: Continuous means all time points included [OK]

Common Mistakes:

• Confusing continuous with discrete signals
• Thinking discrete signals are continuous
• Assuming continuous signals only take integer values

2. Which syntax correctly represents a discrete-time signal sampled from a continuous signal x(t) at integer times n?

easy

A. x[n] = integral of x(t)

B. x(n) = x(t) for all t

C. x[n] = x(t) at t = n

D. x(t) = x[n] for n in integers

Solution

1. Step 1: Identify discrete-time signal notation

   Discrete-time signals use square brackets [n] to denote samples at integer times.

2. Step 2: Match sampling definition

   Sampling means taking values of x(t) at specific times t = n, so x[n] = x(t) at t = n is correct.

3. Final Answer:

   x[n] = x(t) at t = n -> Option C

4. Quick Check:

   Discrete-time uses square brackets and samples at integers [OK]

Hint: Discrete signals use square brackets for indexing [OK]

Common Mistakes:

• Using parentheses instead of square brackets for discrete signals
• Confusing continuous and discrete notation
• Thinking sampling is an integral operation

3. Given the continuous-time signal x(t) = 2t, what is the discrete-time signal x[n] sampled at integer times?

medium

A. x[n] = 2n

B. x[n] = 2t

C. x[n] = n/2

D. x[n] = 2

Solution

1. Step 1: Understand sampling at integer times

   Sampling means replacing t by integer n in x(t).

2. Step 2: Substitute t = n in x(t) = 2t

   We get x[n] = 2n.

3. Final Answer:

   x[n] = 2n -> Option A

4. Quick Check:

   Replace t with n in formula [OK]

Hint: Replace t with n for discrete samples [OK]

Common Mistakes:

• Leaving t as variable in discrete signal
• Dividing instead of multiplying by 2
• Confusing n and t variables

4. Identify the error in this discrete-time signal definition: x[n] = x(t) at t = n + 0.5

medium

A. The notation x[n] is incorrect

B. Sampling at non-integer times is invalid for discrete-time signals

C. The signal should be continuous, not discrete

D. There is no error; this is a valid discrete-time signal

Solution

1. Step 1: Understand discrete-time signal sampling points

   Discrete-time signals are defined at integer time points n.

2. Step 2: Check sampling time t = n + 0.5

   Sampling at half-integers is not standard discrete-time sampling; it breaks the integer indexing rule.

3. Final Answer:

   Sampling at non-integer times is invalid for discrete-time signals -> Option B

4. Quick Check:

   Discrete-time = integer time samples only [OK]

Hint: Discrete-time signals sample only integer times [OK]

Common Mistakes:

• Thinking discrete signals can sample at any time
• Confusing notation with error
• Assuming half-integers are valid discrete points

5. You have a continuous-time signal x(t) = sin(2πt). You sample it at 1 Hz (once per second). Which discrete-time signal x[n] do you get, and what is a potential problem?

hard

A. x[n] = sin(2πn); signal appears constant at zero

B. x[n] = sin(2πn); signal is perfectly captured

C. x[n] = sin(πn); signal is undersampled

D. x[n] = sin(4πn); signal is oversampled

Solution

1. Step 1: Write discrete-time signal from sampling

   Sampling at 1 Hz means t = n seconds, so x[n] = sin(2πn).

2. Step 2: Evaluate sin(2πn) for integer n

   Since sin(2πn) = 0 for all integers n, the sampled signal is always zero.

3. Step 3: Identify the problem

   This is an example of aliasing where the sampled signal loses information and appears constant.

4. Final Answer:

   x[n] = sin(2πn); signal appears constant at zero -> Option A

5. Quick Check:

   Sampling frequency equals signal frequency causes zero samples [OK]

Hint: Sampling at signal period causes zero values [OK]

Common Mistakes:

• Assuming sampling always captures signal perfectly
• Ignoring aliasing effects
• Confusing frequency and sampling rate