Time shifting a signal in Signal Processing

Introduction

Time shifting moves a signal forward or backward in time. It helps us see how signals change when delayed or advanced.

To delay a sound signal in audio processing.

To align sensor data collected at different times.

To simulate echoes by repeating signals with delays.

To analyze how a system responds to inputs at different times.

Syntax

Signal Processing

y(t) = x(t - t0)

t0 is the amount of time to shift.

If t0 is positive, the signal moves right (delay). If negative, it moves left (advance).

Examples

The signal x(t) is delayed by 2 units of time.

Signal Processing

y(t) = x(t - 2)

The signal x(t) is advanced by 3 units of time.

Signal Processing

y(t) = x(t + 3)

Sample Program

This code creates a 5 Hz sine wave and shifts it by 0.2 seconds to the right (delay). The shifted signal is zero before the start time.

The plot shows the original and delayed signals for easy comparison.

Signal Processing

import numpy as np
import matplotlib.pyplot as plt

# Create a simple signal: a sine wave
fs = 100  # samples per second
 t = np.linspace(0, 1, fs, endpoint=False)
x = np.sin(2 * np.pi * 5 * t)  # 5 Hz sine wave

# Time shift amount
shift = 0.2  # seconds

# Shift the signal by shifting the time vector
# y(t) = x(t - shift) means we evaluate x at (t - shift)
t_shifted = t - shift

# For values where t_shifted < 0, set signal to 0 (no data before start)
y = np.where(t_shifted >= 0, np.sin(2 * np.pi * 5 * t_shifted), 0)

# Plot original and shifted signals
plt.plot(t, x, label='Original signal')
plt.plot(t, y, label='Shifted signal (delay)')
plt.xlabel('Time (seconds)')
plt.ylabel('Amplitude')
plt.title('Time Shifting a Signal')
plt.legend()
plt.grid(True)
plt.show()

Important Notes

When shifting signals, values outside the original time range may be undefined. We often set them to zero.

Time shifting is a basic tool in signal processing to study delays and advances.

Summary

Time shifting moves a signal forward or backward in time by changing its time variable.

Positive shift delays the signal; negative shift advances it.

It helps analyze how signals behave when delayed or advanced.

Practice

1. What does a positive time shift do to a signal in signal processing?

easy

A. It advances the signal, moving it backward in time.

B. It delays the signal, moving it forward in time.

C. It changes the signal's amplitude.

D. It reverses the signal in time.

Solution

1. Step 1: Understand time shift direction

   A positive time shift means the signal is moved to a later time, which delays it.

2. Step 2: Compare options with definition

   Only It delays the signal, moving it forward in time. correctly states that a positive shift delays the signal.

3. Final Answer:

   It delays the signal, moving it forward in time. -> Option B

4. Quick Check:

   Positive shift = delay [OK]

Hint: Positive shift means delay, negative means advance [OK]

Common Mistakes:

• Confusing positive shift with advancing the signal
• Thinking time shift changes amplitude
• Mixing up delay and reversal

2. Which of the following is the correct Python syntax to shift a discrete signal array x by 3 units forward (delay)?

easy

A. shifted_x = np.roll(x, 3)

B. shifted_x = np.roll(x, -3)

C. shifted_x = x + 3

D. shifted_x = x.shift(3)

Solution

1. Step 1: Understand np.roll behavior

   In numpy, np.roll(x, 3) shifts elements to the right by 3, delaying the signal.

2. Step 2: Check other options

   shifted_x = x.shift(3) is invalid syntax for numpy arrays, np.roll(x, -3) shifts left (advances), x + 3 adds 3 to values (not time shift).

3. Final Answer:

   shifted_x = np.roll(x, 3) -> Option A

4. Quick Check:

   np.roll with positive shift delays signal [OK]

Hint: np.roll with positive number delays signal [OK]

Common Mistakes:

• Using negative shift to delay signal
• Adding number to array instead of rolling
• Using non-existent shift method

3. Given the signal array x = [1, 2, 3, 4, 5], what is the output of np.roll(x, -2)?

medium

A. [4, 5, 1, 2, 3]

B. [2, 3, 4, 5, 1]

C. [3, 4, 5, 1, 2]

D. [5, 1, 2, 3, 4]

Solution

1. Step 1: Understand np.roll with negative shift

   Negative shift moves elements to the left by 2 positions.

2. Step 2: Apply shift to array

   Original: [1, 2, 3, 4, 5]
   Shift left 2: [3, 4, 5, 1, 2]

3. Final Answer:

   [3, 4, 5, 1, 2] -> Option C

4. Quick Check:

   Negative shift = advance signal [OK]

Hint: Negative np.roll shifts left (advances) [OK]

Common Mistakes:

• Confusing left and right shift directions
• Miscounting shifted elements
• Assuming shift changes values

4. The following code attempts to shift a signal x = [10, 20, 30, 40] by 1 unit forward but produces incorrect output. What is the error?

shifted_x = np.roll(x, -1)

medium

A. The shift value should be positive to delay the signal.

B. np.roll cannot be used for shifting signals.

C. The array x must be converted to a numpy array first.

D. The shift value should be zero for no change.

Solution

1. Step 1: Analyze shift direction

   Negative shift (-1) moves signal left (advances), not forward (delay).

2. Step 2: Correct shift value

   To delay (shift forward), use positive shift value +1.

3. Final Answer:

   The shift value should be positive to delay the signal. -> Option A

4. Quick Check:

   Positive shift delays signal [OK]

Hint: Positive shift delays; negative advances [OK]

Common Mistakes:

• Using negative shift to delay signal
• Thinking np.roll is invalid for shifting
• Forgetting to convert list to numpy array (not required here)

5. You have two signals sampled at the same rate: s1 = [0, 1, 2, 3, 4] and s2 = [3, 4, 0, 1, 2]. Which time shift applied to s1 aligns it best with s2?

hard

A. Shift s1 by +3 units (delay by 3)

B. Shift s1 by +1 units (delay by 1)

C. Shift s1 by -2 units (advance by 2)

D. Shift s1 by -3 units (advance by 3)

Solution

1. Step 1: Compare signals visually

   s1 = [0, 1, 2, 3, 4], s2 = [3, 4, 0, 1, 2]. s2 matches s1 advanced (shifted left) by 3.

2. Step 2: Verify with np.roll

   np.roll(s1, -3) = [3, 4, 0, 1, 2] matches s2 exactly. np.roll(s1, 1) = [4, 0, 1, 2, 3], np.roll(s1, -2) = [2, 3, 4, 0, 1], np.roll(s1, 3) = [2, 3, 4, 0, 1] do not match.

3. Final Answer:

   Shift s1 by -3 units (advance by 3) -> Option D

4. Quick Check:

   np.roll(s1, -3) == s2 [OK]

Hint: Match s2 by rolling s1 left/right to align [OK]

Common Mistakes:

• Choosing wrong shift direction
• Not checking exact element alignment
• Confusing delay with advance