Signal Operations

Why it matters

Signal operations are fundamental in the analysis and processing of signals in various engineering applications. They allow engineers to manipulate signals to extract useful information, design systems, and improve performance in communication, control, and signal processing systems.

Key ideas

• Time Shifting: This operation involves shifting a signal in time. If a signal is shifted to the right, it is delayed; if shifted to the left, it is advanced.
• Time Scaling: This involves compressing or expanding a signal in time. For x(at), |a| > 1 compresses and 0 < |a| < 1 expands; a < 0 also reverses time.
• Time Inversion: Also known as time reversal, this operation flips the signal around the vertical axis, effectively reversing the time.
• Amplitude Scaling: This operation changes the amplitude of a signal by multiplying it by a constant factor.
• Signal Addition: Combining two signals to form a new signal by adding their respective amplitudes at each point in time.
• Signal Multiplication: Forming a new signal by multiplying the amplitudes of two signals at each point in time.

Formulas

• Time Shifting: x(t - t₀)
  • x(t) is the original signal
  • t₀ is the time shift (s)
• Time Scaling: x(at)
  • a is the scaling factor (dimensionless)
• Time Inversion: x(-t)
• Amplitude Scaling: A·x(t)
  • A is the amplitude scaling factor (dimensionless)
• Signal Addition: x₁(t) + x₂(t)
• Signal Multiplication: x₁(t)·x₂(t)

Worked example

Problem: Given a signal x(t) = 2t for 0 ≤ t ≤ 2 and zero elsewhere, perform the following operations:

1. Shift the signal by 1 second to the right.
2. Scale the signal by a factor of 2 in time.
3. Invert the signal in time.

Solution (each operation acts independently on the original signal; output is zero outside the stated interval):

1. Time Shifting: x(t - 1) = 2(t - 1) for 1 ≤ t ≤ 3
2. Time Scaling: x(2t) = 2(2t) = 4t for 0 ≤ t ≤ 1
3. Time Inversion: x(-t) = 2(-t) = -2t for -2 ≤ t ≤ 0

Final Answer: The operations result in the signals x(t - 1), x(2t), and x(-t) with the respective ranges.

Common mistakes

• Confusing time shifting with time scaling.
• Forgetting to adjust the time range when performing time scaling.
• Incorrectly applying the sign in time inversion.
• Overlooking the effect of amplitude scaling on the signal's range.

For GATE EE

Questions often involve performing multiple operations on a given signal and analyzing the effects. Practice problems that require understanding the sequence of operations and their impact on signal properties.

Quick check

1. What is the result of time inverting the signal x(t) = 3t?
2. How does amplitude scaling affect a signal?
3. What happens to a signal when it is shifted to the left by 2 seconds?

Answers: 1. x(-t) = -3t; 2. Changes the amplitude by a constant factor; 3. The signal is advanced by 2 seconds.