📘 Signal Operations – Time Shifting, Scaling & Reversal
Signal operations modify signals in time or amplitude. Understanding these transformations is essential for solving convolution and Fourier problems.
🔹 1. Amplitude Scaling
Multiplying signal by a constant.
y(t) = A x(t)
If A > 1 → Amplification If 0 < A < 1 → Attenuation🔹 2. Time Shifting
Right Shift (Delay)
y(t) = x(t − t₀)
Signal moves to right.Left Shift (Advance)
y(t) = x(t + t₀)
Signal moves to left.🔹 3. Time Scaling
y(t) = x(at)
If |a| > 1 → Compression If 0 < |a| < 1 → Expansion Example:x(2t) → Compressed x(t/2) → Expanded
🔹 4. Time Reversal
y(t) = x(−t)
Mirror of signal about vertical axis.🔹 5. Combined Operations
Example:y(t) = x(2t − 3)
Step-by-step approach: 1. First scaling → x(2t) 2. Then shifting → x(2t − 3) Important rule: Perform scaling first, then shifting.🔹 6. Worked Example 1
Given x(t), find y(t) = x(t − 2)
Solution: Shift original signal 2 units to right.🔹 7. Worked Example 2
Given x(t), find y(t) = x(−2t)
Step 1: Time scaling → x(2t) Step 2: Time reversal → x(−2t) Graph becomes compressed and mirrored.🔹 8. Discrete Time Case
y[n] = x[n − k]
Right shift by k samples. Time scaling not continuous in discrete case.🎯 GATE Important Points
- Scaling done before shifting
- Compression changes frequency
- Time reversal important for convolution
- Discrete signals behave differently for scaling
Signal Transformations Are Foundation of Convolution
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