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Note 2: Elementary Operations on Signals

Focus: Graphical and mathematical manipulation of signals. Examiners aggressively test your ability to combine shifting, scaling, and reversal in a single function. This note includes the exact theory and the step-by-step solutions for the sketching and explanation PYQs.

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1. Core Operations on the Independent Variable (Time)

In signal processing, manipulating the independent variable (time $t$ for continuous, or index $n$ for discrete) alters when and how fast a signal occurs without fundamentally changing its amplitude.

1.1 Time Shifting

Time shifting delays or advances the signal along the horizontal axis. It does not change the shape or width of the signal.

• Time Delay (Right Shift): Mathematically represented as $x(t-t_0)$ where $t_0>0$. The signal occurs later.
• Time Advance (Left Shift): Mathematically represented as $x(t+t_0)$ where $t_0>0$. The signal occurs earlier. [Graph: Original rectangular pulse x(t) centered at origin. To its right, the delayed pulse x(t-t_0). To its left, the advanced pulse x(t+t_0)]

1.2 Time Scaling

Time scaling speeds up or slows down the signal.

• Compression: Mathematically represented as $x(at)$ where $a>1$. The signal occurs faster, so it shrinks horizontally. (e.g., $x(2t)$ is twice as fast, taking half the time).
• Expansion: Mathematically represented as $x(at)$ where $0<a<1$. The signal occurs slower, stretching horizontally. (e.g., $x(0.5t)$ takes twice as long). [Graph: Original triangular signal x(t) spanning from -2 to 2. Below it, the compressed signal x(2t) spanning from -1 to 1. Below that, the expanded signal x(t/2) spanning from -4 to 4]

1.3 Time Reversal (Folding)

Time reversal flips the signal across the vertical $y$-axis ($t=0$).

• Mathematical Representation: $x(-t)$.
• Whatever happened at $t=5$ now happens at $t=-5$. If a signal is an “Even” signal, $x(t)=x(-t)$, so time reversal has no visible effect. [Graph: An asymmetrical signal x(t) strictly in the positive x-axis, and its time-reversed mirror image x(-t) strictly in the negative x-axis]

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2. PYQ Solution: Processing of Discrete Signals

🔥 Exact PYQ: What is signal? Explain the operations performed on the independent variables in the processing of discrete signals mathematically & graphically. [11/12 marks] (2019, 2018, 2017, 2015)

How to answer for full marks: If the examiner specifies discrete signals ($x[n]$), you must explain the operations using discrete sequences and note the critical difference in time scaling (down-sampling/up-sampling) which causes data loss or requires zero-padding.

1. Discrete Time Shifting:

• Mathematical: $y[n]=x[n-k]$. If $k>0$, the sequence is shifted right (delayed) by $k$ samples. If $k<0$, it is shifted left (advanced) by $|k|$ samples.
• Graphical: [Graph: Discrete sequence x[n] = {1, 2, 3} at n=0,1,2. Below it, y[n] = x[n-2] showing {1, 2, 3} at n=2,3,4]

2. Discrete Time Reversal (Folding):

• Mathematical: $y[n]=x[-n]$. The sequence is folded around the origin $n=0$.
• Graphical: [Graph: x[n] with values at n=1,2. Below it, y[n]=x[-n] with the same values mirrored at n=-1,-2]

3. Discrete Time Scaling (Crucial Difference from Continuous):

• Down-sampling (Decimation / Compression): $y[n]=x[Mn]$ where $M$ is an integer $>1$. The system keeps only every $M$-th sample and discards the rest. Data is lost.
• Up-sampling (Interpolation / Expansion): $y[n]=x[n/M]$ only for indices where $n/M$ is an integer. For all other values, $y[n]=0$. This inserts $M-1$ zeros between every sample.
• Graphical: [Graph: Original sequence x[n] from n=0 to 4. Decimated sequence y[n]=x[2n] showing only the 0th, 2nd, and 4th original samples at n=0, 1, 2. Expanded sequence y[n]=x[n/2] showing original samples separated by zeros]

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3. PYQ Solution: Combined Operations & Sketching

When sketching a signal with combined operations like $x(at-b)$, always follow this rigid sequence to avoid critical boundary errors:

1. Shift First: Draw $x(t-b)$.
2. Scale Second: Replace $t$ with $at$ to compress/expand.
3. Reverse Last (if negative): Flip the entire result across the y-axis.

🔥 Exact PYQ: Sketch the following signal: $x(t)=4r(3t-1)$ (2024, 2019) Step-by-Step Solution:

1. Identify Base Signal: The ramp function $r(t)=t\cdot u(t)$. It starts at origin with a slope of 1.
2. Amplitude Scaling: $4r(t)$. The signal still starts at 0, but the slope is now 4.
3. Time Shifting (Delay): $4r(t-1)$. Shift the starting point right to $t=1$. The formula is $4(t-1)$ for $t\ge 1$.
4. Time Scaling (Compression): $4r(3t-1)$.
   • To find the new starting point, set the inside to zero: $3t-1=0⟹t=1/3$.
   • The new slope becomes: Amplitude multiplier $\times$ Time compression factor $=4\times 3=12$.
   • Mathematical verification: $x(t)=4(3t-1)=12t-4$ for $t\ge 1/3$. [Graph: x-y axis. A straight diagonal line starting at exactly x = 1/3 on the t-axis. The line shoots upward very steeply with a calculated slope of 12. Value is 0 for x < 1/3]

🔥 Exact PYQ: Sketch the following signal: $x(t)=r(-0.5t+2)$ (2015) Step-by-Step Solution:

1. Rewrite inside the bracket: $r(-0.5(t-4))$. This prevents shift-scale confusion.
2. Identify Base: $r(t)$, starting at 0, slope 1.
3. Time Reversal: $r(-t)$. The ramp now goes to the left. It is 0 for $t>0$, and grows as you move negatively (slope is -1).
4. Time Scaling (Expansion): $r(-0.5t)$. The ramp expands. The slope becomes shallower: $1\times -0.5=-0.5$.
5. Time Shifting: Delay by 4 units $⟹r(-0.5(t-4))$.
   • Set inside to zero to find the boundary: $-0.5t+2=0⟹t=4$.
   • Because of the time reversal, the signal exists for $t\le 4$.
   • At $t=4$, $x(t)=0$.
   • At $t=0$, $x(t)=r(2)=2$.
   • At $t=-2$, $x(t)=r(3)=3$. [Graph: A ramp that is 0 for all t > 4. Starting from t = 4 and moving left towards negative infinity, the line rises steadily with a slope of -0.5, passing through the y-axis at amplitude 2]

🔥 Exact PYQ: Sketch the signal: $u(t-5)-u(t-7)$ (2021) Step-by-Step Solution: This is the standard formulation of a rectangular gate pulse using step functions.

1. $u(t-5)$ is a step that turns ON (+1 amplitude) at $t=5$.
2. $-u(t-7)$ is a negative step that turns OFF (-1 amplitude subtraction) at $t=7$.
3. When added together, the signal is 1 between $t=5$ and $t=7$, and 0 everywhere else. [Graph: A clean rectangular block starting at x=5, ending at x=7, with a constant height of y=1]