GATE EE

Signal Representation, Shifting, and Scaling : Quick Notes for GATE Signals and Systems Preparation

Lecture Notes

SEC 01

Signal Classification

1Continuous vs. Discrete-Time Signals

Continuous-Time Signals

Defined for all \(t \in \mathbb{R}\)
Notation: \(x(t)\)
Examples: \(\sin(2\pi ft)\), \(e^{-at}u(t)\)
Analog signals

Discrete-Time Signals

Defined only at integer values
Notation: \(x[n]\), \(n \in \mathbb{Z}\)
Examples: \(\cos(2\pi fn)\), \(\delta[n]\)
Digital signals

1GATE Key Point

Always distinguish between \(x(t)\) and \(x[n]\) - parentheses for continuous, brackets for discrete!

1Periodic vs. Non-Periodic Signals

Periodic Signals:

Continuous: \(x(t) = x(t + T)\) for all \(t\), where \(T\) is the fundamental period
Discrete: \(x[n] = x[n + N]\) for all \(n\), where \(N\) is the fundamental period
Examples: \(\sin(t)\), \(\cos(2\pi n/8)\)

Non-Periodic Signals:

Do not repeat with any finite period
Examples: \(e^{-t}u(t)\), \(\delta(t)\), \(t^2\)

1GATE Formula

For \(x(t) = \sin(\omega_0 t)\): Period \(T = \dfrac{2\pi}{\omega_0}\)

For \(x[n] = \cos(2\pi fn)\): Period \(N = \dfrac{2\pi}{2\pi f} = \dfrac{1}{f}\) (if rational)

1Even vs. Odd Signals

Even Signals:

\(x(t) = x(-t)\) or \(x[n] = x[-n]\)
Symmetric about y-axis
Examples: \(\cos(t)\), \(t^2\), \(|t|\)

Odd Signals:

\(x(t) = -x(-t)\) or \(x[n] = -x[-n]\)
Anti-symmetric about origin
Examples: \(\sin(t)\), \(t\), \(\text{sgn}(t)\)

1GATE Important

Any signal can be decomposed as: \(x(t) = x_e(t) + x_o(t)\)

\[\begin{aligned} x_e(t) &= \dfrac{x(t) + x(-t)}{2} \quad \text{(even part)} \\ x_o(t) &= \dfrac{x(t) - x(-t)}{2} \quad \text{(odd part)} \end{aligned}\]

1Energy vs. Power Signals

Energy Signals:

Finite energy: \(0 < E < \infty\)
Average power: \(P = 0\)
\(E = \int_{-\infty}^{\infty} |x(t)|^2 dt\) or \(E = \sum_{n=-\infty}^{\infty} |x[n]|^2\)
Examples: \(e^{-t}u(t)\), \(\delta(t)\), finite duration signals

Power Signals:

Finite power: \(0 < P < \infty\)
Infinite energy: \(E = \infty\)
\(P = \lim_{T \to \infty} \dfrac{1}{2T} \int_{-T}^{T} |x(t)|^2 dt\)
Examples: \(\sin(t)\), \(\cos(t)\), constants, periodic signals

1GATE Rule

A signal cannot be both energy and power signal simultaneously!

SEC 02

Standard Signals

1Unit Step and Unit Impulse

Unit Step Function:

\[u(t) = \begin{cases} 1 & \text{if } t \geq 0 \\ 0 & \text{if } t < 0 \end{cases}\]

\[u[n] = \begin{cases} 1 & \text{if } n \geq 0 \\ 0 & \text{if } n < 0 \end{cases}\]

Unit Impulse Function:

\[\delta(t) = \begin{cases} \infty & \text{if } t = 0 \\ 0 & \text{if } t \neq 0 \end{cases}\]

\(\int_{-\infty}^{\infty} \delta(t) dt = 1\)

\[\delta[n] = \begin{cases} 1 & \text{if } n = 0 \\ 0 & \text{if } n \neq 0 \end{cases}\]

1GATE Relationships

\(\delta(t) = \dfrac{du(t)}{dt}\) and \(u(t) = \int_{-\infty}^{t} \delta(\tau) d\tau\)

Sifting Property: \(\int_{-\infty}^{\infty} x(t)\delta(t-t_0) dt = x(t_0)\)

1Ramp and Exponential Signals

Unit Ramp Function:

\[r(t) = tu(t) = \begin{cases} t & \text{if } t \geq 0 \\ 0 & \text{if } t < 0 \end{cases}\]

Exponential Signals:

Real exponential: \(x(t) = Ae^{at}\)
Complex exponential: \(x(t) = Ae^{(a+j\omega)t}\)
If \(a < 0\): decaying, if \(a > 0\): growing, if \(a = 0\): constant amplitude

Sinusoidal Signals:

\[x(t) = A\cos(\omega t + \phi)\]

where \(A\) = amplitude, \(\omega\) = angular frequency, \(\phi\) = phase

1Eulerâ€™s Formula

\(e^{j\omega t} = \cos(\omega t) + j\sin(\omega t)\)

SEC 03

Time Shifting Properties

1Time Shifting - Continuous Signals

Definition: \(y(t) = x(t - t_0)\)

If \(t_0 > 0\): Right shift (delay)
If \(t_0 < 0\): Left shift (advance)
Shape remains unchanged, only position changes

Examples:

\[\begin{aligned} u(t) &\rightarrow u(t-2) \quad \text{(step delayed by 2 seconds)} \\ \delta(t) &\rightarrow \delta(t+1) \quad \text{(impulse advanced by 1 second)} \\ e^{-t}u(t) &\rightarrow e^{-(t-3)}u(t-3) \quad \text{(exponential delayed by 3)} \end{aligned}\]

1GATE Tip

For \(u(t-a)\): step occurs at \(t = a\), not \(t = -a\)!

1Time Shifting - Discrete Signals

Definition: \(y[n] = x[n - n_0]\)

If \(n_0 > 0\): Right shift (delay by \(n_0\) samples)
If \(n_0 < 0\): Left shift (advance by \(|n_0|\) samples)
Shifting is by integer number of samples only

Examples:

\[\begin{aligned} \delta[n] &\rightarrow \delta[n-2] \quad \text{(impulse at } n = 2\text{)} \\ u[n] &\rightarrow u[n+1] \quad \text{(step starts at } n = -1\text{)} \\ x[n] = \{1, 2, 3, 4\} &\rightarrow x[n-1] = \{0, 1, 2, 3, 4\} \end{aligned}\]

1Memory in Systems

Time shifting represents memory: \(y[n] = x[n-1]\) is a unit delay element

SEC 04

Time Scaling Properties

1Time Scaling - Continuous Signals

Definition: \(y(t) = x(at)\) where \(a \neq 0\)

If \(|a| > 1\): Compression (signal becomes faster)
If \(0 < |a| < 1\): Expansion (signal becomes slower)
If \(a < 0\): Time reversal + scaling

Key Properties:

Duration changes by factor \(\dfrac{1}{|a|}\)
Frequency changes by factor \(|a|\)
Area under curve remains same for \(a > 0\)

Examples:

\[\begin{aligned} x(t) = \sin(t) &\rightarrow x(2t) = \sin(2t) \quad \text{(frequency doubles)} \\ x(t) = e^{-t}u(t) &\rightarrow x(t/2) = e^{-t/2}u(t/2) \quad \text{(slower decay)} \end{aligned}\]

1Time Scaling - Discrete Signals

Decimation: \(y[n] = x[Mn]\) (downsampling by factor \(M\))

Keep every \(M\)-th sample
Reduces data rate by factor \(M\)
Example: \(x[n] = \{1, 2, 3, 4, 5, 6\} \rightarrow x[2n] = \{1, 3, 5\}\)

Interpolation: Insert zeros between samples

\(y[n] = x[n/L]\) for \(n = 0, \pm L, \pm 2L, ...\)
\(y[n] = 0\) otherwise
Increases data rate by factor \(L\)

Time Reversal: \(y[n] = x[-n]\)

Flips signal about \(n = 0\)
Example: \(x[n] = \{1, 2, 3\} \rightarrow x[-n] = \{3, 2, 1\}\)

SEC 05

Amplitude Scaling

1Amplitude Scaling and Reflection

Amplitude Scaling: \(y(t) = Ax(t)\) or \(y[n] = Ax[n]\)

If \(A > 1\): Amplification
If \(0 < A < 1\): Attenuation
If \(A < 0\): Scaling + reflection about time axis

DC Shift: \(y(t) = x(t) + B\)

Shifts entire signal up (B > 0) or down (B < 0)
Changes average value of signal

Combined Operations:

\[y(t) = Ax(at + b) + C\]

Order of operations: Time shift \(\to\) Time scaling \(\to\) Amplitude scaling \(\to\) DC shift

1GATE Common Error

\(x(2t-4) = x(2(t-2))\) NOT \(x(2t-4) = x((2t-4))\)

SEC 06

Combined Transformations

1Step-by-Step Transformation

Given: \(y(t) = 3x(2t - 4) + 1\)

Step 1: Start with \(x(t)\)

Step 2: Apply time shift \(\to\) \(x(t - 2)\) (since \(2t - 4 = 2(t - 2)\))

Step 3: Apply time scaling \(\to\) \(x(2(t - 2)) = x(2t - 4)\)

Step 4: Apply amplitude scaling \(\to\) \(3x(2t - 4)\)

Step 5: Apply DC shift \(\to\) \(3x(2t - 4) + 1\)

1GATE Shortcut

For \(x(at + b)\): First factor out \(a\) to get \(x(a(t + b/a))\)

This represents: shift by \(-b/a\), then scale by \(a\)

1Properties of Transformations

1Linearity

If \(y_1(t) = T\{x_1(t)\}\) and \(y_2(t) = T\{x_2(t)\}\), then:

\[T\{ax_1(t) + bx_2(t)\} = ay_1(t) + by_2(t)\]

1Time Invariance

If \(y(t) = T\{x(t)\}\), then:

\[T\{x(t - t_0)\} = y(t - t_0)\]

1Energy/Power Scaling

For \(y(t) = x(at)\):

Energy: \(E_y = \dfrac{1}{|a|} E_x\)
Power: \(P_y = P_x\) (for \(a \neq 0\))

1Summary

Key Points Covered:

Signal classification: CT/DT, Periodic/Aperiodic, Even/Odd, Energy/Power
Standard signals: Step, Impulse, Ramp, Exponential, Sinusoidal
Time shifting: Delay and advance operations
Time scaling: Compression, expansion, and reversal
Amplitude scaling and DC shift
Combined transformations and their order

1GATE Success Tips

Practice sketching transformed signals
Master the sifting property of impulse function
Remember: Energy and Power signals are mutually exclusive
Always check units in continuous-time problems