Signal Classification
Continuous vs. Discrete-Time Signals
Continuous-Time Signals
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Defined for all \(t \in \mathbb{R}\)
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Notation: \(x(t)\)
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Examples: \(\sin(2\pi ft)\), \(e^{-at}u(t)\)
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Analog signals
Discrete-Time Signals
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Defined only at integer values
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Notation: \(x[n]\), \(n \in \mathbb{Z}\)
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Examples: \(\cos(2\pi fn)\), \(\delta[n]\)
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Digital signals
GATE Key Point
Always distinguish between \(x(t)\) and \(x[n]\) - parentheses for continuous, brackets for discrete!
Periodic vs. Non-Periodic Signals
Periodic Signals:
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Continuous: \(x(t) = x(t + T)\) for all \(t\), where \(T\) is the fundamental period
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Discrete: \(x[n] = x[n + N]\) for all \(n\), where \(N\) is the fundamental period
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Examples: \(\sin(t)\), \(\cos(2\pi n/8)\)
Non-Periodic Signals:
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Do not repeat with any finite period
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Examples: \(e^{-t}u(t)\), \(\delta(t)\), \(t^2\)
GATE 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)
Even vs. Odd Signals
Even Signals:
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\(x(t) = x(-t)\) or \(x[n] = x[-n]\)
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Symmetric about y-axis
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Examples: \(\cos(t)\), \(t^2\), \(|t|\)
Odd Signals:
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\(x(t) = -x(-t)\) or \(x[n] = -x[-n]\)
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Anti-symmetric about origin
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Examples: \(\sin(t)\), \(t\), \(\text{sgn}(t)\)
GATE Important
Any signal can be decomposed as: \(x(t) = x_e(t) + x_o(t)\)
Energy vs. Power Signals
Energy Signals:
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Finite energy: \(0 < E < \infty\)
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Average power: \(P = 0\)
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\(E = \int_{-\infty}^{\infty} |x(t)|^2 dt\) or \(E = \sum_{n=-\infty}^{\infty} |x[n]|^2\)
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Examples: \(e^{-t}u(t)\), \(\delta(t)\), finite duration signals
Power Signals:
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Finite power: \(0 < P < \infty\)
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Infinite energy: \(E = \infty\)
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\(P = \lim_{T \to \infty} \dfrac{1}{2T} \int_{-T}^{T} |x(t)|^2 dt\)
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Examples: \(\sin(t)\), \(\cos(t)\), constants, periodic signals
GATE Rule
A signal cannot be both energy and power signal simultaneously!
Standard Signals
Unit Step and Unit Impulse
Unit Step Function:
Unit Impulse Function:
GATE Relationships
\(\delta(t) = \dfrac{du(t)}{dt}\) and \(u(t) = \int_{-\infty}^{t} \delta(\tau) d\tau\)
Sifting Property: \(\int_{-\infty}^{\infty} x(t)\delta(t-t_0) dt = x(t_0)\)
Ramp and Exponential Signals
Unit Ramp Function:
Exponential Signals:
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Real exponential: \(x(t) = Ae^{at}\)
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Complex exponential: \(x(t) = Ae^{(a+j\omega)t}\)
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If \(a < 0\): decaying, if \(a > 0\): growing, if \(a = 0\): constant amplitude
Sinusoidal Signals:
Euler’s Formula
\(e^{j\omega t} = \cos(\omega t) + j\sin(\omega t)\)
Time Shifting Properties
Time Shifting - Continuous Signals
Definition: \(y(t) = x(t - t_0)\)
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If \(t_0 > 0\): Right shift (delay)
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If \(t_0 < 0\): Left shift (advance)
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Shape remains unchanged, only position changes
Examples:
GATE Tip
For \(u(t-a)\): step occurs at \(t = a\), not \(t = -a\)!
Time Shifting - Discrete Signals
Definition: \(y[n] = x[n - n_0]\)
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If \(n_0 > 0\): Right shift (delay by \(n_0\) samples)
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If \(n_0 < 0\): Left shift (advance by \(|n_0|\) samples)
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Shifting is by integer number of samples only
Examples:
Memory in Systems
Time shifting represents memory: \(y[n] = x[n-1]\) is a unit delay element
Time Scaling Properties
Time Scaling - Continuous Signals
Definition: \(y(t) = x(at)\) where \(a \neq 0\)
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If \(|a| > 1\): Compression (signal becomes faster)
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If \(0 < |a| < 1\): Expansion (signal becomes slower)
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If \(a < 0\): Time reversal + scaling
Key Properties:
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Duration changes by factor \(\dfrac{1}{|a|}\)
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Frequency changes by factor \(|a|\)
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Area under curve remains same for \(a > 0\)
Examples:
Time Scaling - Discrete Signals
Decimation: \(y[n] = x[Mn]\) (downsampling by factor \(M\))
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Keep every \(M\)-th sample
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Reduces data rate by factor \(M\)
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Example: \(x[n] = \{1, 2, 3, 4, 5, 6\} \rightarrow x[2n] = \{1, 3, 5\}\)
Interpolation: Insert zeros between samples
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\(y[n] = x[n/L]\) for \(n = 0, \pm L, \pm 2L, ...\)
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\(y[n] = 0\) otherwise
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Increases data rate by factor \(L\)
Time Reversal: \(y[n] = x[-n]\)
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Flips signal about \(n = 0\)
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Example: \(x[n] = \{1, 2, 3\} \rightarrow x[-n] = \{3, 2, 1\}\)
Amplitude Scaling
Amplitude Scaling and Reflection
Amplitude Scaling: \(y(t) = Ax(t)\) or \(y[n] = Ax[n]\)
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If \(A > 1\): Amplification
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If \(0 < A < 1\): Attenuation
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If \(A < 0\): Scaling + reflection about time axis
DC Shift: \(y(t) = x(t) + B\)
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Shifts entire signal up (B > 0) or down (B < 0)
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Changes average value of signal
Combined Operations:
GATE Common Error
\(x(2t-4) = x(2(t-2))\) NOT \(x(2t-4) = x((2t-4))\)
Combined Transformations
Step-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 \(\to\) \(3x(2t - 4) + 1\)
GATE 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 \(a\)
Properties of Transformations
Linearity
If \(y_1(t) = T\{x_1(t)\}\) and \(y_2(t) = T\{x_2(t)\}\), then:
Time Invariance
If \(y(t) = T\{x(t)\}\), then:
Energy/Power Scaling
For \(y(t) = x(at)\):
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Energy: \(E_y = \dfrac{1}{|a|} E_x\)
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Power: \(P_y = P_x\) (for \(a \neq 0\))
Summary
Key Points Covered:
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Signal classification: CT/DT, Periodic/Aperiodic, Even/Odd, Energy/Power
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Standard signals: Step, Impulse, Ramp, Exponential, Sinusoidal
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Time shifting: Delay and advance operations
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Time scaling: Compression, expansion, and reversal
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Amplitude scaling and DC shift
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Combined transformations and their order
GATE Success Tips
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Practice sketching transformed signals
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Master the sifting property of impulse function
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Remember: Energy and Power signals are mutually exclusive
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Always check units in continuous-time problems