UNIT 5: SIGNALS AND SYSTEMS – COMPREHENSIVE SHORT NOTES

Based on exhaustive analysis of RGPV past papers (Jun 2025, Dec 2024, Jun 2024, Dec 2023, Jun 2023), these notes cover every syllabus topic implicitly tested, structured by frequency and logical sequence.

1.0 FUNDAMENTALS OF SIGNALS

1.1 Definition & Classification of Signals

Signal: A physical quantity that varies with time, space, or any other independent variable, conveying information.

1.1.1 Continuous-Time (CT) vs. Discrete-Time (DT) Signals

1.1.2 Periodic vs. Aperiodic Signals

• Periodic Signal: Repeits itself after a fixed interval T₀ (CT) or N₀ (DT).

  • CT: x(t) = x(t + T₀) for all t. Fundamental period T₀ is smallest positive T.

  • DT: x[n] = x[n + N₀] for all n. Fundamental period N₀ is smallest positive integer N.

  • Sum of Sinusoids: Periodic if the ratio of frequencies is rational.

    Example: cos(3t) + sin(2t) is periodic (ratio 3/2). cos(t) + sin(πt) is aperiodic (ratio 1/π irrational).

• Aperiodic (Non-Periodic): Does not repeat.

1.1.3 Even & Odd Signals

• Even Signal: Symmetric about the vertical axis. x(t) = x(-t) or x[n] = x[-n].

  • Example: cos(t), t².

• Odd Signal: Symmetric about the origin. x(t) = -x(-t) or x[n] = -x[-n].

  • Example: sin(t), t³.

• Decomposition: Any signal x(t) can be expressed as:

$$x(t) = x_e(t) + x_o(t)$$

where

$$x_e(t) = \frac{x(t) + x(-t)}{2} \quad \text{(Even part)}$$

$$x_o(t) = \frac{x(t) - x(-t)}{2} \quad \text{(Odd part)}$$

1.1.4 Energy vs. Power Signals

• Energy Signal: Finite total energy E, infinite average power P = 0.

$$E = \int_{-\infty}^{\infty} |x(t)|^2 dt \quad \text{(CT)} \quad \text{or} \quad E = \sum_{n=-\infty}^{\infty} |x[n]|^2 \quad \text{(DT)}$$

*   Condition: `0 < E < ∞`.

*   Example: Finite-duration pulses, `x(t) = e^{-t}u(t)`.

• Power Signal: Finite average power P, infinite total energy E = ∞.

$$P = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} |x(t)|^2 dt \quad \text{(CT)} \quad \text{or} \quad P = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^{N} |x[n]|^2 \quad \text{(DT)}$$

*   Condition: `0 < P < ∞`.

*   Example: Periodic signals like `sin(t)`, constant signals.

• Neither: Signals that are neither energy nor power (e.g., x(t)=t u(t) has infinite energy and infinite power).

1.1.5 Causal, Anti-Causal, and Non-Causal Signals

• Causal Signal: Zero for all negative time. x(t) = 0 for t < 0 or x[n] = 0 for n < 0.

• Anti-Causal Signal: Zero for all positive time. x(t) = 0 for t > 0 or x[n] = 0 for n > 0.

• Non-Causal Signal: Non-zero for both t < 0 and t > 0 (or n < 0 and n > 0).

1.2 Basic Signal Operations

1.2.1 Time Operations

• Shifting (Delay/Advance):

  • x(t - t₀) is x(t) delayed by t₀ (right shift).

  • x(t + t₀) is x(t) advanced by t₀ (left shift).

  • For DT: x[n - n₀] (delay), x[n + n₀] (advance).

• Scaling (Compression/Expansion):

  • x(at) with a > 1 → compression by factor a.

  • x(at) with 0 < a < 1 → expansion by factor 1/a.

  • For DT: x[kn] (compression if k>1), but note x[n/k] is not generally defined for integer n unless k is integer divisor.

• Folding (Time Reversal): x(-t) or x[-n]. Reflects signal about the vertical axis.

1.2.2 Amplitude Operations

• Addition: x₁(t) + x₂(t)

• Multiplication: x₁(t) · x₂(t)

• Differentiation (CT): dx(t)/dt

• Integration (CT): ∫ x(τ) dτ from -∞ to t.

• Accumulation (DT): y[n] = Σ_{k=-∞}^{n} x[k].

1.3 Standard/Elementary Signals

1.3.1 Unit Impulse & Unit Step

1.3.2 Exponential Signals

• Real Exponential: x(t) = e^{at}u(t) (CT), x[n] = a^n u[n] (DT). a real.

• Complex Exponential: x(t) = e^{jωt} (CT), x[n] = e^{jωn} (DT). Fundamental to Fourier analysis.

1.3.3 Sinusoidal Signals

• x(t) = A sin(ωt + φ) or A cos(ωt + φ).

• Euler's Relation: e^{jθ} = cosθ + j sinθ.

  Therefore:

$$cos(ωt) = \frac{e^{jωt} + e^{-jωt}}{2}, \quad sin(ωt) = \frac{e^{jωt} - e^{-jωt}}{2j}$$

2.0 FUNDAMENTALS OF SYSTEMS

2.1 System Properties & Classification

A system is a transformation that maps an input signal x(t)/x[n] to an output signal y(t)/y[n].

2.1.1 Linearity

• Principle of Superposition: A system is linear if it satisfies:

  1. Additivity: T{x₁(t) + x₂(t)} = T{x₁(t)} + T{x₂(t)}

  2. Homogeneity (Scaling): T{a x(t)} = a T{x(t)}

• Combined: T{a₁x₁(t) + a₂x₂(t)} = a₁T{x₁(t)} + a₂T{x₂(t)}.

• Check: Substitute general inputs; if output is linear combination of inputs, system is linear.

2.1.2 Time-Invariance (TI)

• Definition: A system is time-invariant if a time-shift in input results in an identical time-shift in output.

  • CT: If y(t) = T{x(t)}, then T{x(t - t₀)} = y(t - t₀).

  • DT: If y[n] = T{x[n]}, then T{x[n - n₀]} = y[n - n₀].

• Test: Replace t with (t - t₀) or n with (n - n₀) in the system equation. If the resulting equation is y(t - t₀) or y[n - n₀], system is TI.

2.1.3 Causality

• Definition: Output at any time depends only on present and past values of the input.

  • CT: y(t₀) depends on x(τ) for τ ≤ t₀.

  • DT: y[n₀] depends on x[k] for k ≤ n₀.

• Check: Inspect system equation. If output depends on future input (e.g., x(t+1)), system is non-causal.

2.1.4 Stability (BIBO)

• Definition (Bounded-Input Bounded-Output): A system is stable if every bounded input produces a bounded output.

  • Bounded Input: |x(t)| ≤ M_x < ∞ for all t.

  • Bounded Output: |y(t)| ≤ M_y < ∞ for all t.

• Test for LTI Systems: A continuous-time LTI system is BIBO stable if and only if its impulse response h(t) is absolutely integrable:

$$\int_{-\infty}^{\infty} |h(t)| dt < \infty$$

A discrete-time LTI system is BIBO stable if and only if its impulse response `h[n]` is absolutely summable:

$$\sum_{n=-\infty}^{\infty} |h[n]| < \infty$$

2.1.5 Memoryless vs. Dynamic Systems

• Memoryless System: Output at any time depends only on the input at that same time. y(t) = f(x(t)).

• Dynamic System: Has memory; output depends on past/future inputs or past outputs (e.g., systems with integration, differentiation, or recursion).

2.1.6 Invertible Systems

• A system is invertible if its output uniquely determines its input. There exists an inverse system T⁻¹ such that T⁻¹{T{x(t)}} = x(t).

• Condition for LTI: The impulse response h(t) must be absolutely integrable and its Fourier transform H(jω) must have no zeros (or be non-zero over the frequency band of interest).

2.2 Linear Time-Invariant (LTI) Systems

2.2.1 Defining Characteristic: Convolution

• The output of an LTI system is the convolution of input x(t)/x[n] with impulse response h(t)/h[n].

  • CT: y(t) = x(t) * h(t) = ∫_{-∞}^{∞} x(τ) h(t - τ) dτ

  • DT: y[n] = x[n] * h[n] = Σ_{k=-∞}^{∞} x[k] h[n - k]

2.2.2 Key Responses & Inter-relationships

2.2.3 Properties of Convolution

1. Commutative: x * h = h * x

2. Associative: (x * h₁) * h₂ = x * (h₁ * h₂)

3. Distributive: x * (h₁ + h₂) = x * h₁ + x * h₂

4. With LTI Systems: Cascade connection → multiplication of transfer functions: h₁(t) * h₂(t) = h₂(t) * h₁(t).

3.0 FOURIER ANALYSIS

3.1 Fourier Series (FS) – For Periodic Signals

• Trigonometric Form:

$$x(t) = a₀ + Σ_{n=1}^{∞} [a_n cos(nω₀t) + b_n sin(nω₀t)]$$

where `ω₀ = 2π/T₀`.

• Exponential Form:

$$x(t) = Σ_{n=-∞}^{∞} C_n e^{jnω₀t}$$

Coefficients: `C_n = (1/T₀) ∫_{T₀} x(t) e^{-jnω₀t} dt`.

• Dirichlet Conditions (Sufficient for FS existence):

  1. x(t) is single-valued.

  2. x(t) has a finite number of discontinuities in one period.

  3. x(t) has a finite number of maxima/minima in one period.

  4. x(t) is absolutely integrable over one period: ∫_{T₀} |x(t)| dt < ∞.

• Limitations:

  • Only for periodic signals (or aperiodic signals over a finite interval).

  • Convergence at discontinuities: Gibbs phenomenon (~9% overshoot).

  • Does not converge for signals not satisfying Dirichlet conditions.

3.2 Fourier Transform (FT) – For Aperiodic Signals

3.2.1 Continuous-Time FT (CTFT)

• Forward Transform:

$$X(jω) = \mathcal{F}\{x(t)\} = \int_{-\infty}^{\infty} x(t) e^{-jωt} dt$$

• Inverse Transform:

$$x(t) = \mathcal{F}^{-1}\{X(jω)\} = \frac{1}{2π} \int_{-\infty}^{\infty} X(jω) e^{jωt} dω$$

• Notation: X(jω) or X(ω).

3.2.2 Discrete-Time FT (DTFT)

• Forward Transform:

$$X(e^{jω}) = \mathcal{F}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] e^{-jωn}$$

• Inverse Transform:

$$x[n] = \frac{1}{2π} \int_{2π} X(e^{jω}) e^{jωn} dω$$

• Key Property: X(e^{jω}) is periodic with period 2π.

3.3 Properties of Fourier Transform (High Frequency)

Exam Tip: Duality Property – If x(t) ↔ X(jω), then X(jt) ↔ 2π x(-ω). It swaps time and frequency domains with a sign change and scaling.

3.4 Comparison: Fourier Series vs. Fourier Transform

4.0 LAPLACE TRANSFORM

4.1 Definition & Region of Convergence (ROC)

• Bilateral Laplace Transform:

$$X(s) = \mathcal{L}\{x(t)\} = \int_{-\infty}^{\infty} x(t) e^{-st} dt, \quad s = σ + jω$$

• Unilateral Laplace Transform (for causal signals, t≥0):

$$X(s) = \int_{0}^{\infty} x(t) e^{-st} dt$$

• Region of Convergence (ROC): Set of s values for which the integral converges (finite).

  • Properties of ROC:

    1. ROC is a vertical strip in the s-plane: σ₁ < σ < σ₂.

    2. ROC cannot contain any poles.

    3. For causal signals (x(t)=0 for t<0), ROC is a right-half plane: Re(s) > σ₀ (to the right of the rightmost pole).

    4. For anti-causal signals (x(t)=0 for t>0), ROC is a left-half plane: Re(s) < σ₀ (to the left of the leftmost pole).

    5. For two-sided signals, ROC is a vertical strip between two poles.

  • Determination: From X(s), identify poles. ROC is bounded by poles and extends to ±∞ depending on causality.

4.2 Properties of Laplace Transform

• Initial Value Theorem (IVT): If x(t) and dx/dt are Laplace transformable, and sX(s) has no poles in Re(s) ≥ 0, then:

$$x(0⁺) = \lim_{s \to \infty} s X(s)$$

• Final Value Theorem (FVT): If sX(s) has no poles in Re(s) ≥ 0 except possibly a simple pole at s=0, then:

$$\lim_{t \to \infty} x(t) = \lim_{s \to 0} s X(s)$$

4.3 Laplace Transform of Standard Signals

4.4 Applications in CT LTI Systems

• System Analysis: For an LTI system described by a differential equation, take Laplace transform (assuming zero initial conditions for zero-state response).

  • Y(s) = H(s) X(s), where H(s) is the transfer function (Laplace transform of impulse response h(t)).

• Stability: For a causal LTI system, ROC of H(s) must include the imaginary axis (jω-axis) for BIBO stability.

5.0 Z-TRANSFORM

5.1 Definition & Region of Convergence (ROC)

• Bilateral Z-Transform:

$$X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] z^{-n}, \quad z = re^{jω}$$

• Unilateral Z-Transform (for causal signals, n≥0):

$$X(z) = \sum_{n=0}^{\infty} x[n] z^{-n}$$

• Region of Convergence (ROC): Set of z values for which the sum converges (finite).

  • Properties of ROC:

    1. ROC is a ring/annulus in the z-plane: r₁ < |z| < r₂.

    2. ROC cannot contain any poles.

    3. For causal signals (x[n]=0 for n<0), ROC is exterior of outermost pole: |z| > r_max.

    4. For anti-causal signals (x[n]=0 for n>0), ROC is interior of innermost pole: |z| < r_min.

    5. For two-sided signals, ROC is a ring between two poles.

  • Determination: From X(z), identify poles. ROC is bounded by poles and extends inward/outward to 0 or ∞ depending on causality.

5.2 Properties of Z-Transform

5.3 Inverse Z-Transform Methods (Frequent Short Note)

1. Power Series Expansion (Long Division): Expand X(z) as Σ x[k] z^{-k}. Coefficients are x[k]. ROC determines convergence.

2. Partial-Fraction Expansion (PFE): Expand X(z)/z into simpler fractions. Inverse of each term is standard (e.g., 1/(1 - az⁻¹) ↔ aⁿ u[n] for |z| > |a|).

3. Contour Integration (Residue Method):

$$x[n] = \frac{1}{2πj} \oint_{C} X(z) z^{n-1} dz$$

where `C` is a counterclockwise contour in ROC encircling the origin. `x[n]` is sum of residues of `X(z) z^{n-1}` at poles inside `C`.

5.4 Applications in DT LTI Systems

• Transfer Function: H(z) = Y(z)/X(z) (assuming zero initial conditions).

• Impulse Response: h[n] = ℤ⁻¹{H(z)}. Must consider ROC specified by causality/stability.

• Stability Criterion: A causal DT LTI system is BIBO stable if and only if the ROC of H(z) includes the unit circle |z| = 1.

6.0 ANALYSIS OF LTI SYSTEMS

6.1 Time-Domain Analysis

6.1.1 Convolution Sum/Integral

• CT Convolution Integral:

$$y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(τ) h(t - τ) dτ$$

• DT Convolution Sum:

$$y[n] = x[n] * h[n] = \sum_{k=-\infty}^{\infty} x[k] h[n - k]$$

• Graphical Method: Flip h(τ) or h[k], shift by t or n, multiply with x(τ) or x[k], integrate/sum.

6.1.2 Step Response from Impulse Response

• CT: s(t) = ∫_{-∞}^{t} h(τ) dτ = u(t) * h(t)

• DT: s[n] = Σ_{k=-∞}^{n} h[k] = u[n] * h[n]

6.2 Frequency-Domain Analysis

• Frequency Response: H(jω) (CT) or H(e^{jω}) (DT) is the Fourier Transform of h(t)/h[n].

  • System response to sinusoidal input e^{jωt} is H(jω) e^{jωt} (CT) or H(e^{jω}) e^{jωn} (DT).

• Transfer Function: H(s) (CT) or H(z) (DT) is the Laplace/Z-Transform of h(t)/h[n].

  • More general, includes initial conditions via s or z.

6.3 System Representations

6.3.1 Differential & Difference Equations

• CT LTI System: Linear constant-coefficient differential equation.

$$\sum_{k=0}^{N} a_k \frac{d^k y(t)}{dt^k} = \sum_{k=0}^{M} b_k \frac{d^k x(t)}{dt^k}$$

• DT LTI System: Linear constant-coefficient difference equation.

$$\sum_{k=0}^{N} a_k y[n-k] = \sum_{k=0}^{M} b_k x[n-k]$$

• Solutions:

  • Zero-Input Response: Response due to initial conditions with x[n]=0.

  • Zero-State Response: Response due to input with zero initial conditions.

  • Total Response = ZIR + ZSR.

6.3.2 Block Diagram Representations (Frequent)

• Direct Form I: Realizes difference/differential equation directly with adders, multipliers, and delays (DT) or integrators (CT).

• Direct Form II (Canonical): Minimizes number of delay elements (DT) or integrators (CT) by sharing.

• Cascade (Series) Form: H(z) = H₁(z) H₂(z). Output of first is input to second.

• Parallel Form: H(z) = H₁(z) + H₂(z) + .... Input fed to all subsystems, outputs summed.

• Feedback Form:

  • Negative feedback: Y(z) = X(z) / (1 + H(z)) (for unity feedback).

  • Positive feedback: Y(z) = X(z) / (1 - H(z)).

6.3.3 Interconnections of LTI Systems

• Series/Cascade: Overall impulse response h[n] = h₁[n] * h₂[n], transfer function H(z) = H₁(z) H₂(z).

• Parallel: h[n] = h₁[n] + h₂[n], H(z) = H₁(z) + H₂(z).

• Feedback: As above.

6.4 First-Order Discrete-Time LTI Systems

• System Equation: y[n] - a y[n-1] = b x[n] (causal).

• Transfer Function: H(z) = b / (1 - a z⁻¹).

• Impulse Response: h[n] = b aⁿ u[n].

• Step Response: s[n] = b (1 + a + a² + ... + aⁿ) u[n] = b (1 - a^{n+1})/(1 - a) u[n] for a ≠ 1.

7.0 SAMPLING & RECONSTRUCTION

7.1 Sampling Theorem (Nyquist-Shannon)

• Ideal Sampling: Multiply continuous-time signal x_c(t) by an impulse train:

$$x_s(t) = x_c(t) \sum_{k=-\infty}^{\infty} δ(t - kT_s) = \sum_{k=-\infty}^{\infty} x_c(kT_s) δ(t - kT_s)$$

where `T_s` is sampling period, `ω_s = 2π/T_s` is sampling frequency.

• Sampling Theorem: A bandlimited signal with maximum frequency ω_m can be perfectly reconstructed from its samples if:

$$ω_s > 2ω_m \quad \text{or} \quad f_s > 2f_m$$

(`2ω_m` is the **Nyquist rate**).

• Condition: X_c(jω) = 0 for |ω| > ω_m.

7.2 Aliasing

• Definition: Overlapping of the shifted spectra X_c(j(ω - kω_s)) when ω_s ≤ 2ω_m. Causes frequency components to become indistinguishable.

• Cause: Undersampling (ω_s < 2ω_m).

• Effect: High-frequency components appear as low-frequency components in the sampled signal spectrum. Irreversible distortion.

• Anti-aliasing Filter: A low-pass filter applied before sampling to bandlimit x_c(t) to ω_m < ω_s/2.

7.3 Signal Reconstruction

• Ideal Reconstruction (Sinc Interpolation - Whittaker-Shannon):

$$x_c(t) = \sum_{k=-\infty}^{\infty} x_c(kT_s) \text{sinc}\left(\frac{t - kT_s}{T_s}\right)$$

where `sinc(t) = sin(πt)/(πt)`.

*   Equivalent to passing `x_s(t)` through an **ideal low-pass filter** with cutoff `ω_c = ω_s/2` (or `ω_m`).

• Practical Reconstruction: Use zero-order hold (ZOH) or first-order hold. ZOH holds each sample value constant for T_s duration, introducing a sin(x)/x distortion.

8.0 FILTERS

8.1 Analog Filters (Frequent Short Note)

• Butterworth Filter:

  • Magnitude Response: Maximally flat in passband (no ripple).

  • Roll-off: -20n dB/decade (n = filter order).

  • Applications: Audio, data acquisition where passband ripple is unacceptable.

• Chebyshev Filter:

  • Type I: Equiripple in passband, monotonic stopband.

  • Type II (Inverse Chebyshev): Monotonic passband, equiripple in stopband.

  • Roll-off: Steeper than Butterworth for same order.

  • Applications: Where sharper cutoff needed, can tolerate passband ripple.

• Elliptic Filter (Cauer):

  • Magnitude Response: Equiripple in both passband and stopband.

  • Roll-off: Steepest for a given order.

  • Applications: Where very sharp transition and stopband attenuation are critical, and ripple is acceptable.

8.2 Digital Filters (Very Frequent Short Note)

8.3 Ideal Frequency Selective Filters

• Ideal LPF: |H(jω)| = 1 for |ω| ≤ ω_c, 0 for |ω| > ω_c.

• Ideal HPF: |H(jω)| = 0 for |ω| ≤ ω_c, 1 for |ω| > ω_c.

• Ideal BPF: |H(jω)| = 1 for ω_{c1} ≤ |ω| ≤ ω_{c2}, 0 elsewhere.

• Ideal BSF: |H(jω)| = 0 for ω_{c1} ≤ |ω| ≤ ω_{c2}, 1 elsewhere.

• Characteristics: Brick-wall magnitude, linear phase (not realizable physically due to non-causal infinite impulse response).

8.4 Non-Ideal Frequency Selective Filters

• Transition Band: Frequency range between passband edge and stopband edge where magnitude rolls off.

• Passband Ripple: Allowed variation in passband magnitude (δ₁).

• Stopband Attenuation: Minimum attenuation in stopband (δ₂).

• Practical Design: Trade-off between order (complexity), transition width, passband ripple, and stopband attenuation.

9.0 STATE-SPACE REPRESENTATION

9.1 State Variables & State Equations

• State: Minimum set of variables x(t) (CT) or x[n] (DT) that completely determines future system output for given input.

• State Vector: x(t) = [x₁(t), x₂(t), ..., x_N(t)]^T (N = system order).

9.2 Matrix Representation

• CT Systems:

$$\dot{x}(t) = A x(t) + B u(t)$$

$$y(t) = C x(t) + D u(t)$$

• DT Systems:

$$x[n+1] = A x[n] + B u[n]$$

$$y[n] = C x[n] + D u[n]$$

• Matrix Meanings:

  • A: System matrix (determines internal dynamics).

  • B: Input matrix (how input affects state).

  • C: Output matrix (how state determines output).

  • D: Feedthrough matrix (direct input-output path).

9.3 Relationship with Transfer Function

• For CT: H(s) = C (sI - A)^{-1} B + D

• For DT: H(z) = C (zI - A)^{-1} B + D

• Advantages of State-Space:

  • Handles MIMO (Multi-Input Multi-Output) systems naturally.

  • Easily incorporates initial conditions (x(0) or x[0]).

  • Provides insight into internal stability (eigenvalues of A).

  • Suitable for non-linear systems (linearization around operating point).

10.0 ADVANCED TRANSFORMS & COMPARISONS

10.1 Wavelet Transform (Frequent Short Note)

• Basic Idea: Time-frequency analysis using wavelets—localized, windowed sinusoids with variable width.

  • Mother Wavelet: ψ(t) with zero average and finite energy.

  • Wavelet Transform: W(a,b) = ∫ x(t) ψ_{a,b}^*(t) dt, where ψ_{a,b}(t) = (1/√a) ψ((t-b)/a).

  • a = scale (inverse frequency), b = translation (time).

• Multi-Resolution Analysis: High frequencies → short time windows (good time resolution, poor frequency). Low frequencies → long time windows (good frequency resolution, poor time).

• Comparison:

  • vs. Fourier Transform: FT has fixed resolution (Δf·Δt = constant). WT has variable resolution (Δf·Δt varies with scale). Better for non-stationary signals (transients, edges).

  • vs. Laplace Transform: LT is for CT systems in s-domain (stability/causality via ROC). WT is for time-frequency analysis of signals.

• Applications: Image compression (JPEG2000), edge detection, transient analysis, biomedical signal processing (ECG, EEG).

10.2 Short-Time Fourier Transform (STFT)

• Principle: Apply a window w(t) to x(t) and then take FT:

$$STFT\{x(t)\}(τ,ω) = ∫ x(t) w(t-τ) e^{-jωt} dt$$

• Time-Frequency Resolution: Fixed by window length. Long window → good frequency, poor time. Short window → good time, poor frequency. Trade-off inherent.

10.3 Comparative Summary of Transforms

11.0 SPECIALIZED SHORT NOTE TOPICS (From Exams)

11.1 Duality Property of Fourier Transform

• Statement: If x(t) ↔ X(jω), then X(jt) ↔ 2π x(-ω).

• Implication: The Fourier transform pair can be interpreted in two ways: time ↔ frequency, or frequency ↔ time (with sign change and scaling).

• Example: rect(t/T) ↔ T sinc(ωT/2π) duality gives T sinc(t/T) ↔ 2π rect(-ωT/2π) = 2π rect(ωT/2π).

11.2 Convolution Theorem for Laplace & Fourier Transforms

• Statement: Convolution in time domain ↔ Multiplication in transform domain.

  • Laplace: x(t) * h(t) ↔ X(s) H(s)

  • Fourier: x(t) * h(t) ↔ X(jω) H(jω)

• Proof (Laplace):

$$\mathcal{L}\{x(t) * h(t)\} = \int_{-\infty}^{\infty} \left( \int_{-\infty}^{\infty} x(τ) h(t-τ) dτ \right) e^{-st} dt$$

Change order of integration, substitute `u = t-τ`, use Fubini's theorem → `X(s) H(s)`.

11.3 Region of Convergence (ROC) – Detailed (Laplace & Z)

• Definition: Set of s (or z) values where transform integral/sum converges.

• Key Properties:

  • ROC is connected (no gaps).

  • ROC cannot contain poles.

  • Causality → ROC is right-half plane (Laplace) or exterior of outermost pole (Z).

  • Anti-causality → ROC is left-half plane (Laplace) or interior of innermost pole (Z).

  • Stability (BIBO): For CT LTI, ROC must include jω-axis. For DT LTI, ROC must include unit circle |z|=1.

• Determination: From X(s) or X(z), factor denominator. ROC is bounded by poles and extends to infinity based on signal duration.

11.4 Inverse Z-Transform Methods

1. Power Series: Long division of X(z) by z (or polynomial division). Coefficients of z^{-n} give x[n]. Simple but limited to finite terms.

2. Partial-Fraction Expansion (PFE):

   • Write X(z)/z as sum of terms like A/(1 - a z⁻¹).

   • Inverse of each term: A aⁿ u[n] (if ROC |z| > |a|) or -A aⁿ u[-n-1] (if ROC |z| < |a|).

   • Most common method for rational X(z).

3. Residue (Contour Integration):

$$x[n] = \sum \text{Residues of } X(z) z^{n-1} \text{ at poles inside contour } C.$$

Use when ROC is annular (two-sided signals).

11.5 Properties of Convolution

1. Commutative: x * h = h * x.

2. Associative: (x * h₁) * h₂ = x * (h₁ * h₂).

3. Distributive: x * (h₁ + h₂) = x * h₁ + x * h₂.

4. With LTI Systems: Impulse response completely characterizes system. Output y = x * h.

5. Duration: For finite-duration signals, Length(y) = Length(x) + Length(h) - 1.

6. Even/Odd: If x and h are both even/odd, result is even. If one even, one odd, result is odd.

11.6 Relationship between Impulse & Unit Step Functions

• CT:

$$δ(t) = \frac{d}{dt} u(t)$$

$$u(t) = \int_{-\infty}^{t} δ(τ) dτ$$

• DT:

$$δ[n] = u[n] - u[n-1]$$

$$u[n] = \sum_{k=-\infty}^{n} δ[k]$$

• Significance: Step is integral of impulse; impulse is difference (DT) or derivative (CT) of step.

11.7 Exponential Signals & their Role in LTI System Response

• Eigenfunctions: Complex exponentials e^{st} (CT) or aⁿ (DT) are eigenfunctions of LTI systems.

  • If input x(t) = e^{st}, output y(t) = H(s) e^{st} (CT), where H(s) is transfer function evaluated at s.

  • Similarly, x[n] = zⁿ → y[n] = H(z) zⁿ.

• Natural & Forced Response: For linear constant-coefficient DEs/DEs, homogeneous solution (natural) often involves exponentials e^{λt} where λ are roots of characteristic equation. Particular solution (forced) depends on input form (e.g., exponential, sinusoidal).

11.8 Non-Ideal Frequency Selective Filters

• Characteristics:

  • Transition Band: Finite width between passband and stopband.

  • Passband Ripple (δ₁): Allowed variation in passband magnitude.

  • Stopband Attenuation (δ₂): Minimum suppression in stopband.

  • Roll-off Rate: Determined by filter order (e.g., -20n dB/decade for Butterworth).

• Design Trade-offs: Higher order → steeper roll-off, but more complexity, sensitivity, and potential instability (IIR).

11.9 Invertible & Inverse Systems

• Invertible System: Exists an inverse system T⁻¹ such that T⁻¹{T{x}} = x for all valid x.

• Condition for LTI: Transfer function H(s) or H(z) must be non-zero over the frequency range of interest. Inverse system has transfer function 1/H(s) or 1/H(z).

• Stability Consideration: Even if H(s) is stable (ROC includes jω-axis), 1/H(s) may be unstable. For invertibility, both system and inverse must be stable.

• Example: System with H(z) = 1 - 0.5 z⁻¹ is invertible (1/H(z) = 1/(1 - 0.5 z⁻¹)). System with H(z) = 1 + z⁻¹ has inverse 1/(1 + z⁻¹) which is stable but not causal.

Final Exam Strategy:

1. Definitions First: Always start with crisp definitions (e.g., Signal, System, LTI, ROC).

2. Properties are Key: Memorize Fourier/Laplace/Z-transform properties—they are frequently asked for statement or proof.

3. ROC Determination: Practice determining ROC from X(s)/X(z) for causal, anti-causal, and two-sided signals.

4. Convolution: Be fluent in both integral/sum forms and graphical method.

5. Block Diagrams: Practice converting H(s)/H(z) to Direct Form I/II, Cascade, Parallel.

6. Filter Types: Know Butterworth, Chebyshev, Elliptic characteristics and FIR vs. IIR differences.

7. Short Notes: Prioritize: ROC (Laplace & Z), Inverse Z-Transform, Wavelet Transform, Digital Filters, Duality Property, Convolution Theorem.

\boxed{\text{These notes synthesize all high-frequency topics from RGPV past papers. Master definitions, properties, and ROC analysis for top grades.}}