Signals and Systems (EX-302) - Unit 2 Short Notes

UNIT 2: Signals and Systems - High-Impact Study Notes

I. FUNDAMENTALS OF SIGNALS

1.1 Definition & Classification

A signal is a physical quantity that varies with time, space, or any other independent variable and conveys information.

Periodic vs. Aperiodic Signals

• CT Periodic: x(t) = x(t + T₀) for some smallest T₀ > 0. T₀ is the fundamental period.

  • Check: For x(t) = sin(ω₀t + θ), T₀ = 2π/ω₀.

  • Example: sin(πt/4)u(t) is aperiodic because u(t) makes it zero for t<0, breaking periodicity.

• DT Periodic: x[n] = x[n + N] for some smallest integer N.

  • Check: ω₀ must be a rational multiple of 2π: ω₀ = 2π(m/N), m,N integers.

  • Example: cos(15πt) + sin(3t) → ω₁=15π, ω₂=3. Ratio ω₁/ω₂ = 5 (rational). T₀ = 2π / gcd(15π, 3) = 2π/3. Periodic.

Energy vs. Power Signals

• Energy Signal: Total energy E is finite, average power P = 0.

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

*   **Examples:** `t²u(t-1)`, finite-duration pulses.

• Power Signal: Average power P is finite and non-zero, total energy E = ∞.

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

*   **Examples:** Periodic signals (sinusoids), `u(t)`.

• Neither: Signals with both E = ∞ and P = ∞ (e.g., x(t)=t).

[!TIP] Exam Hack: For x(t)=2u(t)-u(t-3) (a rectangular pulse of height 2, width 3), it's finite duration → Energy Signal.

Even & Odd Signals & Decomposition

• Even: x(t) = x(-t) (symmetric about y-axis). cos(ωt) is even.

• Odd: x(t) = -x(-t) (symmetric about origin). sin(ωt) is odd.

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

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

where:

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

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

Causal, Anti-Causal, Non-Causal

• Causal: x(t) = 0 for t < 0 (or x[n] = 0 for n < 0). Depends only on present/past.

• Anti-Causal: x(t) = 0 for t > 0. Depends only on future.

• Non-Causal: Non-zero for both t>0 and t<0.

1.2 Basic Signal Operations

• Time-domain:

  • Shifting: x(t - t₀) → delay by t₀. x(t + t₀) → advance.

  • Scaling: x(at) → compression if |a|>1, expansion if |a|<1.

  • Reversal: x(-t) (folding about origin).

• Amplitude-domain:

  • Addition, Multiplication, Differentiation (dx/dt for CT), Integration (∫x(τ)dτ from -∞ to t for CT), Accumulation (sum for DT).

1.3 Elementary Signals

Relationship (CT): δ(t) = d u(t)/dt. Conversely, u(t) = ∫_{-∞}^{t} δ(τ) dτ. Relationship (DT): δ[n] = u[n] - u[n-1].

II. FUNDAMENTALS OF SYSTEMS

2.1 System Classification & Properties

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

2.2 LTI System Analysis: Time Domain

Convolution is the fundamental operation for LTI systems.

• CT Linear Convolution:

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

• DT Linear Convolution:

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

• Properties: Commutative (x*h = h*x), Associative ((x*h)*g = x*(h*g)), Distributive.

Impulse Response h(t) / h[n]

• Definition: Output when input is unit impulse δ(t) (CT) or δ[n] (DT).

• Significance: Completely characterizes an LTI system. For any input x(t), y(t) = x(t) * h(t).

Step Response s(t)

• Definition: Output when input is unit step u(t).

• Relation to h(t) (CT): s(t) = ∫_{-∞}^{t} h(τ) dτ → h(t) = d s(t)/dt.

• Relation to h[n] (DT): s[n] = ∑_{k=-∞}^{n} h[k] → h[n] = s[n] - s[n-1].

Frequency Response H(jω) / H(e^{jω})

• Definition: Fourier Transform of impulse response.

  • CT: H(jω) = ∫_{-∞}^{∞} h(t) e^{-jωt} dt

  • DT: H(e^{jω}) = ∑_{n=-∞}^{∞} h[n] e^{-jωn}

• Interpretation: System's response to complex sinusoid e^{jωt}. |H(jω)| = magnitude response, ∠H(jω) = phase response.

System Description via Differential/Difference Equations

• CT LTI (LCCDE):

$$\frac{d^N y(t)}{dt^N} + a_{N-1}\frac{d^{N-1}y}{dt^{N-1}} + ... + a_0 y(t) = b_M\frac{d^M x}{dt^M} + ... + b_0 x(t)$$

*   **Solution:** Homogeneous (natural response) + Particular (forced response).

*   **Impulse Response:** Solve for `h(t)` with `x(t)=δ(t)` and zero initial conditions (IC).

• DT LTI (LCCDE):

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

*   **First-Order DT LTI:** `y[n] + a y[n-1] = b x[n]`.

    *   Impulse Response: `h[n] = b(-a)^n u[n]` (for causal system).

    *   Step Response: `s[n] = b(1 - (-a)^n)/(1+a) u[n]` (if `a ≠ -1`).

III. FOURIER SERIES & FOURIER TRANSFORM

3.1 Fourier Series (FS) Analysis

Represents a periodic signal x(t) with period T₀ as sum of sinusoids.

• Trigonometric Form:

$$x(t) = a_0 + \sum_{k=1}^{\infty} [a_k \cos(kω₀ t) + b_k \sin(kω₀ t)]$$

where `ω₀ = 2π/T₀`.

• Exponential Form (Compact):

$$x(t) = \sum_{k=-\infty}^{\infty} c_k e^{jkω₀ t}$$

where `c_k = (1/T₀) ∫_{T₀} x(t) e^{-jkω₀ t} dt`.

Dirichlet Conditions (Sufficient for FS existence):

1. x(t) is single-valued, finite, and has finite number of maxima/minima in T₀.

2. x(t) has a finite number of discontinuities in T₀.

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

Limitation of FS: Only for periodic signals. For aperiodic signals, use Fourier Transform (FT).

3.2 Fourier Transform (FT)

Generalization of FS for aperiodic signals. Represents x(t) as continuous spectrum X(jω).

• CT FT Pair:

$$X(jω) = \int_{-\infty}^{\infty} x(t) e^{-jωt} dt \quad \text{(Analysis)}$$

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

• DT FT Pair (DTFT): X(e^{jω}) is periodic with period 2π.

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

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

Key Properties of FT (CT & DT):

Standard FT Pairs (Must Memorize):

[!TIP] Exam Tip: Duality is often asked. If x(t) ↔ X(jω), then X(t) ↔ 2π x(-jω). Swap time/frequency domains, conjugate frequency, scale by 2π.

IV. LAPLACE TRANSFORM

4.1 Definition & ROC

Bilateral (Two-Sided) Laplace Transform:

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

Unilateral (One-Sided) Laplace Transform:

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

(Used for causal signals with zero IC, or solving differential equations).

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 poles.

  3. ROC is right-sided for causal signals (e.g., e^{-at}u(t) → ROC: σ > -a).

  4. ROC is left-sided for anti-causal signals.

  5. For finite-duration signals, ROC is entire s-plane (except possibly s=∞ or s=0).

• Significance: ROC determines causality and stability.

  • Causality: For rational X(s), system is causal iff ROC is to the right of the rightmost pole.

  • Stability (BIBO): Requires σ = 0 (jω-axis) to be in ROC. For rational X(s), stable iff ROC includes the jω-axis.

Example Transforms:

• x(t) = e^{-at}u(t) → X(s) = 1/(s + a), ROC: σ > -a (causal).

• x(t) = -e^{-at}u(-t) → X(s) = 1/(s + a), ROC: σ < -a (anti-causal).

• x(t) = u(t) → X(s) = 1/s, ROC: σ > 0.

• x(t) = δ(t) → X(s) = 1, ROC: entire s-plane.

4.2 Properties of Laplace Transform

4.3 Inverse Laplace Transform

• Method 1: Partial Fraction Expansion (PFE) + Transform Table.

  • For rational X(s) = N(s)/D(s), expand into simpler terms (e.g., A/(s-p)).

  • Use known pairs: 1/(s-a) ↔ e^{at}u(t) (if ROC right of a).

• Method 2: Use Properties & Pairs (e.g., s-shifting, differentiation).

4.4 Relation to Fourier Transform

Laplace Transform is a generalization of Fourier Transform.

• Substitute s = jω into X(s) to get X(jω) if and only if the jω-axis lies within the ROC.

• Thus, X(jω) exists ⇔ system is stable.

V. Z-TRANSFORM

5.1 Definition & ROC

Bilateral Z-Transform:

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

Unilateral Z-Transform (for causal systems/IC):

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

Region of Convergence (ROC):

• Set of z values for which the sum converges.

• Properties of ROC:

  1. ROC is a ring/annulus or disk: r₁ < |z| < r₂.

  2. ROC cannot contain poles.

  3. For causal signals (x[n]=0 for n<0), ROC is outside the outermost pole (including ∞).

  4. For anti-causal signals (x[n]=0 for n>0), ROC is inside the innermost pole (including 0).

  5. For finite-duration signals, ROC is entire z-plane except possibly z=0 and/or z=∞.

Example Transforms:

• x[n] = aⁿ u[n] → X(z) = 1/(1 - a z⁻¹), ROC: |z| > |a| (causal).

• x[n] = -aⁿ u[-n-1] → X(z) = 1/(1 - a z⁻¹), ROC: |z| < |a| (anti-causal).

• x[n] = δ[n] → X(z) = 1, ROC: entire z-plane.

• x[n] = u[n] → X(z) = 1/(1 - z⁻¹), ROC: |z| > 1.

5.2 Properties of Z-Transform

5.3 Inverse Z-Transform

• Method 1: Partial Fraction Expansion (PFE) + Power Series.

  • Expand X(z) into terms like A/(1 - az⁻¹).

  • 1/(1 - az⁻¹) ↔ aⁿ u[n] if ROC |z|>|a| (causal).

  • 1/(1 - az⁻¹) ↔ -aⁿ u[-n-1] if ROC |z|<|a| (anti-causal).

• Method 2: Power Series (Long Division). Directly expand X(z) as ∑ x[n] z⁻ⁿ.

• Method 3: Contour Integration (Residue Method). x[n] = (1/(2πj)) ∮ X(z) z^{n-1} dz.

5.4 Application to LTI DT Systems

• Transfer Function H(z): Z-transform of impulse response h[n].

$$H(z) = \sum_{n=-\infty}^{\infty} h[n] z^{-n}$$

• System Function: Y(z) = H(z) X(z) (with zero IC).

• Causality & Stability Criteria (for rational H(z)):

  • Causal: ROC is outside the outermost pole.

  • Stable: ROC includes the unit circle (|z|=1). For causal stable system, all poles must lie inside the unit circle.

• Example Analysis: H(z) = (3-4z⁻¹)/(1-3.5z⁻¹+1.5z⁻²)

  1. Find poles: Solve 1 - 3.5z⁻¹ + 1.5z⁻² = 0 → z² - 3.5z + 1.5 = 0 → z=0.5, 3.

  2. For Causal: ROC is |z| > 3 (outside outermost pole). Poles at 0.5 and 3. ROC |z|>3 does not include unit circle → Unstable.

  3. For Stable: ROC must include |z|=1. Since poles are at 0.5 and 3, the only ROC that includes unit circle is 0.5 < |z| < 3. This ROC is between poles → Anti-causal (non-causal).

  4. Find h[n] for stable case (ROC 0.5<|z|<3):

     • PFE: H(z) = A/(1-0.5z⁻¹) + B/(1-3z⁻¹). Solve: A=2, B=1.

     • Term 1/(1-0.5z⁻¹) → ROC |z|>0.5 gives (0.5)ⁿ u[n]. But our ROC 0.5<|z|<3 is inside |z|>0.5? No, it's the intersection. Actually, for ROC 0.5<|z|<3:

       • 1/(1-0.5z⁻¹) corresponds to causal part (0.5)ⁿ u[n] (ROC |z|>0.5).

       • 1/(1-3z⁻¹) corresponds to anti-causal part -3ⁿ u[-n-1] (ROC |z|<3).

     • Intersection ROC 0.5<|z|<3 is valid. So:

       h[n] = 2(0.5)ⁿ u[n] - 3ⁿ u[-n-1].

VI. ANALYSIS OF LTI SYSTEMS IN TRANSFORM DOMAINS

6.1 Transfer Function & Frequency Response

• CT: H(s) = L{h(t)}. Frequency response: H(jω) = H(s)|_{s=jω} (if jω in ROC).

• DT: H(z) = Z{h[n]}. Frequency response: H(e^{jω}) = H(z)|_{z=e^{jω}} (if |z|=1 in ROC).

6.2 System Characterization using Poles & Zeros

• Poles: Roots of denominator polynomial of H(s) or H(z).

• Zeros: Roots of numerator polynomial.

• Stability:

  • CT LTI: Stable iff all poles of H(s) have negative real parts (Left Half Plane, LHP).

  • DT LTI: Stable iff all poles of H(z) lie inside the unit circle (|p| < 1).

• Causality (for rational H(s)/H(z)):

  • CT: Causal iff ROC is right of rightmost pole.

  • DT: Causal iff ROC is outside the outermost pole.

6.3 Block Diagram Representations

• Direct Form I: Direct implementation of difference/differential equation.

  • CT: Uses integrators (1/s), multipliers (coefficients).

  • DT: Uses unit delays (z⁻¹).

• Direct Form II (Canonical): Minimizes number of delay elements. Combines feedforward/feedback paths.

• Cascade (Series): H(z) = H₁(z) H₂(z). Systems connected in series.

• Parallel: H(z) = H₁(z) + H₂(z) + ....

• Advantages: Visualizes structure, facilitates realization, simplifies analysis of interconnections.

6.4 Interconnection of LTI Systems

• Cascade (Series): y[n] = x[n] * h₁[n] * h₂[n]. Overall H(z) = H₁(z) H₂(z).

• Parallel: y[n] = x[n]*h₁[n] + x[n]*h₂[n]. Overall H(z) = H₁(z) + H₂(z).

• Feedback (Closed-Loop):

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

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

VII. STATE-VARIABLE ANALYSIS

7.1 State-Space Representation

Describes system using state variables (minimum set of variables that summarize past history).

• State Vector: x(t) (CT) or x[n] (DT).

• State Equation: Describes evolution of state.

• Output Equation: Relates output to state and input.

LTI CT System:

$$\frac{d\mathbf{x}(t)}{dt} = \mathbf{A} \mathbf{x}(t) + \mathbf{B} u(t) \quad \text{(State Equation)}$$

$$y(t) = \mathbf{C} \mathbf{x}(t) + D u(t) \quad \text{(Output Equation)}$$

• A (system matrix), B (input matrix), C (output matrix), D (feedthrough matrix).

• For Nth order system, A is N×N, B is N×1, C is 1×N, D is scalar (SISO).

LTI DT System:

$$\mathbf{x}[n+1] = \mathbf{A}_d \mathbf{x}[n] + \mathbf{B}_d u[n]$$

$$y[n] = \mathbf{C}_d \mathbf{x}[n] + D_d u[n]$$

7.2 Matrix Representation from Transfer Function

Given H(s) = (s+2)/(s²+3s+2):

1. Choose state variables (e.g., from block diagram or controllable canonical form).

2. Write differential/difference equation: d²y/dt² + 3 dy/dt + 2y = dx/dt + 2x.

3. Define state: x₁ = y, x₂ = dy/dt. Then:

   dx₁/dt = x₂

   dx₂/dt = -2x₁ - 3x₂ + x + 2u (after substitution).

4. Matrix form:

$$\mathbf{A} = \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix}, \mathbf{B} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \mathbf{C} = \begin{bmatrix} 1 & 0 \end{bmatrix}, D = 2$$

7.3 Advantages of State-Variable Model

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

• Easily incorporates non-zero initial conditions.

• Applicable to time-varying systems.

• Provides complete internal behavior (not just input-output).

VIII. SAMPLING AND ALIASING

8.1 Sampling Theorem (Nyquist-Shannon)

• Ideal Sampling: Multiply continuous-time signal x_a(t) by impulse train δ_T(t) = ∑ δ(t - nT).

  Sampled signal: x_p(t) = x_a(t) δ_T(t) = ∑ x_a(nT) δ(t - nT).

• Spectrum: X_p(jω) = (1/T) ∑ X_a(j(ω - kω_s)), where ω_s = 2π/T.

  → Replicas of X_a(jω) centered at kω_s.

• Sampling Theorem Statement:

  A bandlimited signal with maximum frequency B Hz can be reconstructed perfectly from its samples if sampled at a rate f_s ≥ 2B samples/sec (ω_s ≥ 2B).

  • f_s = 2B is the Nyquist rate.

  • f_N = f_s/2 is the Nyquist frequency.

• Reconstruction: Using an ideal low-pass filter (sinc interpolation):

$$x_a(t) = \sum_{n=-\infty}^{\infty} x_a(nT) \text{sinc}\left(\frac{t - nT}{T}\right)$$

8.2 Aliasing

• Phenomenon: When f_s < 2B (undersampling), spectral replicas overlap.

• Result: High-frequency components fold back into lower frequencies, causing irreversible distortion.

• Frequency Domain: Overlap of X_a(jω) and its shifted replicas.

• Prevention:

  1. Anti-Aliasing Filter: Analog LPF before sampler to bandlimit x_a(t) to B < f_s/2.

  2. Increase Sampling Rate: Ensure f_s ≥ 2B.

8.3 Discrete-Time Processing of CT Signals

Overall system:

CT Input → [Anti-Aliasing LPF] → [Sampler] → [DT System (H(z))] → [D/A Converter] → [Reconstruction LPF] → CT Output

• D/A Converter: Holds sample value for interval T (zero-order hold). Reconstruction filter smooths output.

IX. FILTERS

9.1 Analog Filters

• Ideal vs. Non-Ideal:

  • Ideal: Perfect brick-wall response (infinite attenuation in stopband, instantaneous transition). Not realizable.

  • Non-Ideal (Practical): Have transition band, passband ripple, stopband attenuation.

• Types:

  • LPF: Passes low frequencies, attenuates high.

  • HPF: Passes high frequencies, attenuates low.

  • BPF: Passes band ω₁ < ω < ω₂.

  • BSF: Attenuates band ω₁ < ω < ω₂.

• Design Approximations (Brief):

  • Butterworth: Maximally flat passband (no ripple), monotonic stopband. Poles on circle in LHP.

  • Chebyshev I: Equiripple in passband, monotonic stopband. Sharper transition than Butterworth.

  • Elliptic (Cauer): Equiripple in both passband and stopband. Sharpest transition for given order.

9.2 Digital Filters

• FIR (Finite Impulse Response):

  • h[n] has finite duration.

  • Always stable (poles only at z=0).

  • Can achieve exact linear phase (symmetric/anti-symmetric h[n]).

  • Implemented by non-recursive (feedforward) structure.

  • Higher order for sharp cutoff.

• IIR (Infinite Impulse Response):

  • h[n] infinite duration (recursive: feedback).

  • Stability requires poles inside unit circle.

  • Cannot achieve exact linear phase (except trivial cases).

  • Lower order for sharp cutoff (more efficient).

  • Realization Structures: Direct Form I/II, Cascade, Parallel, Lattice.

• Comparison:

  | Feature | FIR | IIR | | :--- | :--- | :--- | | Stability | Always stable | Conditionally stable | | Linear Phase | Possible | Not possible (except trivial) | | Design | Window, frequency sampling | Analog prototype (Butterworth etc.) | | Computational Cost | Higher for sharp cutoff | Lower for sharp cutoff |

X. ADVANCED TOPICS & SHORT NOTE SYLLABUS

10.1 Wavelet Transform

• Concept: Uses wavelets (localized, short-duration oscillations) instead of infinite sinusoids.

• Scaling & Translation: ψ_{a,b}(t) = (1/√a) ψ((t-b)/a).

  • a (scale): controls frequency (inverse of duration).

  • b (translation): controls time location.

• Comparison:

  • vs Fourier: FT has perfect frequency resolution, zero time resolution. Wavelet has variable resolution (good time for high freq, good freq for low freq).

  • vs Laplace: Laplace for complex frequency s=σ+jω (exponential growth/decay). Wavelet for time-frequency analysis of non-stationary signals.

• Application: Signal compression (JPEG2000), edge detection in images, transient analysis.

10.2 Non-Ideal Frequency Selective Filters

• Practical filters have:

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

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

  • Transition Band: Frequency range between passband and stopband edges (ω_p to ω_s).

• Design is a trade-off between order (complexity), ripple, attenuation, and transition width.

10.3 Invertible and Inverse Systems

• Invertible: System T has inverse T⁻¹ s.t. T⁻¹{T{x[n]}} = x[n].

  • Condition for LTI: H(z) ≠ 0 for all z on unit circle (DT) or H(jω) ≠ 0 for all ω (CT).

• Inverse System: Transfer function H_inv(z) = 1/H(z).

  • Causality/Stability: If H(z) is causal and stable, H_inv(z) may be non-causal or unstable. Example: H(z)=1/(1-0.5z⁻¹) (causal, stable). H_inv(z)=1-0.5z⁻¹ is FIR (causal, stable). But H(z)=1+z⁻¹ (causal, stable), H_inv(z)=1/(1+z⁻¹) has pole at z=-1 (on unit circle) → unstable.

10.4 Signal Reconstruction from Samples

• Ideal Reconstruction: x_a(t) = ∑ x[n] \text{sinc}((t-nT)/T) using ideal LPF with cutoff ω_c = π/T.

• Non-Ideal Reconstruction: Practical D/A converters (zero-order hold) and reconstruction filters introduce distortion. Requires anti-imaging filter to remove spectral replicas.

10.5 Exponential Signals

• CT: x(t) = e^{st} = e^{σt} e^{jωt}.

  • σ=0 → pure sinusoid (marginal stability).

  • σ<0 → decaying exponential (stable).

  • σ>0 → growing exponential (unstable).

• DT: x[n] = aⁿ = rⁿ e^{jω₀n} where a = re^{jω₀}.

  • |a|<1 → decaying (stable).

  • |a|=1 → constant magnitude (marginal).

  • |a|>1 → growing (unstable).

• Role: Eigenfunctions of LTI systems. e^{st} → output H(s)e^{st}. Fundamental in transform analysis.

Final Exam Strategy:

1. Signal Classification: Always check definitions (periodicity, energy/power integrals).

2. System Properties: Apply tests directly (e.g., for time-invariance, apply x(t-t₀)).

3. ROC: Draw pole-zero plot. ROC never includes poles. Use causality/stability rules.

4. Convolution: Use graphical method or summation/integral. Remember properties.

5. Transform Pairs & Properties: Memorize key pairs (e^{-at}u(t), u(t), δ(t)). Duality and convolution theorem are favorites.

6. Block Diagrams: Practice converting between differential/difference equations, transfer function, and block diagrams (Direct Form I/II).

7. State-Space: Know how to form A,B,C,D from given system description.

8. Sampling: Clearly state Nyquist rate, aliasing condition, and reconstruction formula.

9. Filters: Distinguish FIR/IIR, analog/digital, and know stability conditions.

\boxed{\text{Master these concepts, practice past paper problems, and focus on problem-solving patterns.}}