UNIT 3: Signals & Systems - Comprehensive Short Notes

Based on rigorous analysis of RGPV past examination papers (JUN 2025, DEC 2024, JUN 2024, JUN 2023, NOV 2023, JUN 2022).

I. SIGNAL CLASSIFICATION & BASIC OPERATIONS

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

[!TIP] Exam Focus: Distinguish using representation, domain, and practical examples (analog vs. digital).

Standard Signals & Basic Operations (DT)

• Unit Step: u[n] = 1 for n ≥ 0, 0 otherwise.

• Unit Ramp: r[n] = n u[n].

• Unit Impulse: δ[n] = 1 for n = 0, 0 otherwise. Σ_{k=-∞}^{∞} δ[n-k] = 1.

• Basic Operations on x[n]:

  1. Time Shifting: x[n - n₀] (delay if n₀ > 0).

  2. Time Scaling: x[kn] (compression if |k| > 1).

  3. Folding (Time Reversal): x[-n].

  4. Amplitude Operations: a·x[n] + b.

[!CAUTION] Common Pitfall: Time scaling x[2n] does not mean stretching; it means taking every 2nd sample (compression).

Periodicity & Aperiodicity

• CT Signal x(t): Periodic if ∃ T > 0 such that x(t + T) = x(t) for all t. Fundamental period T₀ = smallest T.

  • Condition: T₀ = 2π / ω₀ must be same for all sinusoidal components.

• DT Signal x[n]: Periodic if ∃ N ∈ ℤ⁺ such that x[n + N] = x[n] for all n. Fundamental period N₀ = smallest N.

  • Condition: x[n] periodic iff ∃ N₀ such that ω₀ N₀ = 2πk for some integer k. N₀ = 2πk / ω₀ must be integer.

  • Example: x[n] = cos(0.1πn) is periodic (ω₀=0.1π, N₀=20). x[n] = cos(0.5πn) is periodic (N₀=4). x[n] = cos(πn) is periodic (N₀=2). x[n] = cos(1.5n) is aperiodic.

[!TIP] Key Distinction: For DT, ω₀ must be a rational multiple of 2π for periodicity.

Deterministic vs. Random Signals

• Deterministic: Completely predictable for all t/n. Can be expressed by an explicit mathematical formula. Example: x(t) = 5 sin(2πt).

• Random (Stochastic): Not predictable; described by statistical properties (mean, variance, probability distributions). Example: Thermal noise voltage.

Even & Odd Signals

• Even: x(-t) = x(t) (CT), x[-n] = x[n] (DT). Symmetric about y-axis.

• Odd: x(-t) = -x(t) (CT), x[-n] = -x[n] (DT). Symmetric about origin.

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

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

where

$$ x_e(t) = \frac{1}{2}[x(t) + x(-t)], \quad x_o(t) = \frac{1}{2}[x(t) - x(-t)] $$

Energy & Power Signals

• Total Energy:

  • CT: $$\displaystyle E = \int_{-\infty}^{\infty} |x(t)|^2 dt $$

  • DT: $$\displaystyle E = \sum_{n=-\infty}^{\infty} |x[n]|^2 $$

• Average Power:

  • CT: $$\displaystyle P = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} |x(t)|^2 dt $$

  • DT: $$\displaystyle P = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^{N} |x[n]|^2 $$

• Classification:

  • Energy Signal: E is finite and non-zero, P = 0. Typically time-limited (e.g., pulse).

  • Power Signal: P is finite and non-zero, E = ∞. Typically periodic or persistent (e.g., sinusoid).

  • Neither: Infinite energy and power (e.g., x(t)=t).

[!CAUTION] Exam Trap: x(t) = cos(t) is a power signal (P=1/2, E=∞), not an energy signal.

II. SYSTEM CLASSIFICATION & PROPERTIES

System Definition & Types

A system is a transformation that maps an input signal x to an output signal y: y = T{x}.

[!TIP] Causality Check: For CT: if y(t₀) depends on x(τ) for τ > t₀ → Non-causal. For DT: if y[n₀] depends on x[m] for m > n₀ → Non-causal.

Linearity: Additivity & Homogeneity (Superposition)

A system is Linear if it satisfies Superposition:

$$ T\{a_1 x_1(t) + a_2 x_2(t)\} = a_1 T\{x_1(t)\} + a_2 T\{x_2(t)\} $$

• Test: Apply two inputs x₁ and x₂, check if output for a₁x₁ + a₂x₂ equals a₁y₁ + a₂y₂.

• Non-linear examples: y(t) = x²(t), y(t) = x(t) + 2, y[n] = n·x[n].

Stability (BIBO Stability)

• Definition: A system is BIBO Stable if every bounded input |x(t)| ≤ B_x < ∞ produces a bounded output |y(t)| ≤ B_y < ∞.

• Test for LTI Systems (CT): Impulse response h(t) must be absolutely integrable:

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

• Test for LTI Systems (DT): Impulse response h[n] must be absolutely summable:

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

System Property Analysis Methodology

Given a system equation (e.g., y(t) = ... or differential/difference equation):

1. Linearity: Check for terms like x²(t), |x(t)|, x(t)·t, or non-zero independent terms. Replace x with a₁x₁ + a₂x₂ and verify superposition.

2. Time-Invariance: Apply input x(t-t₀) and check if output is y(t-t₀).

3. Causality: Express y(t₀) or y[n₀]. If it depends on x(τ) for τ > t₀ (CT) or m > n₀ (DT) → Non-causal.

4. Stability: For LTI systems, check absolute summability/integrability of h(t) or h[n]. For general systems, reason about bounded input leading to bounded output.

III. LINEAR TIME-INVARIANT (LTI) SYSTEMS

Impulse Response Representation

• For any LTI system, the output y(t) is the convolution of input x(t) with impulse response h(t):

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

For DT:

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

• Properties:

  • Complete Characterization: h(t) or h[n] fully defines an LTI system.

  • Causality: h(t) = 0 for t < 0 (CT), h[n] = 0 for n < 0 (DT).

  • Stability: As defined in Section II (absolute integrability/summability).

Convolution

• Definition (DT): y[n] = Σ_{k=-∞}^{∞} x[k] h[n-k]. It is a sliding, weighted sum.

• Graphical Method: Flip h[k] → h[-k]. Shift by n, multiply with x[k], sum.

• Properties:

  • Commutative: x * h = h * x

  • Associative: (x * h₁) * h₂ = x * (h₁ * h₂)

  • Distributive: x * (h₁ + h₂) = x*h₁ + x*h₂

• Convolution with Standard Signals:

  • x[n] * δ[n] = x[n]

  • u[n] * u[n] = (n+1)u[n]

  • aⁿu[n] * bⁿu[n] = \frac{a^{n+1} - b^{n+1}}{a-b} u[n] (for a ≠ b)

System Differential/Difference Equations

• CT LTI: Linear Constant-Coefficient Differential Equation (LCCDE):

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

• DT LTI: Linear Constant-Coefficient Difference Equation (LCCDE):

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

• Finding Impulse Response: Assume zero initial conditions. Take LCCDE, replace x[n] with δ[n], solve for y[n] = h[n] (using Z-transform or recursive solution).

Block Diagram Representations

• Direct Form I: Realizes the difference equation directly with adders and delays. For Σ a_k y[n-k] = Σ b_k x[n-k].

• Direct Form II: Combines the two delay lines of Direct Form I into one. More efficient.

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

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

[!DIAGRAM: CANVAS] Direct Form II (DT): Draw a single chain of N delays. At the output side, feed b₀x[n], b₁x[n-1], ..., b_Mx[n-M] into a summing junction. At the input side, feed -a₁y[n-1], -a₂y[n-2], ..., -a_Ny[n-N] into the same summing junction. The output of the summing junction is y[n].

IV. Z-TRANSFORM

Definition & Region of Convergence (ROC)

• Bilateral Z-Transform:

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

• Unilateral Z-Transform (for causal systems/ICs):

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

• ROC: Set of z for which the sum converges (|X(z)| < ∞).

ROC Properties (Critical for Exam)

[!CAUTION] Golden Rule: For a causal sequence, ROC is always exterior of the outermost pole, including ∞.

Z-Transform Properties (with Proof Sketches)

1. Linearity: a₁x₁[n] + a₂x₂[n] ↔ a₁X₁(z) + a₂X₂(z). ROC = Intersection of ROCs.

2. Time Shifting: x[n-k] ↔ z^{-k} X(z). ROC same as X(z).

3. Time Reversal: x[-n] ↔ X(z^{-1}). ROC: If X(z) has ROC R, then X(z^{-1}) has ROC 1/R.

4. Convolution: x[n] * h[n] ↔ X(z) H(z). ROC is intersection of ROC_X and ROC_H (at least).

5. Differentiation in z-domain: n x[n] ↔ -z \frac{dX(z)}{dz}.

6. Initial Value Theorem (Causal): x[0] = \lim_{z \to \infty} X(z).

7. Final Value Theorem: \lim_{n \to \infty} x[n] = \lim_{z \to 1} (z-1)X(z). Condition: Poles of (z-1)X(z) must be inside unit circle, except possibly at z=1.

Inverse Z-Transform Methods

1. Power Series Expansion: Expand X(z) in powers of z^{-1}. Coefficients give x[n]. ROC determines duration (causal/anti-causal).

2. Partial Fraction Expansion (PFE): For rational X(z). Expand into simpler terms, use standard pairs.

3. Contour Integration (Residue Method): x[n] = \frac{1}{2\pi j} \oint_C X(z) z^{n-1} dz. (Theoretical).

Unilateral Z-Transform

• Definition: Sum from n=0 to ∞. Used for solving difference equations with non-zero initial conditions.

• Key Property: For a causal sequence, X_U(z) = X_B(z) (same). But X_U(z) ignores n<0 terms.

• Solving Difference Equations:

  1. Take unilateral Z-transform of both sides (using time-shift property for y[n-k] which introduces initial conditions y[-1], y[-2],...).

  2. Solve for Y(z).

  3. Take inverse unilateral Z-transform.

V. FOURIER ANALYSIS

A. Continuous-Time Fourier Series (CTFS)

• Exponential Form (Synthesis & Analysis):

$$ x(t) = \sum_{k=-\infty}^{\infty} C_k e^{jk\omega_0 t}, \quad C_k = \frac{1}{T_0} \int_{T_0} x(t) e^{-jk\omega_0 t} dt $$

where `ω₀ = 2π/T₀`.

• Properties (for periodic x(t)):

  • Linearity, Time Shift (C_k phase shift), Time Scaling (changes ω₀), Multiplication (Modulation - coefficients convolve), Conjugation (symmetry for real signals: C_{-k} = C_k^*).

• Spectra: |C_k| vs kω₀ (Magnitude Spectrum), ∠C_k vs kω₀ (Phase Spectrum). Discrete lines (line spectrum).

• Power: P = \frac{1}{T_0} \int_{T_0} |x(t)|^2 dt = \sum_{k=-\infty}^{\infty} |C_k|^2 (Parseval's theorem for FS).

B. Continuous-Time Fourier Transform (CTFT)

• Definition:

$$ X(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt, \quad x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(j\omega) e^{j\omega t} d\omega $$

• Existence Conditions: ∫ |x(t)| dt < ∞ (sufficient), or ∫ |x(t)|² dt < ∞ (square integrable).

• Key Properties Table:

• Fourier Transform Pairs (Standard):

  • δ(t) ↔ 1

  • 1 ↔ 2π δ(ω)

  • u(t) ↔ πδ(ω) + 1/(jω)

  • e^{-at}u(t) (a>0) ↔ 1/(a + jω)

  • e^{-a|t|} ↔ 2a/(a² + ω²)

  • rect(t/T) ↔ T sinc(ωT/2π)

  • sinc(t) ↔ rect(ω/2π)

[!TIP] Using Properties: To find FT of t e^{-at} sin(ω₀t)u(t), start from sin(ω₀t)u(t) or e^{-at}u(t), use frequency shift and differentiation in time.

C. Discrete-Time Fourier Transform (DTFT)

• Definition:

$$ X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n}, \quad x[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(e^{j\omega}) e^{j\omega n} d\omega $$

*Note: `ω` is **periodic** with period `2π`.*

• Convergence: For absolutely summable sequences (Σ|x[n]| < ∞), or square summable (Σ|x[n]|² < ∞), or periodic sequences.

• Properties (Analogous to CTFT, with differences):

  • Periodicity: X(e^{j(ω+2π)}) = X(e^{jω}).

  • Conjugation: x^*[n] ↔ X^*(e^{-jω}).

  • Differentiation in Frequency: n x[n] ↔ j \frac{dX(e^{j\omega})}{d\omega}.

• Relationship with Z-Transform:

$$ X(e^{j\omega}) = X(z) \big|_{z = e^{j\omega}} $$

**Condition:** ROC must include the **unit circle** (`|z|=1`).

• Frequency Response of LTI System: H(e^{jω}) is the DTFT of impulse response h[n].

• Applications: Analyzing frequency response, filter design, spectral analysis of DT signals.

VI. STATE-SPACE ANALYSIS

State Variables & State Equations (LTI)

• State: Minimal set of numbers x(t) (CT) or x[n] (DT) that, with input u(t)/u[n], uniquely determines future output.

• State-Space Representation (SISO/MIMO):

$$ \text{CT: } \dot{\mathbf{x}}(t) = \mathbf{A} \mathbf{x}(t) + \mathbf{B} \mathbf{u}(t) \\ \mathbf{y}(t) = \mathbf{C} \mathbf{x}(t) + \mathbf{D} \mathbf{u}(t) $$

$$ \text{DT: } \mathbf{x}[n+1] = \mathbf{A} \mathbf{x}[n] + \mathbf{B} \mathbf{u}[n] \\ \mathbf{y}[n] = \mathbf{C} \mathbf{x}[n] + \mathbf{D} \mathbf{u}[n] $$

*   `x`: state vector (p×1), `u`: input vector (q×1), `y`: output vector (r×1).

*   `A`: system matrix (p×p), `B`: input matrix (p×q), `C`: output matrix (r×p), `D`: feedthrough matrix (r×q).

• Example (3-input, 2-output):

  u[n] = [u₁[n], u₂[n], u₃[n]]ᵀ (3×1), y[n] = [y₁[n], y₂[n]]ᵀ (2×1). A is 2×2, B is 2×3, C is 2×2, D is 2×3.

State Transition Matrix (Φ)

• Definition: Φ(t) (CT) or Φ[n] (DT) is the state response to initial condition x(0)=x₀ and zero input (u(t)=0).

$$ \mathbf{x}(t) = \Phi(t) \mathbf{x}(0) \quad (\text{CT}), \quad \mathbf{x}[n] = \Phi[n] \mathbf{x}[0] \quad (\text{DT}) $$

• CT: Φ(t) = e^{At} (matrix exponential).

• DT: Φ[n] = Aⁿ.

• Properties:

  1. Φ(0) = I (identity).

  2. Φ(t₁ + t₂) = Φ(t₁) Φ(t₂).

  3. Φ(-t) = Φ^{-1}(t).

  4. dΦ(t)/dt = A Φ(t).

• General Solution (with input):

$$ \mathbf{x}(t) = \Phi(t)\mathbf{x}(0) + \int_{0}^{t} \Phi(t-\tau) \mathbf{B} \mathbf{u}(\tau) d\tau $$

Methods to Determine State Transition Matrix

1. Laplace Transform Method (CT):

$$ \Phi(t) = \mathcal{L}^{-1} \{ (s\mathbf{I} - \mathbf{A})^{-1} \} $$

Compute `(sI - A)⁻¹` (resolvent matrix), take inverse Laplace of each element.

2. Cayley-Hamilton Theorem: Since A satisfies its own characteristic equation |λI - A| = 0, Φ(t) can be expressed as a finite series:

$$ \Phi(t) = a_0(t)\mathbf{I} + a_1(t)\mathbf{A} + a_2(t)\mathbf{A}^2 + ... + a_{p-1}(t)\mathbf{A}^{p-1} $$

Coefficients `a_i(t)` found by substituting eigenvalues.

3. Spectral Decomposition (Diagonalizable A): If A = VΛV⁻¹ (Λ diagonal of eigenvalues), then Φ(t) = V e^{Λt} V⁻¹, where e^{Λt} is diagonal with e^{λ_i t}.

4. Direct Computation (Small Systems): For 2×2 or diagonal A, compute e^{At} directly using series or known formulas.

VII. SAMPLING THEOREM & APPLICATIONS

Nyquist-Shannon Sampling Theorem

• Statement: A band-limited continuous-time signal x_a(t) with no frequency components above W Hz (i.e., X_a(jω)=0 for |ω| > ω_m, where ω_m = 2πW) is completely determined by its samples x_a(nT) taken at a rate f_s = 1/T ≥ 2W samples/sec.

• Nyquist Rate: f_N = 2W (samples/sec). T_N = 1/(2W) (max sampling interval).

• Proof Sketch:

  1. Sampling: x_p(t) = x_a(t) \cdot \sum_{n=-\infty}^{\infty} δ(t - nT).

  2. Spectrum: X_p(jω) = (1/T) \sum_{k=-\infty}^{\infty} X_a(j(ω - kω_s)), where ω_s = 2π/T.

  3. For perfect recovery (no overlap/aliasing), shifted spectra must not overlap: ω_s ≥ 2ω_m → T ≤ π/ω_m → f_s ≥ 2W.

  4. Reconstruction (Ideal LPF): x_a(t) = Σ_{n=-∞}^{∞} x_a(nT) \text{sinc}((t-nT)/T).

Aliasing

• Cause: Undersampling (f_s < 2W).

• Effect: High-frequency components fold back into lower frequencies, causing irreversible distortion. The sampled spectrum X_p(jω) has overlapping replicas.

• Illustration: Show X_a(jω) and shifted replicas for f_s > 2W (no overlap) vs f_s < 2W (overlap).

Reconstruction of Original Signal

• Ideal Interpolation (Sinc Interpolation):

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

Requires infinite samples and ideal LPF with cutoff `ω_c = ω_s/2`.

• Practical Reconstruction:

  • Zero-Order Hold (ZOH): Holds sample value constant for interval T. Smoothing required.

  • First-Order Hold (FOH): Linear interpolation between samples.

Applications & Numerical Example

• Applications: Digital Signal Processing, Analog-to-Digital Conversion, Communications (PCM).

• Example (From Past Paper): Signal spectrum from 5.6 MHz to 6.8 MHz.

  • Bandwidth W = 6.8 - 5.6 = 1.2 MHz.

  • Minimum Sampling Rate: f_s(min) = 2W = 2.4 MHz.

  • Maximum Sampling Interval: T_max = 1 / f_s(min) = 1 / (2.4 × 10⁶) ≈ 416.67 ns.

VIII. ADDITIONAL & SPECIFIC TOPICS FROM PAST PAPERS

Convergence of DTFT

• Sufficient Condition: Absolute summability (Σ |x[n]| < ∞) → DTFT converges absolutely.

• Connection to Stability: For an LTI DT system with impulse response h[n], BIBO stability ⇔ Σ |h[n]| < ∞ ⇔ H(e^{jω}) converges absolutely for all ω.

• Other Cases: DTFT of periodic sequences exists in the sense of generalized functions (impulses in frequency).

Block Diagram Realization from Transfer Function

• Given H(s) or H(z), write difference/differential equation.

• Direct Form I: Implement coefficients a_k, b_k directly with delays and adders.

• Direct Form II: Combine delay lines.

• Cascade: Factor H(z) into H₁(z)H₂(z)..., realize each separately.

• Parallel: Use partial fractions: H(z) = D + Σ (Residues)/(1 - p_i z^{-1}).

Finding H(z) and Difference Equation from h[n]

1. Z-Transform: H(z) = Σ_{n=-∞}^{∞} h[n] z^{-n}.

2. Difference Equation: Express H(z) = Y(z)/X(z) as rational function. Cross-multiply, take inverse Z-transform to get:

$$ y[n] + a_1 y[n-1] + ... = b_0 x[n] + b_1 x[n-1] + ... $$

Step Response from Impulse Response

• Relationship: Step response s[n] is cumulative sum of impulse response:

$$ s[n] = \sum_{k=-\infty}^{n} h[k] = u[n] * h[n] $$

• In Z-domain: S(z) = H(z) · (z/(z-1)) (for causal h[n]).

Fourier Series Coefficients for Periodic Waveforms

• Exponential FS: C_k = (1/T) ∫_{<T>} x(t) e^{-jkω₀t} dt.

• Trigonometric FS: x(t) = a₀/2 + Σ_{k=1}^{∞} [a_k cos(kω₀t) + b_k sin(kω₀t)].

$$ a_k = \frac{2}{T} \int_{<T>} x(t) \cos(kω₀t) dt, \quad b_k = \frac{2}{T} \int_{<T>} x(t) \sin(kω₀t) dt $$

• Symmetry for Real Signals: a_k = 2|C_k| cos(∠C_k), b_k = -2|C_k| sin(∠C_k).

Short Notes on Specific Concepts

• CT LTI vs. DT LTI:

  • CT: Convolution integral, h(t) continuous. DT: Convolution sum, h[n] discrete.

  • Frequency variable: ω (rad/s) vs ω (rad/sample, periodic 2π).

  • Analysis tools: CTFT vs DTFT/Z-transform.

• Properties of LTI Systems: Commutative, Associative, Distributive, Memory (if h[0] only → memoryless), Invertible (if H(z)/H(s) has inverse), Stable (if h absolutely summable/integrable).

• Role of State Transition Matrix: Φ(t) propagates state from initial time to any time t. Solves homogeneous state equation. Fundamental in state-space analysis.

• Applications of DTFT: 1) Frequency response analysis of DT LTI systems. 2) Spectral analysis of non-periodic DT signals. 3) Bridge between DT signals and continuous-frequency representation. 4) Filter design (specifying H(e^{jω})).

• Unilateral vs. Bilateral Z-Transform:

  • Bilateral: n from -∞ to ∞. Used for two-sided sequences. ROC is annular.

  • Unilateral: n from 0 to ∞. Used for solving difference equations with initial conditions (ICs). For causal x[n], X_U(z) = X_B(z) but ROC may differ if sequence has anti-causal part. Unilateral transform includes ICs in its z-domain expression.

END OF UNIT 3 NOTES