UNIT 1: Signals and Systems - Comprehensive Short Notes

I. FUNDAMENTALS OF SIGNALS

1.1 Basic Signal Classification & Types

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

[!TIP] In exams, clearly state the domain difference. CT signals are functions of a continuous variable t, while DT signals are sequences indexed by integers n.

Periodic vs. Aperiodic Signals

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

• DT Periodic: x[n] = x[n + N] for some integer N. Fundamental period N is the smallest positive integer.

• Key Check: For x(t) = sin(ω₀t), T₀ = 2π/ω₀. For sum of sinusoids, period exists only if frequency ratio is rational.

• Aperiodic: No such T₀ or N exists.

[!CAUTION] For DT signals, N must be an integer. x[n] = sin(0.5πn) is periodic with N=4, but x[n] = sin(0.6πn) is aperiodic.

Even & Odd Signals

• Even: x(t) = x(-t) (symmetric about y-axis). Example: cos(t), t².

• Odd: x(-t) = -x(t) (symmetric about origin). Example: sin(t), t³.

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

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

where

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

\boxed{x_e(t) = \frac{x(t) + x(-t)}{2}, \quad x_o(t) = \frac{x(t) - x(-t)}{2}}

[!TIP] This decomposition is always possible and unique. Use it to simplify analysis of symmetric systems.

Energy & Power Signals

• Energy Signal: Finite total energy.

$$ E = \int_{-\infty}^{\infty} |x(t)|^2 dt < \infty \quad \text{(CT)} $$

$$ E = \sum_{n=-\infty}^{\infty} |x[n]|^2 < \infty \quad \text{(DT)} $$

Average power P = 0.

• Power Signal: Finite average power (non-zero).

$$ P = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} |x(t)|^2 dt < \infty \quad \text{(CT)} $$

$$ P = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^{N} |x[n]|^2 < \infty \quad \text{(DT)} $$

Total energy E = ∞.

• Neither: Periodic signals like sin(t) are power signals (finite P, infinite E). x(t)=t is neither.

[!CAUTION] Periodic signals are power signals (unless identically zero). Energy signals must have finite duration and be bounded.

Causal, Anti-Causal & Non-Causal Signals

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

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

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

Deterministic vs. Random Signals

• Deterministic: Completely predictable for all t. Example: x(t) = 5cos(2πt).

• Random (Stochastic): Cannot be predicted precisely; described by statistical properties. Example: speech signal, noise.

Right-sided, Left-sided & Two-sided Signals

• Right-sided (CT): x(t) = 0 for t < t₀ for some finite t₀.

• Left-sided (CT): x(t) = 0 for t > t₀.

• Two-sided: Non-zero for both t → -∞ and t → ∞.

1.2 Basic Signal Operations

Time Operations

1. Shifting:

   • CT: x(t - t₀) → delay by t₀ (right shift if t₀ > 0).

   • DT: x[n - n₀] → delay by n₀ samples.

2. Scaling:

   • CT: x(at) → compresses if |a| > 1, expands if |a| < 1. If a < 0, also reflects.

   • DT: x[an] → only defined for integer a (e.g., x[2n] is downsampling).

3. Reflection: x(-t) or x[-n].

Amplitude Operations

• Addition/Multiplication: Point-wise.

• Differentiation (CT): dx(t)/dt. Sharpens transitions.

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

• Differencing (DT): x[n] - x[n-1]. Discrete derivative.

[!TIP] Time operations affect the domain, amplitude operations affect the range. Shifting is not commutative with scaling: x(2(t-1)) ≠ x(2t-1).

1.3 Standard Elementary Signals

Relationship between Unit Step and Unit Impulse

• CT: δ(t) = du(t)/dt, u(t) = ∫_{-∞}^{t} δ(τ) dτ.

• DT: δ[n] = u[n] - u[n-1], u[n] = ∑_{k=-∞}^{n} δ[k].

[!CAUTION] In DT, δ[n] is not the derivative of u[n] (difference operator instead).

II. FUNDAMENTALS OF SYSTEMS

2.1 System Classification & Properties

Linear vs. Non-Linear

• Linear: Satisfies Superposition (Additivity + Homogeneity).

$$ \text{If } x_1(t) \to y_1(t), x_2(t) \to y_2(t), \text{ then } a x_1(t) + b x_2(t) \to a y_1(t) + b y_2(t). $$

• Test: Apply ax₁(t) + bx₂(t) and check if output is ay₁(t) + by₂(t).

Time-Invariant (TI) vs. Time-Variant (TV)

• TI: A time shift in input causes identical shift in output.

$$ x(t - t_0) \to y(t - t_0) $$

• Test: Replace t by t - t₀ in system equation. If equation structure unchanged → TI.

• Example: y(t) = t x(t) is TV (time-varying coefficient).

Causal vs. Non-Causal

• Causal: Output at time t depends only on input values at time t and before.

• Test: Express y(t₀) as function of x(τ) for τ ≤ t₀. If any τ > t₀ appears → non-causal.

Stable (BIBO) vs. Unstable

• BIBO Stable: Every bounded input produces bounded output.

$$ |x(t)| ≤ B_x < ∞ \quad \Rightarrow \quad |y(t)| ≤ B_y < ∞ $$

• Test for LTI: Check if impulse response h(t) is absolutely integrable (CT) or absolutely summable (DT).

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

$$ \sum_{n=-\infty}^{\infty} |h[n]| < \infty \quad \text{(DT)} $$

Static (Memoryless) vs. Dynamic (With Memory)

• Static: Output depends only on current input. Example: y(t) = x²(t).

• Dynamic: Output depends on past/future inputs or internal state. Example: y(t) = x(t-1) (has memory).

Invertible vs. Non-invertible

• Invertible: Distinct inputs produce distinct outputs. Inverse system exists.

• Test: Check if x₁(t) ≠ x₂(t) always implies y₁(t) ≠ y₂(t).

2.2 Linear Time-Invariant (LTI) Systems

Defining Characteristics

• Linear + Time-Invariant.

• Completely characterized by impulse response h(t) (CT) or h[n] (DT).

Impulse Response h(t)/h[n]

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

• Significance: For any input x(t), output y(t) = x(t) * h(t) (convolution).

Step Response s(t)/s[n]

• Output when input is unit step u(t) or u[n].

• Relation: s(t) = ∫_{-∞}^{t} h(τ) dτ (CT), s[n] = ∑_{k=-∞}^{n} h[k] (DT).

• Also, h(t) = ds(t)/dt (CT), h[n] = s[n] - s[n-1] (DT).

Frequency Response H(jω)/H(e^{jω})

• Fourier Transform of impulse response.

$$ H(jω) = \int_{-\infty}^{\infty} h(t) e^{-jωt} dt \quad \text{(CT)} $$

$$ H(e^{jω}) = \sum_{n=-\infty}^{\infty} h[n] e^{-jωn} \quad \text{(DT)} $$

• Describes system's response to complex exponentials e^{jωt}.

Convolution

• CT Linear Convolution:

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

• 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 * h₁) * h₂ = x * (h₁ * h₂)

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

• Connection to LTI Output: y(t) = x(t) * h(t).

[!TIP] Convolution integral/sum is fundamental for LTI system analysis. Always check limits based on signal support.

Differential Equation Representation (CT)

• General form:

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

• Impulse response found by solving with x(t)=δ(t) and zero initial conditions.

• Example: dy/dt + ay(t) = x(t) → h(t) = e^{-at} u(t).

Difference Equation Representation (DT)

• General form:

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

• First-order DT LTI: y[n] + a y[n-1] = b x[n] → h[n] = b (-a)^n u[n] for causal system.

Block Diagram Representation

[!CAUTION] Direct Form II is more efficient (fewer storage elements) but can be numerically sensitive.

Interconnection of LTI Systems

• Series/Cascade: Overall h_total[n] = h₁[n] * h₂[n], H_total(z) = H₁(z) H₂(z).

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

• Feedback: H(z) = H₁(z) / (1 ± H₁(z)H₂(z)) (negative/positive feedback).

III. FOURIER ANALYSIS

3.1 Fourier Series (FS) for Periodic Signals

Dirichlet's Conditions (for FS existence):

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

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

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

If satisfied, FS converges to x(t) at continuity, and to average at discontinuities.

Trigonometric vs. Exponential FS

• Trigonometric:

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

• Exponential:

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

where

$$ c_k = \frac{1}{T₀} \int_{t₀}^{t₀+T₀} x(t) e^{-jkω₀ t} dt $$

\boxed{c_k = \frac{1}{T₀} \int_{t₀}^{t₀+T₀} x(t) e^{-jkω₀ t} dt}

Limitations of Fourier Series

• Only for periodic signals.

• Represents signal in frequency domain but loses time-localization (no time information).

• Convergence issues at discontinuities (Gibbs phenomenon).

FS of Sinusoidal Signal

For x(t) = A sin(ω₀ t) (period T₀ = 2π/ω₀):

• Only one coefficient non-zero: c₁ = -jA/2, c₋₁ = jA/2, others zero.

• Shows sinusoid is a single-frequency component.

3.2 Fourier Transform (FT) for Aperiodic & Periodic Signals

CTFT Definition & Inverse

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

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

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

DTFT Definition & Inverse

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

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

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

Properties of Fourier Transform (CT & DT analogous)

Duality Property

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

• Significance: Symmetry between time and frequency domains. Allows deriving transforms by swapping t and ω.

FT of Standard Signals

[!TIP] FT of sinusoids are impulses in frequency domain—key for spectral analysis.

Comparison: Fourier Series vs. Fourier Transform

IV. LAPLACE TRANSFORM (CT)

4.1 Definition & Region of Convergence (ROC)

Definition

• Bilateral (Two-sided):

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

• Unilateral (One-sided) (for causal signals/initial conditions):

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

Region of Convergence (ROC)

• Set of s values for which integral converges.

• Importance:

  1. Determines existence of X(s).

  2. Determines causality: ROC is right-half plane for causal signals.

  3. Determines stability: For causal LTI, stability ⇔ ROC includes jω-axis.

  4. Determines finite duration: ROC is entire s-plane except possibly s=∞.

Properties of ROC

• ROC is a vertical strip (or half-plane) in s-plane: σ₁ < Re(s) < σ₂.

• No poles in ROC (poles are boundary points).

• For rational X(s), ROC extends to infinity or is bounded by poles.

• ROC cannot contain poles.

Relationship between ROC and Signal Nature

Laplace Transform of Standard Signals

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

• t u(t): X(s) = 1/s², ROC: Re(s) > 0.

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

[!CAUTION] For e^{-at} u(t) and e^{-at} u(-t), ROC never overlaps. Same X(s) but different ROC → different signals.

4.2 Properties of Laplace Transform

Convolution in Laplace Domain

$$ x(t) * h(t) \leftrightarrow X(s) H(s) $$

ROC contains at least the intersection of ROCs of X(s) and H(s).

4.3 Inverse Laplace Transform

Methods:

1. Partial Fraction Expansion (for rational functions):

   • Expand X(s) into simpler terms (e.g., A/(s+a), B/(s+b)²).

   • Use table lookup.

   • For repeated poles: include terms like A/(s+a) + B/(s+a)².

2. Using Laplace Transform Tables: Direct lookup for standard forms.

[!TIP] Always consider ROC when doing inverse. ROC determines time-domain nature (causal/anti-causal).

V. Z-TRANSFORM (DT)

5.1 Definition & Region of Convergence (ROC)

Definition

• Bilateral:

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

• Unilateral (for causal sequences/initial conditions):

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

Region of Convergence (ROC)

• Set of z values for which sum converges.

• Importance:

  1. Existence of X(z).

  2. Causality: For causal sequence, ROC is outside outermost pole (including ∞).

  3. Stability: For causal LTI, stability ⇔ ROC includes unit circle |z|=1.

  4. Finite duration: Entire z-plane except possibly z=0 or z=∞.

Properties of ROC

• ROC is ring/disk centered at origin: r₁ < |z| < r₂.

• No poles in ROC.

• For rational X(z), ROC extends to infinity or is bounded by poles.

• ROC cannot contain poles.

Relationship between ROC and Sequence Nature

Z-Transform of Standard Sequences

• a^n u[n]: X(z) = 1/(1 - a z^{-1}), ROC: |z| > |a| (causal).

• -a^n u[-n-1]: X(z) = 1/(1 - a z^{-1}), ROC: |z| < |a| (anti-causal).

• n a^n u[n]: X(z) = a z^{-1} / (1 - a z^{-1})², ROC: |z| > |a|.

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

[!CAUTION] Same X(z) but different ROC → different sequences. E.g., 1/(1 - 0.5 z^{-1}) with ROC |z|>0.5 is causal (0.5)^n u[n]; with ROC |z|<0.5 is anti-causal -(0.5)^n u[-n-1].

5.2 Properties of Z-Transform

5.3 Inverse Z-Transform

Methods:

1. Partial Fraction Expansion (for rational functions):

   • Expand X(z) as sum of terms like A/(1 - a z^{-1}).

   • Use table lookup for inverse.

   • For repeated poles: include A/(1 - a z^{-1}) + B/(1 - a z^{-1})² + ....

2. Long Division (power series expansion): For ROC |z| > r₀ (causal), divide to get positive powers of z^{-1}.

3. Using Z-Transform Tables: Direct lookup.

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

[!TIP] For causal sequences, use long division or partial fraction with ROC |z| > max pole. For anti-causal, partial fraction with ROC |z| < min pole.

VI. SAMPLING & RECONSTRUCTION

6.1 Sampling of Continuous-Time Signals

Ideal Impulse Sampling

• Sampled signal: x_s(t) = x(t) ∑_{n=-∞}^{∞} δ(t - nT).

• In frequency domain: X_s(jω) = (1/T) ∑_{k=-∞}^{∞} X(j(ω - kωₛ)), where ωₛ = 2π/T.

• Effect: Replication of X(jω) at multiples of ωₛ.

Natural Sampling & Flat-Top Sampling

• Natural: x_s(t) = ∑_{n} x(nT) rect((t-nT)/T) (sample-and-hold).

• Flat-Top: x_s(t) = ∑_{n} x(nT) h(t - nT), where h(t) is rectangular pulse.

• Both are practical (non-ideal) versions of impulse sampling.

Sampling Theorem (Nyquist-Shannon)

• A bandlimited signal x(t) with no frequencies above ω_max can be perfectly reconstructed from its samples if sampling frequency ωₛ > 2ω_max.

• Nyquist Rate: ωₙ = 2ω_max (minimum sampling rate).

• Aliasing: Occurs when ωₛ < 2ω_max. High frequencies fold back into lower frequencies.

$$ \text{Aliased frequency } ω_a = |ω - kωₛ| \text{ for some } k \text{ such that } |ω_a| ≤ ωₛ/2. $$

[!CAUTION] Aliasing is irreversible. Once aliased, original frequencies cannot be recovered. Always use anti-aliasing filter (low-pass with cutoff ω_max) before sampling.

6.2 Signal Reconstruction

Reconstruction from Ideally Sampled Signal

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

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

where sinc(t) = sin(πt)/(πt).

• Ideal LPF with cutoff ω_c = ωₛ/2 (if ωₛ > 2ω_max):

$$ H_r(jω) = T \text{rect}\left(\frac{ω}{ωₛ}\right) \quad \text{for } |ω| ≤ ωₛ/2, \text{ else } 0. $$

VII. FILTERS

7.1 Analog Filters

Classification by Frequency Response

Ideal vs. Non-Ideal Filters

• Ideal: Brick-wall response (abrupt transitions), non-causal, unstable (non-rational).

• Non-Ideal (Realizable): Smooth transitions, causal, stable (rational transfer function). Trade-offs: passband ripple, stopband attenuation, transition width.

Properties of LTI Systems Related to Filtering

• Causality: For analog, requires h(t)=0 for t<0. Ideal filters are non-causal.

• Stability: Requires ∫|h(t)|dt < ∞. Ideal filters are unstable (impulse response not absolutely integrable).

• Phase Linearity: Constant group delay τ_g = -dθ(ω)/dω. FIR filters can have exact linear phase; IIR generally cannot.

7.2 Digital Filters

IIR vs. FIR

Realization Structures for DT LTI Systems

• Direct Form I: From difference equation directly (separate delays for numerator/denominator).

• Direct Form II: Combined delays (fewer delays).

• Cascade Form: H(z) = H₁(z) H₂(z) ... (series of second-order sections).

• Parallel Form: H(z) = ∑ H_k(z) (sum of sections).

• Advantages: Cascade/parallel reduce coefficient sensitivity, allow modular design.

Applications

• IIR: Audio equalization, communications (where phase less critical).

• FIR: Data transmission (linear phase essential), image processing.

VIII. ADVANCED TOPICS & REPRESENTATIONS

8.1 State-Variable Representation

State, State Vector, State Equations

• State: Minimal set of variables x₁(t), x₂(t), ..., xₙ(t) that summarize past history.

• State Vector: 𝐱(t) = [x₁(t) x₂(t) ... xₙ(t)]ᵀ.

• State-Space Model:

  • CT LTI:

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

$$ y(t) = \mathbf{C} \mathbf{x}(t) + \mathbf{D} u(t) $$

• DT LTI:

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

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

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

Advantages over Input-Output Description

1. Handles multiple inputs/outputs (MIMO) naturally.

2. Incorporates initial conditions explicitly.

3. Provides insight into internal behavior (stability via eigenvalues of 𝐀).

4. Suitable for optimal control and estimation (Kalman filter).

5. Easily extends to nonlinear systems.

8.2 Wavelet Transform

Comparison with Fourier & Laplace

Basic Concept

• Wavelet Function ψ(t) (mother wavelet): Zero-mean, localized in time and frequency.

• Scaled & Translated:

$$ ψ_{a,b}(t) = \frac{1}{\sqrt{|a|}} ψ\left(\frac{t - b}{a}\right), \quad a ≠ 0 $$

where a (scale), b (translation).

• Continuous Wavelet Transform (CWT):

$$ W(a,b) = \int_{-\infty}^{\infty} x(t) ψ_{a,b}^*(t) dt $$

Application Example

• Signal Compression: JPEG2000 uses wavelets (Daubechies) for better compression than DCT (JPEG).

• Transient Detection: Wavelets localize sudden changes (edges in images, fault detection).

8.3 Short Notes on Specific Topics

DTFT and its Properties

• Definition: X(e^{jω}) = ∑_{n=-∞}^{∞} x[n] e^{-jωn}, periodic with period 2π.

• Key Properties: Periodicity (X(e^{j(ω+2π)}) = X(e^{jω})), symmetry for real x[n] (X(e^{jω}) = X^*(e^{-jω})), convolution theorem.

• Importance: Frequency analysis of DT signals, filter design.

Aliasing (Detailed)

• Cause: Sampling rate ωₛ < 2ω_max.

• Effect in Frequency Domain: Spectral replicas overlap. Original spectrum X(jω) cannot be separated from aliased components.

• Mathematical: X_s(jω) = (1/T) ∑_{k=-∞}^{∞} X(j(ω - kωₛ)). Overlap when ωₛ < 2ω_max.

• Avoidance:

  1. Anti-aliasing filter: Analog LPF before sampling, cutoff ω_c ≤ ωₛ/2.

  2. Increase sampling rate: ωₛ > 2ω_max.

Even and Odd Signals (Detailed)

• Even: Symmetric about y-axis. x_e(t) = x_e(-t). Fourier series contains only cosine terms (if real).

• Odd: Anti-symmetric about origin. x_o(t) = -x_o(-t). Fourier series contains only sine terms (if real).

• Decomposition: x(t) = x_e(t) + x_o(t) always possible. Useful in simplifying integrals (odd × even = 0 over symmetric limits).

Invertible and Inverse System

• Invertible: One-to-one mapping from input to output. Inverse system S⁻¹ exists such that S⁻¹{S{x(t)}} = x(t).

• Example: y(t) = 2x(t) is invertible (x(t) = y(t)/2). y(t) = x²(t) is non-invertible (sign lost).

• LTI Invertibility: Transfer function H(s) must have no zeros on jω-axis (for stability) and be minimum phase? Actually, invertible if H(s) ≠ 0 for all s in ROC. Inverse system has transfer function 1/H(s).

IX. PROBLEM-SOLVING & DERIVATIONS (High-Frequency Exam Topics)

1. Determining Signal Properties

• Periodicity Check:

  • CT: x(t) = A sin(ω₀ t + φ) → T₀ = 2π/ω₀. For sum x(t) = sin(ω₁t) + sin(ω₂t), periodic if ω₁/ω₂ rational.

  • DT: x[n] = A sin(ω₀ n + φ) → periodic if ω₀/(2π) rational. Find smallest integer N such that ω₀ N = 2π k for integer k.

• Energy/Power:

  • Energy: ∫|x(t)|² dt finite? Check if signal has finite duration and bounded.

  • Power: lim_{T→∞} (1/(2T)) ∫_{-T}^{T} |x(t)|² dt finite? Periodic signals → power.

2. Transform Derivations & Calculations

• FT of e^{-at} u(t):

$$ X(jω) = \int_{0}^{\infty} e^{-at} e^{-jωt} dt = \int_{0}^{\infty} e^{-(a+jω)t} dt = \frac{1}{a + jω}, \quad \text{Re}(a) > 0 $$

• Laplace of t u(t):

$$ X(s) = \int_{0}^{\infty} t e^{-st} dt = \frac{1}{s^2}, \quad \text{Re}(s) > 0 $$

• Z-transform of (n+1) u[n]:

$$ X(z) = \sum_{n=0}^{\infty} (n+1) z^{-n} = \frac{1}{(1 - z^{-1})^2}, \quad |z| > 1 $$

3. System Analysis Problems

• Finding Impulse Response from Differential Equation:

  Given d³y/dt³ + 4 d²y/dt² + 2 dy/dt + (1/3)y(t) = x(t), take Laplace with zero IC:

$$ (s³ + 4s² + 2s + 1/3) Y(s) = X(s) \Rightarrow H(s) = \frac{Y(s)}{X(s)} = \frac{1}{s³ + 4s² + 2s + 1/3} $$

Inverse Laplace → h(t).

• Stability/Causality from H(s) or H(z):

  • Causal: Poles in left half s-plane (CT) or inside unit circle (DT).

  • Stable: ROC includes jω-axis (CT) or unit circle (DT). For causal, check if all poles in LHP (CT) or inside unit circle (DT).

• Finding h[n] from H(z):

  Given H(z) = (3 - 4z⁻¹)/(1 - 3.5z⁻¹ + 1.5z⁻²).

  • Causal: ROC |z| > 1.5 (outside outermost pole at z=1.5). Partial fraction → h[n] = (6)(0.5)^n u[n] - (3)(1)^n u[n]? Actually, poles at z=1.5 and z=1? Solve denominator: 1 - 3.5z⁻¹ + 1.5z⁻² = 0 → z² - 3.5z + 1.5 = 0 → (z-1)(z-1.5)=0. So poles at z=1, 1.5. For causal, ROC |z|>1.5 → both poles inside? Actually |z|>1.5 means outside 1.5, so pole at 1.5 is on boundary? ROC cannot include poles. For causal stable, need ROC |z|>1.5 but pole at 1.5 is on boundary → not stable. For stable causal, need all poles inside unit circle? Here pole at 1.5 > 1 → unstable. For stable system: ROC must include unit circle, so need |z|>1.5? No, unit circle |z|=1 is inside |z|>1.5? Actually |z|>1.5 does not include |z|=1 (since 1 < 1.5). So no stable causal ROC. For causal (not necessarily stable): ROC |z|>1.5. Then h[n] has terms (1.5)^n u[n] and (1)^n u[n] → grows → unstable. For stable (not necessarily causal): ROC |z|<1 (inside innermost pole at z=1)? Poles at 1 and 1.5, so annulus 1<|z|<1.5? But that doesn't include unit circle. Actually unit circle |z|=1 is less than 1? |z|=1 is inside |z|<1? No, |z|<1 is interior of unit circle. But pole at z=1 is on unit circle → not stable. So no ROC includes unit circle → system unstable for any realization. But problem asks: "Determine h(n) for: i) The system is stable (4m)" → implies stable exists? Possibly misprint or different H(z). In exam, carefully compute poles and choose ROC accordingly.

4. Convolution Problems

• Linear Convolution: For finite sequences, use tabular method or graphical.

  Example: x[n] = {1, 2, 3}, h[n] = {1, 1} → y[n] = {1, 3, 5, 3}.

• Using Convolution Theorem: Compute Y(jω) = X(jω) H(jω), then inverse FT.

EXAM STRATEGY & COMMON PITFALLS

[!TIP] Always:

1. State definitions clearly before solving.

2. For ROC problems, draw pole-zero plot and shade ROC.

3. For system properties, show test steps (e.g., for time-invariance: replace t by t-t₀ and compare).

4. For convolution, flip and slide method for CT/DT.

5. For Fourier Series of sin(ω₀t), remember only c₁ and c₋₁ are non-zero.

[!CAUTION] Avoid:

• Confusing bilateral vs unilateral transforms (Laplace/Z).

• Forgetting initial/final value theorem conditions.

• Misidentifying causality from differential/difference equations (check if output depends on future input).

• Assuming all LTI systems are stable (check BIBO).

• In periodicity for DT, forgetting N must be integer.

High-Yield Derivations to Practice:

1. FT of e^{-at} u(t), t u(t), cos(ω₀t).

2. Laplace of e^{-at} u(t), t u(t), sin(ω₀t) u(t).

3. Z-transform of a^n u[n], n a^n u[n], δ[n].

4. Relationship between step response and impulse response.

5. Convolution theorem proof (FT and Laplace).

Short Notes Topics (14m questions):

• Be concise: 1-2 paragraphs + key formulas.

• For Wavelet Transform: Emphasize multi-resolution vs Fourier.

• For DTFT: Stress periodicity 2π and relation to DFS.

• For Digital Filters: Contrast IIR/FIR in table form.

• For Analog Filters: Sketch ideal vs real magnitude responses.

• For Aliasing: Diagram showing frequency folding.

Final Reminder: In RGPV exams, step-wise derivation and clear diagrams (block diagrams, pole-zero plots, filter responses) fetch full marks. Always relate concepts to LTI systems as the central theme.