Signals and Systems (EX-302) - Unit 4 Short Notes

UNIT 4: SIGNALS AND SYSTEMS - EXAM-FOCUSED SHORT NOTES

I. FUNDAMENTALS OF SIGNALS

1.1 Definition & Classification of Signals

A signal is a function of one or more independent variables that conveys information.

1.2 Basic Signal Operations

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

• Time Scaling: x(at) → compression (a>1) or expansion (0<a<1).

• Time Reversal: x(-t) → reflection about origin.

• Amplitude Operations: Addition, Multiplication, Differentiation (CT), Integration (CT), Differencing (DT: x[n] - x[n-1]).

[!TIP] Common Pitfall: x(2t-1) is not first shift then scale. Correct order: x(2(t-½)) → shift by ½, then scale by 2.

1.3 Standard Signals

II. ANALYSIS OF SIGNALS: FOURIER SERIES & TRANSFORM

2.1 Fourier Series (FS)

• Dirichlet's Conditions: For FS to exist, x(t) must be:

  1. Single-valued, finite.

  2. Finite number of discontinuities in one period.

  3. Finite number of maxima/minima in one period.

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

• Trigonometric FS: x(t) = a₀ + Σ_{n=1}^{∞} [a_n cos(nω₀t) + b_n sin(nω₀t)].

  a₀ = (1/T₀)∫_{T₀}x(t)dt, a_n = (2/T₀)∫_{T₀}x(t)cos(nω₀t)dt, b_n = (2/T₀)∫_{T₀}x(t)sin(nω₀t)dt.

• Exponential FS: x(t) = Σ_{n=-∞}^{∞} c_n e^{jnω₀t}, where c_n = (1/T₀)∫_{T₀}x(t)e^{-jnω₀t}dt.

• Example: x(t) = A sin(ω₀t) → c_n = (A/(2j))[δ[n-1] - δ[n+1]] (using Euler).

• Limitations: Only for periodic signals. Convergence at discontinuities (Gibbs phenomenon). Requires infinite terms for perfect reconstruction.

2.2 Fourier Transform (FT)

Definition: X(jω) = ∫_{-∞}^{∞} x(t) e^{-jωt} dt. Inverse: x(t) = (1/2π)∫_{-∞}^{∞} X(jω) e^{jωt} dω.

Properties (MOST FREQUENT):

FT of Standard Signals:

• rect(t/τ) ↔ τ sinc(ωτ/2)

• tri(t/τ) ↔ τ sinc²(ωτ/2)

• u(t) ↔ πδ(ω) + 1/(jω)

• sgn(t) ↔ 2/(jω)

• e^{-at}u(t) (Re(a)>0) ↔ 1/(s+a) with s=jω

• cos(ω₀t) ↔ π[δ(ω-ω₀) + δ(ω+ω₀)]

• sin(ω₀t) ↔ jπ[δ(ω+ω₀) - δ(ω-ω₀)]

[!TIP] Convolution Theorem Proof: Use definition: (x*h)(t) = ∫x(τ)h(t-τ)dτ. Take FT, swap integrals, recognize product.

2.3 Comparison: Fourier Series vs. Fourier Transform

III. ANALYSIS OF SIGNALS: LAPLACE TRANSFORM

3.1 Definition & Region of Convergence (ROC)

Definition: X(s) = ∫_{-∞}^{∞} x(t) e^{-st} dt, where s = σ + jω. ROC: Set of s values for which integral converges. CRITICAL for stability & causality.

Properties of ROC:

1. ROC is a vertical strip in s-plane (for rational X(s)).

2. ROC cannot contain poles.

3. ROC is right-sided (including ∞) for causal signals.

4. ROC is left-sided (including -∞) for anti-causal signals.

5. ROC is two-sided (strip between poles) for non-causal, finite-duration signals.

6. ROC is connected (no gaps).

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

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

Pole-Zero Plot & ROC: Poles (X(s)→∞) must be excluded from ROC. ROC extends from rightmost pole to ∞ for causal systems.

3.2 Properties of Laplace Transform

Theorems:

• Initial Value Theorem (IVT): x(0⁺) = lim_{s→∞} s X(s). (Requires sX(s) has no poles at ∞).

• Final Value Theorem (FVT): x(∞) = lim_{s→0} s X(s). (Requires poles of sX(s) in Re(s)<0 and possibly at s=0).

3.3 Inverse Laplace Transform

Methods:

1. Partial Fraction Expansion (PFE): For rational X(s). Expand into simpler terms (e.g., 1/(s+a), 1/s²), then use table.

2. Convolution Integral: x(t) = ∫_{-∞}^{∞} x₁(τ) x₂(t-τ) dτ.

3. Bromwich Integral: x(t) = (1/2πj) ∫_{σ=j∞}^{σ=-j∞} X(s) e^{st} ds (theoretical).

Example: X(s) = (s+3)/((s+1)(s+2)) → PFE: = 2/(s+1) - 1/(s+2) → x(t) = [2e^{-t} - e^{-2t}]u(t) (assuming ROC: σ > -1).

3.4 Derivation for Standard Signals

• x(t) = e^{-at}u(t): X(s) = ∫_{0}^{∞} e^{-at} e^{-st} dt = ∫_{0}^{∞} e^{-(s+a)t} dt = [ -1/(s+a) e^{-(s+a)t} ]_{0}^{∞} = 1/(s+a). ROC: Re(s+a) > 0 → σ > -a.

• x(t) = t u(t): X(s) = ∫_{0}^{∞} t e^{-st} dt. Use integration by parts: = [-t/s e^{-st}]_{0}^{∞} + (1/s)∫_{0}^{∞} e^{-st} dt = 0 + (1/s)(1/s) = 1/s². ROC: σ > 0.

IV. ANALYSIS OF SIGNALS: Z-TRANSFORM

4.1 Definition & Region of Convergence (ROC)

Definition: X(z) = Σ_{n=-∞}^{∞} x[n] z^{-n}, where z = re^{jω}. ROC: Set of z values for which sum converges.

Properties of ROC:

1. ROC is a ring/annulus in z-plane (for rational X(z)).

2. ROC cannot contain poles.

3. ROC includes |z| = ∞ for right-sided sequences (finite n start).

4. ROC includes |z| = 0 for left-sided sequences (finite n end).

5. ROC is two-sided (ring between poles) for two-sided sequences.

6. ROC is connected.

Example: x[n] = aⁿ u[n] → X(z) = 1/(1 - a z^{-1}) = z/(z-a). ROC: |z| > |a| (right-sided, causal).

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

Pole-Zero Plot & ROC: For causal systems, ROC is outside the outermost pole. For anti-causal, ROC is inside the innermost pole.

4.2 Properties of Z-Transform (MOST FREQUENT)

4.3 Inverse Z-Transform

Methods:

1. Long Division: For |z| > |a| (causal). X(z) = Σ x[n] z^{-n} → read coefficients.

2. Partial Fraction Expansion (PFE): For rational X(z). Expand into terms like 1/(1 - a z^{-1}).

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

Example: X(z) = z/(z-a) → = 1/(1 - a z^{-1}) → x[n] = aⁿ u[n] (ROC: |z|>|a|).

4.4 Z-Transform of Standard Sequences

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

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

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

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

• Example: x[n] = (-1)ⁿ cos(πn/3) u[n] → Use Euler: cos(θ) = (e^{jθ}+e^{-jθ})/2. <br> x[n] = ½[(e^{jπ/3}(-1))ⁿ + (e^{-jπ/3}(-1))ⁿ] u[n]. <br> (-1) = e^{jπ}, so e^{jπ/3} e^{jπ} = e^{j4π/3}, e^{-jπ/3} e^{jπ} = e^{j2π/3}. <br> X(z) = ½[1/(1 - e^{j4π/3} z^{-1}) + 1/(1 - e^{j2π/3} z^{-1})], ROC: |z| > 1 (since |e^{jθ}|=1).

V. LINEAR TIME-INVARIANT (LTI) SYSTEMS

5.1 Definition & Properties

• Linearity: Superposition (T{a x₁ + b x₂} = a T{x₁} + b T{x₂}) & Homogeneity.

• Time-Invariance: Shift in input → identical shift in output. T{x(t-t₀)} = y(t-t₀).

• Causality: Output depends only on present/past inputs. For CT: h(t)=0 for t<0. For DT: h[n]=0 for n<0.

• Stability (BIBO): Bounded-input bounded-output. Condition: ∫_{-∞}^{∞} \|h(t)\| dt < ∞ (CT) or Σ_{n=-∞}^{∞} \|h[n]\| < ∞ (DT). Equivalent to ROC of H(s)/H(z) includes jω-axis/unit circle.

• Memoryless: Output depends only on current input (y(t)=f(x(t))).

• Invertible: Exists inverse system T⁻¹ such that T⁻¹{T{x}} = x.

5.2 System Representations (CT)

• Differential Equation: Σ_{k=0}^{N} a_k d^k y(t)/dt^k = Σ_{k=0}^{M} b_k d^k x(t)/dt^k. Order = N.

  • Example: d³y/dt³ + 4 d²y/dt² + 2 dy/dt + (1/3)y(t) = x(t) → Order = 3.

• Block Diagrams:

  DiagramCANVAS: Show Direct Form I (delays in parallel), Direct Form II (cascade of sections), Cascade (series H₁(z)H₂(z)), Parallel (sum of branches).

• State-Space:

  • State Equation: ẋ(t) = A x(t) + B u(t).

  • Output Equation: y(t) = C x(t) + D u(t).

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

  • Conversion from Transfer Function: H(s) = (s² + 3s + 2)/(s³ + 4s² + 2s + 1/3) → Write in controllable canonical form: A = [[0,1,0],[0,0,1],[-1/3,-2,-4]], B = [0,0,1]^T, C = [1,3,2], D=0.

5.3 System Representations (DT)

• Difference Equation: Σ_{k=0}^{N} a_k y[n-k] = Σ_{k=0}^{M} b_k x[n-k]. Order = N.

• Block Diagrams: Similar forms (Direct I/II, Cascade, Parallel).

• State-Space (DT): x[n+1] = A x[n] + B u[n], y[n] = C x[n] + D u[n].

5.4 System Analysis Using Transforms

• Transfer Function (CT): H(s) = Y(s)/X(s) (zero IC). H(s) = L{h(t)}.

• System Function (DT): H(z) = Y(z)/X(z). H(z) = Z{h[n]}.

• Finding Impulse Response:

  • From Differential/Difference Eq: Apply x(t)=δ(t) or x[n]=δ[n] and solve.

  • From H(s)/H(z): Take inverse transform.

  • Example (from integral): y(t)=∫_{-∞}^{t} e^{-2(t-τ)} x(τ-3) dτ. <br> Rewrite: y(t) = e^{-2t} ∫_{-∞}^{t} e^{2τ} x(τ-3) dτ. Let v = τ-3 → τ = v+3, dτ=dv, limits -∞ to t-3: <br> y(t) = e^{-2t} ∫_{-∞}^{t-3} e^{2(v+3)} x(v) dv = e^{-2(t-3)} ∫_{-∞}^{t-3} e^{-2(t-3-v)} x(v) dv. <br> This is convolution y(t) = [e^{-2t}u(t)] * [x(t-3)]. <br> So h(t) = e^{-2(t-3)}u(t-3) = e^{-6} e^{-2t} u(t-3).

  • Example (from H(z)): H(z) = (3-4z⁻¹)/(1-3.5z⁻¹+1.5z⁻²). <br> Factor denominator: 1 - 3.5z⁻¹ + 1.5z⁻² = (1 - 0.5z⁻¹)(1 - 3z⁻¹). Poles at z=0.5, 3. <br> i) Stable: ROC includes unit circle → |z|=1 inside ROC. Poles at 0.5 (inside) and 3 (outside). For causality, ROC must be |z|>3 (outside outermost pole) but that excludes unit circle. So stable & causal impossible simultaneously. <br> Stable only: ROC is ring between poles: 0.5 < |z| < 3. Then h[n] is anti-causal (two-sided). Use PFE: H(z) = A/(1-0.5z⁻¹) + B/(1-3z⁻¹). Solve: A=2, B=1. <br> h[n] = [2(0.5)ⁿ + 1(3)ⁿ] u[-n-1] (since ROC |z|<3 for second term → anti-causal). <br> ii) Causal: ROC |z|>3. Then h[n] = [2(0.5)ⁿ + 1(3)ⁿ] u[n] (both terms causal). But unstable (pole at 3 outside unit circle).

• Determining System Characteristics from H(s)/H(z):

  • Causality (CT): ROC is right-sided (from Re(s)=σ₀ to ∞). Poles must be in ROC? No, poles excluded. Check: If H(s) proper (deg(N) ≤ deg(D)), and ROC right of rightmost pole → causal.

  • Stability: ROC includes jω-axis (CT) or unit circle (DT).

  • Damping (CT): From poles s = -σ ± jω_d. <br> Over-damped: Real distinct poles (ω_d=0, σ₁≠σ₂). <br> Under-damped: Complex poles (ω_d≠0). <br> Critically damped: Repeated real pole.

5.5 Interconnection of LTI Systems

• Series/Cascade: H_total(s) = H₁(s) H₂(s).

• Parallel: H_total(s) = H₁(s) + H₂(s).

• Feedback: H_total(s) = H₁(s) / (1 ± H₁(s)H₂(s)). Negative feedback uses - in denominator.

VI. CONVOLUTION

6.1 Definition & Significance

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

• DT Linear Convolution: y[n] = Σ_{k=-∞}^{∞} x[k] h[n-k] = (x * h)[n].

• Significance: Output of LTI system for any input x(t) if impulse response h(t) is known.

6.2 Properties

• Commutative: x * h = h * x.

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

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

• x * δ = x.

• Convolution of even/odd: EvenEven=Even, EvenOdd=Odd, Odd*Odd=Even.

6.3 Convolution Theorems

6.4 Methods for Computing Convolution

1. Graphical (Flip-Shift-Multiply-Integrate/Sum):

   DiagramCANVAS: Show `x(τ)` and `h(t-τ)` for fixed `t`, sliding `t`.

2. Tabular (DT): Useful for finite sequences. List x[k] and h[n-k] values, multiply and sum diagonally.

3. Using Transforms: Find X(z), H(z), multiply, then inverse Z-transform.

VII. SAMPLING & ALIASING

7.1 Sampling of CT Signals

• Ideal Sampling (Impulse Sampling): x_s(t) = x(t) Σ_{n=-∞}^{∞} δ(t - nT_s) = Σ_{n=-∞}^{∞} x(nT_s) δ(t - nT_s).

• Spectrum of Sampled Signal: X_s(jω) = (1/T_s) Σ_{k=-∞}^{∞} X(j(ω - kω_s)), where ω_s = 2π/T_s.

  • Replicates of X(jω) shifted by multiples of ω_s, scaled by 1/T_s.

7.2 Nyquist-Shannon Sampling Theorem

A bandlimited signal with maximum frequency ω_m can be reconstructed perfectly from its samples iff ω_s > 2ω_m (or f_s > 2f_m).

• Nyquist Rate: ω_N = 2ω_m.

• Nyquist Frequency: ω_m (highest frequency in signal).

7.3 Aliasing

• Definition: Overlap of shifted replicas in X_s(jω) due to ω_s < 2ω_m. Causes irreversible distortion; high frequencies masquerade as low frequencies.

• Cause: Undersampling (f_s < 2f_m).

• Effect in Frequency Domain: X_s(jω) components from k≠0 overlap with baseband |ω|<ω_m.

• Avoidance: Use anti-aliasing filter (ideal LPF with cutoff ω_c < ω_s/2) before sampling.

7.4 Signal Reconstruction

• Ideal Reconstruction (Sinc Interpolation): x(t) = Σ_{n=-∞}^{∞} x(nT_s) sinc((t - nT_s)/T_s).

• Practical Reconstructions:

  • Zero-Order Hold (ZOH): Holds sample value for T_s. H_r(jω) = T_s sinc(ωT_s/2) e^{-jωT_s/2}.

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

VIII. ANALOG & DIGITAL FILTERS

8.1 Analog Filters

• Need: Remove unwanted frequencies, shape signal spectrum.

• Classification: LPF, HPF, BPF, BSF, All-pass.

• Approximations (Butterworth, Chebyshev, Elliptic):

  | Filter | Passband Ripple | Stopband Ripple | Transition | Key Feature | | :--- | :--- | :--- | :--- | :--- | | Butterworth | 0 dB (maximally flat) | Monotonic | Smooth | No ripple, gradual roll-off | | Chebyshev I | Equiripple | Monotonic | Sharper than Butterworth | Passband ripple for sharper transition | | Chebyshev II | Monotonic | Equiripple | Sharper than Butterworth | Stopband ripple | | Elliptic (Cauer) | Equiripple | Equiripple | Sharpest | Ripple in both bands, complex design |

• Order: Higher order → sharper transition, more components, more sensitive to component variations.

8.2 Digital Filters

• FIR Filters:

  • Structure: Non-recursive (y[n] = Σ_{k=0}^{M} b_k x[n-k]).

  • Properties: Always stable (poles only at z=0), can have exact linear phase (symmetric/anti-symmetric coefficients), no limit cycle oscillations.

  • Design Methods: Window method (truncate IIR impulse response), Frequency sampling.

• IIR Filters:

  • Structure: Recursive (y[n] = Σ_{k=0}^{M} b_k x[n-k] - Σ_{k=1}^{N} a_k y[n-k]).

  • Properties: Can be unstable (poles outside unit circle), cannot have perfect linear phase, more efficient (lower order for same spec), susceptible to quantization noise & limit cycles.

  • Design: Impulse Invariant, Bilinear Transform (from analog prototypes like Butterworth).

• FIR vs. IIR Comparison:

  | Aspect | FIR | IIR | | :--- | :--- | :--- | | Stability | Always stable | Conditional (poles inside unit circle) | | Phase | Can be linear | Non-linear (except all-pass) | | Order | Higher for sharp specs | Lower for same specs | | Implementation | Non-recursive | Recursive (feedback) | | Design | Direct (window, optimal) | From analog prototypes |

8.3 Non-Ideal Frequency Selective Filters

• Specifications:

  • Passband: 0 ≤ ω ≤ ω_p. Ripple δ_p (max deviation).

  • Stopband: ω_s ≤ ω ≤ π. Ripple δ_s (max deviation).

  • Transition Band: ω_p < ω < ω_s.

  • Stopband Attenuation: A_s = -20 log₁₀(δ_s) (dB).

• Design Goal: Minimize order N to meet δ_p, δ_s, ω_p, ω_s.

IX. ADVANCED TOPICS & SPECIAL TRANSFORMS

9.1 Discrete-Time Fourier Transform (DTFT)

Definition: X(e^{jω}) = Σ_{n=-∞}^{∞} x[n] e^{-jωn}. Period: X(e^{j(ω+2π)}) = X(e^{jω}). Properties:

• Periodicity (2π), Linearity, Time Shifting (x[n-n₀] ↔ e^{-jωn₀}X(e^{jω})), Frequency Shifting (e^{jω₀n}x[n] ↔ X(e^{j(ω-ω₀)})).

• Conjugation (x*[n] ↔ X*(e^{-jω})), Convolution (x[n]*h[n] ↔ X(e^{jω})H(e^{jω})).

• Parseval's: Σ\|x[n]\|² = (1/2π) ∫_{-π}^{π} \|X(e^{jω})\|² dω.

• Differentiation in Frequency: n x[n] ↔ j dX(e^{jω})/dω.

• Relationship: X(e^{jω}) = X(z)|_{z=e^{jω}}. DTFT is Z-transform on unit circle.

9.2 Wavelet Transform

• Motivation: FT gives global frequency info. STFT gives local but with fixed window. Wavelet gives multi-resolution analysis (variable window: good for transients).

• Definition (CWT): W(a,b) = (1/√|a|) ∫ x(t) ψ*((t-b)/a) dt.

  • ψ(t): Mother wavelet (zero mean, finite energy).

  • a: Scale (inverse to frequency: large a → low freq, small a → high freq).

  • b: Translation (time shift).

• Comparison:

  • FT: Basis functions: e^{jωt} (infinite support, fixed width). Good for stationary signals.

  • STFT: Basis: windowed complex exponentials (fixed width). Compromise time-frequency.

  • Wavelet: Basis: scaled/shifted wavelets (variable width). Excellent for transients, singularities.

• Example: Haar Wavelet (simplest): ψ(t) = 1 for 0≤t<½, -1 for ½≤t<1, 0 elsewhere.

X. SHORT NOTES ON HIGH-FREQUENCY TOPICS

10.1 Inverse Z-Transform (Methods & Examples)

• Methods: Long Division (causal ROC |z|>r), Partial Fraction (for rational X(z)), Contour Integration.

• Example: X(z) = z/((z+1)(z+2)). PFE: = A/(z+1) + B/(z+2). A = -1, B = 2. <br> X(z) = -1/(z+1) + 2/(z+2) = -z⁻¹/(1+z⁻¹) + 2z⁻¹/(1+2z⁻¹). <br> For ROC |z|>2 (causal): x[n] = [-(-1)ⁿ + 2(-2)ⁿ] u[n] = [(-1)^{n+1} + 2(-2)ⁿ] u[n].

10.2 DTFT and its Properties

• Definition: X(e^{jω}) = Σ_{n=-∞}^{∞} x[n] e^{-jωn}. ω normalized radian freq (-π to π).

• Key Properties: Periodicity (2π), Conjugation, Time reversal (x[-n] ↔ X(e^{-jω})), Real signals (x[n] real → X(e^{-jω}) = X*(e^{jω})), Parseval's.

• Example: x[n] = aⁿ u[n], |a|<1 → X(e^{jω}) = 1/(1 - a e^{-jω}).

10.3 Digital Filters (FIR/IIR)

• FIR: Non-recursive, always stable, linear phase possible (symmetric coefficients). Design: Window, Frequency Sampling, Optimal (Parks-McClellan).

• IIR: Recursive, can be unstable, no perfect linear phase. Design: Impulse Invariant (aliasing), Bilinear Transform (no aliasing, warping).

• Comparison Table: See Section 8.2.

10.4 Analog Filters (Types & Applications)

• Butterworth: Maximally flat passband. Audio, data conversion anti-aliasing.

• Chebyshev I: Equiripple passband, sharper transition. Communications where passband ripple tolerable.

• Chebyshev II: Equiripple stopband. Where stopband attenuation critical.

• Elliptic: Sharpest transition, ripple both bands. Specialized applications.

• Applications: Audio equalization, image processing, biomedical signal filtering, communication channel equalization.

10.5 Aliasing

• Definition: Overlap of spectral replicas during sampling when ω_s < 2ω_m.

• Cause: Undersampling.

• Effect: High-frequency components appear as low-frequency artifacts. Irreversible without original signal.

• Avoidance: Pre-filter with anti-aliasing LPF (ω_c < ω_s/2), sample at ω_s > 2ω_m.

10.6 Even and Odd Signals

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

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

• Decomposition: x(t) = ½[x(t)+x(-t)] + ½[x(t)-x(-t)] = x_e(t) + x_o(t).

• Properties: EvenEven=Even, EvenOdd=Odd, Odd*Odd=Even. Integral of odd over symmetric limits = 0.

10.7 Wavelet Transform

• Motivation: Time-frequency localization. FT: global freq. STFT: fixed window. Wavelet: variable window (multi-resolution).

• CWT: W(a,b) = (1/√|a|) ∫ x(t) ψ*((t-b)/a) dt.

• Advantage: Good for analyzing transients, edges in images, non-stationary signals.

• Example: Haar wavelet (step function). Daubechies, Morlet common.

[!IMPORTANT] Exam Strategy:

1. For 7-mark questions: Always start with definition, then properties (list 4-5), then example/derivation.

2. For "determine" questions: Show step-by-step. For periodicity, find fundamental periods. For energy/power, compute integrals/sums.

3. For ROC problems: Draw pole-zero plot. State causality/stability based on ROC location relative to poles and unit circle/jω-axis.

4. For convolution: Use graphical method for short sequences, transform method for complex ones.

5. For short notes (4-5 marks): Define, give 3-4 key points, one simple example, one application if relevant.

6. Common Pitfalls:

*   Confusing `u(t)` and `δ(t)` relationship.

*   Forgetting ROC conditions for causality/stability.

*   Misapplying convolution theorem (multiplication in time ↔ convolution in frequency, **not** the other way around).

*   Ignoring unit step in exponential signals (`e^{-at}u(t)` vs `e^{-at}`).

*   For DTFT, forgetting periodicity (`-π` to `π`).