Signals and Systems (EX-302) - Unit 3 Short Notes

UNIT 3: SIGNALS AND SYSTEMS - EXAM-FOCUSED SHORT NOTES

1.0 FUNDAMENTALS OF SIGNALS

1.1 Definition and Classification of Signals

A signal is a function of one or more independent variables that conveys information about the state or behavior of a physical system.

[!TIP] Exam Alert: For periodicity of sum x1(t)+x2(t), find T1, T2. If T1/T2 is rational, period = LCM(T1,T2). Else, aperiodic.

1.2 Basic Signal Operations

• Time-domain: x(t ± t0) (shift), x(at) (scale), x(-t) (reversal).

• Amplitude-domain: ax(t), x1(t) ± x2(t), x1(t)·x2(t), dx(t)/dt (CT), ∫x(τ)dτ (CT), ∑_{k=-∞}^n x[k] (DT accumulation).

1.3 Elementary Signals

[!TIP] Common Pitfall: δ(t) is not a finite number but an idealized function. Its integral is 1.

2.0 FUNDAMENTALS OF SYSTEMS

2.1 System Classification & Properties

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

2.2 Linear Time-Invariant (LTI) Systems

• Characterized by Convolution:

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

  • DT: y[n] = (x * h)[n] = ∑_{k=-∞}^{∞} x[k] h[n-k]

  \boxed{y(t) = x(t) * h(t)}

• Impulse Response h(t)/h[n]: Output when input is δ(t)/δ[n]. Complete characterization of LTI system.

• Step Response s(t): Output for u(t). Relation: s(t) = ∫_{-∞}^{t} h(τ) dτ (CT), s[n] = ∑_{k=-∞}^{n} h[k] (DT). Hence, h(t) = ds(t)/dt.

• Frequency Response H(jω)/H(e^(jω)): FT of impulse response. H(jω) = ∫ h(t) e^{-jωt} dt. Describes system's gain/phase at each frequency.

• Properties: Commutative (x*h = h*x), Associative ((x*h)*g = x*(h*g)), Distributive (x*(h1+h2) = x*h1 + x*h2).

[!TIP] Exam Trick: To check time-invariance, replace t with (t-t0) in the system equation. If you get y(t-t0), it's TI. Otherwise, TV.

3.0 ANALYSIS OF CONTINUOUS-TIME SIGNALS & SYSTEMS

3.1 Fourier Series (FS) Analysis

• Purpose: Represent periodic CT signals as sum of sinusoids.

• Trigonometric Form: x(t) = a0 + ∑_{k=1}^∞ [ak cos(kω0 t) + bk sin(kω0 t)]

• Exponential Form: x(t) = ∑_{k=-∞}^∞ Ck e^{jkω0 t}

  \boxed{C_k = \frac{1}{T} \int_{T} x(t) e^{-jk\omega_0 t} dt}

• Dirichlet's Conditions (Sufficiency):

  1. Single-valued, finite number of maxima/minima in period T.

  2. Finite number of discontinuities in T.

  3. Absolutely integrable over T: ∫_T |x(t)| dt < ∞.

• Parseval's Theorem (Power):

  \boxed{P = \frac{1}{T} \int_{T} |x(t)|^2 dt = \sum_{k=-\infty}^{\infty} |C_k|^2}

3.2 Fourier Transform (FT) Analysis

• Definition (from FS limit as T→∞):

  \boxed{X(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt}

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

• Key Properties (MOST ASKED):

  | Property | Time Domain | Frequency Domain | | :--- | :--- | :--- | | Linearity | a x1(t) + b x2(t) | a X1(jω) + b X2(jω) | | Time Shifting | x(t - t0) | X(jω) e^{-jωt0} | | Frequency Shifting | x(t) e^{jω0 t} | X(j(ω-ω0)) | | Time Scaling | x(at) | (1/|a|) X(jω/a) | | Conjugation | x*(t) | X*(-jω) | | Duality | X(t) ↔ 2π x(-ω) | If x(t) ↔ X(jω), then X(t) ↔ 2π x(-ω) | | Convolution | x1(t) * x2(t) | X1(jω) · X2(jω) | | Multiplication | x1(t) x2(t) | (1/2π) X1(jω) * X2(jω) | | Differentiation | dx(t)/dt | jω X(jω) | | Integration | ∫_{-∞}^t x(τ)dτ | X(jω)/(jω) + π X(0) δ(ω) | | Parseval (Energy) | ∫ |x(t)|² dt | (1/2π) ∫ |X(jω)|² dω |

• FT of Standard Signals:

  • e^{-at} u(t) (Re(a)>0): 1/(a + jω)

  • t u(t): 1/(jω)² + π δ(ω) (or 1/(jω)^2 in principal value sense)

  • cos(ω0 t), sin(ω0 t): π[δ(ω-ω0) + δ(ω+ω0)], jπ[δ(ω+ω0) - δ(ω-ω0)]

  • u(t): π δ(ω) + 1/(jω)

• Duality Property Application: If you know FT of a rectangular pulse is sinc, then FT of sinc is rectangular.

3.3 Laplace Transform (LT) Analysis

• Definition (Bilateral):

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

• Region of Convergence (ROC) (FREQUENT TOPIC):

  • Set of s for which integral converges.

  • Properties of ROC:

    1. Right-sided signal (e.g., a^n u[n] CT: e^{at}u(t)): ROC is Re(s) > σ0 (half-plane to right of rightmost pole).

    2. Left-sided signal: ROC is Re(s) < σ0 (half-plane to left of leftmost pole).

    3. Two-sided: ROC is a strip σ1 < Re(s) < σ2.

    4. ROC cannot contain poles.

    5. For causal signals, ROC is right of rightmost pole.

    6. For stable LTI systems, ROC must include jω-axis (σ=0).

• Properties of LT (FREQUENT TOPIC):

  | Property | Time Domain | s-Domain | | :--- | :--- | :--- | | Linearity | a x1(t) + b x2(t) | a X1(s) + b X2(s) | | Time Shifting | x(t - t0)u(t-t0) | e^{-st0} X(s) | | Frequency Shifting | x(t) e^{s0 t} | X(s - s0) | | Scaling in s | t x(t) | -dX(s)/ds | | Differentiation | dx(t)/dt | s X(s) - x(0⁻) | | Integration | ∫_{-∞}^t x(τ)dτ | X(s)/s + (1/s) ∫_{-∞}^{0⁻} x(τ)dτ | | Convolution | x1(t) * x2(t) | X1(s) X2(s) | | Initial Value | x(0⁺) | lim_{s→∞} s X(s) (if poles at ∞ canceled) | | Final Value | x(∞) | lim_{s→0} s X(s) (if poles at 0 canceled, ROC: Re(s)>0) |

• LT of Standard Signals:

  • δ(t): 1, ROC: All s.

  • u(t): 1/s, ROC: Re(s)>0.

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

  • t u(t): 1/s², ROC: Re(s)>0.

• Inverse LT: Use Partial Fraction Expansion (PFE) and match with standard forms. ROC determines time-domain signal (causal, anti-causal, two-sided).

3.4 Interconnection of LTI CT Systems

• Series/Cascade: Overall H(s) = H1(s) H2(s). Output of 1 is input to 2.

• Parallel: Overall H(s) = H1(s) + H2(s). Inputs to both, outputs summed.

• Feedback: Y(s) = X(s) / (1 ± H1(s)H2(s)) for negative/positive feedback.

• Block Diagram Representations:

  • Direct Form I: Realizes difference equation directly (separate adders for each derivative term).

  • Direct Form II: Combines adders, reduces memory elements. Most efficient for implementation.

  [!TIP] Advantage of Block Diagrams: Visualizes system structure, aids in hardware/software implementation, simplifies analysis of interconnections.

4.0 ANALYSIS OF DISCRETE-TIME SIGNALS & SYSTEMS

4.1 Z-Transform (ZT) Analysis

• Definition (Bilateral):

  \boxed{X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}}, where z = r e^{jω}.

• Unilateral ZT: Sum from n=0 to ∞. Used for causal signals with initial conditions.

• ROC (FREQUENT TOPIC):

  • Key Properties: ROC is annular (ring-shaped) or half-plane. Cannot contain poles. For rational X(z), ROC extends to infinity (causal) or is bounded by outermost pole (anti-causal).

  • Causality & Stability Link:

    • Causal x[n] ↔ ROC is outside the outermost pole (includes ∞).

    • Stable LTI system ↔ ROC includes unit circle (|z|=1).

    • Causal & Stable ↔ ROC is outside all poles and includes unit circle.

• Properties of ZT (FREQUENT TOPIC):

  | Property | Time Domain | z-Domain | | :--- | :--- | :--- | | Linearity | a x1[n] + b x2[n] | a X1(z) + b X2(z) | | Time Shifting | x[n - n0] | z^{-n0} X(z) | | Scaling in z | a^n x[n] | X(z/a) | | Conjugation | x*[n] | X*(z*) | | Convolution | x1[n] * x2[n] | X1(z) X2(z) | | Multiplication by n | n x[n] | -z dX(z)/dz | | Differentiation in z | r^n x[n] | (1/r) dX(z)/dz (less common) | | Initial Value | x[0] | lim_{z→∞} X(z) | | Final Value | x[∞] | lim_{z→1} (z-1) X(z) (if poles of (z-1)X(z) inside unit circle) |

• ZT of Standard Sequences:

  • δ[n]: 1, ROC: All z.

  • u[n]: 1/(1-z^{-1}), ROC: |z|>1.

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

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

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

• Inverse ZT Methods (FREQUENT TOPIC):

  1. Partial Fraction Expansion (PFE): Expand X(z)/z into simpler terms. Use long division if deg(N) ≥ deg(D). Inverse each term using standard pairs. ROC determines sequence type (causal, anti-causal).

  2. Power Series Expansion: Expand X(z) as ∑ x[k] z^{-k}. Coefficients are x[k]. Valid within ROC.

  3. Contour Integration (Residue Method): x[n] = (1/(2πj)) ∮ X(z) z^{n-1} dz. (Theoretical, less used in exams).

• System Function H(z) & Impulse Response h[n]: For LTI DT system from LCCDE: ∑_{k=0}^N a_k y[n-k] = ∑_{k=0}^M b_k x[n-k]. Taking ZT (zero ICs): H(z) = Y(z)/X(z) = (∑ b_k z^{-k}) / (∑ a_k z^{-k}). h[n] is inverse ZT of H(z).

4.2 Discrete-Time Fourier Transform (DTFT)

• Definition:

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

  \boxed{x[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(e^{j\omega}) e^{j\omega n} d\omega}

• Periodic Nature: X(e^{j(ω+2π)}) = X(e^{jω}). Periodic with period 2π.

• Properties (FREQUENT TOPIC):

  | Property | Time Domain | Frequency Domain | | :--- | :--- | :--- | | Linearity | a x1[n] + b x2[n] | a X1(e^{jω}) + b X2(e^{jω}) | | Time Shifting | x[n - n0] | X(e^{jω}) e^{-jω n0} | | Frequency Shifting | x[n] e^{jω0 n} | X(e^{j(ω-ω0)}) | | Time Reversal | x[-n] | X(e^{-jω}) | | Conjugation | x*[n] | X*(e^{-jω}) | | Convolution | x1[n] * x2[n] | X1(e^{jω}) X2(e^{jω}) | | Multiplication | x1[n] x2[n] | (1/(2π)) X1(e^{jω}) * X2(e^{jω}) (periodic conv) | | Parseval | ∑ |x[n]|² | (1/(2π)) ∫_{-π}^{π} |X(e^{jω})|² dω |

• DTFT of Standard Sequences:

  • δ[n]: 1.

  • u[n]: 1/(1-e^{-jω}) + π ∑_{k=-∞}^{∞} δ(ω-2πk) (has impulse at ω=0 due to discontinuity).

  • a^n u[n] (|a|<1): 1/(1-a e^{-jω}).

4.3 Difference Equations for LTI DT Systems

• Linear Constant-Coefficient Difference Equation (LCCDE):

  \boxed{\sum_{k=0}^{N} a_k y[n-k] = \sum_{k=0}^{M} b_k x[n-k]}, with a0 ≠ 0.

• Order: N (highest lag difference).

• Direct Form I: Realizes each delay separately. N delays for y[n], M for x[n]. Total max(N,M) delays.

• Direct Form II: Shared delay line. Only N delays needed. More efficient.

• First-Order System: y[n] - a y[n-1] = x[n].

  • Impulse response: h[n] = a^n u[n].

  • Step response: s[n] = ∑_{k=0}^n a^k = (1-a^{n+1})/(1-a) u[n] (for a≠1).

5.0 SYSTEM ANALYSIS & INTERCONNECTION

5.1 State-Variable Representation

• State: Minimum set of variables x1(t), x2(t), ..., xN(t) (state variables) that summarize past history to predict future.

• State Equation: ẋ(t) = A x(t) + B u(t) (CT) / x[n+1] = A x[n] + B u[n] (DT).

• Output Equation: y(t) = C x(t) + D u(t) (CT) / y[n] = C x[n] + D u[n] (DT).

• Matrix Representation (State-Space Model):

  \boxed{\begin{aligned} \dot{\mathbf{x}}(t) &= \mathbf{A} \mathbf{x}(t) + \mathbf{B} u(t) \ y(t) &= \mathbf{C} \mathbf{x}(t) + \mathbf{D} u(t) \end{aligned}}

  where x is state vector, u input, y output. A, B, C, D are system matrices.

• Relationship with Transfer Function: H(s) = C (sI - A)^{-1} B + D (CT). H(z) = C (zI - A)^{-1} B + D (DT).

• Advantages over Input-Output:

  • Handles multiple-input, multiple-output (MIMO) systems naturally.

  • Incorporates initial conditions directly.

  • Essential for controller design (modern control theory).

  • Analyzes internal stability (not just BIBO).

5.2 System Interconnections (DT Focus)

• Series/Cascade: H(z) = H1(z) H2(z).

• Parallel: H(z) = H1(z) + H2(z).

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

• Block Diagram Realizations (from H(z)):

  • Direct Form I: From difference equation. Separate paths for numerator/denominator.

  • Direct Form II: Shared delays. H(z) = (b0 + b1 z^{-1} + ... + bM z^{-M}) / (1 + a1 z^{-1} + ... + aN z^{-N}).

  • Cascade Form: Factor H(z) into second-order sections. H(z) = H1(z) H2(z) .... Improves numerical stability.

  • Parallel Form (Partial Fraction): H(z) = C + Σ (Rk / (1 - pk z^{-1})). Useful for residue calculation.

[!TIP] Example: For H(z) = 1/[(1+z⁻¹/3)(1-z⁻¹/6)], cascade two first-order sections. Direct Form II uses 2 delays (order=2).

6.0 SAMPLING & ALIASING

6.1 Sampling of CT Signals

• Ideal Sampling (Impulse Sampling): x_p(t) = x(t) ∑ δ(t - nTs) = ∑ x(nTs) δ(t - nTs).

• Sampling Theorem (Nyquist-Shannon): A bandlimited signal with max freq B Hz can be perfectly reconstructed from its samples if sampling freq fs > 2B (Nyquist rate). fs = 1/Ts.

• Reconstruction (Ideal Interpolation): x(t) = ∑ x(nTs) sinc((t-nTs)/Ts). Uses sinc filter (ideal LPF).

• Practical Reconstruction: Zero-Order Hold (ZOH). Holds sample value for Ts seconds. Not ideal, but common in DACs.

6.2 Aliasing

• Definition: Overlapping of spectral replicas due to undersampling (fs < 2B). High-frequency components masquerade as low-frequency.

• Cause: X_p(jω) consists of shifted replicas of X(jω) spaced at ωs = 2π/Ts. If ωs < 2B, replicas overlap.

• Effect on Spectrum: Frequencies |ω| > ωs/2 "fold back" into [-ωs/2, ωs/2]. ω_alias = |ω0 - k ωs| for some integer k.

• How to Avoid:

  1. Anti-aliasing Filter: Analog LPF before sampler to bandlimit x(t) to B < fs/2.

  2. Increase Sampling Rate: Ensure fs > 2B.

[!TIP] Key Formula: Aliased frequency f_alias = |f - k fs| where f is original, fs sampling freq, k integer such that f_alias in [0, fs/2].

7.0 FILTERS

7.1 Analog Filters

• Frequency Selective: Pass certain frequency bands, reject others.

• Ideal Types (Magnitude Response):

  • Low-pass (LPF): |H(jω)| = 1 for |ω| < ωc, 0 for |ω| > ωc.

  • High-pass (HPF): 0 for |ω| < ωc, 1 for |ω| > ωc.

  • Band-pass (BPF): Passband ω1 < |ω| < ω2.

  • Band-stop (BSF): Stopband ω1 < |ω| < ω2.

• Non-ideal Characteristics:

  • Passband ripple: Allowed variation in passband gain.

  • Stopband attenuation: Minimum attenuation in stopband.

  • Transition band: Finite roll-off between pass/stop bands.

• Applications: Audio equalization, radio tuning, noise removal.

7.2 Digital Filters

• Advantages over Analog: No component tolerance drift, linear phase possible (FIR), easy to redesign, multiplexing.

• FIR (Finite Impulse Response):

  • h[n] has finite duration. Always stable (poles at z=0).

  • Can have exact linear phase if h[n] symmetric/anti-symmetric.

  • Implemented as non-recursive (no feedback).

• IIR (Infinite Impulse Response):

  • h[n] infinite duration. May be unstable if poles outside unit circle.

  • Recursive (feedback). More efficient (lower order for sharp roll-off).

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

• Design Considerations:

  • FIR: Choose window or optimal method (Parks-McClellan) to meet specs. Order M determines transition width.

  • IIR: Transform analog prototype (Butterworth, Chebyshev, Elliptic) to digital (bilinear transform).

• Applications: Speech processing, biomedical signal filtering, communications.

[!TIP] FIR vs IIR Quick Recall: FIR = Stable, Linear Phase, Higher Order. IIR = Efficient, Potentially Unstable, Non-linear Phase.

8.0 ADVANCED TOPICS & COMPARISONS

8.1 Wavelet Transform (WT)

• Need: FT provides only frequency info. WT provides time-frequency localization.

• Basic Concept: Uses wavelets ψ_{a,b}(t) = (1/√a) ψ((t-b)/a). a (scale) controls frequency, b (translation) controls time.

• Comparison:

  • vs FT: FT uses fixed basis (sinusoids). WT uses scalable/translatable wavelets. WT has variable resolution (good time res at high freq, good freq res at low freq).

  • vs LT: LT is for CT signals, unilateral for ICs. WT is for both CT/DT, multi-resolution.

• Application Example (Denoising):

  1. Compute WT of noisy signal.

  2. Threshold small coefficients (likely noise).

  3. Inverse WT to reconstruct denoised signal.

8.2 Comparative Analysis of Transforms

8.3 Convolution Theorems

• Fourier Transform Convolution Theorem (STATE & PROVE):

  Statement: Convolution in time domain ↔ Multiplication in frequency domain.

  x1(t) * x2(t) ↔ X1(jω) · X2(jω)

  Proof Sketch:

  FT{ x1 * x2 } = ∫ [∫ x1(τ) x2(t-τ) dτ] e^{-jωt} dt

  = ∫ x1(τ) [∫ x2(t-τ) e^{-jωt} dt] dτ (Fubini)

  = ∫ x1(τ) [∫ x2(λ) e^{-jω(λ+τ)} dλ] dτ (let λ=t-τ)

  = [∫ x1(τ) e^{-jωτ} dτ] [∫ x2(λ) e^{-jωλ} dλ]

  = X1(jω) X2(jω)

• Laplace Transform Convolution Theorem: x1(t) * x2(t) ↔ X1(s) X2(s). ROC is intersection of individual ROCs.

• Z-Transform Convolution Theorem: x1[n] * x2[n] ↔ X1(z) X2(z). ROC is intersection of individual ROCs.

• Properties of Convolution (LTI Systems):

  • Commutative: x * h = h * x

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

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

  • Shift: x(t-t0) * h(t) = y(t-t0) if system is TI.

  • Differentiation/Integration: Can be moved into/out of convolution for LTI systems.

[!TIP] Proof Strategy: For convolution theorem, always change integration/summation order (Fubini's theorem) and use change of variables (λ = t-τ). ROC intersection is critical for LT/ZT.

END OF UNIT 3 NOTES
Aligned with RGPV past papers (2023-2025). Focus on bolded terms, boxed formulas, and properties marked as FREQUENT TOPIC.