Signals & Systems (EC-402) - Unit 1 Short Notes

UNIT 1: FUNDAMENTALS OF SIGNALS & SYSTEMS

I. SIGNAL CLASSIFICATION & BASIC OPERATIONS

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

[!TIP] Exam Focus: Distinguish by domain (continuous t vs. discrete n). CT signals are functions of a real variable; DT signals are sequences.

B. Standard Signals & Basic Operations

• Unit Step: u(t) = {1, t≥0; 0, t<0} (CT) / u[n] = {1, n≥0; 0, n<0} (DT)

• Unit Ramp: r(t) = t u(t) (CT) / r[n] = n u[n] (DT). Key Distinction: Ramp = integral of step.

• Unit Impulse (Dirac Delta): δ(t) (CT) with ∫δ(t)dt = 1, δ(t)=0 for t≠0. δ[n] (DT) with ∑δ[n]=1, δ[0]=1, δ[n]=0 for n≠0. Sifting Property: ∫x(t)δ(t-t₀)dt = x(t₀) / ∑x[n]δ[n-n₀] = x[n₀].

• Exponential: x(t)=e^{at} (CT) / x[n]=a^n (DT).

• Sinusoidal: x(t)=A sin(ω₀t+φ) (CT) / x[n]=A sin(Ω₀n+φ) (DT).

Basic Operations:

1. Time-Shifting: x(t-t₀) / x[n-n₀] (delay if t₀>0/n₀>0).

2. Time-Scaling: x(at) (CT, a>1 → compression) / x[kn] (DT, k integer).

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

4. Amplitude Operations: Ax(t), x(t)+y(t), x(t)y(t).

[!TIP] Common Pitfall: For DT scaling x[kn], k must be integer. For CT scaling x(at), a is real.

C. Signal Classification by Nature

• Periodic vs. Aperiodic:

  • CT: x(t) periodic if ∃ T>0 s.t. x(t+T)=x(t) ∀t. Fundamental period T₀ = smallest T.

  • DT: x[n] periodic if ∃ N∈ℤ⁺ s.t. x[n+N]=x[n] ∀n. Fundamental period N₀ = smallest N.

  • Check: For sum x(t)=A₁sin(ω₁t)+A₂sin(ω₂t), periodic if ω₁/ω₂ is rational.

• Deterministic vs. Random:

  • Deterministic: Completely specified for all t/n (e.g., x(t)=e^{-t}).

  • Random: Described probabilistically (e.g., noise voltage).

• Even & Odd Symmetry:

  • Even: x(t)=x(-t) / x[n]=x[-n] (symmetry about y-axis).

  • Odd: x(t)=-x(-t) / x[n]=-x[-n] (symmetry about origin).

  • Decomposition: Any x(t) = xₑ(t) + xₒ(t), where xₑ(t)=½[x(t)+x(-t)], xₒ(t)=½[x(t)-x(-t)].

D. Signal Classification by Energy & Power

\boxed{\text{For a periodic signal with period } T_0 \text{ (CT) or } N_0 \text{ (DT): } P = \frac{1}{T_0} \int_{0}^{T_0} |x(t)|^2 dt \text{ or } P = \frac{1}{N_0} \sum_{n=0}^{N_0-1} |x[n]|^2}

Examples from Past Papers:

• x(t)=t u(t): Neither (Energy ∞, Power finite but non-zero? Actually, E=∞, P=∞ → Neither).

• x(t)=u(t)e^{-at}, a>0: Energy Signal (E = 1/(2a)).

• x[n]=(1/2)^n u[n]: Energy Signal (E = 4/3).

• x(t)=cos(t): Power Signal (P = ½).

II. SYSTEM CLASSIFICATION & PROPERTIES

A. System Properties (Testing Method)

[!TIP] Common Pitfalls:

• Linearity: Presence of constants (not multiplied by input) or non-linear operations (squaring, absolute value) breaks linearity. E.g., y(t)=x(t)+3 is non-linear.

• Causality: y(t)=x(t+1) is non-causal (depends on future).

• Stability: h(t)=u(t) is unstable (integral diverges).

B. System Representation

• CT LTI: Linear constant-coefficient differential equation.

  aₙ dⁿy/dtⁿ + ... + a₁ dy/dt + a₀ y(t) = bₘ dᵐx/dtᵐ + ... + b₀ x(t)

• DT LTI: Linear constant-coefficient difference equation.

  ∑_{k=0}^{N} a_k y[n-k] = ∑_{k=0}^{M} b_k x[n-k]

• Block Diagrams:

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

  • Direct Form II: Reduced delay elements (canonical form).

  • Cascade: H(z)=H₁(z)H₂(z)... (series connection).

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

III. LINEAR TIME-INVARIANT (LTI) SYSTEMS

A. Impulse Response h(t) / h[n]

• Definition: Output when input is Dirac impulse δ(t) / unit impulse δ[n].

• Significance for LTI: Complete characterization. Output for any input x(t) is convolution: y(t)=x(t)*h(t).

• Properties:

  1. Causality: h(t)=0 for t<0 / h[n]=0 for n<0.

  2. Stability (BIBO): ∫_{-∞}^{∞} \|h(τ)\| dτ < ∞ (CT) / ∑_{n=-∞}^{∞} \|h[n]\| < ∞ (DT).

  3. Memory: If h(t) is non-zero over an interval ⇒ system has memory. If h(t)=kδ(t) ⇒ memoryless.

B. Convolution

• CT Convolution Integral:

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

• DT Convolution Sum:

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

• Properties: Commutative, Associative, Distributive.

• Step Response s(t)/s[n]: Output to u(t)/u[n]. Related to impulse response:

  • CT: s(t) = ∫_{-∞}^{t} h(τ) dτ (if h(t) causal).

  • DT: s[n] = ∑_{k=-∞}^{n} h[k] (if h[n] causal).

  • Alternatively, s(t) = u(t) * h(t) / s[n] = u[n] * h[n].

C. Analysis of LTI Systems

• Finding Impulse Response from Differential/Difference Equation:

  1. Take Laplace/Z-transform (assuming zero initial conditions for zero-state response).

  2. H(s) = Y(s)/X(s) or H(z) = Y(z)/X(z).

  3. Find h(t) via inverse Laplace (h(t)=ℒ⁻¹{H(s)}) or h[n] via inverse Z-transform.

• Determining Properties from h(t)/h[n]:

  • Causality: Check support (non-zero only for t≥0/n≥0).

  • Stability: Check absolute integrability/summability.

• System Function: H(s) (CT, Laplace) / H(z) (DT, Z-transform). H(s)=ℒ{h(t)}, H(z)=ℤ{h[n]}.

IV. Z-TRANSFORM ANALYSIS

A. Definition & Region of Convergence (ROC)

• Bilateral Z-transform:

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

• ROC: Set of z for which sum converges. Crucial Properties:

  1. ROC is a ring/annulus in z-plane (cannot be empty).

  2. ROC cannot contain poles (diverges at poles).

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

  4. For right-sided (causal) sequences (n₀ ≥ 0), ROC is ** exterior** of outermost pole (|z| > r_max).

  5. For left-sided sequences, ROC is interior of innermost pole (|z| < r_min).

  6. ROC determines causality & stability:

     • Causal ⇔ ROC is exterior of outermost pole and includes ∞.

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

\boxed{\text{For a causal LTI system: Stability } \iff \text{ All poles of } H(z) \text{ lie inside } |z|=1.}

B. Z-Transform Properties (Key Proofs Required)

C. Inverse Z-Transform Methods

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

2. Partial Fraction Expansion: 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.

D. Unilateral (One-sided) Z-Transform

• Definition: X₊(z) = ∑_{n=0}^{∞} x[n] z^{-n} (sum from n=0).

• ROC: Exterior of outermost pole (|z| > r_max), always.

• Application: Solving causal difference equations with non-zero initial conditions. Incorporates initial conditions directly.

V. FOURIER ANALYSIS

A. Continuous-Time Fourier Series (CTFS)

• For periodic x(t) with period T₀, fundamental frequency ω₀=2π/T₀.

• Exponential Fourier Series:

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

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

• Magnitude Spectrum: \|C_k\| vs kω₀. Phase Spectrum: ∠C_k vs kω₀.

• Properties: Linearity, Time-shift (phase shift), Time-scaling (frequency scaling), Convolution (in time domain ↔ product in frequency domain for FS coefficients), Parseval's theorem.

B. Continuous-Time Fourier Transform (CTFT)

• Definition:

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

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

• Key Properties (with Proofs Expected):

  • Linearity: a₁x₁(t)+a₂x₂(t) ↔ a₁X₁(jω)+a₂X₂(jω)

  • Time-Shifting: x(t-t₀) ↔ e^{-jωt₀} X(jω)

  • Frequency-Shifting: x(t)e^{jω₀t} ↔ X(j(ω-ω₀))

  • Time-Scaling: x(at) ↔ (1/|a|) X(jω/a)

  • Convolution: x(t)*h(t) ↔ X(jω)H(jω)

  • Differentiation in Time: dⁿx(t)/dtⁿ ↔ (jω)ⁿ X(jω)

  • Integration in Time: ∫_{-∞}^{t} x(τ)dτ ↔ \frac{X(jω)}{jω} + πX(0)δ(ω)

  • Parseval's Theorem: ∫_{-∞}^{∞} \|x(t)\|² dt = \frac{1}{2π} ∫_{-∞}^{∞} \|X(jω)\|² dω

• Standard FT Pairs (Memorize):

  • u(t) ↔ πδ(ω) + 1/(jω)

  • e^{-at}u(t), a>0 ↔ 1/(a+jω)

  • rect(t/τ) ↔ τ sinc(ωτ/2π)

  • e^{-α\|t\|} ↔ 2α/(α²+ω²)

[!TIP] Exam Strategy: Use differentiation-in-time property to find FT of tⁿ x(t) if X(jω) of x(t) is known. E.g., if x(t)=t e^{-2t} sin(4t) u(t), express as t * [e^{-2t} sin(4t) u(t)], find FT of bracket, then differentiate.

C. Discrete-Time Fourier Transform (DTFT)

• Definition:

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

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

*Note:* `ω` is **normalized radian frequency** (periodic with `2π`).

• Convergence: Exists if ∑_{n=-∞}^{∞} \|x[n]\| < ∞ (absolutely summable). Periodic sequences (power signals) have DTFT in terms of impulses.

• Properties:

  • Linearity, Time-Shift: x[n-n₀] ↔ e^{-jωn₀} X(e^{jω})

  • Frequency-Shift: x[n]e^{jω₀n} ↔ X(e^{j(ω-ω₀)}) (periodic shift).

  • Time-Reversal: x[-n] ↔ X(e^{-jω}) = X(e^{jω}) (if x[n] real & even).

  • Convolution: x[n]*h[n] ↔ X(e^{jω})H(e^{jω})

  • Differentiation in Frequency: n x[n] ↔ j \frac{d}{dω} X(e^{jω})

  • Parseval's Theorem: ∑_{n=-∞}^{∞} \|x[n]\|² = \frac{1}{2π} ∫_{2π} \|X(e^{jω})\|² dω

• Standard DTFT Pairs:

  • aⁿ u[n], |a|<1 ↔ 1/(1-a e^{-jω})

  • δ[n] ↔ 1

  • u[n] ↔ πδ(ω) + 1/(1-e^{-jω}) (for -π<ω<π)

• Expressing DTFT of Modified Sequences: Use properties. E.g., if y[n]=(n-1)² x[n], then:

  Y(e^{jω}) = \frac{d²}{dω²} [e^{-jω} X(e^{jω})] + \text{lower order terms}.

VI. STATE-SPACE ANALYSIS

A. State Variable Representation

• State: Minimum set of variables x₁(t), x₂(t),..., xₙ(t) (or x[n]) that completely specifies system's future output given present state and future input.

• State-Space Model:

  • CT:

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

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

  • DT:

    State Equation: x[n+1] = A x[n] + B u[n]

    Output Equation: y[n] = C x[n] + D u[n]

  • Matrices: A (system matrix, n×n), B (input matrix, n×m), C (output matrix, p×n), D (feedthrough, p×m). For m inputs, p outputs.

• MIMO Representation: Directly generalized. u(t) is m×1 vector, y(t) is p×1 vector.

B. State Transition Matrix Φ(t) / Φ[n]

• Definition: Matrix that propagates state from time t₀ to t (or n₀ to n).

  • CT: Φ(t, t₀) = e^{A(t-t₀)} (if A time-invariant).

  • DT: Φ[n, n₀] = A^{n-n₀} (if A time-invariant).

• Properties:

  1. Φ(t₀, t₀) = I (identity).

  2. Φ(t, t₁)Φ(t₁, t₀) = Φ(t, t₀) (composition).

  3. Φ⁻¹(t, t₀) = Φ(t₀, t).

  4. dΦ(t,t₀)/dt = A Φ(t,t₀).

• Solution of State Equations (Zero-state, x(t₀)=0):

  x(t) = ∫_{t₀}^{t} Φ(t, τ) B u(τ) dτ

  y(t) = C ∫_{t₀}^{t} Φ(t, τ) B u(τ) dτ + D u(t)

• Methods of Evaluation:

  1. Cayley-Hamilton Theorem: Use characteristic equation to express e^{At} as polynomial in A.

  2. Laplace Transform (CT): Φ(t) = ℒ⁻¹{ (sI - A)⁻¹ }.

  3. Z-Transform (DT): Φ[n] = ℤ⁻¹{ (zI - A)⁻¹ z }.

  4. Series Expansion: e^{At} = I + At + (A²t²)/2! + ....

VII. SAMPLING THEORY

A. Sampling Theorem (Nyquist-Shannon)

• Statement: A band-limited continuous-time signal x(t) with no frequency components above f_max Hz can be completely reconstructed from its samples x(nT_s) if the sampling frequency f_s satisfies:

$$f_s \geq 2 f_{\text{max}}$$

where `T_s = 1/f_s` is the sampling interval. `2f_max` is the **Nyquist rate**.

• Ideal Sampling & Reconstruction:

  • Sampling: x_s(t) = x(t) ∑_{n=-∞}^{∞} δ(t-nT_s).

  • Spectrum: X_s(jω) = (1/T_s) ∑_{k=-∞}^{∞} X(j(ω - kω_s)), where ω_s = 2πf_s.

  • No Aliasing Condition: ω_s ≥ 2ω_max ⇒ replicas do not overlap.

  • Ideal Reconstruction (Sinc Interpolation):

$$x(t) = \sum_{n=-∞}^{∞} x(nT_s) \text{sinc}\left(\frac{t - nT_s}{T_s}\right)$$

    where `sinc(t) = sin(πt)/(πt)`. Equivalent to passing `x_s(t)` through an ideal LPF with cutoff `f_c = f_max`.

\boxed{f_s \geq 2 B \quad \text{where } B = \text{bandwidth (Hz)}} \quad \text{and} \quad T_s \leq \frac{1}{2B}

B. Implications & Effects

• Aliasing: Occurs if f_s < 2f_max (undersampling). High-frequency components fold back into lower frequencies, causing irreversible distortion. The spectrum replicas overlap.

• Minimum Sampling Rate: f_s(min) = 2f_max (Nyquist rate).

• Maximum Sampling Interval: T_s(max) = 1/(2f_max).

• Practical Consideration: Use anti-aliasing filter (analog LPF) before sampling to band-limit x(t) to f_max < f_s/2.

C. Reconstruction of Signal from Samples

• Ideal Reconstruction: Using sinc interpolation as above. Requires infinite support and perfect LPF → impractical.

• Practical Reconstruction (Hold Circuits):

  • Zero-Order Hold (ZOH): Holds sample value constant for T_s. Output: x(t) = ∑ x(nT_s) rect((t-nT_s)/T_s). Frequency response: H_zoh(jω) = T_s \frac{\sin(ωT_s/2)}{(ωT_s/2)} e^{-jωT_s/2}.

  • First-Order Hold (FOH): Linear interpolation between samples. Better approximation but more complex.

[!TIP] Exam Focus: Be able to derive the condition for no aliasing from the sampled spectrum expression. Remember: Aliasing is permanent; once samples are taken with f_s < 2f_max, original signal cannot be recovered perfectly.