UNIT 2: Signals & Systems – Short Notes

I. Signal Classification and Basic Operations

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

[!TIP]

Key Distinction: CT signals are functions of a continuous variable t; DT signals are sequences indexed by integers n. A DT signal can be obtained by sampling a CT signal: x[n] = x(nT_s).

B. Periodic vs. Aperiodic Signals

• Periodic CT Signal: x(t) = x(t + T₀) for all t, smallest T₀ > 0 is fundamental period.

  • Example: x(t) = sin(2πt) → T₀ = 1.

• Periodic DT Signal: x[n] = x[n + N] for all n, smallest integer N > 0 is fundamental period.

  • Condition: ω₀N = 2πk (k integer). If ω₀/2π is irrational → aperiodic.

• Aperiodic (Non-periodic): No T₀ or N satisfies the condition.

  • Example: x(t) = e^{-t}u(t).

[!TIP]

For composite signals: x(t) = A₁ sin(ω₁t) + A₂ sin(ω₂t) is periodic iff ω₁/ω₂ is rational. For DT, x[n] = cos(ω₁n) + cos(ω₂n) periodic if ω₁/(2π) and ω₂/(2π) are rational.

C. Deterministic vs. Random Signals

• Deterministic: Completely predictable; described by explicit mathematical expression.

  • Example: x(t) = 5 cos(2πt + π/4).

• Random (Stochastic): Uncertain; described by statistical properties (mean, variance, probability distributions).

  • Example: Thermal noise voltage, speech signal.

D. Energy vs. Power Signals

E. Basic Signal Operations (CT & DT)

Let x(t) (CT) or x[n] (DT), a, b constants.

[!TIP]

Order Matters: For x(at - b), first scale by a, then shift by b/a. Folding: x(-t) reflects about y-axis; x(t - t₀) shifts right by t₀.

F. Standard Signals

1. Unit Step

   CT: u(t) = {1, t≥0; 0, t<0}

   DT: u[n] = {1, n≥0; 0, n<0}

2. Unit Ramp

   CT: r(t) = t·u(t)

   DT: r[n] = n·u[n]

3. Unit Impulse (Delta)

   CT: δ(t) = 0 (t≠0), ∫_{-∞}^{∞} δ(t) dt = 1, sifting: ∫ x(t)δ(t-t₀) dt = x(t₀)

   DT: δ[n] = {1, n=0; 0, n≠0}, Σ x[n]δ[n-n₀] = x[n₀]

4. Exponential

   CT: x(t) = e^{at} u(t) (a real/complex)

   DT: x[n] = a^n u[n]

5. Sinusoidal

   CT: x(t) = A cos(ω₀t + φ)

   DT: x[n] = A cos(ω₀n + φ)

II. System Classification and Properties

A. Linear vs. Non-linear Systems

• Linear: Satisfies superposition (additivity + homogeneity).

  • T[a₁x₁(t) + a₂x₂(t)] = a₁T[x₁(t)] + a₂T[x₂(t)]

  • Example: y(t) = 2x(t), y[n] = x[n] + 0.5x[n-1]

• Non-linear: Violates superposition.

  • Example: y(t) = x²(t), y[n] = n·x[n]

[!TIP]

Test for Linearity: Check if output is a linear combination of inputs. Presence of powers, products, or functions like sin(x(t)) usually indicates non-linearity.

B. Time-Invariant (TI) vs. Time-Varying (TV) Systems

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

  • T[x(t - t₀)] = y(t - t₀) (CT)

  • T[x[n - n₀]] = y[n - n₀] (DT)

• TV: Does not satisfy above.

  • Example: y(t) = t·x(t) (CT), y[n] = n·x[n] (DT)

[!TIP]

Test: Replace t with t - t₀ (or n with n - n₀) in the system equation. If output becomes y(t - t₀) (or y[n - n₀]) without extra terms → TI.

C. Causal, Non-causal, and Anti-causal Systems

• Causal: Output at time t₀ (or n₀) depends only on present and past inputs.

  • CT: y(t₀) depends on x(τ) for τ ≤ t₀.

  • DT: y[n₀] depends on x[k] for k ≤ n₀.

  • Example: y(t) = x(t) + x(t-1) (causal); y(t) = x(t+1) (non-causal).

• Non-causal: Depends on future inputs.

• Anti-causal: Depends only on future inputs.

  • Example: y(t) = x(t+1) (anti-causal).

[!TIP]

Physical realizability: Real-time systems must be causal. Non-causal systems can be implemented with offline processing (buffering).

D. Stable vs. Unstable Systems (BIBO Stability)

• BIBO Stable: Every bounded input produces bounded output.

  • CT: |x(t)| ≤ M_x < ∞ ⇒ |y(t)| ≤ M_y < ∞.

  • Condition: Impulse response h(t) absolutely integrable: ∫_{-∞}^{∞} |h(t)| dt < ∞.

  • DT: Σ_{n=-∞}^{∞} |h[n]| < ∞.

• Unstable: Some bounded input yields unbounded output.

  • Example: y(t) = t·x(t) (unstable); h(t) = e^{at}u(t) with a>0 (unstable).

[!TIP]

Check Stability: For LTI systems, test absolute summability/integrability of impulse response. For non-LTI, use direct BIBO test.

E. Memoryless vs. Dynamic Systems

• Memoryless: Output at t₀ depends only on input at t₀.

  • Example: y(t) = 2x(t), y[n] = x²[n].

• Dynamic (With Memory): Output depends on past/future inputs.

  • Example: y(t) = x(t) + x(t-1), y[n] = 0.5y[n-1] + x[n].

F. Properties of LTI Systems

1. Commutative: x(t) * h(t) = h(t) * x(t) (CT), x[n] * h[n] = h[n] * x[n] (DT).

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

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

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

5. Stability: As per BIBO condition above.

6. Memory: If h(t) (or h[n]) has more than one non-zero sample → dynamic.

G. Comparison between CT and DT LTI Systems

H. Impulse Response Representation

• Definition: Output h(t) (CT) or h[n] (DT) when input is δ(t) or δ[n].

• Significance: Completely characterizes an LTI system via convolution.

  • y(t) = x(t) * h(t)

  • y[n] = x[n] * h[n]

• For systems described by differential/difference equations: Find h(t) or h[n] by solving with x(t)=δ(t) or x[n]=δ[n] and zero initial conditions.

III. LTI System Analysis: Time Domain

A. Convolution

1. Convolution Integral (CT)

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

Steps:

1. Express x(τ) and h(τ).

2. Replace τ with t-τ in h(·) → h(t-τ).

3. Flip and shift: For fixed t, multiply x(τ) and h(t-τ) over τ.

4. Integrate over all τ.

2. Convolution Sum (DT)

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

Steps:

1. For each n, multiply x[k] with flipped/shifted h[n-k].

2. Sum over all k.

3. Properties of Convolution

• Commutative: x * h = h * x

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

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

• Associative with Scalar Multiplication: a (x * h) = (a x) * h = x * (a h)

• Convolution with δ: x * δ = x

• Convolution with u[n]: x[n] * u[n] = Σ_{k=-∞}^{n} x[k] (running sum)

4. Step Response from Impulse Response

• Step response s(t) (CT) or s[n] (DT) is output when input is u(t) or u[n].

• Since u(t) = ∫_{-∞}^{t} δ(τ) dτ (CT) or u[n] = Σ_{k=-∞}^{n} δ[k] (DT),

$$ s(t) = (u * h)(t) = \int_{-\infty}^{t} h(\tau) d\tau \quad \text{(CT)} $$

$$ s[n] = (u * h)[n] = \sum_{k=-\infty}^{n} h[k] \quad \text{(DT)} $$

• Alternative: s(t) = ∫_{-∞}^{t} h(τ) dτ; s[n] = Σ_{k=-∞}^{n} h[k].

[!TIP]

Convolution Shortcut for DT with Finite Support: Use tabular method. For CT with piecewise signals, find intervals where signals overlap.

B. Solving Differential/Difference Equations for Impulse Response

CT LTI System:

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

1. Take Laplace transform (zero ICs): (a_N s^N + ... + a_0) Y(s) = (b_M s^M + ... + b_0) X(s).

2. System function: H(s) = Y(s)/X(s) = (b_M s^M + ... + b_0)/(a_N s^N + ... + a_0).

3. Impulse response: h(t) = L^{-1}{H(s)}.

DT LTI System:

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

1. Take Z-transform (zero ICs): (a_0 + a_1 z^{-1} + ... + a_N z^{-N}) Y(z) = (b_0 + b_1 z^{-1} + ... + b_M z^{-M}) X(z).

2. System function: H(z) = Y(z)/X(z).

3. Impulse response: h[n] = Z^{-1}{H(z)}.

C. Block Diagram Representations

1. Direct Form I & II

• Direct Form I: Realizes difference equation directly with adders, multipliers, and delays.

  
  x[n] → [b₀] → (+) → y[n]
  
            ↑     |
  
            [b₁]  [z⁻¹]
  
            ↑     |
  
            [b₂]  [z⁻¹]
  
            ...   ...
  
            [a₁]  [z⁻¹]
  
            ↑     |
  
            [a₂]  [z⁻¹]
  
            ...   ...
  
  

  (Feedforward: b_k; Feedback: a_k)

• Direct Form II: Combines delays to reduce memory elements.

  
  x[n] → [b₀] → (+) → y[n]
  
            ↑     |
  
            [b₁]  [z⁻¹]
  
            ↑     |
  
            [b₂]  [z⁻¹]
  
            ...   ...
  
            [a₁]  [z⁻¹]
  
            ↑     |
  
            [a₂]  [z⁻¹]
  
            ...   ...
  
  

  (Same structure but shared delay line)

2. Cascade and Parallel Forms

• Cascade (Series): H(z) = H₁(z) H₂(z) ... H_k(z). Blocks in series.

• Parallel: H(z) = H₁(z) + H₂(z) + ... + H_k(z). Blocks in parallel, outputs summed.

[!TIP]

Direct Form II is more efficient (fewer delays). For higher-order systems, cascade of second-order sections preferred for numerical stability.

IV. Z-Transform

A. Definition and Region of Convergence (ROC)

• Definition (Bilateral):

$$ X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}, \quad z \in \mathbb{C} $$

• ROC: Set of z values for which sum converges absolutely (Σ |x[n] z^{-n}| < ∞).

• Unilateral (One-sided): X⁺(z) = Σ_{n=0}^{∞} x[n] z^{-n} (used for causal sequences with ICs).

B. Properties of Z-Transform

Let x[n] ↔ X(z), ROC: R_x.

C. ROC Properties

1. ROC of Finite Duration Sequences: Entire z-plane except possibly z=0 and/or z=∞.

2. ROC and Causality:

   • Right-sided (causal) sequence → ROC: |z| > r (outside outermost pole, includes ∞).

   • Left-sided (anti-causal) sequence → ROC: |z| < r (inside innermost pole, includes 0).

   • Two-sided → ROC: annular region between two poles.

3. ROC and Stability:

   • For causal LTI system: BIBO stable iff ROC includes unit circle (|z|=1).

   • For anti-causal: ROC includes unit circle iff stable.

   • General: System stable iff ROC includes unit circle.

[!TIP]

ROC Rules: Never contains poles; for rational X(z), ROC bounded by poles. Causality ⇒ ROC outside outermost pole. Stability ⇒ ROC includes |z|=1.

D. Inverse Z-Transform

1. Partial Fraction Expansion (PFE):

   • For rational X(z) = N(z)/D(z).

   • Expand into simpler terms (e.g., A/(1 - p₁z^{-1})).

   • Use known pairs: a^n u[n] ↔ 1/(1 - az^{-1}), |z|>|a|.

   • For two-sided sequences, consider both causal and anti-causal parts based on ROC.

2. Power Series Expansion:

   • Expand X(z) as Taylor series in z^{-1}: X(z) = Σ x[n] z^{-n}.

   • Coefficients x[n] read directly.

   • Useful for finite-duration or when ROC is ring.

E. Unilateral Z-Transform

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

• Used for: Solving difference equations with initial conditions (ICs).

• Property: Z⁺{x[n-k]u[n]} = z^{-k} X⁺(z) + Σ_{m=0}^{k-1} x[m] z^{-m} (for k>0).

• Initial Value Theorem: x[0] = lim_{z→∞} X⁺(z).

• Final Value Theorem: Same as bilateral if applicable.

F. System Function H(z) from Impulse Response

• H(z) = Z{h[n]} with ROC determined by h[n].

• For causal LTI system: H(z) = Y(z)/X(z) from difference equation.

• Poles: Roots of denominator polynomial → determine stability/ROC.

• Zeros: Roots of numerator polynomial.

V. Fourier Analysis

A. Continuous-Time Fourier Transform (CTFT)

1. Definition:

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

Inverse:

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

2. Convergence: ∫ |x(t)| dt < ∞ (absolute integrability). For periodic signals, use Fourier series (impulse train spectrum).

3. Properties (let x(t) ↔ X(jω)):

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

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

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

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

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

   • Parseval’s Theorem: ∫ |x(t)|² dt = (1/(2π)) ∫ |X(jω)|² dω

   • Duality: If x(t) ↔ X(jω), then X(t) ↔ 2π x(-ω) (up to scaling).

[!TIP]

Duality: Swap time and frequency (with 2π factor). Useful to derive transforms (e.g., sinc ↔ rect).

B. Discrete-Time Fourier Transform (DTFT)

1. Definition:

$$ X(e^{jω}) = \sum_{n=-\infty}^{\infty} x[n] e^{-jωn}, \quad ω \in [-π, π] \text{ (periodic with } 2π\text{)} $$

Inverse:

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

2. Convergence: Σ |x[n]| < ∞ (absolute summability). For periodic signals, DTFT is impulse train.

3. Properties (similar to CTFT, but frequency periodic):

   • Linearity, time shifting (x[n-n₀] ↔ e^{-jωn₀} X(e^{jω})), frequency shifting (e^{jω₀n}x[n] ↔ X(e^{j(ω-ω₀)})), etc.

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

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

4. Applications:

   • Frequency analysis of DT signals.

   • Filter design (ideal filters have rectangular DTFT).

   • Solving difference equations (by taking DTFT).

C. Fourier Series

1. Continuous-Time Fourier Series (CTFS)

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

• Exponential Form:

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

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

• Trigonometric Form:

$$ x(t) = a₀ + \sum_{k=1}^{∞} [a_k \cos(kω₀ t) + b_k \sin(kω₀ t)] $$

where a₀ = c₀, a_k = c_k + c_{-k}, b_k = j(c_k - c_{-k}).

2. Discrete-Time Fourier Series (DTFS)

For periodic DT signal x[n] with period N.

• Exponential Form:

$$ x[n] = \sum_{k=0}^{N-1} c_k e^{j(2πkn/N)} $$

$$ c_k = \frac{1}{N} \sum_{n=0}^{N-1} x[n] e^{-j(2πkn/N)} $$

• Properties: Periodic in k with period N. Parseval: (1/N) Σ_{n=0}^{N-1} |x[n]|² = Σ_{k=0}^{N-1} |c_k|².

VI. State-Space Analysis

A. State Variables and State Equations

• State: Minimal set of variables q₁(t), ..., q_n(t) (or q[n]) that uniquely determine future system behavior given present state and input.

• State Vector: q(t) = [q₁(t), ..., q_n(t)]^T (CT) or q[n] (DT).

• State Equations (first-order vector differential/difference equations):

  • CT: dq(t)/dt = A(t) q(t) + B(t) x(t)

  • DT: q[n+1] = A[n] q[n] + B[n] x[n]

B. Matrix Representation of Systems

• State Equation: q̇(t) = A q(t) + B x(t) (CT, LTI case) or q[n+1] = A q[n] + B x[n] (DT).

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

• Matrices:

  • A: system matrix (n×n)

  • B: input matrix (n×m) for m inputs

  • C: output matrix (p×n) for p outputs

  • D: feedthrough matrix (p×m)

C. State Transition Matrix Φ(t)

1. Definition: For homogeneous system q̇(t) = A q(t), Φ(t) satisfies:

   • dΦ(t)/dt = A Φ(t)

   • Φ(0) = I (identity)

   • Solution: q(t) = Φ(t) q(0).

2. Properties:

   • Φ(0) = I

   • Φ(t₁ + t₂) = Φ(t₁) Φ(t₂) (for time-invariant A)

   • Φ(-t) = Φ^{-1}(t)

   • Φ(t) is nonsingular for all t.

3. Methods of Computation:

   • Eigenvalue Method: If A diagonalizable, A = V Λ V^{-1}, then Φ(t) = V e^{Λ t} V^{-1}, where e^{Λ t} is diagonal with e^{λ_i t}.

   • Cayley-Hamilton Theorem: Express Φ(t) as polynomial in A: Φ(t) = α₀(t)I + α₁(t)A + ... + α_{n-1}(t)A^{n-1}, solve for α_i(t) using e^{At} series or inverse Laplace.

   • Laplace Transform: Φ(t) = L^{-1}{(sI - A)^{-1}}.

D. Representation of Multi-input, Multi-output (MIMO) Systems

• State-space naturally handles MIMO:

  • x(t) is m×1 input vector.

  • y(t) is p×1 output vector.

  • B is n×m, C is p×n, D is p×m.

• Example: 3-input, 2-output system:

  
  q̇(t) = A q(t) + [b₁ b₂ b₃] x(t)   (B is n×3)
  
  y(t) = [c₁^T; c₂^T] q(t) + [d₁₁ d₁₂ d₁₃; d₂₁ d₂₂ d₂₃] x(t)   (C is 2×n, D is 2×3)
  
  

VII. Sampling Theorem

A. Nyquist-Shannon Sampling Theorem

• Statement: A band-limited CT signal x(t) with no frequency components above B Hz (X(jω)=0 for |ω|>ω_m, ω_m=2πB) can be perfectly reconstructed from its samples x(nT_s) if sampling frequency ω_s = 2π/T_s > 2ω_m (i.e., f_s > 2B).

• Nyquist Rate: 2B samples/sec (or ω_s = 2ω_m rad/sec).

• Reconstruction Formula (Sinc interpolation):

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

where sinc(t/T_s) = sin(π t/T_s)/(π t/T_s).

B. Aliasing and its Effects

• Aliasing: Occurs when f_s ≤ 2B. Higher frequencies |f| > f_s/2 fold back (alias) into baseband [-f_s/2, f_s/2].

• Effect: Different continuous-time signals can have identical sampled sequences → irreversible distortion.

• Anti-aliasing Filter: Low-pass filter before sampling to band-limit signal to B < f_s/2.

C. Reconstruction of Signals from Samples

• Ideal Reconstruction: Using sinc function as interpolation kernel (as above).

• Practical Reconstruction: Use zero-order hold (ZOH) or first-order hold (FOH), which approximate sinc but introduce distortion.

• Steps:

  1. Sample x(t) at t=nT_s → x[n].

  2. Convert to impulses: x_s(t) = Σ x[n] δ(t - nT_s).

  3. Pass through ideal low-pass filter H_r(f) with gain T_s and cutoff f_c = f_s/2 → x(t).

D. Applications and Implications

• Digital Signal Processing: Foundation of analog-to-digital conversion.

• Communications: Sampling of bandpass signals (undersampling/IF sampling).

• Implications:

  • Must know signal bandwidth to choose f_s.

  • Anti-aliasing filter essential.

  • Reconstruction filter needed for DAC.

VIII. Additional Frequently Tested Topics

A. Convergence of DTFT

• Condition: Σ_{n=-∞}^{∞} |x[n]| < ∞ (absolute summability) → DTFT converges absolutely.

• If not absolutely summable, DTFT may still exist in mean-square sense (e.g., for finite-energy signals).

• Periodic signals: DTFT is periodic impulse train (not absolutely convergent).

B. Basic Operations on DT Signals

• Folding: x[-n] → reflect about n=0.

• Shifting: x[n - n₀] → delay by n₀ (right shift if n₀>0).

• Combined: x[an - b] → scale then shift, or x[a(n - b/a)].

C. Periodicity Determination for Composite Signals

For x[n] = Σ A_k cos(ω_k n + φ_k):

• Fundamental period N: Smallest positive integer such that ω_k N = 2π m_k for all k (m_k integers).

• If any ω_k/(2π) is irrational → aperiodic.

• Example: x[n] = cos(0.1π n) + cos(0.3π n) → ω₁/(2π)=0.05, ω₂/(2π)=0.15 → rational? 0.1π N = 2π m₁ ⇒ N=20m₁; 0.3π N = 2π m₂ ⇒ N=20m₂/3 → LCM(20, 20/3) = 20 → N=20.

D. Short Notes (as per frequent questions)

1. Convolution

• Definition: Operation measuring overlap between two signals as one is shifted.

• CT: Integral; DT: Sum.

• Significance: LTI system output = input * impulse response.

• Properties: Commutative, associative, distributive.

• Graphical Method: Flip, shift, multiply, integrate/sum.

2. Unilateral Z-Transform

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

• Use: Solving difference equations with initial conditions (ICs included as extra terms).

• Property: Z⁺{x[n-k]u[n]} = z^{-k} X⁺(z) + Σ_{m=0}^{k-1} x[m] z^{-m}.

• Initial/Final Value Theorems apply.

3. Applications of DTFT

• Frequency Response: H(e^{jω}) of LTI system.

• Filter Design: Ideal low-pass, high-pass, band-pass filters have rectangular DTFTs.

• Spectral Analysis: Plot magnitude/phase spectra of DT signals.

• Solving Difference Equations: Convert to algebraic in ω-domain.

4. State Space Analysis

• Advantages: Handles MIMO, non-zero ICs, time-varying systems naturally.

• Components: State vector q(t), state equation q̇ = Aq + Bx, output equation y = Cq + Dx.

• Solution: q(t) = Φ(t) q(0) + ∫₀^t Φ(t-τ) B x(τ) dτ (CT).

• State Transition Matrix Φ(t) key for homogeneous solution.

Final Note: Always check causality and stability via impulse response or system function. For Z-transform, ROC determines causality and stability. For Fourier transforms, use properties to avoid direct integration.