Signals & Systems (EC-402) - Unit 4 Short Notes

UNIT 4: Signals & Systems - Short Notes

I. FUNDAMENTALS OF SIGNALS

Classification by Nature

[!TIP] Exam Focus: Be ready to convert between CT and DT representations (e.g., x(t) → x[n] = x(nT_s) where T_s is sampling period).

Standard Signals & Basic Operations

1. Standard Signals:

• Unit Impulse: δ(t) (CT), δ[n] (DT).

  • δ(t) = 0 ∀ t ≠ 0, ∫δ(t)dt = 1. δ[n] = 1 for n=0, 0 otherwise.

  • Sifting property: ∫x(t)δ(t-t₀)dt = x(t₀), ∑x[n]δ[n-n₀] = x[n₀].

• Unit Step: u(t) (CT), u[n] (DT).

  • u(t) = 1 for t ≥ 0, 0 otherwise. u[n] = 1 for n ≥ 0, 0 otherwise.

  • Relation to Ramp: r(t) = ∫u(τ)dτ, r[n] = ∑u[k].

• Unit Ramp: r(t) = t u(t), r[n] = n u[n].

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

2. Basic Operations on DT Signals (x[n]):

[!CAUTION] Common Pitfall: x[2n] is compression by 2, not 2x[n]. x[n/2] is expansion (requires interpolation for integer n).

Classification by Characteristics

1. Periodic vs. Aperiodic:

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

• DT Periodic: x[n] = x[n + N₀] for smallest integer N₀ > 0. Fundamental period N₀.

• Condition: For x(t) = A cos(ω₀t + θ), T₀ = 2π/ω₀. For x[n] = A cos(Ω₀n + θ), N₀ = 2π/Ω₀ must be rational (Ω₀/2π = K/N).

2. Energy vs. Power Signals:

Formulas:

• CT: E = ∫_{-∞}^{∞} |x(t)|² dt, P = lim_{T→∞} (1/(2T)) ∫_{-T}^{T} |x(t)|² dt

• DT: E = ∑_{n=-∞}^{∞} |x[n]|², P = lim_{N→∞} (1/(2N+1)) ∑_{n=-N}^{N} |x[n]|²

3. Deterministic vs. Random:

• Deterministic: Completely predictable for all t/n. No uncertainty. (e.g., x(t)=sin(t)).

• Random (Stochastic): Described probabilistically. Exact value cannot be predicted. (e.g., thermal noise voltage).

4. Even & Odd Signals:

• 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 signal x can be written as x = x_e + x_o, where:

  x_e(t) = ½[x(t) + x(-t)], x_o(t) = ½[x(t) - x(-t)].

5. Causality (for Signals):

• Causal Signal: x(t)=0 for t<0, x[n]=0 for n<0.

• Anti-causal Signal: x(t)=0 for t>0, x[n]=0 for n>0.

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

II. FUNDAMENTALS OF SYSTEMS

System Properties

A system is a transformation T mapping input x to output y: y = T{x}.

[!TIP] Linearity Test: Always check T{0}=0 first. If y(t) = x(t) + 1, it's non-linear (non-zero output for zero input).

Linear Time-Invariant (LTI) Systems

• Definition: System that is both Linear and Time-Invariant.

• Significance: Cornerstone of signals & systems. Analysis is simple via convolution. Characterized completely by impulse response h(t) (CT) or h[n] (DT).

• Impulse Response: h(t) = T{δ(t)}, h[n] = T{δ[n]}.

• Representation (Convolution):

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

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

Properties of Impulse Response ↔ System Properties:

III. ANALYSIS OF LTI SYSTEMS

Representation by Differential/Difference Equations

• Linear Constant-Coefficient Differential Equation (CT):

  ∑_{k=0}^{N} a_k (d^k y(t)/dt^k) = ∑_{k=0}^{M} b_k (d^k x(t)/dt^k)

  • N = order of system.

  • Recursive: Output depends on past outputs (a_k ≠ 0 for k>0).

  • Non-recursive: Output depends only on present/past inputs (a_k = 0 for k>0).

• Linear Constant-Coefficient Difference Equation (DT):

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

  • Similar classification: recursive vs. non-recursive.

Block Diagram Representations

1. Direct Form-I (DF-I): Direct implementation of difference/differential equation. Uses adders, multipliers, and delays (DT) / integrators (CT). 2. Direct Form-II (DF-II / Canonical): Minimizes number of delay elements. States are the contents of delay units. 3. Cascade (Series): Systems connected in series: H(z) = H₁(z) H₂(z). Overall impulse response is convolution: h[n] = h₁[n] * h₂[n]. 4. Parallel: Systems connected in parallel: H(z) = H₁(z) + H₂(z). Overall impulse response is sum: h[n] = h₁[n] + h₂[n].

[!TIP] DF-I vs DF-II: DF-I has separate paths for x and y (more delays). DF-II shares delay elements (canonical form, minimal states).

Convolution

1. Definitions:

• CT Convolution Integral:

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

• DT Convolution Sum:

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

2. Graphical/Computational Method (DT Example):

1. Flip (fold) h[k] to get h[-k].

2. Shift by n to get h[n-k].

3. Multiply x[k] with shifted h[n-k] for all k.

4. Sum the products to get y[n].

3. Key Properties:

4. Step Response s(t)/s[n] in terms of h(t)/h[n]:

\boxed{s(t) = u(t) * h(t) = \int_{-∞}^{t} h(τ) dτ}

\boxed{s[n] = u[n] * h[n] = \sum_{k=-∞}^{n} h[k]}

For causal LTI: s(t) = ∫_{0}^{t} h(τ) dτ, s[n] = ∑_{k=0}^{n} h[k].

IV. Z-TRANSFORM (DISCRETE-TIME FOCUS)

Definition & Region of Convergence (ROC)

• Bilateral (Two-sided) Z-Transform:

  \boxed{X(z) = \sum_{n=-∞}^{∞} x[n] z^{-n}}

  • z = re^{jω} (complex variable).

• Unilateral (One-sided) Z-Transform:

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

  • Used for causal signals/systems with non-zero initial conditions.

• Region of Convergence (ROC): Set of z values for which the sum converges (|X(z)| < ∞).

  • ROC is always a ring/annulus (or disk) in the z-plane: R₁ < |z| < R₂.

  • Never contains poles.

  • For rational X(z), ROC extends outward from outermost pole for causal sequences, inward from innermost pole for anti-causal sequences.

Properties of Z-Transform

Inverse Z-Transform Methods

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

2. Partial Fraction Expansion (PFE):

   • For causal sequences (ROC: |z| > max pole), expand in z^{-1}.

   • For anti-causal sequences (ROC: |z| < min pole), expand in z.

3. Contour Integration (Residue Method):

   x[n] = (1/(2πj)) ∮_{C} X(z) z^{n-1} dz

   • C is a counterclockwise contour within ROC encircling origin.

   • x[n] = ∑ Residues of X(z) z^{n-1} at poles inside C.

Special Theorems & Analysis

1. Initial Value Theorem (IVT):

If x[n] = 0 for n < 0 and X(z) has ROC including |z| > ∞ (causal), then:

\boxed{x[0] = \lim_{z→∞} X(z)}

2. Final Value Theorem (FVT):

If all poles of (1 - z^{-1})X(z) lie inside unit circle (ROC includes z=1), then:

\boxed{\lim_{n→∞} x[n] = \lim_{z→1} (1 - z^{-1}) X(z)}

3. System Function H(z):

H(z) = Z{h[n]}. Transfer function of LTI system.

• Poles: Roots of denominator polynomial. Determine ROC boundaries.

• Zeros: Roots of numerator polynomial.

• Pole-Zero Plot: Visualizes system stability/causality.

  • Causality: All poles inside unit circle? No! For causal h[n], ROC is outside the outermost pole. So poles must be inside unit circle for causal & stable system.

  • Stability: ROC must include unit circle (|z|=1).

V. DISCRETE-TIME FOURIER ANALYSIS

Discrete-Time Fourier Series (DTFS)

• For: Periodic DT signals with fundamental period N₀ (x[n] = x[n+N₀]).

• Exponential Form:

  \boxed{x[n] = \sum_{k=0}^{N₀-1} C_k e^{j k ω₀ n}}, where ω₀ = 2π/N₀.

  \boxed{C_k = (1/N₀) \sum_{n=0}^{N₀-1} x[n] e^{-j k ω₀ n}}

• Properties: Linearity, time-shift (phase shift), frequency-shift, conjugation, Parseval's theorem.

• Application: Spectral analysis of periodic DT signals. |C_k| is magnitude spectrum, ∠C_k is phase spectrum.

Discrete-Time Fourier Transform (DTFT)

• For: Aperiodic DT signals (or periodic, but DTFS is better).

• Definition:

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

  • ω is continuous frequency (rad/sample).

  • X(e^{jω}) is periodic with period 2π.

• Inverse DTFT:

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

• Convergence Condition: Sufficient if ∑ |x[n]| < ∞ (absolutely summable).

• Key Properties:

  | Property | Expression | | :--- | :--- | | Linearity | a x₁[n] + b x₂[n] ↔ a X₁(e^{jω}) + b X₂(e^{jω}) | | Time Shifting | x[n-n₀] ↔ e^{-jω n₀} X(e^{jω}) | | Frequency Shifting | e^{jω₀ n} x[n] ↔ X(e^{j(ω-ω₀)}) | | Time Reversal | x[-n] ↔ X(e^{-jω}) = X(e^{jω}) (if x[n] real) | | Differentiation in freq | n x[n] ↔ j (d/dω) X(e^{jω}) | | Convolution | x[n] * h[n] ↔ X(e^{jω}) H(e^{jω}) | | Multiplication | x[n] y[n] ↔ (1/(2π)) ∫_{-π}^{π} X(e^{jθ}) Y(e^{j(ω-θ)}) dθ | | Parseval's | ∑ x[n] y^*[n] = (1/(2π)) ∫_{-π}^{π} X(e^{jω}) Y^*(e^{jω}) dω |

• Applications: Frequency response analysis of LTI systems (H(e^{jω})), spectral analysis.

VI. CONTINUOUS-TIME FOURIER ANALYSIS

Fourier Series (FS)

• For: Periodic CT signals with period T₀, fundamental frequency ω₀ = 2π/T₀.

• Exponential Form:

  \boxed{x(t) = \sum_{k=-∞}^{∞} C_k e^{j k ω₀ t}}

  \boxed{C_k = (1/T₀) \int_{T₀} x(t) e^{-j k ω₀ t} dt}

• Trigonometric Form:

  x(t) = a₀ + ∑_{k=1}^{∞} [a_k cos(kω₀t) + b_k sin(kω₀t)]

  a₀ = C₀, a_k = C_k + C_{-k}, b_k = j(C_k - C_{-k}).

• Magnitude & Phase Spectra: Plots of |C_k| vs kω₀ and ∠C_k vs kω₀.

Fourier Transform (FT)

• For: Aperiodic CT signals (energy signals).

• Definition:

  \boxed{X(jω) = \int_{-∞}^{∞} x(t) e^{-jω t} dt}

• Inverse FT:

  \boxed{x(t) = (1/(2π)) \int_{-∞}^{∞} X(jω) e^{jω t} dω}

• Key Properties:

  | Property | Expression | | :--- | :--- | | Linearity | a x₁(t) + b x₂(t) ↔ a X₁(jω) + b X₂(jω) | | Time Shifting | x(t-t₀) ↔ e^{-jω t₀} X(jω) | | Frequency Shifting | e^{jω₀ t} x(t) ↔ X(j(ω-ω₀)) | | Time Scaling | x(at) ↔ (1/|a|) X(jω/a) | | Differentiation in time | dⁿx(t)/dtⁿ ↔ (jω)ⁿ X(jω) | | Integration | ∫_{-∞}^{t} x(τ) dτ ↔ X(jω)/(jω) + π X(0) δ(ω) | | Convolution | x(t)*h(t) ↔ X(jω) H(jω) | | Multiplication | x(t) y(t) ↔ (1/(2π)) ∫ X(jθ) Y(j(ω-θ)) dθ | | Duality | X(t) ↔ 2π x(-ω) | | Parseval's | ∫ x(t) y^*(t) dt = (1/(2π)) ∫ X(jω) Y^*(jω) dω |

• Computation Strategy: Use properties and known FT pairs (e.g., e^{-at}u(t) ↔ 1/(a+jω), rect(t/T) ↔ T sinc(ωT/2)).

• Differentiation-in-time property: Useful for finding FT of triangular pulse from rectangular pulse (derivative of triangle is square pulses).

VII. STATE-SPACE ANALYSIS

State-Space Representation

• Concept: Models system using state variables (minimum set of variables x₁(t), ..., xₙ(t) that summarize past history).

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

• State Equation: Describes evolution of state.

  \boxed{\dot{𝐱}(t) = 𝐀 𝐱(t) + 𝐁 𝐮(t)} (CT)

  \boxed{𝐱[n+1] = 𝐀 𝐱[n] + 𝐁 𝐮[n]} (DT)

• Output Equation: Relates output to state and input.

  \boxed{𝐲(t) = 𝐂 𝐱(t) + 𝐃 𝐮(t)} (CT/DT)

  • 𝐮(t): Input vector (p inputs).

  • 𝐲(t): Output vector (q outputs).

  • 𝐀 (n×n), 𝐁 (n×p), 𝐂 (q×n), 𝐃 (q×p) are system matrices.

• For SISO: 𝐮, 𝐲 are scalars u(t), y(t).

State Transition Matrix Φ(t) / Φ[n]

• Definition: Matrix function that gives solution to homogeneous state equation (𝐮(t)=0).

  \boxed{𝐱_h(t) = Φ(t) 𝐱(0)} (CT, zero-input response)

  \boxed{𝐱_h[n] = Φ[n] 𝐱[0]}

• Properties:

  1. Φ(0) = 𝐈 (Identity matrix).

  2. Φ(t₁ + t₂) = Φ(t₁) Φ(t₂) (for CT, t₁,t₂ ≥ 0).

  3. Φ(t) is invertible: Φ^{-1}(t) = Φ(-t).

  4. dΦ(t)/dt = 𝐀 Φ(t), Φ[n+1] = 𝐀 Φ[n].

• Methods of Determination:

  1. Inverse Laplace Transform:

     Φ(t) = ℒ^{-1} {(s𝐈 - 𝐀)^{-1}} (CT).

  2. Eigenvalue Decomposition (Diagonalization):

     If 𝐀 = 𝐏 𝐃 𝐏^{-1} (𝐃 diagonal of eigenvalues), then Φ(t) = 𝐏 e^{𝐃 t} 𝐏^{-1}.

  3. Cayley-Hamilton Theorem: Express Φ(t) as polynomial in 𝐀 using characteristic equation.

System Representations

• Transfer Function from State-Space (SISO):

  \boxed{H(s) = 𝐂 (s𝐈 - 𝐀)^{-1} 𝐁 + 𝐃} (CT)

  \boxed{H(z) = 𝐂 (z𝐈 - 𝐀)^{-1} 𝐁 + 𝐃} (DT)

• Relationship: State-space is a first-principles model (physical variables). Transfer function is input-output description (ignores internal states).

• Block Diagram: Standard form with integrators (CT) or delays (DT) representing 𝐀 matrix.

VIII. SAMPLING THEORY & APPLICATIONS

Sampling Theorem (Nyquist-Shannon)

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

  \boxed{f_s ≥ 2B} (samples/second) or \boxed{T_s ≤ 1/(2B)} (sampling interval).

  • 2B is the Nyquist rate.

  • f_s/2 = B is the Nyquist frequency.

• Ideal Sampling: Multiply x_a(t) by impulse train δ_T(t) = ∑ δ(t - nT_s).

  Sampled spectrum: X_s(jΩ) = (1/T_s) ∑ X_a(j(Ω - kΩ_s)), where Ω_s = 2πf_s.

• Reconstruction (Sinc Interpolation):

  \boxed{x_a(t) = \sum_{n=-∞}^{∞} x_a(nT_s) \text{sinc}((t - nT_s)/T_s)}

  where sinc(t) = sin(πt)/(πt).

Practical Aspects & Aliasing

• Aliasing: If f_s < 2B, spectral replicas overlap. High frequencies fold back as lower frequencies. Irreversible distortion.

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

• Minimum Sampling Rate & Max Interval: Given spectral band [f₁, f₂] (bandpass), f_s ≥ 2(f₂ - f₁)? Not always! Bandpass sampling (undersampling) possible if f_s ≥ 2(f₂ - f₁) and f_s chosen so replicas don't overlap. But for baseband [0, B], f_s ≥ 2B is mandatory.

Applications

1. Analog-to-Digital Conversion (ADC): Sampling → Quantization → Encoding.

2. Digital Signal Processing (DSP): Process discrete-time signal x[n] (from samples) using digital hardware/filters, then convert back via DAC.

3. Discrete-time processing of CT signals: Design discrete filter H(e^{jω}) to approximate desired CT filter H_a(jΩ) using bilinear transform or impulse invariance.

[!TIP] Exam Alert: Be prepared to calculate minimum f_s for a signal with given bandwidth (e.g., 5.6 MHz to 6.8 MHz → Bandwidth B = 1.2 MHz → f_s(min) = 2.4 MHz, T_s(max) = 1/(2.4e6) ≈ 416.67 ns). For bandpass, concept of undersampling is key.