Digital Signal Processin (EX-703 (C)) - Unit 4 Short Notes

UNIT 4: Digital Signal Processing (Based on Nov 2023 Exam)

1.0 Fundamentals of Discrete-Time Signals & Systems

1.1 Representation of Sequences

• A discrete-time signal is represented as a sequence x[n], where n is an integer (time index).

• Length of a signal: For a sequence defined over n₁ ≤ n ≤ n₂, its length is L = n₂ - n₁ + 1.

  Example: x[n] = [2, -1, 5] for 0 ≤ n ≤ 2 has length L = 3.

1.2 Basic Signal Operations

• Addition: y[n] = x₁[n] + x₂[n] (element-wise, requires same length/defined range).

• Multiplication: y[n] = x₁[n] * x₂[n] (element-wise).

  Example from Exam:

  Given g[n] = [-2, 1, 5] (0≤n≤2), c[n] = [3.2, 41, 36, -9.5, 0] (0≤n≤4).

  • Addition: Align indices. y[0]=-2+3.2=1.2, y[1]=1+41=42, y[2]=5+36=41, y[3]=0+(-9.5)=-9.5, y[4]=0+0=0.

  • Multiplication: y[0]=(-2)(3.2)=-6.4, y[1]=(1)(41)=41, y[2]=(5)(36)=180, y[3]=0, y[4]=0.

1.3 Classification of Discrete-Time Signals

• Symmetry (w.r.t. n=0):

  • Even: x[n] = x[-n] for all n.

  • Odd: x[n] = -x[-n] for all n.

• Periodicity: x[n] is periodic with period N if x[n] = x[n+N] for all n, where N is the smallest positive integer satisfying this.

• Energy & Power:

  • Energy Signal: Finite energy E = Σ_{n=-∞}^{∞} |x[n]|² < ∞. Zero average power.

  • Power Signal: Finite average power P = lim_{N→∞} (1/(2N+1)) Σ_{n=-N}^{N} |x[n]|² < ∞. Infinite energy.

  • Periodic signals are power signals; finite-duration non-zero signals are energy signals.

1.4 System Properties (LTI Focus)

• Causality: An LTI system is causal if its output depends only on present and past inputs. For transfer function H(z), causality requires ROC to be exterior of the outermost pole (for causal h[n]).

• Stability (BIBO): An LTI system is BIBO stable if every bounded input produces a bounded output. Necessary & Sufficient: Σ_{n=-∞}^{∞} |h[n]| < ∞. For rational H(z), stability requires ROC includes the unit circle (|z|=1).

• LTI System Representation: Output y[n] = x[n] * h[n] (convolution).

Exam Tip: To determine causality & stability from H(z), first find poles. For a causal system, ROC is |z| > max|poles|. Check if this ROC includes |z|=1 for stability.

2.0 Z-Transform and Its Applications

2.1 Definition & Region of Convergence (ROC)

• Bilateral Z-Transform: X(z) = Σ_{n=-∞}^{∞} x[n] z^{-n}. ROC is the set of z where the sum converges (annular region in z-plane).

• Unilateral Z-Transform: X(z) = Σ_{n=0}^{∞} x[n] z^{-n}. ROC is exterior of the outermost pole (including ∞).

• ROC Properties: Cannot contain poles; for finite-length sequences, ROC is entire z-plane except possibly z=0 and/or z=∞.

2.2 Properties of Z-Transform (for causal signals)

2.3 Inverse Z-Transform Methods

• Power Series Expansion: Expand X(z) as Σ x[k] z^{-k}. Coefficients give x[k]. Valid within ROC.

• Partial Fraction Expansion (PFE): For rational X(z), expand into simpler terms. Use causal/anti-causal inverse based on ROC.

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

2.4 System Analysis using Z-Transform

• Transfer function from difference equation: Take Z-transform (with zero initial conditions for causal system).

• Stability Criterion: For a causal LTI system, all poles must lie inside the unit circle (|p| < 1). Equivalently, ROC must include the unit circle.

• Causal vs. Anti-causal: From ROC and pole-zero plot.

  • Causal: ROC is exterior of outermost pole (|z| > r₀). h[n]=0 for n<0.

  • Anti-causal: ROC is interior of innermost pole (|z| < r₀). h[n]=0 for n>0.

2.5 Z-Transform of Finite-Length Sequence (Exam Problem)

Given: x[n] = α^n for M ≤ n ≤ N-1, 0 otherwise. Solution:

X(z) = Σ_{n=M}^{N-1} α^n z^{-n} = Σ_{n=M}^{N-1} (α/z)^n

This is a finite geometric series.

X(z) = (α/z)^M * [1 - (α/z)^{N-M}] / [1 - (α/z)] for α/z ≠ 1.

\boxed{X(z) = \frac{\alpha^M z^{-M} - \alpha^N z^{-N}}{1 - \alpha z^{-1}}} ROC: Entire z-plane except z=0 and/or z=∞ (since finite duration). If α ≠ 0, ROC is 0 < |z| < ∞.

3.0 Frequency Domain Analysis: Fourier Transforms

3.1 Discrete-Time Fourier Transform (DTFT)

• Definition: X(e^{jω}) = Σ_{n=-∞}^{∞} x[n] e^{-jωn}. Periodic with period 2π.

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

3.2 Discrete Fourier Transform (DFT)

• Definition (N-point): For a sequence x[n] defined for 0 ≤ n ≤ N-1:

  X[k] = Σ_{n=0}^{N-1} x[n] e^{-j(2π/N)kn}, k = 0, 1, ..., N-1.

• Inverse DFT (IDFT): x[n] = (1/N) Σ_{k=0}^{N-1} X[k] e^{j(2π/N)kn}.

• Relationship to DTFT: DFT is a sampled version of DTFT: X[k] = X(e^{jω}) |_{ω = 2πk/N}.

• Computation (Exam Problem): For x[n] = δ[n] (1 at n=0, 0 for 1≤n≤N-1):

  X[k] = Σ_{n=0}^{N-1} δ[n] e^{-j(2π/N)kn} = e^{0} = 1 for all k.

  \boxed{X[k] = 1, \quad k = 0, 1, ..., N-1}

3.3 Fast Fourier Transform (FFT) - Conceptual

• Need: Direct DFT computation requires O(N²) operations. FFT reduces to O(N log₂ N).

• Key Idea: Divide-and-conquer (e.g., Cooley-Tukey).

• Types:

  • Decimation-in-Time (DIT): Input sequence is split into even/odd indexed samples.

  • Decimation-in-Frequency (DIF): Output sequence is split into even/odd indexed frequencies.

4.0 Orthogonal Transforms

4.1 General Concept of Orthogonal Transform

For a length-N real sequence x[n], an orthogonal transform produces coefficients X[k]:

X[k] = Σ_{n=0}^{N-1} x[n] · φ_k[n], k=0,...,N-1

where {φ_k[n]} is a set of orthogonal basis vectors.

• Orthogonality: Σ_{n=0}^{N-1} φ_k[n] φ_m[n] = 0 for k ≠ m.

• Inverse Transform: x[n] = (1/N) Σ_{k=0}^{N-1} X[k] · φ_k[n].

• Parseval's Theorem (Energy Preservation): Σ_{n=0}^{N-1} |x[n]|² = (1/N) Σ_{k=0}^{N-1} |X[k]|².

4.2 Specific Transforms (Brief)

• DFT: Basis vectors are complex exponentials φ_k[n] = e^{j(2π/N)kn}. Orthogonal but not orthonormal (energy scaling factor N).

• Discrete Cosine Transform (DCT): Uses only cosines, real-valued, excellent for energy compaction (JPEG, MP3).

• Walsh-Hadamard Transform (WHT): Basis vectors are ±1 square waves. Computationally simple.

5.0 Digital Filter Structures & Realization

5.1 FIR Filter Structures

• Direct Form (Transversal): y[n] = Σ_{k=0}^{M} b_k x[n-k]. Simple, M multipliers, M delays.

• Cascade Form: H(z) = Π_{i=1}^{K} H_i(z). Each H_i(z) often a second-order section. Improves numerical stability.

5.2 IIR Filter Structures

• Direct Form I: Separate FIR and IIR sections. 2M multipliers (for order M), 2M delays.

• Direct Form II (Canonical): Shares delay elements. Minimum number of delays (M). M multipliers for feedback, M for feedforward.

  Block Diagram (Standard):

  x[n] -->[b0]--+

           |
  

         (+)--> y[n]
  

           |       |
  

      [b1]--[z⁻¹]--+
  

           |       |
  

         (+)------+
  

           |       |
  

      [b2]--[z⁻¹]--+
  

           :       :
  

           :       :
  

         (-)------+
  

           |       |
  

      [-a1]--[z⁻¹]--+
  

           |       |
  

         (+)------+
  

           |       |
  

      [-a2]--[z⁻¹]--+
  

• Cascade Form: H(z) = Π_{i=1}^{K} (b_{0i} + b_{1i}z^{-1} + b_{2i}z^{-2}) / (1 + a_{1i}z^{-1} + a_{2i}z^{-2}). Each section is a biquad.

• Parallel Form: H(z) = Σ_{i=1}^{K} (c_i + d_i z^{-1}) / (1 + a_{1i}z^{-1} + a_{2i}z^{-2}).

• Lattice Structure: Good for quantization analysis, used in speech coding.

5.3 Realization Example (Direct Form II)

Given: H(z) = (0.44z² + 0.362z + 0.02) / (z² + 0.4z + 0.18z - 0.2). Note: Denominator appears misprinted in exam (0.4z² likely 0.4z). Assuming standard form: H(z) = (0.44z² + 0.362z + 0.02) / (z² + 0.4z + 0.18z - 0.2) is inconsistent. Let's assume correct form is:

H(z) = (0.44z² + 0.362z + 0.02) / (z² + 0.4z - 0.2)? Or (0.44z² + 0.362z + 0.02) / (z² + 0.4z + 0.18)? The exam has +0.18z -0.2. Let's use as written but normalize by dividing numerator and denominator by z²:

H(z) = (0.44 + 0.362z^{-1} + 0.02z^{-2}) / (1 + 0.4z^{-1} + 0.18z^{-1} - 0.2z^{-2}) -> still inconsistent. Best approach for exam: Write in standard Direct Form II form y[n] = b₀x[n] + b₁x[n-1] + b₂x[n-2] - a₁y[n-1] - a₂y[n-2].

From H(z) = (0.44z² + 0.362z + 0.02) / (z² + 0.4z + 0.18z - 0.2), the denominator likely is z² + 0.4z + 0.18? The -0.2 might be a constant term. Let's assume:

H(z) = (0.44z² + 0.362z + 0.02) / (z² + 0.4z + 0.18) is plausible. Then:

b₀=0.44, b₁=0.362, b₂=0.02; a₀=1, a₁=0.4, a₂=0.18. Direct Form II Realization:

1. Two delay elements (z⁻¹).

2. Feedback path: -a₁ = -0.4, -a₂ = -0.18.

3. Feedforward path: b₀=0.44, b₁=0.362, b₂=0.02.

4. Summation nodes as per standard block diagram.

5.4 Reducing Multipliers

• FIR Linear Phase: For symmetric/anti-symmetric coefficients (h[n]=h[M-n] or h[n]=-h[M-n]), number of multipliers ≈ (M+2)/2 (about half).

• IIR Cascade/Parallel: Can share multipliers between sections if coefficients are reused.

• Exploiting Zero Coefficients: Skip multipliers for zero-valued coefficients.

6.0 Advanced Topics & Applications

6.1 Optimal Filtering

• Concept: Design a filter that minimizes a cost function (e.g., mean-square error between desired and actual output).

• Wiener Filter: Optimal in mean-square sense for filtering noisy signals. Two types:

  • Wiener Filter for Signal Estimation: Estimates a desired signal d[n] from an observed signal x[n].

  • Wiener Filter for Noise Cancellation: Uses a reference noise signal to cancel noise from primary signal.

  Example: Noise cancellation in ECG: x₁[n] = s[n] + n₁[n] (primary), x₂[n] = n₂[n] (reference, correlated with n₁). Wiener filter estimates n₁[n] from x₂[n] and subtracts.

6.2 Random Processes

• Strict Sense Stationary (SSS): All statistical properties (pdf, moments) invariant to time shift. p(x[t₁],...,x[tₖ]) = p(x[t₁+τ],...,x[tₖ+τ]) for all τ.

• Wide Sense Stationary (WSS): Weaker condition:

  1. Constant mean: E[x[n]] = μ_x (independent of n).

  2. Autocorrelation depends only on lag: R_x[m] = E[x[n] x^*[n-m]].

• Significance in DSP: Many signal processing theorems (e.g., Wiener-Khintchine) assume WSS. Optimal filters (Wiener) are derived for WSS signals.

6.3 Filter Design Methods (Short Notes Scope)

• Window Method:

  1. Start with ideal impulse response h_d[n] (e.g., ideal low-pass).

  2. h_d[n] is infinite and non-causal.

  3. Apply a window w[n] (e.g., Hamming, Hanning) to obtain a finite, causal sequence: h[n] = h_d[n] w[n].

  4. Trade-off: Main lobe width (transition band) vs. side lobe level (ripple).

• Parks-McClellan (Remez) Algorithm:

  1. Designs optimal equiripple FIR filters in the minimax sense (minimize maximum error).

  2. Alternation theorem: Error alternates between +δ and -δ at (L+2)/2 frequencies (for order L).

  3. Iterative algorithm to find filter coefficients that satisfy this condition.

  4. Superior to window method: sharper transition for same order, controlled ripple.

6.4 General Applications of DSP

• Speech/Audio: Compression (MP3), noise reduction, recognition.

• Image/Video: Compression (JPEG, MPEG), enhancement, analysis.

• Radar/Sonar: Target detection, tracking, beamforming.

• Communications: Modulation/demodulation, equalization, channel coding.

• Biomedical: ECG/EEG analysis, MRI, hearing aids.

• Consumer Electronics: Mobile phones, digital cameras, music players.

Key Exam-Centric Formulas & Results

Final Reminder: For Direct Form II realization, always:

1. Write transfer function in negative powers of z (standard form).

2. Identify b_k (feedforward) and a_k (feedback, with a₀=1).

3. Draw block diagram with shared delays (minimum M delays for order M).