Digital Signal Processin (EX-703 (C)) - Unit 2 Short Notes

UNIT 2: DIGITAL SIGNAL PROCESSING - CORE CONCEPTS & ANALYSIS

1.0 Discrete-Time Signals & Sequences

1.1 Representation of Sequences

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

• Graphical: Plot of amplitude vs. integer index n.

• Mathematical: Functional definition (e.g., x[n] = α^n u[n]).

• Tabular: List of values for specific n (e.g., x[n] = {2, 1, -1, 0} for n=0,1,2,3).

1.2 Length of a Discrete-Time Signal

For a finite-duration sequence, the length L is the number of non-zero samples.

Example: x[n] = {1, 2, 3} for n=0,1,2 has length L = 3.

1.3 Classification of Discrete-Time Signals

1.4 Fundamental Sequence Operations

• Addition: y[n] = x₁[n] + x₂[n] (sample-by-sample).

• Multiplication: y[n] = x₁[n] * x₂[n].

• Time-Shifting: x[n - n₀] → delay if n₀>0, advance if n₀<0.

• Time-Reversal: x[-n] (fold about n=0).

• Scaling: α * x[n] (amplitude scaling).

1.5 Nyquist Sampling Theorem

To perfectly reconstruct a band-limited continuous-time signal x_c(t) with maximum frequency f_max (Hz) from its samples x[n] = x_c(nT_s), the sampling frequency f_s = 1/T_s must satisfy:

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

The minimum rate f_s = 2f_max is the Nyquist rate. f_Nyquist = 2f_max is the Nyquist frequency. Aliasing occurs if f_s < 2f_max.

2.0 Z-Transform & System Analysis

2.1 Definition

The bilateral (two-sided) Z-transform of a sequence x[n] is:

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

where z is a complex variable. The unilateral (one-sided) Z-transform uses sum from n=0 to ∞.

2.2 Region of Convergence (ROC)

• Definition: Set of z values for which the Z-transform sum converges (is finite).

• Properties: ROC is an annular region (ring) in the z-plane, bounded by poles. It never contains poles.

• Determination: Find |z| values that make the infinite sum converge (often using geometric series).

2.3 Causal & Anti-Causal Systems

• Causal System: x[n] = 0 for n < 0. ROC is exterior of the outermost pole (|z| > r_max).

• Anti-Causal System: x[n] = 0 for n > 0. ROC is interior of the innermost pole (|z| < r_min).

• Two-Sided System: ROC is an annular ring between poles.

2.4 Properties of Z-Transform (Causal Signals)

Let x[n] ↔ X(z), ROC: |z| > r₀.

1. Linearity: a x₁[n] + b x₂[n] ↔ a X₁(z) + b X₂(z), ROC ≥ intersection.

2. Time-Shifting: x[n - n₀] ↔ z^{-n₀} X(z), ROC same as X(z).

3. Scaling in z: a^n x[n] ↔ X(z/a), ROC scaled: |z| > |a|r₀.

4. Convolution: x₁[n] * x₂[n] ↔ X₁(z) X₂(z), ROC is intersection of individual ROCs.

5. Initial Value Theorem (IVT): For causal x[n], x[0] = lim_{z→∞} X(z).

6. Final Value Theorem (FVT): If poles of (z-1)X(z) are inside unit circle, lim_{n→∞} x[n] = lim_{z→1} (z-1) X(z).

2.5 Inverse Z-Transform Methods

• Long Division (Power Series): Expand X(z) as Σ x[k] z^{-k}. ROC determines causality.

• Partial Fraction Expansion: For rational X(z), decompose into simpler terms and use known pairs.

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

2.6 Stability Determination

An LTI system is BIBO stable iff its impulse response h[n] is absolutely summable: Σ_{n=-∞}^{∞} |h[n]| < ∞.

Using Z-transform: For causal systems, stability ⇔ ROC includes the unit circle (|z|=1). For general systems, ROC must include |z|=1.

3.0 Discrete Fourier Transform (DFT)

3.1 Definition of N-point DFT

For a finite-length sequence x[n], 0 ≤ n ≤ N-1:

$$ X[k] = \sum_{n=0}^{N-1} x[n] e^{-j(2π/N)kn} = \sum_{n=0}^{N-1} x[n] W_N^{kn} $$

where W_N = e^{-j(2π/N)} is the twiddle factor. k = 0, 1, ..., N-1.

3.2 DFT as Orthogonal Transform Pair

The DFT and its inverse (IDFT) form an orthogonal transform pair:

$$ x[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k] e^{j(2π/N)kn} = \frac{1}{N} \sum_{k=0}^{N-1} X[k] W_N^{-kn} $$

The basis vectors {W_N^{kn}} are orthogonal over n=0,...,N-1.

3.3 Computation of DFT for Specific Sequences

• Unit Impulse: δ[n] → X[k] = 1 for all k (DC value only).

• Unit Step (within length): u[n] for 0≤n≤N-1 → X[k] = N for k=0, 0 for k≠0 (after considering periodicity).

• Constant: x[n]=1 → X[0]=N, X[k]=0 for k≠0.

3.4 Relationship: DFT vs. DTFT

The Discrete-Time Fourier Transform (DTFT) of x[n] is X(e^{jω}) = Σ_{n=-∞}^{∞} x[n] e^{-jωn}, periodic with 2π.

• DFT samples the DTFT at N equally spaced frequencies over [0, 2π):

$$ X[k] = X(e^{jω}) \big|_{\omega = (2πk/N)} $$

• For a finite-length x[n], the DTFT is continuous and periodic. The DFT provides a discrete-frequency representation.

4.0 Digital Filter Structures & Realizations

4.1 FIR Filters

• Direct Form (Transversal): Implements the difference equation y[n] = Σ_{k=0}^{M} b_k x[n-k].

  • Structure: M delay elements, M+1 multipliers, M adders.

  • Complexity: M+1 multipliers, M adders.

  
  graph LR
  
  X[n] --> B0[ b0 ];
  
  B0 --> Add;
  
  Delay1[ z^-1 ] --> B1[ b1 ];
  
  B1 --> Add;
  
  Delay2[ z^-1 ] --> BM[ bM ];
  
  BM --> Add;
  
  Add --> Y[n];
  
  Delay1 --> Delay2;
  
  

• Cascade Form: Factors transfer function H(z) = Π H_i(z). Each H_i(z) is a small FIR section (often 1st or 2nd order). Better for fixed-point implementation (less sensitivity to coefficient quantization).

4.2 IIR Filters

Transfer function: H(z) = Σ_{k=0}^{M} b_k z^{-k} / (1 - Σ_{k=1}^{N} a_k z^{-k}).

• Direct Form I: Separately implements numerator (FIR) and denominator (IIR) sections, then adds. Requires max(M,N) delays.

• Direct Form II: Shares delay elements between numerator and denominator. More efficient.

  • Complexity: M+N+1 multipliers, M+N adders, max(M,N) delays.

  Example Realization (Direct Form II): For H(z) = (b0 + b1 z^{-1} + b2 z^{-2}) / (1 - a1 z^{-1} - a2 z^{-2}):

  
  graph LR
  
  X[n] --> Add1;
  
  Add1 --> Delay1[ z^-1 ];
  
  Delay1 --> Delay2[ z^-1 ];
  
  Delay2 --> Add2;
  
  Add2 --> Mult_b2[b2];
  
  Mult_b2 --> Add3;
  
  Delay1 --> Mult_b1[b1];
  
  Mult_b1 --> Add3;
  
  Add1 --> Mult_b0[b0];
  
  Mult_b0 --> Add3;
  
  Add3 --> Y[n];
  
  Add2 --> Mult_a2[-a2];
  
  Mult_a2 --> Add1;
  
  Delay2 --> Mult_a1[-a1];
  
  Mult_a1 --> Add1;
  
  

• Cascade/Parallel Forms: Similar to FIR, but sections are 1st/2nd order IIR sections (poles/zeros). Cascade is common for stability (each section stable).

4.3 Reduction of Computational Complexity

• Exploit Symmetry: For linear-phase FIR, h[n] = h[M-n]. Reduces multipliers by ~half.

• Use Efficient Structures: Direct Form II for IIR uses fewer delays than Direct Form I.

• Polyphase Decomposition: For decimation/interpolation, reduces computation rate.

• Example: For M=4 symmetric FIR (h[0]=h[4], h[1]=h[3], h[2]), direct form needs 5 mults. Symmetric form: y[n] = h[2]*(x[n-2]+x[n-2]) + h[1]*(x[n-1]+x[n-3]) + h[0]*(x[n]+x[n-4]) → 3 multipliers.

4.4 Realization Example (Given IIR)

Given: H(z) = (0.44z^2 + 0.362z + 0.02) / (z^2 + 0.4z + 0.18z - 0.2) → Correct denominator: likely z^2 + 0.4z + 0.18? Assuming H(z) = (0.44z^2 + 0.362z + 0.02) / (z^2 + 0.4z + 0.18).

• Direct Form II:

  1. Write in negative powers: H(z) = (0.44 + 0.362 z^{-1} + 0.02 z^{-2}) / (1 + 0.4 z^{-1} + 0.18 z^{-2}).

  2. Difference equation: y[n] = -0.4 y[n-1] - 0.18 y[n-2] + 0.44 x[n] + 0.362 x[n-1] + 0.02 x[n-2].

  3. Draw structure with 2 delays (shared), 5 multipliers (-0.4, -0.18, 0.44, 0.362, 0.02), 2 adders.

5.0 Random Signals & Optimal Filtering

5.1 Classification of Stochastic Processes

5.2 Optimal Filtering (Wiener Filter)

• Concept: Design a filter H(z) to estimate a desired signal d[n] from an observed signal x[n], minimizing the mean square error (MSE) E\{|e[n]|^2\}, where e[n] = d[n] - ŷ[n].

• Wiener-Hopf Equations: For FIR Wiener filter of length M:

$$ \mathbf{R}_x \mathbf{h} = \mathbf{p}_{xd} $$

where `R_x` is the Toeplitz autocorrelation matrix of `x[n]`, `p_xd` is the cross-correlation vector between `x[n]` and `d[n]`, `h` is the optimal coefficient vector.

• Illustrative Example (Simplest Case): Estimate d[n] = x[n-1] from x[n] corrupted by noise v[n] (WSS, uncorrelated with x). Optimal FIR (1-tap) Wiener filter: h_opt = R_x[1] / (R_x[0] + R_v[0]).

6.0 Digital Filter Design Techniques

6.1 Window Method for FIR Design

• Principle:

  1. Start with an ideal frequency response H_d(e^{jω}) (e.g., ideal low-pass with cutoff ω_c).

  2. Find its infinite-duration impulse response h_d[n] via IDFT (often sinc function).

  3. Truncate h_d[n] to finite length M using a window w[n]: h[n] = h_d[n] w[n], 0 ≤ n ≤ M-1.

• Common Windows & Gibbs Phenomenon:

  • Rectangular: Simple truncation. Highest sidelobe (~-13 dB), slow roll-off. Severe Gibbs (ringing).

  • Hamming: w[n] = 0.54 - 0.46 cos(2πn/(M-1)). Sidelobe ~-41 dB, better stopband attenuation, wider transition band.

  • Hanning: w[n] = 0.5 - 0.5 cos(2πn/(M-1)). Sidelobe ~-31 dB.

  Gibbs Phenomenon: Undershoot/overshoot near discontinuities in H_d(e^{jω}) due to truncation. Window choice trades transition width for stopband attenuation.

6.2 Parks-McClellan (Equiripple) Algorithm

• Principle: Minimax optimization. Designs FIR filter that minimizes the maximum error (ripple) in both passband and stopband simultaneously, subject to specified ripple magnitudes δ_p, δ_s.

• Process: Uses Remez exchange algorithm. Alternation theorem: optimal error alternates between ±δ at (L+2) frequencies (for L-th order filter).

• Comparison with Window Method:

  | Aspect | Window Method | Parks-McClellan | | :--- | :--- | :--- | | Design Control | Indirect (via window & M). Transition width fixed by M. | Direct: specify ω_p, ω_s, δ_p, δ_s. | | Optimality | Suboptimal. | Optimal (minimizes maximum ripple). | | Ripple | Passband/stopband ripples not independently controllable. | Equiripple in both bands (by design). | | Transition | Wider for low sidelobes. | Narrowest possible for given δ_p, δ_s. | | Complexity | Simple, analytical. | Iterative, requires software (MATLAB firpm). |

7.0 Applications of DSP

Broad application areas include:

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

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

• Image/Video Processing: Compression (JPEG, MPEG), enhancement, object detection.

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

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

• Control Systems: Digital controllers, system identification.

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

[!TIP]

Exam Focus: Be prepared to derive Z-transform of finite sequences, determine ROC and deduce causality/stability, compute N-point DFT for simple sequences (impulse, step), draw Direct Form II for given IIR H(z), and differentiate SSS vs WSS with definitions. For design questions, compare Window vs Parks-McClellan in terms of optimality and control. Always state assumptions (e.g., causal signal for IVT/FVT).