Digital Signal Processin (EX-703 (C)) - Unit 5 Short Notes

UNIT 5: DIGITAL SIGNAL PROCESSING

Based on Nov 2023 DSP Paper and blueprint. Focus on definitions, derivations, and problem-solving.

1.0 Discrete-Time Signals & Sequences

1.1 Representation & Length

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

• Length of a finite-duration signal: Number of samples. For $x[n]$ defined from $$\displaystyle n = N_1 $$ to $$\displaystyle n = N_2 $$, length $$\displaystyle L = N_2 - N_1 + 1 $$.

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

1.3 Sequence Operations

• Addition: $$\displaystyle y[n] = x_1[n] + x_2[n] $$. Performed sample-by-sample. Sequences must be defined over a common range (zero-pad otherwise).

• Multiplication: $$\displaystyle y[n] = x_1[n] \cdot x_2[n] $$. Element-wise product.

  Example (from Nov 2023):

$$g[n]=[-2, 1, 5] \text{ for } 0\le n\le 2$$

> 

$$c[n]=[3.2, 41, 36, -9.5, 0] \text{ for } 0\le n\le 4$$

> **Addition** (zero-pad $g[n]$ to length 5): $$\displaystyle y[n] = [1.2, 42, 41, -4.5, 0] $$.

> **Multiplication** (zero-pad $g[n]$): $$\displaystyle y[n] = [-6.4, 41, 180, -47.5, 0] $$.

1.4 Classification by Symmetry (w.r.t. n=0)

[!TIP] Any signal $x[n]$ can be decomposed: $$\displaystyle x[n] = x_e[n] + x_o[n] $$, where $$\displaystyle x_e[n] = \frac{1}{2}[x[n]+x[-n]] $$, $$\displaystyle x_o[n] = \frac{1}{2}[x[n]-x[-n]] $$.

2.0 Sampling & Frequency Domain Fundamentals

2.1 Nyquist Rate

• Definition: The minimum sampling rate required to avoid aliasing. For a signal with maximum frequency $$\displaystyle f_m $$ (Hz), Nyquist rate $$\displaystyle f_s \ge 2f_m $$.

• Significance: Ensures perfect reconstruction of the original continuous-time signal from its samples. $$\displaystyle f_s/2 $$ is the folding frequency.

$$\boxed{f_N = 2f_m}$$

2.2 Applications of DSP (Overview)

• Speech Processing (compression, recognition)

• Image/Video Processing (compression, enhancement)

• Radar/Sonar (target detection, tracking)

• Biomedical Signal Analysis (ECG, EEG)

• Audio Processing (noise reduction, equalization)

• Communications (modulation, channel equalization)

3.0 Z-Transform & System Analysis

3.1 Definition of Z-Transform

For a sequence $x[n]$, the bilateral Z-Transform is:

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

For causal sequences ($$\displaystyle x[n]=0 $$ for $$\displaystyle n<0 $$), the unilateral form is used:

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

3.2 Region of Convergence (ROC)

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

• Key Properties:

  1. ROC is a ring/annulus in the z-plane: $$\displaystyle r_1 < |z| < r_2 $$.

  2. Cannot contain poles.

  3. For finite-length sequences, ROC is entire z-plane except possibly $$\displaystyle z=0 $$ and/or $$\displaystyle z=\infty $$.

  4. ROC must include the unit circle ($$\displaystyle |z|=1 $$) for a stable system.

3.3 Properties of Z-Transform for Causal Signals

3.4 Causal vs. Anti-Causal Systems

• Causal System: Output depends only on present and past inputs. Impulse response $$\displaystyle h[n]=0 $$ for $$\displaystyle n<0 $$.

  • ROC of $H(z)$ is exterior of the outermost pole: $$\displaystyle |z| > r_{\max} $$.

• Anti-Causal System: Output depends only on future inputs. $$\displaystyle h[n]=0 $$ for $$\displaystyle n>0 $$.

  • ROC is interior of the innermost pole: $$\displaystyle |z| < r_{\min} $$.

3.5 Z-Transform & ROC for Finite-Length Sequence

Given: $$\displaystyle x[n] = \alpha^n $$ for $M \le n \le N-1$, zero elsewhere.

$$X(z) = \sum_{n=M}^{N-1} \alpha^n z^{-n} = \sum_{n=M}^{N-1} (\alpha z^{-1})^n$$

This is a finite geometric series.

• Case 1: $$\displaystyle \alpha z^{-1} \neq 1 $$

$$X(z) = \frac{(\alpha z^{-1})^M - (\alpha z^{-1})^N}{1 - \alpha z^{-1}} = \frac{\alpha^M z^{-M} - \alpha^N z^{-N}}{1 - \alpha z^{-1}}$$

• Case 2: $$\displaystyle \alpha z^{-1} = 1 $$ (i.e., $$\displaystyle z = \alpha $$), then $$\displaystyle X(z) = N-M $$.

• ROC: Entire z-plane except $$\displaystyle z=0 $$ and/or $$\displaystyle z=\infty $$ depending on $M,N$:

  • If $M \ge 0$ and $N-1 \ge 0$ (causal-like), ROC: $$\displaystyle |z| > 0 $$ (excludes $$\displaystyle z=0 $$).

  • If $$\displaystyle M < 0 $$ and $$\displaystyle N-1 < 0 $$ (anti-causal-like), ROC: $$\displaystyle |z| < \infty $$ (excludes $$\displaystyle z=\infty $$).

  • If $$\displaystyle M < 0 $$ and $N-1 \ge 0$, ROC: $$\displaystyle 0 < |z| < \infty $$ (entire plane except origin & infinity).

3.6 Stability Determination

• A causal LTI system is BIBO stable if and only if the ROC of its transfer function $H(z)$ includes the unit circle ($$\displaystyle |z|=1 $$).

• For a rational $$\displaystyle H(z) = \frac{B(z)}{A(z)} $$, this means all poles must lie inside the unit circle ($$\displaystyle |p_i| < 1 $$).

$$\boxed{\text{Stability (Causal)} \iff \text{All poles inside } |z|=1}$$

4.0 Discrete Fourier Transform (DFT)

4.1 DFT of an N-point Sequence

For a finite-length sequence $x[n]$, $0 \le n \le N-1$:

$$X[k] = \sum_{n=0}^{N-1} x[n] e^{-j\frac{2\pi}{N}kn}, \quad k = 0, 1, ..., N-1$$

Inverse DFT (IDFT):

$$x[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k] e^{j\frac{2\pi}{N}kn}$$

4.2 Orthogonal Transform Pair

• The DFT basis functions $$\displaystyle \{W_N^{kn}\} = \{e^{-j\frac{2\pi}{N}kn}\} $$ are orthogonal over $$\displaystyle n=0,...,N-1 $$.

$$\sum_{n=0}^{N-1} W_N^{kn} W_N^{* -km} = N \delta[k-m]$$

• This orthogonality ensures the transform pair is invertible and energy-preserving (Parseval's theorem).

4.3 Computation for Specific Sequence

Example (from Nov 2023): Unit impulse $$\displaystyle x[n] = \delta[n] $$, $0 \le n \le N-1$.

$$X[k] = \sum_{n=0}^{N-1} \delta[n] e^{-j\frac{2\pi}{N}kn} = e^{0} = 1, \quad \forall k$$

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

5.0 Digital Filter Structures & Realization

5.1 FIR Filter Structures

• Direct Form: Implements difference equation directly. For $$\displaystyle y[n] = \sum_{k=0}^{M} b_k x[n-k] $$, uses $M$ delays and $M+1$ multipliers.

  DiagramCANVAS: A chain of M delay elements (z^{-1}) with multipliers b0...bM feeding into a summer. Input x[n] enters first multiplier.

• Cascade Form: Realized as a series of second-order sections (SOS). Each SOS: $$\displaystyle H_i(z) = b_{i0} + b_{i1}z^{-1} + b_{i2}z^{-2} $$. Improves numerical robustness.

5.2 IIR Filter Realization: Direct Form II

• Minimizes number of delays. Combines numerator and denominator polynomials into a single set of delays.

• Given (from Nov 2023): $$\displaystyle H(z)=\frac{0.44z^{2}+0.362z+0.02}{z^{2}+0.4z^{2}+0.18z-0.2} $$.

  • Note: Denominator likely typo in paper. Assume standard form: $$\displaystyle H(z)=\frac{0.44z^{2}+0.362z+0.02}{z^{2}+0.4z+0.18z-0.2} $$? Wait, check: $$\displaystyle 0.4z^{2}+0.18z $$ suggests $$\displaystyle 0.4z^2 + 0.18z $$. Let's assume:

$$H(z) = \frac{0.44z^{2} + 0.362z + 0.02}{z^{2} + 0.4z + 0.18} \quad \text{(Correcting apparent typo)}$$

*   Standard form: $$\displaystyle H(z) = \frac{b_0 + b_1 z^{-1} + b_2 z^{-2}}{1 + a_1 z^{-1} + a_2 z^{-2}} $$.

*   Divide numerator & denominator by $$\displaystyle z^2 $$:

$$H(z) = \frac{0.44 + 0.362 z^{-1} + 0.02 z^{-2}}{1 + 0.4 z^{-1} + 0.18 z^{-2}}$$

*   **Direct Form II Diagram**:

    
DiagramCANVAS: Two parallel paths from input: one through b0, another through a feedback loop. The feedback loop: two delays (z^{-1}) in series, with multipliers a1 and a2 feeding back to a summer before the first delay. The feedforward path: multipliers b1, b2 after the first and second delays respectively, summed with b0. All summed at output.

*   Requires **2 multipliers** in feedback, **2** in feedforward, and **2 delays** total.

5.3 Reduction of Total Multipliers

• Exploit symmetry in filter coefficients (linear phase FIR).

• Example: Type-I FIR (odd length $M$, even symmetry): $$\displaystyle h[n] = h[M-1-n] $$.

  • Direct Form needs $M$ multipliers.

  • With symmetry, can pair terms: $$\displaystyle y[n] = h[0](x[n]+x[n-M+1]) + h[1](x[n-1]+x[n-M+2]) + ... $$

  • Multipliers reduced to $$\displaystyle \frac{M+1}{2} $$.

  • Result: Nearly 50% reduction in multipliers for long linear-phase FIR filters.

6.0 Filter Design Techniques

6.1 Window Method for FIR Design

1. Start with ideal impulse response $$\displaystyle h_d[n] $$ (e.g., ideal LPF: $$\displaystyle \frac{\sin(\omega_c (n-M))}{\pi(n-M)} $$).

2. $$\displaystyle h_d[n] $$ is infinite and non-causal. Truncate to $2M+1$ samples: $$\displaystyle h[n] = h_d[n] \cdot w[n] $$, where $w[n]$ is a window (e.g., Rectangular, Hamming, Hanning).

3. Trade-off: Main lobe width $\propto$ transition bandwidth. Side lobe level $\propto$ passband ripple/stopband attenuation.

   • Rectangular: Narrow main lobe, high side lobes (~-13 dB).

   • Hamming: Wider main lobe, lower side lobes (~-41 dB).

6.2 Parks-McClellan (Equiripple) Method

• Iterative algorithm (Remez exchange) to design optimal linear-phase FIR filters.

• Objective: Minimize the maximum error (equiripple) in passband and stopband.

• Advantage over Window Method: For given filter specs (transition width, stopband attenuation), uses fewest coefficients (lowest order).

• Result: Filter has equal ripple in both bands. Transition width is sharper for same $M$.

7.0 Stochastic Processes & Optimal Filtering

7.1 Classification of Random Processes

7.2 Optimal Filtering (Concept)

• Goal: Estimate a desired signal $d[n]$ from an observed signal $x[n]$ by minimizing an error criterion.

• Wiener Filter: Optimal in mean-square error (MSE) sense for WSS processes.

  • Filter Types:

    1. Filtering (smoothing): $d[n]$ is past of $x[n]$.

    2. Prediction: $d[n]$ is future of $x[n]$.

    3. Estimation: $d[n]$ is unrelated to $x[n]$.

  • Solution: Solve Wiener-Hopf equations: $$\displaystyle \mathbf{R}_{xx} \mathbf{w} = \mathbf{r}_{xd} $$, where $$\displaystyle \mathbf{R}_{xx} $$ is autocorrelation matrix of $x[n]$, $$\displaystyle \mathbf{r}_{xd} $$ is cross-correlation vector between $x[n]$ and $d[n]$.

  • Result: Optimal filter coefficients $$\displaystyle \mathbf{w}_{opt} $$.

  Example: Noise cancellation. $$\displaystyle x[n] = s[n] + v[n] $$, desire $$\displaystyle d[n]=s[n] $$. Wiener filter estimates $s[n]$ from $x[n]$ using known statistics of $s[n]$ and $v[n]$.

\boxed{\text{END OF UNIT 5 NOTES}}