Digital Signal Processin (EX-703 (C)) - Unit 3 Short Notes

UNIT 3: DISCRETE-TIME SIGNALS, TRANSFORMS, AND FILTER ANALYSIS

1.0 Discrete-Time Signals and 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 x[n] vs. n (stem plot).

• Functional: Explicit formula, e.g., x[n] = αⁿ u[n].

• Tabular: List of (n, x[n]) pairs.

Example: x[n] = {2, 1, -1, 0, 3} for n = 0,1,2,3,4.

1.2 Signal Length

• Finite-length: Non-zero values over a finite range M ≤ n ≤ N-1. Length L = N-M.

• Infinite-length: Non-zero values over an infinite range (e.g., u[n], aⁿu[n]).

1.3 Basic Sequence Operations

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

2. Multiplication: y[n] = x₁[n] * x₂[n] (sample-by-sample).

3. Time-Shifting: x[n-k] → delay by k samples if k>0.

4. Time-Reversal: x[-n] (folding about n=0).

5. Scaling: A * x[n] (amplitude multiplication).

1.4 Nyquist Sampling Theorem

To perfectly reconstruct a bandlimited analog signal with maximum frequency ω_m (rad/s) or f_m (Hz) from its samples, the sampling frequency ω_s must satisfy:

$$ \omega_s > 2\omega_m \quad \text{or} \quad f_s > 2f_m $$

The Nyquist Rate is 2ω_m (or 2f_m). Sampling below this causes aliasing.

1.5 Classification of Discrete-Time Signals

1.6 Standard Sequences

2.0 Z-Transform and 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 (z = re^{jω}).

The unilateral (one-sided) Z-transform uses sum from n=0 to ∞ (for causal signals).

2.2 Region of Convergence (ROC)

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

• Properties:

  1. ROC is a ring/annulus in the z-plane: R₁ < \|z\| < R₂.

  2. ROC cannot contain poles.

  3. ROC is connected.

  4. For finite-length sequences, ROC is entire z-plane except possibly z=0 and/or z=∞.

• Determination: Find values of z making the sum finite. Depends on signal duration and \|a\| in aⁿ.

2.3 Causal and Anti-causal Sequences

• Causal: x[n]=0 for n<0. ROC: Outside the outermost pole (\|z\| > R_max), including ∞.

• Anti-causal: x[n]=0 for n>0. ROC: Inside the innermost pole (\|z\| < R_min), including 0.

[!TIP] Exam Tip: For a rational X(z), causality is determined by ROC. If ROC is exterior of outermost pole → causal. If interior of innermost pole → anti-causal.

2.4 Properties of Z-Transform (for Causal Signals)

2.5 Inverse Z-Transform Methods

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

2. Partial Fraction Expansion: For rational X(z), expand into simpler terms (e.g., A/(1-az^{-1})), then use known transforms.

3. Contour Integration (Residue Method): x[n] = (1/(2πj)) ∮ X(z) z^{n-1} dz over a contour inside ROC.

2.6 System Function H(z)

For an LTI system with input x[n], output y[n], and impulse response h[n]:

$$ H(z) = \mathcal{Z}\{h[n]\} = \frac{Y(z)}{X(z)} $$

If system is described by a linear constant-coefficient difference equation:

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

Taking Z-transform (assuming zero initial conditions for causal system):

$$ H(z) = \frac{Y(z)}{X(z)} = \frac{\sum_{k=0}^{N} b_k z^{-k}}{\sum_{k=0}^{M} a_k z^{-k}} = \frac{B(z)}{A(z)} $$

2.7 Stability Determination

A causal LTI system is BIBO stable if and only if:

1. Its ROC includes the unit circle (\|z\|=1).

2. All poles of H(z) lie inside the unit circle (\|p_i\| < 1).

For causal systems: Stability ⇔ ROC includes unit circle ⇔ all poles inside unit circle.

2.8 Z-Transform of Finite-Length Sequence

For x[n] = αⁿ for M ≤ n ≤ N-1, and 0 otherwise:

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

ROC: Entire z-plane except z=0 and/or z=∞ (depends on M,N). If M≥0 and N-1≥0 (causal finite), ROC is all z except z=0 (if M>0) and includes ∞.

3.0 Fourier Analysis of Discrete-Time Signals

3.1 Discrete-Time Fourier Transform (DTFT)

$$ X(e^{jω}) = \sum_{n=-\infty}^{\infty} x[n] e^{-jωn} $$

• Inverse DTFT: x[n] = (1/(2π)) ∫_{2π} X(e^{jω}) e^{jωn} dω

• Properties: Linearity, time-shift, frequency-shift, convolution, Parseval's theorem.

• Periodicity: X(e^{j(ω+2π)}) = X(e^{jω}). X(e^{jω}) is periodic with period 2π.

3.2 Discrete Fourier Transform (DFT)

For an N-point sequence x[n] (0 ≤ n ≤ N-1): DFT:

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

IDFT:

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

• DFT as Sampled DTFT: X[k] = X(e^{jω})|_{ω = 2πk/N}.

• Computation Example: For x[n] = δ[n] (length N), X[k] = 1 for all k.

3.3 Orthogonal Transform Pair

For an N-point orthogonal transform with basis vectors {φ_k[n]}:

• Forward: X[k] = Σ_{n=0}^{N-1} x[n] φ_k^*[n]

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

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

DFT is a special case where φ_k[n] = e^{j(2π/N)kn}.

3.4 Fast Fourier Transform (FFT)

• Concept: Efficient algorithm to compute DFT by exploiting symmetry and periodicity of e^{-j(2π/N)kn}.

• Advantage: Reduces computational complexity from O(N²) (direct DFT) to O(N log₂ N).

• Most common: Radix-2 FFT (requires N = 2^m).

4.0 Digital Filter Structures

4.1 FIR Filter Structures

• Direct Form (Transversal):

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

Requires `M` delays, `M+1` multipliers, `M` adders.

**Diagram:** `x[n] → [b₀] → (+) → y[n]`, with `M` unit-delay (`z^{-1}`) branches tapping `b₁` to `b_M`.

• Cascade Form (Series): H(z) = H₁(z) H₂(z) .... Each H_i(z) is a smaller FIR section (often linear phase). Reduces sensitivity to coefficient quantization.

• Linear Phase Condition: For type-I FIR (even M, symmetric coefficients b_k = b_{M-k}):

$$ H(e^{jω}) = e^{-jωM/2} A(ω) $$

where `A(ω)` is real-valued. **Importance:** Guarantees constant group delay `M/2`, no phase distortion.

4.2 IIR Filter Structures

• Direct Form I: Separate recursive (denominator) and non-recursive (numerator) parts.

  • Requires max(N,M) delays.

• Direct Form II (Canonical): Shares delay elements. Minimum number of delays = max(N,M).

  Realization Steps:

  1. Write H(z) = (b₀ + b₁z⁻¹ + ... + b_Nz⁻ᴺ) / (1 + a₁z⁻¹ + ... + a_Mz⁻ᴹ).

  2. Implement as a single lattice of delays with feedback.

  DiagramCANVAS: Direct Form II structure with M=N=2 case: Input x[n] goes to b0 adder and via z^{-1} to b1 adder and feedback from a1. Output y[n] from final adder. Feedback from y[n] via a1 and a2 into the delay chain.

• Cascade Form: H(z) = Π H_i(z), where each H_i(z) is a second-order section (biquad). Improves numerical stability.

• Parallel Form: H(z) = Σ H_i(z), useful for partial fraction expansion.

4.3 Computational Efficiency

• FIR Linear Phase: Exploit symmetry (b_k = b_{M-k}). Reduces multipliers from M+1 to ~M/2 + 1.

• Shared Terms: In cascade/parallel forms, common sub-expressions can be computed once.

• Example: For H(z) = (1 + 2z⁻¹ + 3z⁻² + 2z⁻³ + z⁻⁴), symmetric. Compute w[n] = x[n] + x[n-4], v[n] = x[n-1] + x[n-3]. Then y[n] = w[n] + 2v[n] + 3x[n-2]. Uses 3 multipliers instead of 5.

5.0 Digital Filter Design Techniques

5.1 FIR Filter Design by Window Method

• Ideal Filter (Brick-wall): Has perfect passband/stopband but non-causal, infinite impulse response h_d[n].

  Example: Ideal LPF H_d(e^{jω}) = 1 for |ω| ≤ ω_c, 0 otherwise.

  h_d[n] = (ω_c/π) sinc(ω_c n).

• Gibbs Phenomenon: Truncating h_d[n] to finite length (multiplying by window) causes ripples in passband/stopband that do not vanish with increased length.

• Window Application: h[n] = h_d[n] w[n], where w[n] is a window (e.g., Rectangular, Hamming).

• Effect of Window:

  | Window | Main Lobe Width | Side Lobe Level (dB) | | :--- | :--- | :--- | | Rectangular | 4π/N | -13 | | Hamming | 8π/N | -41 | | Hanning | 8π/N | -31 | | Blackman | 12π/N | -58 |

  Trade-off: Wider main lobe → better stopband attenuation (lower side lobes) but poorer transition width.

5.2 FIR Filter Design by Park-McClellan (Remez) Algorithm

• Concept: Equiripple optimal design. Minimizes the maximum error (ripple) in passband and stopband simultaneously, subject to specifications.

• Specifications: Passband edge ω_p, stopband edge ω_s, passband ripple δ_p, stopband ripple δ_s.

• Advantages over Window Method:

  1. Optimal: For given N, it yields the minimum possible maximum ripple.

  2. Flexible: Can specify different ripples in passband/stopband.

  3. Efficient Transition: Achieves sharper transition for same N compared to most windows.

• Result: Produces an N-tap FIR filter with equiripple behavior in both bands.

5.3 Basic IIR Filter Design Concept

• Approach: Design an analog prototype (Butterworth, Chebyshev I/II, Elliptic) with desired specifications, then transform to digital using bilinear transform or impulse invariance.

• Bilinear Transform: s = (2/T) * (1 - z⁻¹)/(1 + z⁻¹). Maps jΩ axis to unit circle, warps frequency axis. Prevents aliasing.

• Butterworth: Maximally flat magnitude in passband. Monotonic stopband.

• Chebyshev I: Equiripple in passband, monotonic stopband.

• Chebyshev II: Monotonic passband, equiripple in stopband.

6.0 Random Signals and Optimal Filtering

6.1 Classification of Random Processes

• Strict-Sense Stationary (SSS): All statistical properties (pdf, moments) invariant to time shift. p(x(t₁),...,x(t_k)) = p(x(t₁+τ),...,x(t_k+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 k: R_x[m,n] = R_x[n-m].

• Ergodicity: Time averages equal ensemble averages. For WSS process, (1/N) Σ x[n] → μ_x as N→∞.

6.2 Optimal Filtering (Wiener Filter)

• Goal: Estimate a desired signal d[n] from an observed signal x[n] contaminated by noise, minimizing Mean Square Error (MSE).

• Wiener-Hopf Equations: For FIR Wiener filter of length M with coefficients w[k]:

$$ \sum_{k=0}^{M-1} w[k] R_x[m-k] = R_{dx}[m], \quad m=0,1,...,M-1 $$

where `R_x[·]` is autocorrelation of `x[n]`, `R_{dx}[·]` is cross-correlation between `d[n]` and `x[n]`.

• Solution: w = R_x^{-1} p, where R_x is Toeplitz autocorrelation matrix, p is cross-correlation vector.

• Application Example: Noise Cancellation: x[n] = s[n] + v[n] (signal + noise). Use a reference noise v₁[n] correlated with v[n] to estimate and subtract noise.

• Difference from Deterministic Filtering: Wiener filter uses statistical properties (correlations) of signals, not just deterministic input-output relations. It is non-causal in general (uses future samples), but can be made causal by approximation.

7.0 Applications and System Properties

7.1 Broad Applications of DSP

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

• Audio: Noise reduction, echo cancellation, music synthesis.

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

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

• Control: Adaptive control, system identification.

• Radar/Sonar: Target detection, tracking.

7.2 Relationship Between System Properties

• Causality & ROC: For causal system, ROC is exterior of outermost pole, includes ∞.

• Stability & ROC: For stable system, ROC must include unit circle (\|z\|=1).

• Causality & Stability for Rational H(z): A causal LTI system is stable iff all poles of H(z) lie strictly inside the unit circle.

• Linear Phase for FIR: Condition: h[n] = h[M-1-n] (symmetric) or h[n] = -h[M-1-n] (antisymmetric). Ensures constant group delay.

7.3 Problem-Solving Integration

• Example 1 (DFT): Compute N-point DFT of x[n] = δ[n]:

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

• Example 2 (Stability): Given H(z) = 1 / (1 - 0.5z⁻¹)(1 - 2z⁻¹). Poles at z=0.5 and z=2. For causal realization, ROC is \|z\| > 2. Does not include unit circle → unstable. For anti-causal, ROC is \|z\| < 0.5 → stable but non-causal.

• Example 3 (Z-transform of finite sequence): As derived in 2.8.

[!TIP] Common Pitfall: Confusing ROC for causal vs. anti-causal. Always check pole locations and whether ROC includes ∞ (causal) or 0 (anti-causal). For stability, ROC must include unit circle.