Digital Signal Processing (EC-601) - Unit 4 Short Notes

1.0 Discrete-Time Signals and Systems

1.1 Representations of Discrete-Time Signals

• Unit impulse: $$\displaystyle \delta[n] = \begin{cases} 1, & n=0 \\ 0, & n \neq 0 \end{cases} $$

• Unit step: $$\displaystyle u[n] = \begin{cases} 1, & n \geq 0 \\ 0, & n < 0 \end{cases} $$

• Exponential sequence: $$\displaystyle a^n u[n] $$ (causal), $$\displaystyle a^n u[-n-1] $$ (anticausal)

• Sinusoidal: $$\displaystyle A \cos(\omega_0 n + \phi) $$ or $$\displaystyle A e^{j\omega_0 n} $$

• Graphical representation: Plot magnitude vs. sample index $n$.

1.2 Classification of Discrete-Time Systems

1.3 Linear Time-Invariant (LTI) Systems

• Convolution sum: $$\displaystyle y[n] = x[n] * h[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k] $$

  • Properties: commutative, associative, distributive.

• Difference equation: General form: $$\displaystyle \sum_{k=0}^{N} a_k y[n-k] = \sum_{k=0}^{M} b_k x[n-k] $$

  • Solution for LTI (zero-state): $$\displaystyle Y(z) = H(z) X(z) $$, where $$\displaystyle H(z) = \frac{\sum_{k=0}^{M} b_k z^{-k}}{\sum_{k=0}^{N} a_k z^{-k}} $$

  • Impulse response: $$\displaystyle h[n] = Z^{-1}\{H(z)\} $$. For recursive systems, solve using:

    • Homogeneous solution (natural response)

    • Particular solution (forced response)

    • Apply initial conditions.

• Example: Solve $$\displaystyle y[n] - 0.5 y[n-1] = x[n] $$ for $$\displaystyle x[n]=\delta[n] $$, $$\displaystyle y[-1]=1 $$.

  • $$\displaystyle H(z) = \frac{1}{1 - 0.5 z^{-1}} $$, ROC: $$\displaystyle |z| > 0.5 $$ (causal).

  • $$\displaystyle h[n] = (0.5)^n u[n] $$.

  • Total response: $$\displaystyle y[n] = (0.5)^{n+1} u[n+1] + (0.5)^n u[n] $$.

1.4 System Implementations (Structures)

• Direct-Form I:

  • Separate delays for input and output sides.

  • Requires $N+M$ delays for order $N$ (denominator), $M$ (numerator).

  • DiagramDirect-Form I block diagram: feedforward and feedback paths with delays

• Direct-Form II:

  • Shared delay line (canonical form).

  • Requires $\max(N,M)$ delays.

  • More memory efficient.

  • DiagramDirect-Form II block diagram: single delay line with feedback/feedforward

• Key Difference: Direct-Form II uses fewer delays but may be more prone to overflow in fixed-point; Direct-Form I separates paths.

1.5 State-Space Representation

• State equations:

$$ \mathbf{x}[n+1] = \mathbf{A} \mathbf{x}[n] + \mathbf{B} u[n], \quad y[n] = \mathbf{C} \mathbf{x}[n] + D u[n] $$

$\mathbf{x}[n]$: state vector, $u[n]$: input, $y[n]$: output.

• Signal Flow Graph (SFG):

  • Nodes: state variables $$\displaystyle x_i[n] $$, input $u[n]$, output $y[n]$.

  • Branches: coefficients from $\mathbf{A}, \mathbf{B}, \mathbf{C}$.

  • Transposition: Reverse all branches, swap input/output.

• Mason’s Gain Formula: $$\displaystyle T = \frac{\sum_{k} P_k \Delta_k}{\Delta} $$, where $$\displaystyle P_k $$ is forward path gain, $\Delta$ is determinant of graph, $$\displaystyle \Delta_k $$ is cofactor.

[!TIP]

Exam Focus: Converting state-space to SFG is frequent. Draw nodes for each state, connect with A-matrix coefficients as branches. For Mason’s formula, identify all loops and forward paths.

2.0 Z-Transform Analysis

2.1 Definition and Properties

• Bilateral ZT: $$\displaystyle X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n} $$, $$\displaystyle z = re^{j\omega} $$

• Unilateral ZT: $$\displaystyle X(z) = \sum_{n=0}^{\infty} x[n] z^{-n} $$ (for causal systems with ICs).

• Key Properties:

  | Property | Time Domain | Z-Domain | ROC | |--------------------|--------------------------------|---------------------------------------|----------------------------------| | Linearity | $$\displaystyle a x_1[n] + b x_2[n] $$ | $$\displaystyle a X_1(z) + b X_2(z) $$ | At least intersection of ROCs | | Time-shifting | $$\displaystyle x[n-n_0] $$ | $$\displaystyle z^{-n_0} X(z) $$ | Same as $X(z)$ | | Scaling in z | $$\displaystyle a^n x[n] $$ | $$\displaystyle X(a^{-1} z) $$ | Scaled by $|a|$ | | Convolution | $x[n] * h[n]$ | $X(z) H(z)$ | At least intersection | | Initial Value | $$\displaystyle x[0] = \lim_{z \to \infty} X(z) $$ | — | — | | Final Value | $$\displaystyle \lim_{n \to \infty} x[n] = \lim_{z \to 1} (z-1) X(z) $$ | — | Must include $$\displaystyle z=1 $$ for stable |

• Proof of Convolution:

$$ Z\{x[n] * h[n]\} = \sum_{n} \left( \sum_{k} x[k] h[n-k] \right) z^{-n} = \sum_{k} x[k] z^{-k} \sum_{n} h[n-k] z^{-(n-k)} = X(z) H(z) $$

2.2 Region of Convergence (ROC)

• Significance: Determines causality, stability, and uniqueness of sequence.

• For rational $$\displaystyle X(z) = \frac{N(z)}{D(z)} $$:

  • Right-sided (causal): ROC is $$\displaystyle |z| > r_{\max} $$ (outside outermost pole).

  • Left-sided (anticausal): ROC is $$\displaystyle |z| < r_{\min} $$ (inside innermost pole).

  • Two-sided: ROC is annular: $$\displaystyle r_1 < |z| < r_2 $$ between poles.

• Causality: ROC is exterior of rightmost pole (including $\infty$).

• Stability: ROC includes unit circle $$\displaystyle |z|=1 $$.

• Example: $$\displaystyle X(z) = \frac{1}{1 - 0.5 z^{-1}} $$ has pole at $$\displaystyle z=0.5 $$. ROC $$\displaystyle |z|>0.5 $$ → causal and stable.

2.3 Inverse Z-Transform

• Partial Fraction Expansion (PFE):

  • For rational $X(z)$, write as $$\displaystyle \sum \frac{A_i}{1 - p_i z^{-1}} $$ (causal) or $$\displaystyle \sum \frac{B_i}{1 - q_i z} $$ (anticausal).

  • Inverse using $$\displaystyle Z^{-1}\{\frac{1}{1 - a z^{-1}}\} = a^n u[n] $$, $$\displaystyle Z^{-1}\{\frac{1}{1 - a z}\} = -a^n u[-n-1] $$.

• Example: $$\displaystyle H(z) = \frac{z(z+2)}{(z-0.2)(z+0.6)} = \frac{z^2 + 2z}{z^2 + 0.4z - 0.12} $$. Perform PFE in $$\displaystyle z^{-1} $$ form:

$$ H(z) = \frac{1 + 2z^{-1}}{1 + 0.4z^{-1} - 0.12z^{-2}} = \frac{A}{1 - 0.2 z^{-1}} + \frac{B}{1 + 0.6 z^{-1}} $$

Solve for $A, B$, then $$\displaystyle h[n] = A (0.2)^n u[n] + B (-0.6)^n u[n] $$.

2.4 Rational Z-Transform

• Poles: Roots of $$\displaystyle D(z)=0 $$.

• Zeros: Roots of $$\displaystyle N(z)=0 $$.

• Pole-Zero Plot: Plot on $z$-plane; zeros ($\circ$), poles ($\times$). ROC cannot contain poles.

• Frequency Response: $$\displaystyle H(e^{j\omega}) = H(z)|_{z=e^{j\omega}} $$ (evaluate on unit circle if ROC includes it).

2.5 System Analysis in Z-Domain

• Transfer function: $$\displaystyle H(z) = \frac{Y(z)}{X(z)} $$ (zero-state).

• Stability: All poles inside unit circle ($$\displaystyle |p_i| < 1 $$) and ROC includes unit circle.

• Causality: ROC is exterior of rightmost pole.

• Minimum-phase: All zeros inside unit circle.

• Example: $$\displaystyle H(z) = \frac{1}{1 - 1.2 z^{-1}} $$ has pole at $$\displaystyle z=1.2 $$. ROC $$\displaystyle |z|>1.2 $$ → causal but unstable (pole outside unit circle).

[!TIP]

Common Pitfall: Confusing ROC for causal vs. anticausal. Always check pole locations and sequence type. For stability, all poles must be inside unit circle and ROC must include $$\displaystyle |z|=1 $$.

3.0 Frequency Domain Representations: DFS and DFT

3.1 Discrete Fourier Series (DFS)

• For periodic $x[n]$ with period $N$:

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

• Properties (prove from definition):

  • Time-shifting: $$\displaystyle x[n-n_0] \leftrightarrow c_k e^{-j\frac{2\pi}{N}kn_0} $$

  • Frequency-shifting: $$\displaystyle x[n] e^{j\frac{2\pi}{N}k_0 n} \leftrightarrow c_{k-k_0} $$

  • Time reversal: $$\displaystyle x[-n] \leftrightarrow c_{-k} = c_{N-k} $$

  • Convolution: Periodic convolution: $$\displaystyle x[n] \circledast y[n] \leftrightarrow N c_k d_k $$

3.2 Discrete Fourier Transform (DFT)

• For finite-length $x[n]$, $$\displaystyle n=0,...,N-1 $$:

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

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

• Relationship to DFS: DFT coefficients are DFS coefficients of periodic extension with period $N$.

• Properties (state and prove):

  • Linearity: $$\displaystyle a x_1[n] + b x_2[n] \leftrightarrow a X_1[k] + b X_2[k] $$

  • Circular time-shifting: $$\displaystyle x[(n-m)_N] \leftrightarrow X[k] e^{-j\frac{2\pi}{N}km} $$

  • Circular frequency-shifting: $$\displaystyle x[n] e^{j\frac{2\pi}{N}k_0 n} \leftrightarrow X[(k-k_0)_N] $$

  • Time reversal: $$\displaystyle x[(-n)_N] \leftrightarrow X[(-k)_N] $$

  • Duality: If $$\displaystyle x[n] \leftrightarrow X[k] $$, then $$\displaystyle X[n] \leftrightarrow N x[(-k)_N] $$

  • Circular convolution: $$\displaystyle x[n] \circledast y[n] \leftrightarrow X[k] Y[k] $$

3.3 Circular Convolution

• Definition: $$\displaystyle (x \circledast y)[n] = \sum_{m=0}^{N-1} x[m] y[(n-m)_N] $$

• Computation:

  • Concentric circles method: Plot $x[n]$ on outer circle, $y[n]$ on inner circle; rotate inner, multiply, sum.

  • Matrix method: $$\displaystyle \mathbf{y} = \mathbf{X}_c \mathbf{y} $$, where $$\displaystyle \mathbf{X}_c $$ is circulant matrix.

• Example: $$\displaystyle x_1 = \{1,2,3,4\}, x_2 = \{1,5,1,3\} $$:

$$ x_1 \circledast x_2 = \{1\cdot1+2\cdot3+3\cdot5+4\cdot1, \; ...\} = \{1+6+15+4, 1\cdot5+2\cdot1+3\cdot3+4\cdot1, ...\} = \{26, 24, 28, 32\} $$

3.4 Two-Dimensional DFT (2D DFT)

• For $M \times N$ image $x[m,n]$:

$$ X[k,l] = \sum_{m=0}^{M-1} \sum_{n=0}^{N-1} x[m,n] e^{-j\frac{2\pi}{M}km} e^{-j\frac{2\pi}{N}ln} $$

• Compute by row-wise then column-wise (or vice versa).

• Example: $$\displaystyle x = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} $$, $$\displaystyle M=N=2 $$:

$$ X[0,0] = 1+2+3+4 = 10, \; X[0,1] = 1\cdot1 + 2\cdot(-1) + 3\cdot1 + 4\cdot(-1) = -2, \text{ etc.} $$

[!TIP]

Circular vs. Linear Convolution: DFT of linear convolution requires zero-padding to length $L+M-1$ to avoid aliasing. Always check sequence lengths.

4.0 Fast Fourier Transform (FFT) Algorithms

4.1 Need for FFT & Computational Complexity

• Direct DFT: $$\displaystyle N^2 $$ complex multiplies, $N(N-1)$ adds.

• FFT: Exploits symmetry/periodicity of twiddle factors $$\displaystyle W_N^k = e^{-j\frac{2\pi}{N}k} $$.

• Complexity: $$\displaystyle \frac{N}{2} \log_2 N $$ complex multiplies, $$\displaystyle N \log_2 N $$ adds → $$\displaystyle O(N \log_2 N) $$.

4.2 Decimation-in-Time (DIT) FFT

• Radix-2 DIT: Decompose $x[n]$ into even and odd indices:

$$ X[k] = \sum_{m=0}^{N/2-1} x[2m] W_N^{2mk} + \sum_{m=0}^{N/2-1} x[2m+1] W_N^{(2m+1)k} = E[k] + W_N^k O[k] $$

where $$\displaystyle E[k] = \sum_{m} x[2m] W_{N/2}^{mk} $$, $O[k]$ similarly.

• Butterfly structure:

  
  Input: A, B
  
  Output: A + B·W, A - B·W
  
  

• Bit-reversal permutation: Input indices reversed in binary order.

• Example for $$\displaystyle N=8 $$:

  • Stage 1: Combine $(x[0], x[4]), (x[1], x[5]), ...$ with $$\displaystyle W_8^0=1 $$.

  • Stage 2: Combine with $$\displaystyle W_8^0, W_8^2 $$.

  • Stage 3: Combine with $$\displaystyle W_8^0, W_8^1, W_8^2, W_8^3 $$.

4.3 Decimation-in-Frequency (DIF) FFT

• Decompose output into first/second half:

$$ X[k] = \sum_{n=0}^{N/2-1} [x[n] + x[n+N/2]] W_N^{nk} + \sum_{n=0}^{N/2-1} [x[n] - x[n+N/2]] W_N^{n(k+N/2)} $$

• Butterfly: Combine $x[n]$ and $x[n+N/2]$.

• No bit-reversal at input; output may need reordering.

• Example: $$\displaystyle x[n] = \{1,-1,1,-1,0,0,0,0\} $$:

  • Stage 1: Compute sums/diffs of $(x[0],x[4]), (x[1],x[5]), ...$

  • Subsequent stages with twiddle factors.

4.4 Radix-2 FFT Implementation

• Algorithm:

  1. Bit-reverse input order.

  2. For $$\displaystyle s=1 $$ to $$\displaystyle \log_2 N $$:

     • $$\displaystyle m = 2^s $$, $m/2$ butterflies per group.

     • Twiddle factor $$\displaystyle W_m^k $$, $$\displaystyle k=0,...,m/2-1 $$.

• Example: $$\displaystyle x[n] = \{1,1,1,0,0,0,0,0\} $$, $$\displaystyle N=8 $$:

  • Bit-reversed input: $\{1,0,0,0,1,0,0,0\}$ (indices 0,4,2,6,1,5,3,7).

  • Compute stages, final $X[k]$.

4.5 FFT for Composite N (Cooley-Tukey)

• Mixed-radix: $$\displaystyle N = N_1 N_2 $$. Write $$\displaystyle n = n_1 + n_2 N_1 $$, $$\displaystyle k = k_2 + k_1 N_2 $$.

• Decomposition:

$$ X[k] = \sum_{n_2=0}^{N_2-1} \left[ \sum_{n_1=0}^{N_1-1} x[n_1 + n_2 N_1] W_{N_1}^{n_1 k} \right] W_{N_2}^{n_2 k} $$

• Compute $$\displaystyle N_2 $$ DFTs of length $$\displaystyle N_1 $$.

• Multiply by twiddles $$\displaystyle W_N^{n_2 k} $$.

• Compute $$\displaystyle N_1 $$ DFTs of length $$\displaystyle N_2 $$.

• Example $$\displaystyle N=6 $$ ($$\displaystyle N_1=2, N_2=3 $$): $$\displaystyle x[n]=\{1,2,3,4,5,6\} $$

  • Step 1: DFT of length 2 on groups: $(x[0],x[2],x[4])$ and $(x[1],x[3],x[5])$.

  • Step 2: Multiply by $$\displaystyle W_6^{n_2 k} $$.

  • Step 3: DFT of length 3 on resulting sequences.

[!TIP]

DIT vs. DIF: DIT: bit-reversal at input, natural order output. DIF: natural order input, bit-reversal at output. Butterfly structure differs: DIT uses $E[k], O[k]$; DIF uses sums/diffs of $x[n]$ and $x[n+N/2]$.

5.0 Digital Filter Design

5.1 FIR Filter Design (Window Method)

• Ideal low-pass filter:

$$ H_d(e^{j\omega}) = \begin{cases} 1, & |\omega| \leq \omega_c \\ 0, & \omega_c < |\omega| \leq \pi \end{cases} $$

Impulse response: $$\displaystyle h_d[n] = \frac{\omega_c}{\pi} \text{sinc}\left(\frac{\omega_c n}{\pi}\right) $$, $$\displaystyle -\infty < n < \infty $$.

• Gibbs phenomenon: Truncation causes ripples in passband/stopband.

• Window method:

  1. Choose cutoff $$\displaystyle \omega_c $$, length $N$ (odd for symmetry at $$\displaystyle n=0 $$).

  2. Compute $$\displaystyle h_d[n] $$ for $$\displaystyle n = -(N-1)/2, ..., (N-1)/2 $$.

  3. Choose window $w[n]$ (length $N$).

  4. $$\displaystyle h[n] = h_d[n] w[n] $$.

  5. Shift by $(N-1)/2$ to make causal: $$\displaystyle h_{\text{causal}}[n] = h[n - (N-1)/2] $$.

• Common windows:

  | Window | Time-domain $w[n]$ | Sidelobe attenuation | Mainlobe width | |--------------|------------------------|-------------------------|-------------------| | Rectangular | $1$ | $-13$ dB | $4\pi/N$ | | Hanning | $$\displaystyle 0.5 - 0.5\cos(\frac{2\pi n}{N-1}) $$ | $-31$ dB | $8\pi/N$ | | Hamming | $$\displaystyle 0.54 - 0.46\cos(\frac{2\pi n}{N-1}) $$ | $-41$ dB | $8\pi/N$ | | Blackman | $$\displaystyle 0.42 - 0.5\cos(\frac{2\pi n}{N-1}) + 0.08\cos(\frac{4\pi n}{N-1}) $$ | $-57$ dB | $12\pi/N$ |

• Symmetric FIR Filters (Linear Phase):

  • Conditions: $$\displaystyle h[n] = h[N-1-n] $$ (symmetric) or $$\displaystyle h[n] = -h[N-1-n] $$ (antisymmetric).

  • $N$ odd:

    • Symmetric: $$\displaystyle H(e^{j\omega}) = e^{-j\omega (N-1)/2} \cdot \text{real function} $$

    • Antisymmetric: $$\displaystyle H(e^{j\omega}) = j e^{-j\omega (N-1)/2} \cdot \text{imaginary function} $$, $$\displaystyle H(0)=0 $$ or $$\displaystyle H(\pi)=0 $$.

  • Magnitude response: For symmetric type I ($N$ odd), $$\displaystyle H(\omega) = h[(N-1)/2] + 2\sum_{n=0}^{(N-3)/2} h[n] \cos(\omega n) $$.

• Example: Design low-pass FIR with $$\displaystyle \omega_c = 0.4\pi $$, $$\displaystyle N=21 $$ (rectangular).

  • $$\displaystyle h_d[n] = \frac{0.4\pi}{\pi} \text{sinc}(0.4 n) = 0.4 \cdot \frac{\sin(0.4\pi n)}{0.4\pi n} $$ for $$\displaystyle n=-10,...,10 $$.

  • $$\displaystyle h[n] = h_d[n] $$ (rectangular window is 1).

  • Shift: $$\displaystyle h_{\text{causal}}[n] = h[n+10] $$, $$\displaystyle n=0,...,20 $$.

5.2 IIR Filter Design

• Impulse Invariant Method:

  • Sample analog impulse response: $$\displaystyle h[n] = h_a(nT) $$.

  • Steps:

    1. Design analog prototype $$\displaystyle H_a(s) $$ with cutoff $$\displaystyle \Omega_c $$.

    2. Find $$\displaystyle h_a(t) = Z^{-1}\{H_a(s)\} $$.

    3. Sample: $$\displaystyle h[n] = h_a(nT) $$.

    4. $$\displaystyle H(z) = Z\{h[n]\} $$.

  • Aliasing problem: Periodic repetition of $$\displaystyle H_a(j\Omega) $$ in digital frequency. Avoid by making analog filter bandlimited.

  • Example: $$\displaystyle H(s) = \frac{s+0.2}{(s+0.2)^2 + 9} $$, $$\displaystyle T=1 $$.

    • Partial fractions: $$\displaystyle H(s) = \frac{0.2}{s+0.2} + \frac{s}{(s+0.2)^2+9} $$.

    • $$\displaystyle h_a(t) = 0.2 e^{-0.2t} u(t) + e^{-0.2t} \sin(3t) u(t) $$.

    • Sample: $$\displaystyle h[n] = 0.2 (e^{-0.2})^n u[n] + (e^{-0.2})^n \sin(3n) u[n] $$.

    • $$\displaystyle H(z) = \frac{0.2}{1 - e^{-0.2}z^{-1}} + \frac{e^{-0.2} \sin(3) z^{-1}}{1 - 2e^{-0.2}\cos(3) z^{-1} + e^{-0.4} z^{-2}} $$.

• Bilinear Transformation (BLT):

  • Mapping: $$\displaystyle s = \frac{2}{T} \frac{1 - z^{-1}}{1 + z^{-1}} $$.

  • Pre-warping: $$\displaystyle \Omega = \frac{2}{T} \tan(\frac{\omega}{2}) $$.

  • Steps:

    1. Pre-warp digital $$\displaystyle \omega_c $$ to analog $$\displaystyle \Omega_c $$.

    2. Design analog Butterworth/Chebyshev $$\displaystyle H_a(s) $$ with $$\displaystyle \Omega_c $$.

    3. Apply BLT: replace $s$ with $$\displaystyle \frac{2}{T} \frac{1 - z^{-1}}{1 + z^{-1}} $$.

  • Example: Design high-pass IIR with $$\displaystyle \omega_c = 0.5\pi $$, Butterworth order 2, $$\displaystyle T=1 $$.

    • Pre-warp: $$\displaystyle \Omega_c = 2 \tan(0.5\pi/2) = 2 \tan(\pi/4) = 2 $$ rad/s.

    • Analog HP Butterworth order 2: $$\displaystyle H_a(s) = \frac{s^2}{s^2 + \sqrt{2} \Omega_c s + \Omega_c^2} = \frac{s^2}{s^2 + 2\sqrt{2} s + 4} $$.

    • BLT: $$\displaystyle s = 2 \frac{1 - z^{-1}}{1 + z^{-1}} $$.

    • Substitute: $$\displaystyle H(z) = \frac{(2 \frac{1 - z^{-1}}{1 + z^{-1}})^2}{(2 \frac{1 - z^{-1}}{1 + z^{-1}})^2 + 2\sqrt{2} (2 \frac{1 - z^{-1}}{1 + z^{-1}}) + 4} $$.

    • Simplify to polynomial in $$\displaystyle z^{-1} $$.

5.3 Filter Comparisons

• Butterworth vs. Chebyshev:

  • Butterworth: Monotonic passband/stopband, gradual roll-off.

  • Chebyshev I: Ripple in passband, sharper roll-off.

  • Chebyshev II: Ripple in stopband, monotonic passband.

[!TIP]

FIR vs. IIR: For linear phase requirement, use FIR. For computationally efficient sharp cutoff, use IIR. In BLT, always pre-warp critical frequencies. In window design, increasing $N$ reduces mainlobe width but increases sidelobe ripple.

6.0 Special Topics & Recurring Themes

6.1 System Analysis Examples

• Stability from $H(z)$: Check poles inside unit circle.

• Causality from $H(z)$: ROC is exterior of rightmost pole.

• Example: $$\displaystyle y[n] = 0.6 y[n-1] + x[n] $$

  • $$\displaystyle H(z) = \frac{1}{1 - 0.6 z^{-1}} $$, pole at $$\displaystyle z=0.6 $$.

  • ROC $$\displaystyle |z|>0.6 $$ → causal and stable.

6.2 Z-Transform Applications

• Use properties to find ZT of composite signals.

• Example: $$\displaystyle x[n] = (0.5)^n \cos(\pi n) u[n] $$

  • $$\displaystyle \cos(\pi n) = \frac{1}{2}(e^{j\pi n} + e^{-j\pi n}) $$

  • $$\displaystyle x[n] = \frac{1}{2} (0.5 e^{j\pi})^n u[n] + \frac{1}{2} (0.5 e^{-j\pi})^n u[n] $$

  • $$\displaystyle X(z) = \frac{1}{2} \frac{1}{1 - 0.5 e^{j\pi} z^{-1}} + \frac{1}{2} \frac{1}{1 - 0.5 e^{-j\pi} z^{-1}} $$, ROC $$\displaystyle |z| > 0.5 $$.

6.3 State-Space to SFG

• Procedure:

  1. Nodes: $$\displaystyle x_i[n] $$, $u[n]$, $y[n]$.

  2. Branches from $$\displaystyle x_i[n] $$ to $$\displaystyle x_j[n+1] $$ with gain $$\displaystyle A_{ji} $$.

  3. Branches from $u[n]$ to $$\displaystyle x_i[n+1] $$ with gain $$\displaystyle B_i $$.

  4. Branches from $$\displaystyle x_i[n] $$ to $y[n]$ with gain $$\displaystyle C_i $$.

  5. Optional self-loop for $D$.

• Example: $$\displaystyle \mathbf{A} = \begin{bmatrix} 0.5 & 1 \\ 0 & 0.5 \end{bmatrix}, \mathbf{B} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \mathbf{C} = [1, 1] $$

  • Nodes: $$\displaystyle x_1[n], x_2[n], u[n], y[n] $$.

  • $$\displaystyle x_1[n+1] = 0.5 x_1[n] + 1 x_2[n] + 1 u[n] $$.

  • $$\displaystyle x_2[n+1] = 0 x_1[n] + 0.5 x_2[n] + 0 u[n] $$.

  • $$\displaystyle y[n] = 1 x_1[n] + 1 x_2[n] $$.

6.4 DFT/FFT Problem-Solving

• Compute $N$-point DFT:

  • Use DIT/DIF for $$\displaystyle N=2^m $$.

  • For non-power-of-2, use mixed-radix or zero-pad to next power.

• Example: $$\displaystyle x[n] = \{1,2,3,4,4,3,2,1\} $$ using DIF:

  • Stage 1: Compute sums/diffs of $(x[0],x[4]), (x[1],x[5]), ...$

  • Stage 2: Combine with $$\displaystyle W_8^0, W_8^2 $$.

  • Stage 3: Combine with $$\displaystyle W_8^0, W_8^1, W_8^2, W_8^3 $$.

• IDFT using FFT: Compute $$\displaystyle X^*[k] $$, FFT, conjugate, scale by $1/N$.

6.5 Circular Convolution Techniques

• Concentric circles method:

  1. Plot $x[n]$ on outer circle (clockwise), $y[n]$ on inner (counterclockwise).

  2. For each shift $n$, multiply corresponding points, sum.

  3. Rotate inner sequence by one sample counterclockwise each step.

• Example: $$\displaystyle x_1 = \{1,2,3,4\}, x_2 = \{1,5,1,3\} $$:

  • $$\displaystyle n=0 $$: $$\displaystyle 1\cdot1 + 2\cdot3 + 3\cdot5 + 4\cdot1 = 26 $$

  • $$\displaystyle n=1 $$: $$\displaystyle 1\cdot5 + 2\cdot1 + 3\cdot3 + 4\cdot1 = 24 $$, etc.

[!TIP]

Past Paper Patterns:

• Highest: FFT (DIT/DIF steps), circular convolution (concentric circles), filter design (window, BLT), ZT properties/inverse.

• Medium: State-space to SFG, DFS properties, 2D DFT.

• Always: Stability/causality analysis from difference equation or $H(z)$.

Exam Strategy: For FFT, draw butterfly diagrams. For filter design, show coefficient calculations. For ZT, clearly state ROC and sequence type.