7.4 Z-Transform and Digital Systems

7.4 Z-Transform and Digital Systems

1. Sampling and Reconstruction

1.1 Sampling Theorem (Nyquist-Shannon Theorem)

1. Statement: A continuous-time signal with frequencies no higher than $f_{max}$ can be completely reconstructed from its samples if sampled at a rate: $f_s > 2f_{max}$

2. Nyquist Rate: $f_{Nyquist} = 2f_{max}$

   • Minimum sampling rate to avoid aliasing.

3. Nyquist Frequency: $f_{Nyq} = \frac{f_s}{2}$

   • Highest frequency that can be uniquely represented.

4. Mathematical Representation: $x_s(t) = \sum_{n=-\infty}^{\infty} x(nT) \delta(t - nT)$ where $T = 1/f_s$ is sampling interval.

1.2 Aliasing

1. Definition: Higher frequencies appear as lower frequencies when undersampled.

2. Aliased Frequency: $f_{alias} = |f - kf_s| \quad \text{for some integer } k$

3. Anti-aliasing Filter:

   • Low-pass filter with cutoff at $f_s/2$.

   • Applied before sampling to prevent aliasing.

1.3 Signal Reconstruction

1. Ideal Reconstruction: $x_r(t) = \sum_{n=-\infty}^{\infty} x[n] \frac{\sin[\pi(t-nT)/T]}{\pi(t-nT)/T}$

2. Zero-Order Hold (ZOH):

   • Most common practical reconstruction.

   • Holds sample value constant until next sample.

   • Transfer function: $H_{ZOH}(s) = \frac{1 - e^{-sT}}{s}$

   • Frequency response: $|H_{ZOH}(j\omega)| = T \left| \frac{\sin(\omega T/2)}{\omega T/2} \right|$

3. First-Order Hold:

   • Linear interpolation between samples.

   • Better reconstruction than ZOH but more complex.

2. Z-Transform Fundamentals

2.1 Definition of Z-Transform

1. Bilateral (Two-sided) Z-Transform: $X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] z^{-n}$

2. Unilateral (One-sided) Z-Transform: $X(z) = \sum_{n=0}^{\infty} x[n] z^{-n}$

   • Assumes $x[n] = 0$ for $n < 0$.

   • Used for causal systems and difference equations.

3. Complex Variable: $z = re^{j\Omega}$

   • $r$: Magnitude

   • $\Omega$: Normalized frequency (rad/sample)

2.2 Region of Convergence (ROC)

1. Definition: Set of $z$ values for which the Z-transform sum converges.

2. ROC Properties:

   • ROC is always an annular region: $R_1 < |z| < R_2$

   • For finite-duration sequences: ROC is entire z-plane except possibly $z=0$ or $z=\infty$.

   • ROC never contains poles.

3. ROC for Common Sequences:

   • Right-sided (causal): $|z| > R_1$ (outside circle)

   • Left-sided (anticausal): $|z| < R_2$ (inside circle)

   • Two-sided: Annular region $R_1 < |z| < R_2$

2.3 Z-Transform Properties

2.3.1 Linearity

$\mathcal{Z}\{a x_1[n] + b x_2[n]\} = a X_1(z) + b X_2(z)$ ROC: Intersection of individual ROCs.

2.3.2 Time Shifting

$\mathcal{Z}\{x[n - n_0]\} = z^{-n_0} X(z)$ ROC: Same as $X(z)$ except possibly $z=0$ or $z=\infty$.

2.3.3 Time Reversal

$\mathcal{Z}\{x[-n]\} = X(z^{-1})$ ROC: $R_2^{-1} < |z| < R_1^{-1}$

2.3.4 Scaling in z-Domain

$\mathcal{Z}\{a^n x[n]\} = X\left(\frac{z}{a}\right)$ ROC: $|a|R_1 < |z| < |a|R_2$

2.3.5 Differentiation in z-Domain

$\mathcal{Z}\{n x[n]\} = -z \frac{d}{dz} X(z)$ ROC: Same as $X(z)$.

2.3.6 Convolution

$\mathcal{Z}\{x[n] * h[n]\} = X(z) H(z)$ ROC: Intersection of ROCs of $X(z)$ and $H(z)$.

2.3.7 Initial Value Theorem

For causal $x[n]$: $x[0] = \lim_{z \to \infty} X(z)$

2.3.8 Final Value Theorem

For causal $x[n]$ with poles inside unit circle (except possibly $z=1$): $\lim_{n \to \infty} x[n] = \lim_{z \to 1} (z-1)X(z)$

3. Inverse Z-Transform Methods

3.1 Inspection Method

Using known Z-transform pairs and properties.

3.2 Partial Fraction Expansion

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

1. Case 1: Distinct Poles $X(z) = \sum_{k=1}^{N} \frac{A_k}{1 - p_k z^{-1}}$ where $A_k = (1 - p_k z^{-1})X(z)\big|_{z=p_k}$

2. Case 2: Multiple Poles For pole at $z=p$ with multiplicity $m$: $X(z) = \sum_{k=1}^{m} \frac{B_k}{(1 - p z^{-1})^k} + \text{other terms}$ where $B_k = \frac{1}{(m-k)!} \frac{d^{m-k}}{d(z^{-1})^{m-k}}[(1-p z^{-1})^m X(z)]\big|_{z^{-1}=p^{-1}}$

3. Case 3: Complex Poles Combine complex conjugate pairs into second-order terms.

3.3 Power Series Expansion (Long Division)

1. Divide numerator by denominator.

2. For right-sided sequences: Arrange in descending powers of $z^{-1}$.

3. For left-sided sequences: Arrange in ascending powers of $z$.

3.4 Contour Integration (Residue Method)

$x[n] = \frac{1}{2\pi j} \oint_C X(z) z^{n-1} dz$ where $C$ is counterclockwise contour within ROC.

4. System Response and Transfer Function H(z)

4.1 Difference Equation Representation

Nth-order LTI system: $\sum_{k=0}^{N} a_k y[n-k] = \sum_{k=0}^{M} b_k x[n-k]$

4.2 Transfer Function

Taking Z-transform with zero initial conditions: $H(z) = \frac{Y(z)}{X(z)} = \frac{\sum_{k=0}^{M} b_k z^{-k}}{\sum_{k=0}^{N} a_k z^{-k}}$

4.3 Frequency Response

$H(e^{j\Omega}) = H(z)\big|_{z=e^{j\Omega}}$

• Evaluated on unit circle ($|z|=1$).

• Periodic with period $2\pi$.

4.4 Impulse Response

$h[n] = \mathcal{Z}^{-1}\{H(z)\}$

4.5 System Response to Input

$Y(z) = H(z)X(z)$ $y[n] = h[n] * x[n]$

5. Sinusoidal Steady-State Response

5.1 Response to Complex Exponential

For input $x[n] = e^{j\Omega n}$: $y[n] = H(e^{j\Omega}) e^{j\Omega n}$

5.2 Response to Real Sinusoids

For input $x[n] = \cos(\Omega n + \phi)$: $y[n] = |H(e^{j\Omega})| \cos(\Omega n + \phi + \angle H(e^{j\Omega}))$

5.3 Magnitude and Phase Response

1. Magnitude Response: $M(\Omega) = |H(e^{j\Omega})|$

2. Phase Response: $\theta(\Omega) = \angle H(e^{j\Omega})$

3. Group Delay: $\tau_g(\Omega) = -\frac{d}{d\Omega} \theta(\Omega)$

6. Pole-Zero Relationships and Stability Analysis

6.1 Pole-Zero Plot

1. Zeros: Roots of numerator $\sum b_k z^{-k} = 0$

2. Poles: Roots of denominator $\sum a_k z^{-k} = 0$

6.2 Stability Criteria

1. BIBO Stability: System is stable if and only if ROC includes unit circle.

2. For Causal Systems:

   • Stable iff all poles inside unit circle ($|p_k| < 1$).

   • Marginally stable if poles on unit circle ($|p_k| = 1$) are simple.

   • Unstable if any pole outside unit circle ($|p_k| > 1$).

3. For Anticausal Systems:

   • Stable iff all poles outside unit circle.

6.3 Minimum Phase Systems

1. Definition: All poles and zeros inside unit circle.

2. Properties:

   • Minimum energy delay.

   • Stable and causal.

   • Invertible with causal inverse.

6.4 All-Pass Systems

1. Definition: $|H(e^{j\Omega})| = \text{constant}$ for all $\Omega$.

2. Pole-Zero Pattern: For each pole at $z = p$, there's a zero at $z = 1/p^*$.

6.5 Linear Phase Systems

1. Definition: Phase response is linear: $\theta(\Omega) = -\alpha\Omega$

2. Impulse Response: Symmetric or antisymmetric.

3. Constant Group Delay.

7. Discrete Fourier Transform (DFT)

7.1 Definition of DFT

For finite-length sequence $x[n]$ of length $N$:

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

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

7.2 DFT Properties

1. Linearity: $\text{DFT}\{ax[n] + by[n]\} = aX[k] + bY[k]$

2. Circular Time Shift: $\text{DFT}\{x[(n-n_0)_N]\} = e^{-j\frac{2\pi}{N}kn_0} X[k]$

3. Circular Frequency Shift: $\text{DFT}\{e^{j\frac{2\pi}{N}k_0n} x[n]\} = X[(k-k_0)_N]$

4. Circular Convolution: $\text{DFT}\{x[n] \circledast_N y[n]\} = X[k]Y[k]$

5. Parseval's Theorem: $\sum_{n=0}^{N-1} |x[n]|^2 = \frac{1}{N} \sum_{k=0}^{N-1} |X[k]|^2$

7.3 Relationship with DTFT and Z-Transform

1. DFT as samples of DTFT: $X[k] = X(e^{j\Omega})\big|_{\Omega = \frac{2\pi k}{N}}$

2. DFT as Z-transform on unit circle: $X[k] = X(z)\big|_{z=e^{j\frac{2\pi k}{N}}}$

7.4 Circular Convolution vs. Linear Convolution

1. Linear Convolution Length: $L = N_x + N_h - 1$

2. Circular Convolution = Linear Convolution if $N \geq L$

3. Zero-padding: Add zeros to make sequences length $L$.

8. Fast Fourier Transform (FFT)

8.1 FFT Algorithm Concept

1. Cooley-Tukey Algorithm: Most common FFT.

2. Complexity:

   • DFT: $O(N^2)$ operations

   • FFT: $O(N\log_2 N)$ operations

8.2 Radix-2 FFT

For $N = 2^m$:

1. Decimation-in-Time (DIT):

   • Separate even and odd time indices.

2. Decimation-in-Frequency (DIF):

   • Separate even and odd frequency indices.

3. Butterfly Computation: Basic computation unit: $X = A + BW$ $Y = A - BW$ where $W = e^{-j\frac{2\pi}{N}k}$

8.3 FFT Applications

1. Spectral Analysis

2. Filter Implementation

3. Convolution Computation

4. Correlation Analysis

9. Important Formulas Summary

9.1 Sampling and Reconstruction

1. Sampling Theorem: $f_s > 2f_{max}$

2. Aliased Frequency: $f_{alias} = |f - kf_s|$

3. Zero-Order Hold: $H_{ZOH}(s) = \frac{1 - e^{-sT}}{s}$

9.2 Z-Transform Fundamentals

1. Definition: $X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}$

2. Inverse Z-transform: $x[n] = \frac{1}{2\pi j} \oint_C X(z) z^{n-1} dz$

9.3 System Analysis

1. Transfer Function: $H(z) = \frac{Y(z)}{X(z)} = \frac{\sum b_k z^{-k}}{\sum a_k z^{-k}}$

2. Frequency Response: $H(e^{j\Omega}) = H(z)\big|_{z=e^{j\Omega}}$

3. Stability: Causal system stable iff all poles inside unit circle.

9.4 DFT/FFT Formulas

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

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

3. FFT Complexity: $O(N\log_2 N)$

9.5 Common Z-Transform Pairs

9.5 Common Z-Transform Pairs

Time Domain $x[n]$	Z-Transform $X(z)$	ROC
$\delta[n]$	$1$	All z
$u[n]$	$\frac{1}{1-z^{-1}}$	|z| > 1
$a^n u[n]$	$\frac{1}{1-az^{-1}}$	|z| > |a|
$n a^n u[n]$	$\frac{az^{-1}}{(1-az^{-1})^2}$	|z| > |a|
$\cos(\Omega_0 n) u[n]$	$\frac{1 - z^{-1}\cos\Omega_0}{1 - 2z^{-1}\cos\Omega_0 + z^{-2}}$	|z| > 1
$\sin(\Omega_0 n) u[n]$	$\frac{z^{-1}\sin\Omega_0}{1 - 2z^{-1}\cos\Omega_0 + z^{-2}}$	|z| > 1

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9.6 Stability Analysis

1. Causal System: All poles satisfy $|p_i| < 1$

2. Unit Circle Test: ROC includes $|z| = 1$

3. Jury's Test: Tabular method for stability determination.