3.3 Fourier Series, Fourier Transforms & Z-Transforms

3.3 Fourier Series, Fourier Transforms & Z-Transforms

Spectral frequency analysis allows communications and signal processing engineers to evaluate how signals distribute energy across the frequency spectrum. This section covers continuous-time periodic representations using Fourier Series, non-periodic continuous-time signals via the Continuous-Time Fourier Transform (CTFT), and discrete-time signals using the Z-Transform.

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1. Fourier Series Representation of Periodic Signals

Any physically realizable periodic signal $f(t)$ with period $T$ and fundamental frequency $\omega_0 = \frac{2\pi}{T}$ satisfying the Dirichlet conditions can be expanded into an infinite series of sinusoids.

1.1 Trigonometric Fourier Series Form

The Trigonometric Fourier Series expansion is defined as:

$$f(t) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t) \right)$$

Where the Fourier coefficients are evaluated via the integrals:

$$a_0 = \frac{1}{T} \int_{0}^{T} f(t) dt \quad (\text{DC Average Value})$$ $$a_n = \frac{2}{T} \int_{0}^{T} f(t) \cos(n\omega_0 t) dt \quad (n = 1, 2, 3, \dots)$$ $$b_n = \frac{2}{T} \int_{0}^{T} f(t) \sin(n\omega_0 t) dt \quad (n = 1, 2, 3, \dots)$$

1.2 Waveform Symmetry Simplifications

Recognizing waveform symmetry drastically simplifies coefficient computation:

1.3 Exponential Fourier Series Form

Using Euler's identity, the Exponential Fourier Series offers a compact representation:

$$f(t) = \sum_{n=-\infty}^{\infty} c_n e^{j n \omega_0 t}$$

Where complex coefficients $c_n$ are given by:

$$c_n = \frac{1}{T} \int_{0}^{T} f(t) e^{-j n \omega_0 t} dt$$

Relationship to trigonometric coefficients: $c_0 = a_0$, and for $n > 0$, $c_n = \frac{a_n - j b_n}{2}$.

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2. Continuous-Time Fourier Transform (CTFT)

For non-periodic continuous signals ($T \to \infty$), the discrete line spectrum becomes a continuous spectrum governed by the Continuous-Time Fourier Transform (CTFT) pair:

$$\text{Forward CTFT: } F(\omega) = \mathcal{F}\{f(t)\} = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt$$ $$\text{Inverse CTFT: } f(t) = \mathcal{F}^{-1}\{F(\omega)\} = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{j\omega t} d\omega$$

2.1 Essential Properties of the Fourier Transform

• Linearity: $\mathcal{F}\{a f_1(t) + b f_2(t)\} = a F_1(\omega) + b F_2(\omega)$
• Time Scaling: $\mathcal{F}\{f(at)\} = \frac{1}{|a|} F\left(\frac{\omega}{a}\right)$
• Frequency Shifting: $\mathcal{F}\{f(t) e^{j\omega_0 t}\} = F(\omega - \omega_0)$
• Duality: If $\mathcal{F}\{f(t)\} = F(\omega)$, then $\mathcal{F}\{F(t)\} = 2\pi f(-\omega)$
• Time Convolution Theorem: $\mathcal{F}\{f(t) * g(t)\} = F(\omega) \cdot G(\omega)$
• Frequency Convolution Theorem: $\mathcal{F}\{f(t) \cdot g(t)\} = \frac{1}{2\pi} [F(\omega) * G(\omega)]$
• Parseval's Theorem: Energy equivalence between domains: $$E = \int_{-\infty}^{\infty} |f(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |F(\omega)|^2 d\omega$$

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3. Z-Transform & Discrete-Time Analysis

The Z-Transform is the discrete-time counterpart of the Laplace transform, fundamental to Digital Signal Processing (DSP) and sampled-data control systems.

3.1 Unilateral and Bilateral Z-Transforms

For a discrete sequence $x[n]$, the bilateral Z-transform is defined as:

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

Where $z = r e^{j\Omega}$ is a complex variable in the $z$-plane.

3.2 Region of Convergence (ROC)

The Region of Convergence (ROC) specifies the set of $z$ values for which the summation converges absolutely ($\sum |x[n] z^{-n}| < \infty$).

                    ROC Characterization in the Z-Plane
                    
      Causal Sequence          Anti-Causal Sequence        Stable LTI System
     +---------------+          +---------------+          +---------------+
     |  |z| > |r_max| |          |  |z| < |r_min| |          | Includes Unit |
     | Exterior of   |          | Interior of   |          | Circle (|z|=1)|
     |  Outer Pole   |          |  Inner Pole   |          |   in the ROC  |
     +---------------+          +---------------+          +---------------+

3.3 Essential Z-Transform Pairs

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4. Difference Equations & Discrete System Stability

Linear constant-coefficient difference equations (LCCDE) model discrete-time systems:

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

Taking the Z-transform yields the system transfer function $H(z)$:

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

BIBO Stability Criterion: A causal discrete-time LTI system is Bounded-Input Bounded-Output (BIBO) stable if and only if all poles of $H(z)$ lie strictly inside the unit circle in the $z$-plane ($|p_i| < 1$).

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5. PRC ECE Board Exam Worked Examples

Worked Example 1: Z-Transform & ROC Determination

Problem: Find the Z-transform $X(z)$ and ROC for $x[n] = 3(0.5)^n u[n] - 2(0.8)^n u[n]$.

Solution:

1. Apply linearity and standard transform pairs: $$X(z) = 3 \cdot \frac{z}{z - 0.5} - 2 \cdot \frac{z}{z - 0.8}$$
2. Determine individual ROCs:
   • For $3(0.5)^n u[n] \implies |z| > 0.5$
   • For $2(0.8)^n u[n] \implies |z| > 0.8$
3. Find the combined ROC (intersection): $$\text{ROC} = \{z: |z| > 0.5\} \cap \{z: |z| > 0.8\} \implies |z| > 0.8$$
4. Simplify $X(z)$ into a single fraction: $$X(z) = \frac{3z(z - 0.8) - 2z(z - 0.5)}{(z - 0.5)(z - 0.8)} = \frac{3z^2 - 2.4z - 2z^2 + z}{(z - 0.5)(z - 0.8)} = \frac{z^2 - 1.4z}{(z - 0.5)(z - 0.8)}$$

Worked Example 2: Discrete System Stability Evaluation

Problem: A discrete filter has $H(z) = \frac{z}{z^2 - 0.6z + 0.25}$. Determine if the system is stable.

Solution:

1. Find system poles by setting denominator to zero: $z^2 - 0.6z + 0.25 = 0$.
2. Use quadratic formula: $$z = \frac{0.6 \pm \sqrt{(-0.6)^2 - 4(1)(0.25)}}{2} = \frac{0.6 \pm \sqrt{0.36 - 1.0}}{2} = 0.3 \pm j 0.4$$
3. Calculate the magnitude of the complex poles $|z_p|$: $$|z_p| = \sqrt{(0.3)^2 + (0.4)^2} = \sqrt{0.09 + 0.16} = \sqrt{0.25} = 0.5$$
4. Check stability: Since $|z_p| = 0.5 < 1.0$, all poles lie strictly inside the unit circle. The filter is BIBO stable.