Signals and Systems

I. Signals

1. Classification of Signals

Amplitude \ Time	Continuous Time	Discrete Time
Continuous Amplitude	Analog Signal	Sampled Signal
Discrete Amplitude	Quantized Signal	Digital Signal

2. Signal Decomposition

① Convergence Properties (Spectral & Basis Expansions in Dirac Bra-Ket Notation):

1. Continuous-Time, Continuous-Spectrum Decomposition: $$f(t) = \int d\lambda \, |a_\lambda\rangle \langle a_\lambda | f(t)\rangle$$ Examples: Time domain ($t / \delta(t)$); Laplace Transform ($\text{LT} / e^{st}$); Fourier Transform ($\text{FT} / e^{j\omega t}$).
2. Continuous-Time, Discrete-Spectrum (Periodic) Decomposition: $$f(t) = \sum_i |a_i\rangle \langle a_i | f(t + mT)\rangle$$ Examples: Time domain ($t / \delta(t)$); Fourier Series ($\text{FS} / e^{j n (\frac{2\pi}{T}) t}$); $\text{LT}, \text{FT}$.
3. Discrete-Time, Continuous-Spectrum Decomposition: $$f(n) = \int d\lambda \, |a_\lambda\rangle \langle a_\lambda | f(n)\rangle$$ Examples: Discrete time ($n / \delta(n)$); $z$-Transform ($\text{ZT} / z^{n-1}$); Discrete-Time Fourier Transform ($\text{DTFT} / e^{j n \omega}$).
4. Discrete-Time, Discrete-Spectrum (Periodic) Decomposition: $$f(n) = \sum_i |a_i\rangle \langle a_i | f(n + rN)\rangle$$ Examples: Discrete time ($n / \delta(n)$); Discrete Fourier Series ($\text{DFS} / e^{j (\frac{2\pi}{N}) n k}$); $\text{ZT}, \text{DTFT}$.

Orthogonality and Completeness (Parseval's Identity):

• Orthogonality: $\langle a_\lambda | a_{\lambda'} \rangle = \delta(\lambda - \lambda')$, $\quad \langle a_i | a_j \rangle = \delta_{ij}$
• Completeness: $$\int dt \, \langle f(t) | f(t) \rangle = \int d\lambda \, \langle a_\lambda | a_\lambda \rangle, \quad \int dt \, \langle f(t) | f(t) \rangle = \sum_i \langle a_i | a_i \rangle$$ $$\sum_n \langle f(n) | f(n) \rangle = \int d\lambda \, \langle a_\lambda | a_\lambda \rangle, \quad \sum_n \langle f(n) | f(n) \rangle = \sum_i \langle a_i | a_i \rangle$$

② Correlation Properties:

• Energy Signals:
  1. Continuous-time: $R_{12}(\tau) = \langle f_2(t - \tau) | f_1(t) \rangle = R_{21}^*(-\tau)$, with energy $E_1 = R_{11}(0)$.
  2. Discrete-time: $R_{12}(m) = \langle f_2(n - m) | f_1(n) \rangle = R_{21}^*(-m)$, with energy $E_1 = R_{11}(0)$.
• Power Signals:
  3. Continuous-time: $R_{12}(\tau) = \lim_{T \to \infty} \frac{1}{T} \langle f_2(t - \tau) | f_1(t) \rangle_T = R_{21}^*(-\tau)$, with average power $P_1 = R_{11}(0)$.
  4. Discrete-time: $R_{12}(m) = \lim_{N \to \infty} \frac{1}{N} \langle f_2(n - m) | f_1(n) \rangle_N = R_{21}^*(-m)$, with average power $P_1 = R_{11}(0)$.

II. Systems

1. Classification of Systems

Property	Class I	Class II
Parameter Distribution	Lumped: ODE / Difference Eq.	Distributed: PDE / Partial Diff. Eq.
Time Domain	Continuous: Differential Eq.	Discrete: Difference Eq.
Memory	Memoryless: Algebraic Eq.	With Memory: Differential / Difference Eq.
Linearity	Linear: Superposition & Homogeneity	Nonlinear: Non-superposition or Non-homogeneous
Time-Invariance	Time-Varying: $x(t - t_0) \not\to y(t - t_0)$	Time-Invariant: $x(t - t_0) \to y(t - t_0)$
Invertibility	Invertible: Injective & Surjective (Bijective)	Non-invertible: Non-injective or Non-surjective
Causality	Causal: $x(t - t_0) \to y(t)$ ($t_0 \ge 0$)	Non-causal: $x(t + t_0) \to y(t)$ ($t_0 > 0$)
Stability	Stable: $\text{BIBO}$	Unstable: $\overline{\text{BIBO}}$

Primary Focus: Linear Time-Invariant (LTI) systems with deterministic inputs, lumped parameters, linearity, and time-invariance.

• Models: Input-output external models vs. state-variable internal models.
• Transform Domains:
  • Continuous: $t \to s \to \Omega \to \Omega_0$ (Time $\to$ Laplace $\to$ Continuous Fourier $\to$ Fourier Series)
  • Discrete: $n \to z \to \omega \to \omega_0$ (Discrete Time $\to$ $z$-Transform $\to$ DTFT $\to$ DFS/DFT)

2. System Decomposition & Representation

• ① Block diagrams
• ② Signal flow graphs and Mason's Gain Formula

Appendix A1: Generalized Functions (Distributions) and Impulse Properties

1. The Dirac Delta Distribution $\delta(t)$

Defined as a functional acting on a test function $\varphi(t)$:

$$N_\delta[\varphi(t)] = \langle \delta(t), \varphi(t) \rangle = \varphi(0)$$

• Distribution (Generalized Function) $\delta(t)$: Assigns the value $\varphi(0)$ to $\varphi(t)$.
• Test Function $\varphi(t)$: Smooth function of compact support $\varphi(t) \in C^\infty(\mathbb{R})$ (i.e., $D(\mathbb{R})$), where $\varphi(t) \neq 0$ only on a bounded open interval $a < t < b$ ($-\infty < a, b < \infty$).

Fundamental Properties of $\delta(t)$:

1. Linearity: $a\delta(t) + b\delta(t) = (a + b)\delta(t)$
2. Sampling (Multiplication) Property: $f(t)\delta(t - t_0) = f(t_0)\delta(t - t_0)$
3. Scaling Property: $\delta(at) = \frac{1}{|a|}\delta(t) \implies \delta(t)$ is an even function ($\delta(-t) = \delta(t)$).
4. Composition with a Smooth Function $f(t)$: For simple roots $t_i$ where $f(t_i) = 0$ and $f'(t_i) \neq 0$: $$\delta[f(t)] = \sum_i \frac{1}{|f'(t_i)|} \delta(t - t_i)$$

2. Higher-Order Derivatives of the Impulse $\delta^{(k)}(t)$

$$N_{\delta^{(k)}}[\varphi(t)] = \langle \delta^{(k)}(t), \varphi(t) \rangle = (-1)^k \varphi^{(k)}(0)$$

This identity is proved via integration by parts and the compact support of $\varphi(t)$. The strength of $\delta(t)$ corresponds to the area enclosed during the limiting approximation process.

• ① Multiplication by a smooth function $f(t)$: Can be evaluated to arbitrary order $k$ via integration by parts: $$\langle f(t)\delta^{(k)}(t), \varphi(t) \rangle = \langle \delta^{(k)}(t), f(t)\varphi(t) \rangle = (-1)^k \left.\frac{d^k}{dt^k}[f(t)\varphi(t)]\right|_{t=0}$$
• ② Scaling of $\delta^{(k)}(t)$: $$\delta^{(k)}(at) = \frac{1}{|a|} \frac{1}{a^k} \delta^{(k)}(t) \implies \delta^{(k)}(t) \text{ is even for even } k \text{ and odd for odd } k.$$

3. Riemann–Lebesgue Lemma in Distribution Sense

$$N_{e^{-j\omega t}}[\varphi(t)] = \lim_{\omega \to \infty} \int_a^b e^{-j\omega t} \varphi(t) \, dt = 0 \quad (-\infty < a, b < \infty)$$

• ① As $\omega \to \infty$, in the distributional (weak) sense, $e^{-j\omega t} = e^{j\omega t} = \cos(\omega t) = \sin(\omega t) = 0$.

III. Continuous-Time Signals and Systems ($t$-Domain)

1. Signal Representation via Impulses (Sifting Property)

• ① $f(\tau) = \int_{-\infty}^{\infty} f(t) \delta(\tau - t) \, dt$
• ② $f(t) = \int_{-\infty}^{\infty} f(\tau) \delta(t - \tau) \, d\tau$

2. System Response Solution

① Natural (Free) Response + Forced Response (Classical ODE Solution):

• Given a linear constant-coefficient ordinary differential equation (ODE):
  • Homogeneous Equation ($e(t) = 0$): Solve the characteristic equation to obtain the homogeneous solution (natural/free response): $r_h(t) = \sum_k A_k e^{\alpha_k t}$.
  • Non-homogeneous Equation ($t > 0_+$): Use the method of undetermined coefficients corresponding to the input form to obtain a particular solution (forced response): $B(t)$.
• Summing the two gives the total response containing undetermined constants $A_k$: $r(t) = \sum_k A_k e^{\alpha_k t} + B(t)$.
• Using the initial states $x(0_-)$ and the impulse matching method on the ODE at $t = 0$, determine the post-impulse initial conditions $x(0_+)$, which then uniquely determine the coefficients $A_k$ for the complete response.

② Zero-State Response ($r_{zs}$) + Zero-Input Response ($r_{zi}$):

• Zero-State Response ($e(t) \to H[e(t)]$): $$r_{zs}(t) = \sum_k A_{zsk} e^{\alpha_k t} + B(t)$$ Subject to initial conditions: $$\begin{cases} x_{zs}(0_-) = 0 \\ x_{zs}(0_+) \text{ determined via the impulse matching method} \end{cases}$$ Note: $x_{zs}(t)$ contains $\delta(t)$ (and its derivatives) when the highest derivative order of $e(t)$ (or $\delta$ order in $e(t)$) $\ge$ the order of the differential equation; determined via impulse matching.
• Zero-Input Response ($x(0_+) \to H[x(0_+)]$): $$r_{zi}(t) = \sum_k A_{zik} e^{\beta_k t}$$ Subject to initial conditions: $$\begin{cases} x_{zi}(0_-) \text{ is given} \\ x_{zi}(0_+) = x_{zi}(0_-) \quad (\text{since } x_{zi}(t) \text{ contains no } \delta) \end{cases}$$
• Total Response Decomposition & Linearity: $$\begin{aligned} r(t) &= [r_{zs}(t) + r_{zi}(t)] u(t) = H[e(t)] + H[x(0_+)] \\ &= H\left[\int_{-\infty}^{\infty} e(\tau) \delta(t - \tau) \, d\tau\right] + H[x(0_+)] \\ &= \int_{-\infty}^{\infty} e(\tau) h(t - \tau) \, d\tau + H[x(0_+)] \\ &= e(t) * h(t) + H[x(0_+)] \end{aligned}$$ Exhibits zero-state linearity (with respect to excitation $e(t)$) and zero-input linearity (with respect to initial state $x(0_+)$).

3. System Impulse Response $h(t)$ Properties

• ① Invertibility: $h(t) * \tilde{h}(t) = \delta(t)$
• ② Causality: $h(t) = h(t) u(t)$ (i.e., $h(t) = 0$ for $t < 0$)
• ③ Stability (BIBO): Suppose $|e(t)| \le B < \infty$: $$|r(t)| = \left|\int_{-\infty}^{\infty} h(\tau) e(t - \tau) \, d\tau\right| \le \int_{-\infty}^{\infty} |h(\tau)| \, |e(t - \tau)| \, d\tau \le B \int_{-\infty}^{\infty} |h(\tau)| \, d\tau$$ Therefore, a necessary and sufficient condition for BIBO stability is absolute integrability: $$\int_{-\infty}^{\infty} |h(\tau)| \, d\tau < \infty$$

IV. Laplace Domain Signals and Systems ($s$-Domain)

1. Signal Representation (Bilateral Laplace Transform)

• ① Analysis Equation (Direct Transform): $$F(s) = \int_{-\infty}^{\infty} f(t) e^{-st} \, dt, \quad s = \sigma + j\omega$$ Region of Convergence (ROC): $\text{ROC} = \left\{ s \;\middle|\; \int_{-\infty}^{\infty} |f(t)| e^{-\sigma t} \, dt < \infty \right\}$, which forms a vertical strip in the $s$-plane: $$\begin{cases} \text{Left-sided signal:} & \text{ROC} = \{ s \mid -\infty < \sigma < \sigma_r \} \\ \text{Right-sided signal:} & \text{ROC} = \{ s \mid \sigma_l < \sigma < \infty \} \\ \text{Two-sided signal:} & \text{ROC} = \{ s \mid \sigma_l < \sigma < \sigma_r \} \end{cases}$$
• ② Synthesis Equation (Inverse Transform): $$f(t) = \frac{1}{2\pi j} \int_{\sigma - j\infty}^{\sigma + j\infty} F(s) e^{st} \, ds = \sum_{p_i} \text{Res}[F(s)e^{st}, p_i] \quad (\sigma \in \text{ROC})$$ For the same algebraic expression $F(s)$, the time-domain signal depends on the ROC: $$\begin{cases} \text{If } \text{ROC} = \{ s \mid -\infty < \sigma < \sigma_r \}, & f(t) = f(t) u(-t) \quad (\text{left-sided}) \\ \text{If } \text{ROC} = \{ s \mid \sigma_l < \sigma < \infty \}, & f(t) = f(t) u(t) \quad (\text{right-sided}) \\ \text{If } \text{ROC} = \{ s \mid \sigma_l < \sigma < \sigma_r \}, & f(t) = f_r(t) u(t) + f_l(t) u(-t) \quad (\text{two-sided}) \end{cases}$$

2. System Response & Partial Fraction Expansion

Under zero-state conditions, $r(t) = e(t) * h(t)$, so in the $s$-domain:

$$R(s) = E(s) \cdot H(s), \quad \text{ROC}(R) \supseteq \text{ROC}(H) \cap \text{ROC}(E)$$

For a system described by a linear constant-coefficient ODE, $H(s)$ is a rational function. By the Fundamental Theorem of Algebra:

$$H(s) = \frac{\prod_{i=1}^m (s - z_i)}{\prod_{j=1}^n (s - p_j)}$$

• ① Distinct Real Poles: $$\begin{cases} m < n \text{ (Proper):} & \text{Partial fraction expansion with residues } K_i = \left.(s - p_i) H(s)\right|_{s=p_i} \\ m \ge n \text{ (Improper):} & \text{Long division first; note } \mathcal{L}^{-1}(s^k) = \delta^{(k)}(t) \end{cases}$$
• ② Complex Conjugate Poles: Can be expanded either in real quadratic form or complex conjugate first-order terms: $$\begin{cases} \dfrac{1}{s^2 + p_1^2} \cdot \dfrac{1}{s + p_2} = \dfrac{as + b}{s^2 + p_1^2} + \dfrac{k}{s + p_2} \\[1.2ex] \dfrac{1}{s^2 + p_1^2} \cdot \dfrac{1}{s^2 + p_2^2} = \dfrac{k_1}{s + jp_1} + \dfrac{k_1^*}{s - jp_1} + \dfrac{k_2}{s + jp_2} + \dfrac{k_2^*}{s - jp_2} \end{cases}$$
• ③ Multiple (Repeated) Poles of Order $k$ at $s = p_1$: $$H(s) = \frac{K_{11}}{(s - p_1)^k} + \frac{K_{12}}{(s - p_1)^{k-1}} + \cdots + \frac{K_{1k}}{s - p_1} + \frac{E(s)}{P(s)}$$ where the coefficients are computed recursively (or via derivatives): $$K_{11} = \left.H(s)(s - p_1)^k\right|_{s=p_1} = \left.H_1(s)\right|_{s=p_1}$$ $$K_{12} = \left.\frac{H_1(s) - K_{11}}{s - p_1}\right|_{s=p_1} = \left.H_2(s)\right|_{s=p_1}, \quad K_{13} = \left.\frac{H_2(s) - K_{12}}{s - p_1}\right|_{s=p_1} = \left.H_3(s)\right|_{s=p_1}, \dots$$ Inverse transform pair: $\mathcal{L}^{-1}\left(\frac{1}{s^n}\right) = \pm \frac{t^{n-1}}{(n-1)!} u(\pm t)$.

3. System Function $H(s)$ Properties

• ① Invertibility: $H(s) \cdot \tilde{H}(s) = 1$
• ② Causality: For a rational $H(s)$, the ROC is a right half-plane ($\text{Re}(s) > \sigma_{\max}$).
• ③ Stability:
  1. Stable $\Longleftrightarrow$ ROC contains the $j\omega$-axis ($\text{Re}(s) = 0$): $$\int_{-\infty}^{\infty} |h(t)| \, dt < \infty \Longleftrightarrow \left.\int_{-\infty}^{\infty} |h(t) e^{-st}| \, dt\right|_{s=j\omega} < \infty \implies |H(s)|\Big|_{s=j\omega} < \infty$$
  2. Marginally Stable $\Longleftrightarrow$ ROC has the $j\omega$-axis as its boundary (with only simple poles on the $j\omega$-axis).
  3. Unstable $\Longleftrightarrow$ ROC does not contain the $j\omega$-axis.

Appendix A3: Bilateral Laplace Transform Properties

Property	Time Domain $f(t)$	$s$-Domain $F(s)$	ROC
① Differentiation / Integration	$\frac{d}{dt} \to s, \quad \int_{-\infty}^t \to \frac{1}{s}$	$\frac{d}{ds} \to -t, \quad \int_s^\infty \to \frac{1}{t}$	Contains $R$ (or $R \cap \{\sigma > 0\}$)
② Time Shifting ($t$-move)	$f(t - t_0)$	$e^{-s t_0} F(s)$	$R$
③ Frequency Shifting ($s$-move)	$e^{s_0 t} f(t)$	$F(s - s_0)$	$R$ shifted by $\text{Re}(s_0)$
④ Scaling	$f(at)$	$\frac{1}{|a|} F\left(\frac{s}{a}\right)$	$R / a$
⑤ Conjugation	$f^*(t)$	$F^*(s^*)$	$R$
⑥ Convolution	$f_1(t) * f_2(t)$	$F_1(s) \cdot F_2(s)$	At least $R_1 \cap R_2$
⑦ Initial Value Theorem	If $f(t) = f(t)u(t)$ and $f(t)$ contains $k\delta(t)$, then $f(0_+) = \lim_{s \to \infty} [s F(s) - k s]$
⑧ Final Value Theorem	If $f(t) = f(t)u(t)$ and $F(s)$ is stable or has at most a simple pole at $p = 0$, then $f(\infty) = \lim_{s \to 0} s F(s)$
⑨ Correlation	$f_1(t) * f_2(-t)$	$F_1(s) \cdot F_2(-s)$	At least $R_1 \cap (-R_2)$

Appendix A4: Response Classification, All-Pass, and Minimum-Phase Systems

• ① Transient, Steady-State, and Divergent Responses:
  • Transient (Decaying) Response: $R(s)$ stable $\implies$ both $H(s)$ and $E(s)$ are stable.
  • Steady-State Response: $R(s)$ marginally stable $\implies$ $H(s)$ or $E(s)$ is marginally stable while the other is stable.
  • Divergent Response: $R(s)$ unstable $\implies$ $H(s)$ or $E(s)$ is unstable.
  • Special Case: Pole-zero cancellation between $H(s)$ and $E(s)$.
• ② Natural (Free) vs. Forced Response in $s$-Domain:
  • Natural Response: Contributed by the poles of $H(s)$.
  • Forced Response: Contributed by the poles of $E(s)$.
  • Special Case: When $H(s)$ and $E(s)$ share common poles, in the partial fraction expansion of $R(s)$, the terms whose pole orders match the pole orders in $H(s)$ belong to the natural response.
• ③ All-Pass and Minimum-Phase Systems:
  • First-order all-pass: $H_1(s) = \frac{s - s_p}{s + s_p^*}$; Second-order all-pass: $H_2(s) = \frac{s - s_p}{s + s_p^*} \cdot \frac{s - s_p^*}{s + s_p}$; General all-pass: $H_{ap}(s) = \prod_i H_{1i}(s) \prod_j H_{2j}(s)$.
  • All-pass group delay property: $-\frac{d}{d\omega}\varphi_{ap}(\omega) > 0$.
  • Minimum-Phase System $H_{\min}(s)$: All poles and zeros lie in the left half-plane (or on the $j\omega$-axis).
  • Canonical Decomposition: Any stable, causal non-minimum-phase system can be factored as: $$H(s) = H_{ap}(s) \cdot H_{\min}(s)$$
  • Properties: Group delay satisfies $-\frac{d}{d\omega}\varphi(\omega) > -\frac{d}{d\omega}\varphi_{\min}(\omega)$ (minimum-phase has minimum group delay). Furthermore, if $|H(j\omega)| = |H_{\min}(j\omega)|$, partial energy satisfies $\int_0^t |h_{\min}(\tau)|^2 d\tau \ge \int_0^t |h(\tau)|^2 d\tau$ (written $\langle h_{\min}(t) | h_{\min}(t) \rangle \ge \langle h(t) | h(t) \rangle$).

Appendix A5: Unilateral Laplace Transform Properties

Definition: $F(s) = \int_{0_-}^{\infty} f(t) e^{-st} \, dt = \int_{-\infty}^{\infty} f(t) u(t) e^{-st} \, dt$, with inverse $f(t) = \frac{1}{2\pi j} \int_{\sigma - j\infty}^{\sigma + j\infty} F(s) e^{st} \, ds$.

• ① Differentiation & Integration: $$\mathcal{L}\left[\frac{d^n}{dt^n} f(t)\right] = s^n F(s) - s^{n-1} f(0_-) - s^{n-2} f'(0_-) - \cdots - s^0 f^{(n-1)}(0_-)$$ $$\mathcal{L}\left[\int_{-\infty}^t f(\tau) \, d\tau\right] = \frac{F(s)}{s} + \frac{\int_{-\infty}^{0_-} f(\tau) \, d\tau}{s}$$
• ② Convolution: When both $f_1(t)$ and $f_2(t)$ are causal, $f_1(t) * f_2(t) \to F_1(s) F_2(s)$.
• ③ Time Shifting ($t$-move): $f(t - t_0) u(t - t_0) \to F(s) e^{-s t_0}$ ($t_0 > 0$).

V. Continuous-Time Fourier Domain ($\omega$-Domain Signals & Systems)

1. Signal Representation (Continuous-Time Fourier Transform)

• ① Direct Transform: $$F(j\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} \, dt \quad (\text{Laplace transform evaluated on the } j\omega\text{-axis})$$ Relation to $F(s)$ based on $\text{ROC}(F(s))$: $$\begin{cases} \text{ROC}(F(s)) \text{ contains } j\omega\text{-axis}: & F(j\omega) = \left.F(s)\right|_{s=j\omega} \\ \text{ROC}(F(s)) \text{ does not contain } j\omega\text{-axis}: & F(j\omega) \text{ does not exist} \\ \text{ROC}(F(s)) \text{ has } j\omega\text{-axis as boundary}: & \text{If there is a } k\text{-th order pole on the } j\omega\text{-axis at } s = j\omega_0: \end{cases}$$ $$F(s) = F_0(s) + \frac{K_0}{(s - j\omega_0)^k} \implies F(j\omega) = \left.F_0(s)\right|_{s=j\omega} + \frac{K_0 \pi j^{k-1}}{(k-1)!} \delta^{(k-1)}(\omega - \omega_0)$$
• ② Inverse Transform: $$f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(j\omega) e^{j\omega t} \, d\omega$$

2. System Response & Spectral Density

Under zero-state conditions, $r(t) = e(t) * h(t)$, so:

$$R(j\omega) = E(j\omega) \cdot H(j\omega) \quad \text{if } \text{ROC}(E) \text{ and } \text{ROC}(H) \text{ contain the } j\omega\text{-axis.}$$

• Energy Signals (Energy Spectral Density $\mathcal{E}(\omega)$ & Autocorrelation): $$\mathcal{E}_r(j\omega) = \mathcal{E}_e(j\omega) \cdot |H(j\omega)|^2, \quad R_r(t) = R_e(t) * h(t) * h^*(-t) = R_e(t) * R_h(t)$$
• Power Signals (Power Spectral Density $P(\omega)$ & Autocorrelation): $$P_r(j\omega) = P_e(j\omega) \cdot |H(j\omega)|^2, \quad R_r(t) = R_e(t) * h(t) * h^*(-t) = R_e(t) * R_h(t)$$

3. System Function $H(j\omega)$ Properties

• ① Invertibility: $H(j\omega) \cdot \tilde{H}(j\omega) = 1$
• ② Causality: For rational $\left.H(j\omega)\right|_{j\omega=s}$, the ROC is the right half-plane containing the $j\omega$-axis.
• ③ Stability: Always BIBO stable (since $H(j\omega)$ exists on the $j\omega$-axis).

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VI. Continuous-Time Periodic Fourier Series Domain ($\omega_0$-Domain Signals & Systems)

1. Signal Representation (Fourier Series, $\omega_0 = \frac{2\pi}{T}$)

• ① Fourier Series Coefficients: $$F_n = \frac{1}{T} \int_{-T/2}^{T/2} f(t) e^{-j n \left(\frac{2\pi}{T}\right) t} \, dt = \frac{1}{2}\sqrt{a_n^2 + b_n^2} \, e^{-j \arctan\frac{b_n}{a_n}}$$
• ② Fourier Series Expansion: $$f(t) = \sum_{n=-\infty}^{\infty} F_n e^{j n \left(\frac{2\pi}{T}\right) t} = a_0 + \sum_{n=1}^{\infty} \left[ a_n \cos\left(n \frac{2\pi}{T} t\right) + b_n \sin\left(n \frac{2\pi}{T} t\right) \right]$$

2. System Response via Periodic Extension

Under zero-state conditions, $r(t) = e(t) * h(t)$, so for periodic/extended signals:

$$R_n = E_n \cdot H_n \cdot T \quad \left(\text{or } R_n = E_n \cdot H_n \cdot \frac{1}{T} \text{ depending on normalization}\right)$$

Suppose $e(t)$ has finite duration $T_1$ and $h(t)$ has finite duration $T_2$; then $r(t) = e(t) * h(t)$ has duration $T_3 = T_1 + T_2$. Periodically extending $e(t), h(t), r(t)$ with period $T \ge T_3$ yields $e_T(t), h_T(t), r_T(t)$, whose Fourier Series with fundamental frequency $\omega_0 = \frac{2\pi}{T}$ give $E_n, H_n, R_n$.

3. System Function $H_n$ Properties

• ① Invertibility: $H_n \cdot \tilde{H}_n = T$ (or $1$), with period $T \ge T_h + T_{\tilde{h}}$.
• ② Causality: For rational $\left.H_n\right|_{j\frac{2\pi}{T}n = s}$, the ROC is a right half-plane containing the $j\omega$-axis.
• ③ Stability: Always stable (since $h(t)$ is the periodic extension of a time-limited impulse response $h$).

Appendix A6: Fourier Transform of Periodic Signals & Wiener–Khinchin Derivation

Let $f(t)$ be periodic with period $T$ ($\omega_0 = \frac{2\pi}{T}$). Then:

$$f(t) = \sum_{n=-\infty}^{\infty} F_n e^{j n \left(\frac{2\pi}{T}\right) t}, \quad F(\omega) = 2\pi \sum_{n=-\infty}^{\infty} F_n \delta\left(\omega - n\frac{2\pi}{T}\right), \quad F_n = \frac{1}{T} \int_{-T/2}^{T/2} f(t) e^{-j n \left(\frac{2\pi}{T}\right) t} \, dt = \frac{1}{T} \left.F_T(\omega)\right|_{\omega = n\frac{2\pi}{T}}$$

Power Spectral Density $P(\omega) = \mathcal{F}[R(\tau)]$ of a Periodic Signal:

$$\begin{aligned} P(\omega) &= \mathcal{F}[R(\tau)] = \mathcal{F}\left\{ \lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{T/2} f(t) f^*(t - \tau) \, dt \right\} \\ &= \mathcal{F}\left\{ \lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{T/2} \sum_{p=-\infty}^{\infty} F_p e^{j p \left(\frac{2\pi}{T}\right) t} \sum_{q=-\infty}^{\infty} F_q^* e^{-j q \left(\frac{2\pi}{T}\right) (t - \tau)} \, dt \right\} \\ &= \mathcal{F}\left\{ \sum_{p,q} F_p F_q^* e^{j q \left(\frac{2\pi}{T}\right) \tau} \left[ \lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{T/2} e^{j (p - q) \left(\frac{2\pi}{T}\right) t} \, dt \right] \right\} \\ &= \mathcal{F}\left\{ \sum_{p,q} F_p F_q^* e^{j q \left(\frac{2\pi}{T}\right) \tau} \delta_{p,q} \right\} = \mathcal{F}\left\{ \sum_{q=-\infty}^{\infty} |F_q|^2 e^{j q \left(\frac{2\pi}{T}\right) \tau} \right\} = 2\pi \sum_{n=-\infty}^{\infty} |F_n|^2 \delta\left(\omega - n\frac{2\pi}{T}\right) \end{aligned}$$

Appendix A7: Continuous-Time Fourier Transform Properties

1. Duality (Symmetry): $\mathcal{F}[F(t)] = 2\pi f(-\omega) \implies \left.\mathcal{F}^{-1}[F(\omega)]\right|_{t=-\omega} = \frac{1}{2\pi} \mathcal{F}[F(t)] \implies \mathcal{F}^{-1}[f(\omega)] = \frac{1}{2\pi} \left.\mathcal{F}[f(t)]\right|_{\omega=-t}$
2. Scaling: $\mathcal{F}[f(at)] = \frac{1}{|a|} F\left(\frac{\omega}{a}\right)$
3. Time Shifting ($t$-move): $\mathcal{F}[f(t - t_0)] = F(\omega) e^{-j\omega t_0} \implies \mathcal{F}[f(at - t_0)] = \frac{1}{|a|} F\left(\frac{\omega}{a}\right) e^{-j\omega t_0 / a}$
4. Frequency Shifting ($\omega$-move): $\mathcal{F}[f(t) e^{j\omega_0 t}] = F(\omega - \omega_0)$
5. Differentiation & Integration: $$\frac{d^n}{dt^n} \to (j\omega)^n, \quad \frac{d^n}{d\omega^n} \to (-jt)^n, \quad \int_{-\infty}^t \to \cdot\,\mathcal{F}[u(t)] = \frac{1}{j\omega} + \pi\delta(\omega), \quad \int_{-\infty}^\omega \to \cdot\,2\pi \mathcal{F}^{-1}[u(\omega)]$$
6. Convolution & Multiplication: $$\mathcal{F}[f_1 * f_2] = F_1 \cdot F_2, \quad \mathcal{F}[f_1 \cdot f_2] = \frac{1}{2\pi} F_1 * F_2$$ $$\mathcal{F}^{-1}[F_1(\omega) \cdot F_2(\omega)] = \mathcal{F}^{-1}[F_1(\omega)] * \mathcal{F}^{-1}[F_2(\omega)], \quad \mathcal{F}^{-1}[F_1 * F_2] = 2\pi \mathcal{F}^{-1}[F_1] \cdot \mathcal{F}^{-1}[F_2]$$
7. Correlation (Wiener–Khinchin): $\mathcal{F}[R_{12}(\tau)] = F_1(\omega) \cdot F_2^*(\omega) \implies \mathcal{F}[R(\tau)] = |F(\omega)|^2$

Appendix A8: Reconstructing Continuous Spectrum from Fourier Series

Let $f(t)$ be a time-limited signal of duration $T_0$ (supported on $[0, T]$). Periodically extending $f(t)$ with period $T \ge T_0$ yields $f_T(t)$:

• FT of $f(t)$: $F(j\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} \, dt$
• FS of $f_T(t)$: $F_n = \frac{1}{T} \int_T f_T(t) e^{-j n \left(\frac{2\pi}{T}\right) t} \, dt$

Since $f(t) = f_T(t)[u(t) - u(t - T)]$, convolving in the frequency domain gives the interpolation formula:

$$F(j\omega) = \frac{1}{2\pi} \left[ \sum_{n=-\infty}^{\infty} F_n \cdot 2\pi \delta\left(\omega - n\frac{2\pi}{T}\right) \right] * \left[ T \cdot \text{Sa}\left(\frac{\omega T}{2}\right) e^{-j\omega \frac{T}{2}} \right] = T \sum_{n=-\infty}^{\infty} F_n \, \text{Sa}\left[\frac{T}{2}\left(\omega - n\frac{2\pi}{T}\right)\right] e^{-j\frac{T}{2}\left(\omega - n\frac{2\pi}{T}\right)}$$

where $\text{Sa}(x) = \frac{\sin x}{x}$.

VII. Discrete-Time Signals and Systems ($n$-Domain)

1. Signal Representation via Unit Impulses

• ① $x(n) = \sum_{m=-\infty}^{\infty} x(m) \delta(n - m)$
• ② $x(m) = \sum_{n=-\infty}^{\infty} x(n) \delta(m - n)$

2. Difference Equation Solution

① Natural (Free) Response + Forced Response:

• Given an $N$-th order linear constant-coefficient difference equation:
  • Homogeneous Equation ($e(n) = 0$): Solve the characteristic equation to get the homogeneous solution (natural response): $\sum_k C_k \alpha_k^n$.
  • Particular Solution ($n \ge 0$): Use undetermined coefficients matching the input form to obtain the forced response: $D(n)$.
• Summing gives the complete response with undetermined constants $C_k$: $r(n) = \sum_k C_k \alpha_k^n + D(n)$.
• From pre-initial states $y(-N) \sim y(-1)$, iterate the difference equation to find $y(0) \sim y(N-1)$, which uniquely determine $C_k$.

② Zero-State Response ($r_{zs}$) + Zero-Input Response ($r_{zi}$):

• Zero-State Response ($e(n) \to H[e(n)]$): $$r_{zs}(n) = \sum_k C_{zsk} \alpha_k^n + D(n), \quad \begin{cases} y_{zs}(-N) \sim y_{zs}(-1) = 0 \\ y_{zs}(0) \sim y_{zs}(N-1) \text{ obtained by iteration} \end{cases}$$
• Zero-Input Response ($y(0) \sim y(N-1) \to H[\cdots]$): $$r_{zi}(n) = \sum_k C_{zik} \beta_k^n, \quad \begin{cases} y_{zi}(-N) \sim y_{zi}(-1) \text{ are given} \\ y_{zi}(0) \sim y_{zi}(N-1) \text{ (or solved directly from } y_{zi}(-N) \sim y_{zi}(-1)\text{)} \end{cases}$$ Note: The response to $\delta(n)$ and $\delta(n - n_0)$ in $e(n)$ can be found using LTI time-invariance and the zero-input method (since $h(0) = y(0)$ determines the coefficients).
• Total Response Decomposition & Linearity: $$\begin{aligned} r(n) &= [r_{zs}(n) + r_{zi}(n)] u(n) = H[e(n)] + H[y(0) \sim y(N-1)] \\ &= H\left[\sum_{m=-\infty}^{\infty} e(m) \delta(n - m)\right] + H[y(0) \sim y(N-1)] \\ &= \sum_{m=-\infty}^{\infty} e(m) h(n - m) + H[y(0) \sim y(N-1)] \\ &= e(n) * h(n) + H[y(0) \sim y(N-1)] \end{aligned}$$ Exhibits both zero-state linearity and zero-input linearity.

3. Discrete-Time System Impulse Response $h(n)$ Properties

• ① Invertibility: $h(n) * \tilde{h}(n) = \delta(n)$
• ② Causality: $h(n) = h(n) u(n)$ (i.e., $h(n) = 0$ for $n < 0$)
• ③ Stability (BIBO): For bounded input $|e(n)| \le B < \infty$: $$|r(n)| = \left|\sum_{m=-\infty}^{\infty} h(m) e(n - m)\right| \le \sum_{m=-\infty}^{\infty} |h(m)| \, |e(n - m)| \le B \sum_{m=-\infty}^{\infty} |h(m)|$$ Thus, BIBO stability holds if and only if $h(n)$ is absolutely summable: $$\sum_{m=-\infty}^{\infty} |h(m)| < \infty$$

Appendix A9: Hilbert Transform Relations for Causal Signals and Sequences

① Causal Continuous-Time Signal ($h(t) = h(t) u(t)$)

Let $H(\omega) = R(\omega) + j X(\omega) = \mathcal{F}[h(t)]$. Since $h(t) = h(t) u(t)$ (assuming no impulse at $t = 0$):

$$\begin{aligned} H(\omega) &= \frac{1}{2\pi} \mathcal{F}[h(t)] * \mathcal{F}[u(t)] = \frac{1}{2\pi} [R(\omega) + j X(\omega)] * \left[\pi \delta(\omega) + \frac{1}{j\omega}\right] \\ &= \frac{1}{2\pi} \left[\pi R(\omega) + X(\omega) * \frac{1}{\omega}\right] + \frac{j}{2\pi} \left[\pi X(\omega) - R(\omega) * \frac{1}{\omega}\right] \\ &= \left[\frac{R(\omega)}{2} + \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{X(\xi)}{\omega - \xi} \, d\xi\right] + j\left[\frac{X(\omega)}{2} - \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{R(\xi)}{\omega - \xi} \, d\xi\right] \end{aligned}$$

Equating real and imaginary parts gives the Kramers–Kronig / Hilbert Transform relations:

$$R(\omega) = \frac{1}{\pi} \int_{-\infty}^{\infty} \frac{X(\xi)}{\omega - \xi} \, d\xi, \quad X(\omega) = -\frac{1}{\pi} \int_{-\infty}^{\infty} \frac{R(\xi)}{\omega - \xi} \, d\xi$$

② Causal Discrete-Time Sequence ($x(n) = x(n) u(n)$)

A causal sequence can be recovered from its even part $x_e(n)$ or odd part $x_o(n)$:

$$x(n) = 2x_e(n)u(n) - x_e(0)\delta(n) = 2x_o(n)u(n) + x(0)\delta(n)$$

Taking the DTFT, where $\text{DTFT}[u(n)] = \pi \sum_{k=-\infty}^{\infty} \delta(\omega - 2\pi k) + \frac{1}{1 - e^{-j\omega}} = \pi \sum_{k=-\infty}^{\infty} \delta(\omega - 2\pi k) + \frac{1}{2} - \frac{j}{2}\cot\left(\frac{\omega}{2}\right)$:

$$\begin{aligned} X(e^{j\omega}) &= \text{DTFT}[2x_e(n)u(n) - x_e(0)\delta(n)] \\ &= \frac{1}{2\pi} \left[ \text{Re}\,X(e^{j\omega}) * 2\pi \sum_{k=-\infty}^{\infty} \delta(\omega - 2\pi k) + \int_{-\pi}^{\pi} \text{Re}\,X(e^{j\omega}) \, d\omega - \text{Re}\,X(e^{j\omega}) * j\cot\frac{\omega}{2} \right] - \frac{1}{2\pi}\int_{-\pi}^{\pi} \text{Re}\,X(e^{j\omega}) \, d\omega \\ &= \text{Re}\,X(e^{j\omega}) - \frac{j}{2\pi} \text{Re}\,X(e^{j\omega}) * \cot\frac{\omega}{2} \end{aligned}$$

Similarly, from the odd part:

$$X(e^{j\omega}) = \text{DTFT}[2x_o(n)u(n) + x(0)\delta(n)] = \frac{1}{2\pi} \left[ j\,\text{Im}\,X(e^{j\omega}) * 2\pi \sum_{k=-\infty}^{\infty} \delta(\omega - 2\pi k) + j\int_{-\pi}^{\pi} \text{Im}\,X(e^{j\omega}) \, d\omega - j\,\text{Im}\,X(e^{j\omega}) * j\cot\frac{\omega}{2} \right] + x(0)$$

Therefore, the real and imaginary parts of $X(e^{j\omega})$ satisfy the discrete periodic Hilbert relations (where $\mathcal{P}$ denotes the Cauchy principal value):

$$\text{Re}[X(e^{j\omega})] = x(0) + \frac{1}{2\pi} \mathcal{P} \int_{-\pi}^{\pi} \text{Im}[X(e^{j\theta})] \cot\left(\frac{\omega - \theta}{2}\right) d\theta$$ $$\text{Im}[X(e^{j\omega})] = -\frac{1}{2\pi} \mathcal{P} \int_{-\pi}^{\pi} \text{Re}[X(e^{j\theta})] \cot\left(\frac{\omega - \theta}{2}\right) d\theta$$

③ Causal Spectrum (Analytic Signal $f_a(t) = f(t) + j\hat{f}(t)$ with One-Sided Spectrum)

$$f(t) = -\frac{1}{\pi} \int_{-\infty}^{\infty} \frac{\hat{f}(\tau)}{t - \tau} \, d\tau, \quad \hat{f}(t) = \frac{1}{\pi} \int_{-\infty}^{\infty} \frac{f(\tau)}{t - \tau} \, d\tau$$

④ Causal Discrete Line Spectrum

Analogous discrete Hilbert relations hold by duality between the time and frequency domains.

VIII. $z$-Domain Signals and Systems ($z$-Transform)

1. Signal Representation (Bilateral $z$-Transform)

• ① Direct Transform: $$X(z) = \sum_{n=-\infty}^{\infty} x(n) z^{-n}, \quad z = r e^{j\omega}$$ Region of Convergence (ROC): $\text{ROC} = \left\{ z \;\middle|\; \sum_{n=-\infty}^{\infty} |x(n) z^{-n}| < \infty \right\}$, forming an annulus (ring) in the $z$-plane: $$\begin{cases} \text{Left-sided sequence:} & \text{ROC} = \{ z \mid 0 \le |z| < r_o \} \quad (\text{may exclude } z = 0) \\ \text{Right-sided sequence:} & \text{ROC} = \{ z \mid r_i < |z| \le \infty \} \quad (\text{may exclude } z = \infty) \\ \text{Two-sided sequence:} & \text{ROC} = \{ z \mid r_i < |z| < r_o \} \end{cases}$$
• ② Inverse $z$-Transform (Contour Integration & Residue Theorem): $$x(n) = \frac{1}{2\pi j} \oint_C X(z) z^{n-1} \, dz = \sum_{p_i \text{ inside } C} \text{Res}[X(z) z^{n-1}, p_i] \quad (C \subset \text{ROC})$$ For the same rational $X(z)$, the sequence depends on the ROC: $$\begin{cases} \text{If } \text{ROC} = \{ z \mid 0 \le |z| < r_i \}, & x(n) = x(n) u(-n-1) \quad (\text{left-sided}) \\ \text{If } \text{ROC} = \{ z \mid |z| > r_o \}, & x(n) = x(n) u(n) \quad (\text{right-sided}) \\ \text{If } \text{ROC} = \{ z \mid r_i < |z| < r_o \}, & x(n) = x_r(n) u(n) + x_l(n) u(-n-1) \quad (\text{two-sided}) \end{cases}$$

2. System Response & Partial Fraction Expansion

Under zero-state conditions, $r(n) = e(n) * h(n)$, so:

$$R(z) = E(z) \cdot H(z), \quad \text{ROC}(R) \supseteq \text{ROC}(H) \cap \text{ROC}(E)$$

For a linear constant-coefficient difference equation, $H(z)$ is rational. Expanding $\frac{H(z)}{z} = \frac{\prod_{i=1}^m (z - z_i)}{\prod_{j=1}^n (z - p_j)}$:

• ① Distinct Real Poles: $$\begin{cases} m < n: & \text{Partial fractions on } \frac{H(z)}{z} \text{ with } K_i = \left.(z - p_i)\frac{H(z)}{z}\right|_{z=p_i} \\ m \ge n: & \text{Long division; note } \mathcal{Z}^{-1}(z^k) = \delta(n + k) \end{cases}$$
• ② Complex Conjugate Poles on/near Circle: $$\begin{cases} \dfrac{z \sin(\omega_0 T)}{z^2 - 2z\cos(\omega_0 T) + 1} \longleftrightarrow \pm \sin(n\omega_0 T) u(n / -n-1) \\[1.2ex] \dfrac{z(z - \cos(\omega_0 T))}{z^2 - 2z\cos(\omega_0 T) + 1} \longleftrightarrow \pm \cos(n\omega_0 T) u(n / -n-1) \end{cases}$$
• ③ Multiple Poles of Order $k$ at $z = p$: $$H(z) = \sum_{r=1}^k \frac{C_r z}{(z - p)^r} + \frac{E(z)}{P(z)}$$ where coefficients are found via: $$C_1 = \left.\frac{X(z)}{z}(z - p)^k\right|_{z=p} = \left.X_1(z)\right|_{z=p}, \quad C_2 = \left.\frac{X_1(z) - C_1}{z - p}\right|_{z=p} = \left.X_2(z)\right|_{z=p}, \quad C_3 = \left.\frac{X_2(z) - C_2}{z - p}\right|_{z=p} = \left.X_3(z)\right|_{z=p}, \dots$$ Inverse transform pair: $\mathcal{Z}^{-1}\left[\frac{z}{(z - 1)^m}\right] = \pm \binom{n + m - 1}{m - 1} u(n / -n-1)$ (written $\pm C_n^m u(n / -n-1)$).

3. System Function $H(z)$ Properties

• ① Invertibility: $H(z) \cdot \tilde{H}(z) = 1$
• ② Causality: For rational $H(z)$, the ROC is the exterior of a circle including $z = \infty$ ($|z| > r_{\max}$), and the numerator degree in $z$ is $\le$ the denominator degree.
• ③ Stability:
  1. Stable $\Longleftrightarrow$ ROC contains the unit circle $|z| = 1$: $$\sum_{n=-\infty}^{\infty} |h(n)| < \infty \Longleftrightarrow \left.\sum_{n=-\infty}^{\infty} |h(n) z^{-n}|\right|_{|z|=1} < \infty \implies |H(z)|\Big|_{|z|=1} < \infty$$
  2. Marginally Stable $\Longleftrightarrow$ ROC has the unit circle $|z| = 1$ as its boundary (with simple poles on $|z| = 1$).
  3. Unstable $\Longleftrightarrow$ ROC does not contain the unit circle $|z| = 1$.

Appendix A10: Bilateral $z$-Transform Properties

Property	Sequence $x(n)$	$z$-Transform $X(z)$	ROC
① Multiplication by $n^m$	$n^m x(n)$	$\left(-z\frac{d}{dz}\right)^m X(z)$	$R$
② Time Shifting ($n$-move)	$x(n - n_0)$	$z^{-n_0} X(z)$	$R$ (may add or remove $0, \infty$)
③ Scaling in $z$-Domain	$z_0^n x(n)$	$X\left(\frac{z}{z_0}\right)$	$|z_0| R$
④ Time Reversal	$x(-n)$	$X(z^{-1})$	$R^{-1}$
⑤ Conjugation	$x^*(n)$	$X^*(z^*)$	$R$
⑥ Convolution	$x_1(n) * x_2(n)$	$X_1(z) \cdot X_2(z)$	At least $R_1 \cap R_2$
⑦ Initial Value Theorem	If $x(n) = x(n)u(n)$ (causal), then $x(0) = \lim_{z \to \infty} X(z)$, and generally $x(k) = \lim_{z \to \infty} z^k \left[X(z) - \sum_{r=0}^{k-1} x(r) z^{-r}\right]$
⑧ Final Value Theorem	If $x(n) = x(n)u(n)$ and $X(z)$ is stable (or has at most a simple pole at $p = 1$), then $x(\infty) = \lim_{z \to 1} (z - 1) X(z)$
⑨ Correlation	$x_1(n) * x_2(-n)$	$X_1(z) \cdot X_2(z^{-1})$	At least $R_1 \cap R_2^{-1}$

Appendix A11: Discrete-Time Response Classification, All-Pass, and Minimum-Phase Systems

• ① Transient, Steady-State, and Divergent Responses:
  • Transient Response: $R(z)$ stable $\implies$ $H(z)$ and $E(z)$ are both stable.
  • Steady-State Response: $R(z)$ marginally stable $\implies$ $H(z)$ or $E(z)$ is marginally stable while the other is stable.
  • Divergent Response: $R(z)$ unstable $\implies$ $H(z)$ or $E(z)$ is unstable.
  • Special Case: Pole-zero cancellation between $H(z)$ and $E(z)$.
• ② Natural vs. Forced Response in $z$-Domain:
  • Natural Response: Produced by poles of $H(z)$.
  • Forced Response: Produced by poles of $E(z)$.
  • Special Case: When $H(z)$ and $E(z)$ share identical poles, the partial fraction terms of $R(z)$ whose pole orders match those of $H(z)$ constitute the natural response.
• ③ Discrete-Time All-Pass and Minimum-Phase Systems:
  • First-order all-pass: $H_1(z) = \frac{z^{-1} - z_p^*}{1 - z_p z^{-1}}$; Second-order all-pass: $H_2(z) = \frac{z^{-1} - z_p^*}{1 - z_p z^{-1}} \cdot \frac{z^{-1} - z_p}{1 - z_p^* z^{-1}}$; General all-pass: $H_{ap}(z) = \prod_i H_{1i}(z) \prod_j H_{2j}(z)$.
  • All-pass group delay property: $-\frac{d}{d\omega}\varphi_{ap}(\omega) > 0$.
  • Minimum-Phase System $H_{\min}(z)$: All poles and zeros lie inside (or on) the unit circle $|z| \le 1$.
  • Factorization: Any stable, causal non-minimum-phase system can be decomposed as $H(z) = H_{ap}(z) \cdot H_{\min}(z)$.
  • Properties: $-\frac{d}{d\omega}\varphi(\omega) > -\frac{d}{d\omega}\varphi_{\min}(\omega)$; and if $|H(e^{j\omega})| = |H_{\min}(e^{j\omega})|$, partial energy satisfies $\sum_{n=0}^M |h_{\min}(n)|^2 \ge \sum_{n=0}^M |h(n)|^2$ (written $\langle h_{\min}(n) | h_{\min}(n) \rangle \ge \langle h(n) | h(n) \rangle$).

Appendix A12: Unilateral $z$-Transform Properties

Definition: $X(z) = \sum_{n=0}^{\infty} x(n) z^{-n} = \sum_{n=-\infty}^{\infty} x(n) u(n) z^{-n}$, with inverse $x(n) = \frac{1}{2\pi j} \oint_C X(z) z^{n-1} \, dz$.

• ① Convolution: When both $x_1(n)$ and $x_2(n)$ are causal, $x_1(n) * x_2(n) \to X_1(z) X_2(z)$.
• ② Time Shifting ($n$-move, Advance & Delay): $$\mathcal{Z}[x(n + m) u(n)] = z^m \left[ X(z) - \sum_{k=0}^{m-1} x(k) z^{-k} \right] \quad (m > 0)$$ $$\mathcal{Z}[x(n - m) u(n)] = z^{-m} \left[ X(z) + \sum_{k=-m}^{-1} x(k) z^{-k} \right] \quad (m > 0)$$

IX. Discrete-Time Fourier Transform Domain ($\omega$-Domain / DTFT)

1. Signal Representation

• ① Direct Transform (DTFT): $$X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x(n) e^{-j n \omega} \quad (\text{$z$-Transform evaluated on the unit circle } |z| = 1)$$ Relation to $X(z)$ based on $\text{ROC}(X(z))$: $$\begin{cases} \text{ROC}(X(z)) \text{ contains } |z| = 1: & X(e^{j\omega}) = \left.X(z)\right|_{z=e^{j\omega}} \\ \text{ROC}(X(z)) \text{ does not contain } |z| = 1: & X(e^{j\omega}) \text{ does not exist} \\ \text{ROC}(X(z)) \text{ has } |z| = 1 \text{ as boundary}: & \text{If } |z| = 1 \text{ has } k\text{-th order poles, } X(e^{j\omega}) \text{ includes }\delta^{(k-1)} \text{ impulses} \end{cases}$$
• ② Inverse Transform (IDTFT): $$x(n) = \frac{1}{2\pi} \int_{2\pi} X(e^{j\omega}) e^{j n \omega} \, d\omega$$

2. System Response & Spectral Density

Under zero-state conditions, $r(n) = e(n) * h(n)$, so:

$$R(e^{j\omega}) = E(e^{j\omega}) \cdot H(e^{j\omega}) \quad \text{if } \text{ROC}(E) \text{ and } \text{ROC}(H) \text{ contain } |z| = 1.$$

• Energy Sequences: $$\mathcal{E}_r(e^{j\omega}) = \mathcal{E}_e(e^{j\omega}) \cdot |H(e^{j\omega})|^2, \quad R_r(n) = R_e(n) * h(n) * h^*(-n) = R_e(n) * R_h(n)$$
• Power Sequences: $$P_r(e^{j\omega}) = P_e(e^{j\omega}) \cdot |H(e^{j\omega})|^2, \quad R_r(n) = R_e(n) * h(n) * h^*(-n) = R_e(n) * R_h(n)$$

3. System Function $H(e^{j\omega})$ Properties

• ① Invertibility: $H(e^{j\omega}) \cdot \tilde{H}(e^{j\omega}) = 1$
• ② Causality: For rational $\left.H(e^{j\omega})\right|_{e^{j\omega}=z}$, the ROC is the exterior of a circle containing $|z| = 1$ (and $z = \infty$).
• ③ Stability: Always BIBO stable.

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X. Discrete Fourier Series / Transform Domain ($\omega_0$-Domain, $\omega_0 = \frac{2\pi}{N}$)

1. Signal Representation (DFS / DFT, where $W_N = e^{-j\frac{2\pi}{N}}$)

• ① Inverse DFS / IDFT: $$x(n) = \frac{1}{N} \sum_{k=0}^{N-1} X(k) e^{j\left(\frac{2\pi}{N}\right) n k} = \frac{1}{N} \sum_{k=0}^{N-1} X(k) W_N^{-nk}$$
• ② Direct DFS / DFT: $$X(k) = \sum_{n=0}^{N-1} x(n) e^{-j\left(\frac{2\pi}{N}\right) n k} = \sum_{n=0}^{N-1} x(n) W_N^{nk}$$

2. System Response via Periodic Extension (Linear Convolution via Circular/DFS)

Under zero-state conditions, $r(n) = e(n) * h(n)$, so:

$$R(k) = E(k) \cdot H(k)$$

Suppose $e(n)$ has finite length $N_1$ and $h(n)$ has finite length $N_2$; then the linear convolution $r(n) = e(n) * h(n)$ has length $N_3 = N_1 + N_2 - 1$. Periodically extending $e(n), h(n), r(n)$ with period $N \ge N_3$ yields $e_N(n), h_N(n), r_N(n)$, whose DFS with fundamental frequency $\omega_0 = \frac{2\pi}{N}$ give $E(k), H(k), R(k)$.

3. System Function $H(k)$ Properties

• ① Invertibility: $H(k) \cdot \tilde{H}(k) = 1$, for $N > N_h + N_{\tilde{h}}$.
• ② Causality: For rational $\left.H(k)\right|_{e^{j\frac{2\pi}{N}k}=z}$, the ROC is the exterior of $|z| = 1$.
• ③ Stability: Always stable (since $h(n)$ is a periodic extension of a finite-duration impulse response).

Appendix A13: DTFT of Periodic Sequences

Let $x(n)$ be periodic with period $N$. Then:

$$x(n) = \frac{1}{N} \sum_{k=0}^{N-1} X(k) e^{j\frac{2\pi}{N}nk}, \quad X(e^{j\omega}) = \frac{2\pi}{N} \sum_{r=-\infty}^{\infty} \sum_{k=0}^{N-1} X(k) \delta\left(\omega - \frac{2\pi}{N}k - 2\pi r\right), \quad X(k) = \sum_{n=0}^{N-1} x(n) e^{-j\frac{2\pi}{N}nk} = \left.X_N(e^{j\omega})\right|_{\omega = \frac{2\pi}{N}k}$$

The power spectral density is $P(e^{j\omega}) = \mathcal{F}[R(m)] = \frac{2\pi}{N^2} \sum_{r=-\infty}^{\infty} \sum_{k=0}^{N-1} |X(k)|^2 \delta\left(\omega - \frac{2\pi}{N}k - 2\pi r\right)$.

Appendix A14: DTFT Properties

1. Time Shifting ($n$-move): $x(n - n_0) \to e^{-j\omega n_0} X(e^{j\omega})$
2. Frequency Shifting ($\omega$-move): $e^{j\omega_0 n} x(n) \to X\left(e^{j(\omega - \omega_0)}\right)$
3. Frequency Differentiation: $n x(n) \to j \frac{d}{d\omega} X(e^{j\omega})$
4. Convolution & Multiplication: $$x(n) * y(n) \to X(e^{j\omega}) \cdot Y(e^{j\omega}), \quad x(n) \cdot y(n) \to \frac{1}{2\pi} \int_{2\pi} X(e^{j\theta}) Y\left(e^{j(\omega - \theta)}\right) d\theta$$
5. Correlation: $R_{xy}(m) = x(m) * y^*(-m) \to X(e^{j\omega}) Y^*(e^{j\omega})$

Appendix A15: Reconstructing DTFT Spectrum from DFS / DFT

Let $x(n)$ be a finite-length sequence of length $N_0$ (supported on $0 \le n \le N_0 - 1$). Periodically extending $x(n)$ with period $N \ge N_0$ gives $x_N(n)$:

• DTFT of $x(n)$: $X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x(n) e^{-j n \omega}$
• DFS of $x_N(n)$: $X(k) = \sum_{n=0}^{N-1} x_N(n) e^{-j\left(\frac{2\pi}{N}\right) n k}$

Since $x(n) = x_N(n)[u(n) - u(n - N)]$, periodic convolution in the frequency domain yields the exact interpolation formula:

$$X(e^{j\omega}) = \frac{1}{2\pi} \left[ \frac{2\pi}{N} \sum_{r=-\infty}^{\infty} \sum_{k=0}^{N-1} X(k) \delta\left(\omega - \frac{2\pi}{N}k - 2\pi r\right) \right] * \frac{1 - e^{-j N \omega}}{1 - e^{-j\omega}} = \frac{1}{N} \sum_{k=0}^{N-1} X(k) \frac{1 - e^{-j N \omega}}{1 - e^{-j\left(\omega - \frac{2\pi}{N}k\right)}}$$

XI. Sampling Theory (Time & Frequency Domains)

1. Time-Domain Sampling of Continuous Aperiodic Signals

• ① Discrete-Time Sequence Representation ($f(t) \to f(nT) = x(n)$): $$\begin{aligned} X(e^{j\omega}) &= \sum_{n=-\infty}^{\infty} x(n) e^{-j n \omega} = \sum_{n=-\infty}^{\infty} f(nT) e^{-j n \omega} = \sum_{n=-\infty}^{\infty} \left[ \frac{1}{2\pi} \int_{-\infty}^{\infty} F(j\Omega) e^{j\Omega n T} \, d\Omega \right] e^{-j n \omega} \\ &= \frac{1}{2\pi} \int_{-\infty}^{\infty} F(j\Omega) \, d\Omega \sum_{n=-\infty}^{\infty} e^{j n (\Omega T - \omega)} = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(j\Omega) \, d\Omega \cdot 2\pi \sum_{r=-\infty}^{\infty} \delta(\omega - \Omega T - 2\pi r) \\ &= \frac{1}{T} \int_{-\infty}^{\infty} F(j\Omega) \, d\Omega \sum_{r=-\infty}^{\infty} \delta\left(\Omega - \frac{\omega}{T} - \frac{2\pi}{T}r\right) = \frac{1}{T} \sum_{r=-\infty}^{\infty} F\left(j\frac{\omega}{T} + j\frac{2\pi}{T}r\right) \end{aligned}$$
• ② Impulse-Train Modulated Representation ($f(t) \to f_s(t) = f(t) \sum_{n=-\infty}^{\infty} \delta(t - nT)$): $$F_s(j\Omega) = \mathcal{F}\left[ f(t) \cdot \sum_{n=-\infty}^{\infty} \delta(t - nT) \right] = \frac{1}{2\pi} F(j\Omega) * \frac{2\pi}{T} \sum_{r=-\infty}^{\infty} \delta\left(\Omega + \frac{2\pi}{T}r\right) = \frac{1}{T} \sum_{r=-\infty}^{\infty} F\left(j\Omega + j\frac{2\pi}{T}r\right)$$

2. Time-Domain Sampling of Continuous Periodic Signals

Assume the signal period $T$ is an integer multiple $N$ of the sampling interval $T_s$ ($T = N T_s$):

• ① Discrete Periodic Sequence ($f(t) \to f(n T_s) = x(n)$): $$\begin{aligned} X(k) &= \sum_{n=0}^{N-1} x(n) e^{-j\left(\frac{2\pi}{N}\right) n k} = \sum_{n=0}^{N-1} f(n T_s) e^{-j\left(\frac{2\pi}{N}\right) n k} = \sum_{n=0}^{N-1} \sum_{m=-\infty}^{\infty} F_m e^{j m \left(\frac{2\pi}{T}\right) n T_s} e^{-j\left(\frac{2\pi}{N}\right) n k} \\ &= \sum_{m=-\infty}^{\infty} F_m \sum_{n=0}^{N-1} e^{j n \left(\frac{2\pi}{N}m - \frac{2\pi}{N}k\right)} = \sum_{m=-\infty}^{\infty} F_m \cdot N \sum_{r=-\infty}^{\infty} \delta(m - k - rN) = N \sum_{r=-\infty}^{\infty} F_{k + rN} \end{aligned}$$
• ② Impulse-Train Sampled Periodic Signal ($f(t) \to f(t) \sum_{n=-\infty}^{\infty} \delta(t - n T_s)$): $$\begin{aligned} F_m &= \frac{1}{T} \int_T \left[ f(t) \sum_{n=-\infty}^{\infty} \delta(t - n T_s) \right] e^{-j m \left(\frac{2\pi}{T}\right) t} \, dt = \frac{1}{T} \sum_{\ell=-\infty}^{\infty} F_\ell \sum_{n=-\infty}^{\infty} \int_T \delta(t - n T_s) e^{j \frac{2\pi}{T} n T_s (\ell - m)} \, dt \\ &= \frac{1}{T} \sum_{\ell=-\infty}^{\infty} F_\ell \cdot 2\pi \sum_{r=-\infty}^{\infty} \delta\left[\frac{2\pi}{N}(\ell - m) - 2\pi r\right] = \frac{2\pi}{T} \sum_{r=-\infty}^{\infty} \sum_{\ell=-\infty}^{\infty} F_\ell \frac{N}{2\pi} \delta(\ell - m - rN) = \frac{1}{T_s} \sum_{r=-\infty}^{\infty} F_{m + rN} \end{aligned}$$

3. Frequency-Domain Sampling of Continuous Aperiodic Signals

• ① Fourier Series from Spectral Samples ($F(j\omega) \to F(j n \Omega) = F_n$): $$\begin{aligned} f_s(t) &= \sum_{n=-\infty}^{\infty} F_n e^{j n \Omega t} = \sum_{n=-\infty}^{\infty} \left[ \int_{-\infty}^{\infty} f(\tau) e^{-j n \Omega \tau} \, d\tau \right] e^{j n \Omega t} = \int_{-\infty}^{\infty} f(\tau) \, d\tau \sum_{n=-\infty}^{\infty} e^{j n \Omega (t - \tau)} \\ &= \int_{-\infty}^{\infty} f(\tau) \, d\tau \cdot 2\pi \sum_{r=-\infty}^{\infty} \delta[\Omega(t - \tau) - 2\pi r] = \frac{2\pi}{\Omega} \int_{-\infty}^{\infty} f(\tau) \, d\tau \sum_{r=-\infty}^{\infty} \delta\left(\tau - t + \frac{2\pi}{\Omega}r\right) = \frac{2\pi}{\Omega} \sum_{r=-\infty}^{\infty} f\left(t - \frac{2\pi}{\Omega}r\right) \end{aligned}$$
• ② Impulse-Train Sampled Spectrum ($F(j\omega) \to F(j\omega) \sum_{n=-\infty}^{\infty} \delta(\omega - n\Omega)$): $$\tilde{f}_s(t) = \mathcal{F}^{-1}\left[ F(j\omega) \cdot \sum_{n=-\infty}^{\infty} \delta(\omega - n\Omega) \right] = f(t) * \frac{2\pi}{\Omega} \sum_{m=-\infty}^{\infty} \delta\left(t - m\frac{2\pi}{\Omega}\right) = \frac{2\pi}{\Omega} \sum_{m=-\infty}^{\infty} f\left(t - \frac{2\pi}{\Omega}m\right)$$

Appendix A16: Fast Fourier Transform (FFT) Algorithms

Let $W_N = e^{-j\frac{2\pi}{N}}$. Using $W_N^{2} = W_{N/2}$, $W_N^{4} = W_{N/4}$, and $W_N^{N/2} = -1$:

① Radix-2 Decimation-in-Time (DIT) FFT

Split $x(n)$ into even ($n = 2r$) and odd ($n = 2r + 1$) indices:

$$X(k) = \sum_{r=0}^{\frac{N}{2}-1} x(2r) W_N^{2rk} + \sum_{r=0}^{\frac{N}{2}-1} x(2r + 1) W_N^{(2r + 1)k} = \sum_{r=0}^{\frac{N}{2}-1} x(2r) W_{N/2}^{rk} + W_N^k \sum_{r=0}^{\frac{N}{2}-1} x(2r + 1) W_{N/2}^{rk}$$ $$X\left(k + \frac{N}{2}\right) = \sum_{r=0}^{\frac{N}{2}-1} x(2r) W_{N/2}^{rk} + W_N^{k + \frac{N}{2}} \sum_{r=0}^{\frac{N}{2}-1} x(2r + 1) W_{N/2}^{rk} = \sum_{r=0}^{\frac{N}{2}-1} x(2r) W_{N/2}^{rk} - W_N^k \sum_{r=0}^{\frac{N}{2}-1} x(2r + 1) W_{N/2}^{rk}$$

② Radix-2 Decimation-in-Frequency (DIF) FFT

Split the summation interval into first half ($0 \le n < N/2$) and second half ($n + N/2$), then evaluate at even ($k = 2r$) and odd ($k = 2r + 1$) frequencies:

$$X(2r) = \sum_{n=0}^{N-1} x(n) W_N^{2rn} = \sum_{n=0}^{\frac{N}{2}-1} x(n) W_{N/2}^{rn} + \sum_{n=0}^{\frac{N}{2}-1} x\left(n + \frac{N}{2}\right) W_{N/2}^{r(n + \frac{N}{2})} = \sum_{n=0}^{\frac{N}{2}-1} \left[ x(n) + x\left(n + \frac{N}{2}\right) \right] W_{N/2}^{rn}$$ $$X(2r + 1) = \sum_{n=0}^{N-1} x(n) W_N^{(2r + 1)n} = \sum_{n=0}^{\frac{N}{2}-1} x(n) W_N^n W_{N/2}^{rn} + \sum_{n=0}^{\frac{N}{2}-1} x\left(n + \frac{N}{2}\right) W_N^{(2r + 1)(n + \frac{N}{2})} = \sum_{n=0}^{\frac{N}{2}-1} \left\{ \left[ x(n) - x\left(n + \frac{N}{2}\right) \right] W_N^n \right\} W_{N/2}^{rn}$$

③ Radix-4 Decimation-in-Time (DIT) FFT

Decompose into 4 subsequences $x(4r + i)$ ($i = 0, 1, 2, 3$):

$$\begin{aligned} X(k) &= \sum_{r=0}^{\frac{N}{4}-1} x(4r) W_N^{4rk} + \sum_{r=0}^{\frac{N}{4}-1} x(4r + 1) W_N^{(4r + 1)k} + \sum_{r=0}^{\frac{N}{4}-1} x(4r + 2) W_N^{(4r + 2)k} + \sum_{r=0}^{\frac{N}{4}-1} x(4r + 3) W_N^{(4r + 3)k} \\ &= \sum_{r=0}^{\frac{N}{4}-1} x(4r) W_{N/4}^{rk} + W_N^k \sum_{r=0}^{\frac{N}{4}-1} x(4r + 1) W_{N/4}^{rk} + W_N^{2k} \sum_{r=0}^{\frac{N}{4}-1} x(4r + 2) W_{N/4}^{rk} + W_N^{3k} \sum_{r=0}^{\frac{N}{4}-1} x(4r + 3) W_{N/4}^{rk} \end{aligned}$$

And similarly for $X\left(k + \frac{N}{4}\right), X\left(k + \frac{N}{2}\right), X\left(k + \frac{3N}{4}\right)$ using $W_N^{m(k + qN/4)} = (-j)^{mq} W_N^{mk}$.

④ Radix-4 Decimation-in-Frequency (DIF) FFT

Split the input into 4 quarters $x\left(n + i\frac{N}{4}\right)$ ($i = 0, 1, 2, 3$):

$$\begin{aligned} X(4r) &= \sum_{n=0}^{\frac{N}{4}-1} x(n) W_N^{4nr} + \sum_{n=\frac{N}{4}}^{\frac{N}{2}-1} x(n) W_N^{4nr} + \sum_{n=\frac{N}{2}}^{\frac{3N}{4}-1} x(n) W_N^{4nr} + \sum_{n=\frac{3N}{4}}^{N-1} x(n) W_N^{4nr} \\ &= \sum_{n=0}^{\frac{N}{4}-1} \left[ x(n) + x\left(n + \frac{N}{4}\right) + x\left(n + \frac{N}{2}\right) + x\left(n + \frac{3N}{4}\right) \right] W_{N/4}^{rn} \end{aligned}$$

With analogous expressions for $X(4r + 1), X(4r + 2), X(4r + 3)$.

⑤ Split-Radix DIT FFT & ⑥ Split-Radix DIF FFT

Combines Radix-2 for even indices ($n = 2r$) and Radix-4 for odd indices ($n = 4\ell + 1, 4\ell + 3$):

$$\begin{aligned} X(k) &= \sum_{r=0}^{\frac{N}{2}-1} x(2r) W_N^{2rk} + \sum_{\ell=0}^{\frac{N}{4}-1} x(4\ell + 1) W_N^{(4\ell + 1)k} + \sum_{\ell=0}^{\frac{N}{4}-1} x(4\ell + 3) W_N^{(4\ell + 3)k} \\ &= \sum_{r=0}^{\frac{N}{2}-1} x(2r) W_{N/2}^{rk} + W_N^k \sum_{\ell=0}^{\frac{N}{4}-1} x(4\ell + 1) W_{N/4}^{\ell k} + W_N^{3k} \sum_{\ell=0}^{\frac{N}{4}-1} x(4\ell + 3) W_{N/4}^{\ell k} \end{aligned}$$

(And dual formulation for ⑥ Split-Radix DIF FFT.)

XI (cont.). Frequency-Domain Sampling & Multirate Signal Processing

4. Frequency-Domain Sampling of Periodic Spectrum $X(e^{j\omega})$

Assume the frequency period $2\pi$ is an integer multiple $N$ of the frequency sampling interval $\Omega_s$ ($2\pi = N \Omega_s$, i.e., $\Omega_s = \frac{2\pi}{N}$):

• ① IDFS from DTFT Samples ($X(e^{j\omega}) \to X(e^{j k \Omega_s}) = X(k)$): $$\begin{aligned} x_s(n) &= \frac{1}{N} \sum_{k=0}^{N-1} X(k) e^{j\left(\frac{2\pi}{N}\right) n k} = \frac{1}{N} \sum_{k=0}^{N-1} \left[ \sum_{m=-\infty}^{\infty} x(m) e^{-j m k \Omega_s} \right] e^{j \Omega_s n k} \\ &= \frac{1}{N} \sum_{m=-\infty}^{\infty} x(m) \sum_{k=0}^{N-1} e^{j \Omega_s k (n - m)} = \sum_{m=-\infty}^{\infty} x(m) \sum_{r=-\infty}^{\infty} \delta(n - m + rN) = \sum_{r=-\infty}^{\infty} x(n + rN) \end{aligned}$$
• ② Impulse-Train Sampled DTFT ($X(e^{j\omega}) \to X(e^{j\omega}) \sum_{k=-\infty}^{\infty} \delta(\omega - k\Omega_s)$): $$\begin{aligned} \tilde{x}_s(n) &= \frac{1}{2\pi} \int_{2\pi} X(e^{j\omega}) \sum_{k=-\infty}^{\infty} \delta(\omega - k\Omega_s) e^{j\omega n} \, d\omega = \frac{1}{2\pi} \sum_{m=-\infty}^{\infty} x(m) \sum_{k=-\infty}^{\infty} e^{j k \Omega_s (n - m)} \\ &= \sum_{m=-\infty}^{\infty} x(m) \sum_{r=-\infty}^{\infty} \delta[\Omega_s(n - m) + 2\pi r] = \sum_{m=-\infty}^{\infty} x(m) \frac{1}{\Omega_s} \sum_{r=-\infty}^{\infty} \delta(m - n - rN) = \frac{1}{\Omega_s} \sum_{r=-\infty}^{\infty} x(n + rN) \end{aligned}$$

5. Downsampling (Decimation) by Integer Factor $M$

Anti-aliasing filter followed by decimation: $x(n) \xrightarrow{h(n)} w(n) \xrightarrow{\downarrow M} y(n)$.

$$\begin{cases} w(n) = \sum_k h(k) x(n - k) \\ y(n) = w(Mn) \end{cases} \implies y(n) = \sum_k h(k) x(Mn - k) = \sum_k h(Mn - k) x(k)$$

Using the identity $\frac{1}{M} \sum_{\ell=0}^{M-1} e^{j\frac{2\pi}{M}\ell n} = \begin{cases} 1, & n = mM \\ 0, & \text{otherwise} \end{cases}$:

$$Y(z) = \sum_m y(m) z^{-m} = \sum_m w(Mm) z^{-m} = \sum_{n=-\infty}^{\infty} w(n) z^{-n/M} \frac{1}{M} \sum_{\ell=0}^{M-1} e^{j\frac{2\pi}{M}\ell n} = \frac{1}{M} \sum_{\ell=0}^{M-1} W\left(e^{-j\frac{2\pi}{M}\ell} z^{1/M}\right)$$

Evaluating on the unit circle ($z = e^{j\omega}$):

$$Y(e^{j\omega}) = \frac{1}{M} \sum_{\ell=0}^{M-1} H\left(e^{j\left(\frac{\omega - 2\pi\ell}{M}\right)}\right) X\left(e^{j\left(\frac{\omega - 2\pi\ell}{M}\right)}\right) \implies \frac{1}{M} X\left(e^{j\frac{\omega}{M}}\right), \quad |\omega| \le \pi \quad (\text{with ideal lowpass } H)$$

6. Upsampling (Interpolation) by Integer Factor $L$

Zero-insertion followed by anti-imaging filter: $x(n) \xrightarrow{\uparrow L} w(n) \xrightarrow{h(n)} y(n)$.

$$\begin{cases} w(n) = x(n / L) \quad (\text{for } n = mL, \text{ else } 0) \\ y(n) = \sum_k h(n - k) w(k) \end{cases} \implies y(n) = \sum_k h(n - k) x(k / L) = \sum_k h(n - kL) x(k)$$

In the $z$-domain and frequency domain:

$$Y(z) = H(z) W(z) = H(z) \sum_m w(m) z^{-m} = H(z) \sum_m x(m) z^{-mL} = H(z) X(z^L)$$ $$Y(e^{j\omega}) = H(e^{j\omega}) X(e^{j\omega L}) \implies L X(e^{j\omega L}), \quad |\omega| \le \pi \quad (\text{with ideal lowpass } H \text{ of gain } L)$$

7. Rational Sampling Rate Conversion by $L / M$ (Interpolate by $L$, then Decimate by $M$)

$$y(n) = \sum_{r=-\infty}^{\infty} h(Mn - Lr) x(r)$$ $$Y(e^{j\omega}) = \frac{1}{M} \sum_{\ell=0}^{M-1} H\left(e^{j\left(\frac{\omega - 2\pi\ell}{M}\right)}\right) X\left(e^{j L\left(\frac{\omega - 2\pi\ell}{M}\right)}\right) \implies \frac{L}{M} X\left(e^{j\frac{L}{M}\omega}\right), \quad |\omega| \le \min\left\{\pi, \pi\frac{M}{L}\right\}$$

Header Note: Minimum-Phase System: Among all systems sharing the same magnitude response, the minimum-phase system has the minimum group delay (and minimum phase lag).

XII. Analog and Digital Filters

1. Analog Filter Synthesis (Doubly-Terminated Lossless $LC$ Network — Darlington Synthesis)

Consider a source $E_s$ with internal resistance $R_s$ driving a two-port lossless $LC$ network (ports $1\text{-}1'$ and $2\text{-}2'$, input impedance $Z_{11}(s)$, input voltage $V_i$, input current $I_1$) terminated in load resistance $R_L$ (output voltage $V_o$, current $I_2$).

• Problem Statement: Given that $H_a(s) = \frac{V_o(s)}{E_s(s)}$ is stable, causal, minimum-phase, and conjugate-symmetric, and given $|H_a(j\Omega)|^2$, determine the driving-point impedance $Z_{11}(s)$ and synthesize the $LC$ ladder network.
• Solution Procedure:
  • From $\left.|H_a(j\Omega)|^2\right|_{j\Omega = s} = H_a(s) H_a(-s)$ and the stability/minimum-phase conditions, identify $H_a(s)$.
  • Available (Rated) Source Power: $P_m = \frac{1}{2} \frac{|E_s|^2}{4 R_s}$
  • Load Power: $P_L = \frac{1}{2} \frac{|V_o|^2}{R_L}$
  • Transducer Power Gain (Transmission Function): $|K|^2 = \frac{P_L}{P_m} = \frac{4 R_s}{R_L} \left|\frac{V_o}{E_s}\right|^2 = \frac{4 R_s}{R_L} |H_a(j\Omega)|^2$
  • Reflection Coefficient: $|\rho|^2 = 1 - \frac{4 R_s}{R_L} |H_a(s)|^2$
  • Since the $LC$ network is lossless, input power at port $1\text{-}1'$ equals load power ($P_1 = P_2 = P_L$): $$\frac{1}{2} \text{Re}[Z_{11}(j\Omega)] \, |I_1(j\Omega)|^2 = \frac{1}{2} \frac{|V_o(j\Omega)|^2}{R_L} \implies \left|\frac{V_o}{I_1}\right|^2 = R_L \cdot \text{Re}[Z_{11}(s)]$$
  • Also, since $\frac{E_s}{I_1} = R_s + Z_{11}(s)$: $$|H_a(s)|^2 = \left|\frac{V_o}{I_1}\right|^2 \left|\frac{I_1}{E_s}\right|^2 = \frac{R_L \cdot \text{Re}[Z_{11}(s)]}{|R_s + Z_{11}(s)|^2}$$ $$|\rho(s)|^2 = \frac{|Z_{11} - R_s|^2}{|Z_{11} + R_s|^2} \implies \rho(s) = \pm \frac{Z_{11} - R_s}{Z_{11} + R_s} \implies Z_{11}(s) = R_s \frac{1 \pm \rho(s)}{1 \mp \rho(s)}$$
• Network Realization Methods from $Z_{11}(s)$:
  • ① Foster Canonical Forms: Series form ($Z_{11}(s) = \sum \cdots$ partial fractions) and Parallel form ($Y_{11}(s) = \sum \cdots$).
  • ② Cauer Canonical (Ladder) Forms: T-type ($s$ continued fraction expansion around $\infty$) and $\Pi$-type ($\frac{1}{s}$ continued fraction expansion around $0$).

2. Analog Filter Frequency Transformations (Normalized Prototype $s'$, $L'$, $C'$)

Filter Type	Complex Frequency Mapping $s' \leftarrow$	Inductor $s'L' \leftarrow$	Capacitor $\frac{1}{s'C'} \leftarrow$
Prototype	$s'$	$s'L'$	$\frac{1}{s'C'}$
Lowpass (LP)	$\frac{s}{\Omega_c}$	$s \frac{R_s}{\Omega_c} L'$	$\frac{\Omega_c R_s}{s C'}$
Highpass (HP)	$\frac{\Omega_c}{s}$	$\frac{1}{s \frac{1}{\Omega_c R_s L'}}$	$s \frac{R_s}{C' \Omega_c}$
Bandpass (BP)	$\frac{s^2 + \Omega_0^2}{s \Omega_r}$	$s \frac{R_s L'}{\Omega_r} + \frac{\Omega_r}{s \Omega_0^2 R_s L'}$ (Series $LC$)	$\left(s \frac{\Omega_r R_s}{\Omega_0^2 C'}\right) \parallel \left(\frac{C'}{s \Omega_r R_s}\right)$ (Parallel $LC$)
Bandstop (BS)	$\frac{s \Omega_r}{s^2 + \Omega_0^2}$	$\left(s \frac{\Omega_r R_s L'}{\Omega_0^2}\right) \parallel \left(\frac{1}{s \Omega_r R_s L'}\right)$ (Parallel $LC$)	$s \frac{R_s}{\Omega_r C'} + \frac{\Omega_r C'}{s \Omega_0^2 R_s}$ (Series $LC$)

(where $\Omega_r = \Omega_2 - \Omega_1$ is the bandwidth and $\Omega_0^2 = \Omega_1 \Omega_2$ is the center frequency squared.)

3. IIR Digital Filter Structures

$$H(z) = \frac{\sum_{r=0}^M b_r z^{-r}}{1 + \sum_{k=1}^N a_k z^{-k}}, \quad y(n) = \sum_{r=0}^M b_r x(n - r) - \sum_{k=1}^N a_k y(n - k)$$

• ① Direct Form Structures (Direct Form I & Canonical Direct Form II):
  • Direct Form II uses the minimum number of delay elements ($\max(M, N)$).
  • Poles and zeros are difficult to tune independently; highly sensitive to finite word-length (coefficient quantization) effects.
• ② Cascade Form Structure:
  • Factored into second-order sections (biquads). Uses minimum delay elements; zeros and poles can be tuned independently.
  • Pairing & Ordering Rules: Pair poles with their closest zeros; order sections by decreasing gain (from the perspective of minimizing quantization noise) or by decreasing pole distance to the unit circle (from the perspective of signal attenuation / preventing overflow).
• ③ Parallel Form Structure: (Continued on Page 19)

Appendix A17: Butterworth Analog Filter Design & Synthesis

1. Magnitude-Squared Response & Pole Distribution

Given the $N$-th order Butterworth magnitude-squared specification:

$$|H_a(j\Omega)|^2 = \left( \frac{1}{\sqrt{1 + \left(\frac{\Omega}{\Omega_c}\right)^{2N}}} \right)^2 = \frac{1}{1 + \left(\frac{\Omega}{\Omega_c}\right)^{2N}} \implies |H_a(s)|^2 = H_a(s) H_a(-s) = \frac{(j\Omega_c)^{2N}}{s^{2N} + (j\Omega_c)^{2N}}$$

The $2N$ poles lie on a circle of radius $\Omega_c$ in the $s$-plane:

$$s_k = \Omega_c \, e^{j\left[\frac{1}{2N}(2k - 1)\pi + \frac{\pi}{2}\right]}, \quad k = 1, 2, \dots, 2N$$

2. Stable Transfer Function $H_a(s)$

Selecting all $N$ left-half-plane (LHP) poles ($k = 1, 2, \dots, N$) to form a stable and causal transfer function $H_a(s)$:

$$H_a(s) = \frac{\Omega_c^N}{\prod_{k=1}^N (s - s_k)} = \begin{cases} \displaystyle \Omega_c^N \Big/ \left\{ (s + \Omega_c) \prod_{k=1}^{(N-1)/2} \left[ s^2 - 2\Omega_c \cos\left(\frac{2k - 1}{2N}\pi + \frac{\pi}{2}\right) s + \Omega_c^2 \right] \right\}, & N \text{ odd} \\[2.5ex] \displaystyle \Omega_c^N \Big/ \prod_{k=1}^{N/2} \left[ s^2 - 2\Omega_c \cos\left(\frac{2k - 1}{2N}\pi + \frac{\pi}{2}\right) s + \Omega_c^2 \right], & N \text{ even} \end{cases}$$

With normalized frequency $s' = s / \Omega_c$, the normalized Butterworth polynomial $B_N(s')$ gives:

$$H_a(s') = \frac{1}{B_N(s')} = \begin{cases} \displaystyle 1 \Big/ \left\{ (s' + 1) \prod_{k=1}^{(N-1)/2} \left[ {s'}^2 - 2\cos\left(\frac{2k - 1}{2N}\pi + \frac{\pi}{2}\right) s' + 1 \right] \right\}, & N \text{ odd} \\[2.5ex] \displaystyle 1 \Big/ \prod_{k=1}^{N/2} \left[ {s'}^2 - 2\cos\left(\frac{2k - 1}{2N}\pi + \frac{\pi}{2}\right) s' + 1 \right], & N \text{ even} \end{cases}$$

3. Darlington Impedance Synthesis & Element Denormalization

Given passband and stopband specifications, solve for $N$ and $\Omega_c$ to determine $H_a(s)$. Typically, source and load resistances are matched ($R_s = R_L$), so $H_a(j0) = \frac{1}{2}$. Then:

$$\rho(s') \rho(-s') = 1 - \frac{4 R_s}{R_L} \cdot \frac{1/2}{\sqrt{1 + (s'/j)^{2N}}} \cdot \frac{1/2}{\sqrt{1 + (-s'/j)^{2N}}} = 1 - \frac{1}{1 + (-1)^N (s')^{2N}} = \frac{(s')^N (-s')^N}{B_N(s') B_N(-s')}$$

Therefore, the reflection coefficient and driving-point impedance are:

$$\rho(s') = \frac{(s')^N}{B_N(s')} \implies Z_{11}(s') = R_s \frac{1 \pm \rho(s')}{1 \mp \rho(s')} = R_s \frac{B_N(s') \pm (s')^N}{B_N(s') \mp (s')^N}$$

Normalized with respect to source resistance $R_s$:

$$\frac{Z_{11}(s')}{R_s} = \frac{B_N(s') \pm (s')^N}{B_N(s') \mp (s')^N}$$

Denormalizing to actual physical inductance, capacitance, and resistance values:

$$\begin{cases} L = \dfrac{R_s}{\Omega_c} L' \\[1.2ex] C = \dfrac{1}{\Omega_c R_s} C' \\[1.2ex] R_L = R_s \end{cases}$$

XII (cont.). IIR Design Methods & FIR Filter Structures

3 (cont.). Parallel Form IIR Structure

Realized via partial fraction expansion of $H(z)$ into parallel first- and second-order sections:

• High computational speed (parallel execution); poles can be tuned independently; quantization/arithmetic errors in individual parallel branches do not accumulate across sections.
• Zeros are difficult to tune directly.

4. IIR Digital Filter Design Methods

① Impulse Invariance Method:

• Principle: Sample the analog impulse response: $h(n T_s) = \left.h_a(t)\right|_{t = n T_s}$ (or $h(n) = T_s h_a(n T_s)$).
• Mapping Result: $$\frac{1}{s - s_k} \implies \frac{1}{1 - e^{s_k T_s} z^{-1}}, \quad H(e^{j\omega}) = \frac{1}{T_s} \sum_{r=-\infty}^{\infty} H_a\left(j\frac{\omega}{T_s} + j\frac{2\pi}{T_s}r\right)$$
• Conclusion: Suffers from frequency aliasing; high-frequency response is distorted (unsuitable for highpass or bandstop filters).

② Bilinear Transformation Method:

• Principle: Two-step conformal mapping ($s \to s_1$-plane via $\tanh$, then $s_1 \to z$-plane):
  • Step 1 ($s \to s_1$-plane, compressed strip $[-\pi/T_s, \pi/T_s]$): $$\begin{cases} s = \frac{2}{T_s} \frac{1 - e^{-s_1 T_s}}{1 + e^{-s_1 T_s}} \\[1ex] \Omega = \frac{2}{T_s} \tan\left(\frac{\Omega_1 T_s}{2}\right) \Longleftrightarrow \Omega_1 = \frac{2}{T_s} \arctan\left(\frac{\Omega T_s}{2}\right) \end{cases}$$
  • Step 2 ($s_1 \to z$-plane): $$\begin{cases} z = e^{s_1 T_s} \\ \omega = \Omega_1 T_s \end{cases}$$
• Combined Mapping Result: $$s = \frac{2}{T_s} \frac{1 - z^{-1}}{1 + z^{-1}}, \quad \Omega = \frac{2}{T_s} \tan\left(\frac{\omega}{2}\right)$$
• Conclusion: Completely eliminates aliasing (one-to-one mapping of $j\Omega$-axis to unit circle), but introduces nonlinear frequency warping distortion (requires pre-warping at critical frequencies).

5. FIR Digital Filter Structures

$$H(z) = \sum_{n=0}^{N-1} h(n) z^{-n}, \quad y(n) = \sum_{r=0}^{N-1} h(r) x(n - r)$$

• ① Direct Form Structure (Transversal / Tapped-Delay-Line Filter): Delay line $z^{-1}$ with tap weights $h(0), h(1), \dots, h(N-1)$. Zeros are difficult to control directly.
• ② Cascade Form Structure: Product of second-order sections $(1 + \beta_{1k} z^{-1} + \beta_{2k} z^{-2})$. Zeros can be tuned independently.
• ③ Fast Convolution Structure (Overlap-Add / Overlap-Save): $x(n) \to \text{FFT} \to \times H(k) \to \text{IDFT} \to y(n)$. High computational speed for long $N$.
• ④ Frequency Sampling Structure (Comb Filter + Parallel Resonators): $$H(z) = \sum_{n=0}^{N-1} h(n) z^{-n} = \sum_{n=0}^{N-1} \left( \frac{1}{N} \sum_{k=0}^{N-1} H(k) W_N^{-nk} \right) z^{-n} = \frac{1}{N} (1 - z^{-N}) \sum_{k=0}^{N-1} \frac{H(k)}{1 - W_N^{-k} z^{-1}}$$
• ⑤ Linear-Phase FIR Structures:

  For a linear-phase filter $H(e^{j\omega}) = |H(e^{j\omega})| e^{-j(\beta + \alpha\omega)}$ (or $\hat{H}(e^{j\omega}) e^{-j(\beta + \alpha\omega)}$):

  $$\begin{cases} \text{Even Symmetry (Type I & II):} & \beta = 0, \quad \alpha = \frac{N-1}{2}, \quad h(n) = h(N - 1 - n) \\[1ex] \text{Odd / Anti-Symmetry (Type III & IV):} & \beta = \pm\frac{\pi}{2}, \quad \alpha = \frac{N-1}{2}, \quad h(n) = -h(N - 1 - n) \end{cases}$$

  1) Direct Form Linear-Phase Realization: Folds the delay line to add symmetric/anti-symmetric taps $x(n - k) \pm x(n - (N - 1 - k))$ before multiplying by $h(k)$, halving the number of multipliers (detailed for all 4 types on Page 21).

Appendix A18: Finite Word-Length Quantization Effects & Electronic Circuit Noise

1. Quantization Error Statistical Models (Step Size $q = 2^{-b}$)

Error Type	Distribution	Mean $\mu_e$	Variance $\sigma_e^2$
Truncation Error (Two's Complement)	$U(-q, 0)$	$-\frac{q}{2}$	$\frac{q^2}{12} = \frac{1}{12} \times 2^{-2b}$
Rounding Error	$U\left(-\frac{q}{2}, \frac{q}{2}\right)$	$0$	$\frac{q^2}{12} = \frac{1}{12} \times 2^{-2b}$

Generally, two's complement fixed-point rounding is used to analyze quantization noise. The Signal-to-Quantization-Noise Ratio (SQNR) is:

$$\frac{S}{N} = 10 \lg \frac{\sigma_x^2}{\sigma_e^2} = 10 \lg \left( \frac{\sigma_x^2}{\frac{1}{12} \times 2^{-2b}} \right) = \left( 6.02 b + 10.79 + 10 \lg \sigma_x^2 \right) \text{ dB}$$

2. Quantization Noise Propagation & Dynamic Range Scaling

• By the theory of random processes through linear systems, the quantization noise variance at the output of a digital filter with impulse response $h(m)$ from noise source to output is: $$\sigma_f^2 = \sigma_e^2 \sum_{m=0}^{\infty} h^2(m) = \sigma_e^2 \frac{1}{2\pi} \int_{-\pi}^{\pi} |H(e^{j\omega})|^2 \, d\omega$$
• For multiple independent quantization noise sources ($i = 1, \dots, N$), superposition holds at the output: $$\sigma_{\text{total}}^2 = \sigma_e^2 \sum_{i=1}^N \sum_{m=0}^{\infty} h_i^2(m)$$
• Scaling to Prevent Overflow: The above noise analysis is valid only when no addition overflow occurs at any internal node. To prevent fixed-point overflow, every internal node variable $w(n)$ must satisfy $|w(n)| \le 1$. Suppose the input signal is bounded by $|x(n)| \le 1$, and let $f(n)$ denote the unit impulse response from the input to node variable $w(n)$: $$w(n) = f(n) * x(n) = \sum_{m=0}^{\infty} f(m) x(n - m)$$ We scale the input to node $w(n)$ by dividing by $\beta$ (and multiply by $\beta$ at the output), where the scaling factor $\beta$ is chosen according to: $$\begin{cases} \ell_1\text{-norm criterion (strict):} & \beta = \sum_{n=0}^{\infty} |f(n)| = \|f\|_1 \\[1.2ex] \ell_2\text{-norm criterion (energy/RMS):} & \beta = \sqrt{\sum_{n=0}^{\infty} f^2(n)} = \|f\|_2 \end{cases}$$ Trade-off: After introducing the compression scaling factor $\beta$, for the same input signal, the output quantization noise increases (because scaling requires multiplying the output or internal branches by $\beta$, amplifying the noise sources).

3. Electronic Device Noise Power Spectral Densities (PSD)

• Resistor Thermal (Johnson–Nyquist) Noise: $$W_v(f) = 4 k T R(f), \quad W_i(f) = 4 k T G(f) \quad (\text{Power Spectral Density})$$
• Bipolar Junction Transistor (BJT) Noise:
  • Base resistance thermal noise: $W_v(f) = 4 k T r_{bb'}$
  • Shot noise: $W_i(f) = 2 q I_{E/CO}$
  • Partition noise: $W_i(f) = 2 q I_C \left(1 - \frac{|\alpha(f)|^2}{\alpha_0}\right)$
  • Flicker ($1/f$) noise: $W_i(f) = \frac{K}{f} \cdot \frac{I}{A}$
• MOSFET Noise & Linear System Output Noise PSD: $$W_{\text{out}}(f) = |H(f)|^2 W_{\text{in}}(f)$$

XII (cont.). Linear-Phase FIR Types, Design, and Digital Frequency Transformations

5 (cont.). Four Types of Linear-Phase FIR Filters

• a. Type I: $h(n)$ Even-Symmetric, $N$ Odd: $$H(e^{j\omega}) = e^{-j\frac{N-1}{2}\omega} \left\{ \sum_{n=0}^{\frac{N-3}{2}} 2 h(n) \cos\left[\omega\left(\frac{N-1}{2} - n\right)\right] + h\left(\frac{N-1}{2}\right) \right\}$$ Constraint: No fixed zeros at $\omega = 0$ or $\pi$; suitable for all filter types (LP, HP, BP, BS).
• b. Type II: $h(n)$ Even-Symmetric, $N$ Even: $\left.H(e^{j\omega})\right|_{\omega = \pi} = 0$ $$H(e^{j\omega}) = e^{-j\frac{N-1}{2}\omega} \left\{ \sum_{n=0}^{\frac{N}{2}-1} 2 h(n) \cos\left[\omega\left(\frac{N-1}{2} - n\right)\right] \right\}$$ Constraint: Always has a zero at $\omega = \pi$ ($z = -1$); cannot be used for highpass or bandstop filters.
• c. Type III: $h(n)$ Odd-Symmetric (Anti-Symmetric), $N$ Odd: $\left.H(e^{j\omega})\right|_{\omega = 0, \pi} = 0$ $$H(e^{j\omega}) = e^{-j\frac{N-1}{2}\omega} \left\{ j \sum_{n=0}^{\frac{N-3}{2}} 2 h(n) \sin\left[\omega\left(\frac{N-1}{2} - n\right)\right] \right\}$$ Constraint: Fixed zeros at both $\omega = 0$ and $\omega = \pi$; used for differentiators, Hilbert transformers, and bandpass filters.
• d. Type IV: $h(n)$ Odd-Symmetric (Anti-Symmetric), $N$ Even: $\left.H(e^{j\omega})\right|_{\omega = 0} = 0$ $$H(e^{j\omega}) = e^{-j\frac{N-1}{2}\omega} \left\{ j \sum_{n=0}^{\frac{N}{2}-1} 2 h(n) \sin\left[\omega\left(\frac{N-1}{2} - n\right)\right] \right\}$$ Constraint: Fixed zero at $\omega = 0$ ($z = 1$); used for differentiators, Hilbert transformers, and highpass filters.
• 2) Cascade Structure of Linear-Phase FIR Filters: Zeros occur in mirror-image conjugate reciprocal quadruplets ($z_k, z_k^*, 1/z_k, 1/z_k^*$) with respect to the unit circle.

6. FIR Filter Design Methods

• ① Window Design Method ($h(n) = h_d(n) \cdot w(n)$):
  • Mainlobe width $\uparrow \Longleftrightarrow$ Peak sidelobe level $\downarrow \Longleftrightarrow$ Transition band width $\uparrow \Longleftrightarrow$ Minimum stopband attenuation $\uparrow$ (stopband ripple $\downarrow$).
• ② Frequency Sampling Design Method: $$H(e^{j\omega}) = \frac{1}{N} e^{-j\omega\frac{N-1}{2}} \sum_{k=0}^{N-1} H(k) e^{-j\frac{k\pi}{N}(N-1)} \frac{\sin\left[\frac{N}{2}\left(\omega - \frac{2\pi}{N}k\right)\right]}{\sin\left[\frac{1}{2}\left(\omega - \frac{2\pi}{N}k\right)\right]}$$ Matches the desired response exactly at the frequency sampling points $\omega_k = \frac{2\pi}{N}k$. At abrupt transitions between passband and stopband, Gibbs ripples are large; typically one inserts several unconstrained transition samples between passband and stopband to optimize stopband attenuation.

7. Digital Filter Frequency Transformations (Constantinides All-Pass Mapping $Z \leftarrow G(z)$)

Mapping requirements from prototype $Z$-plane to target $z$-plane:

• ① Unit circle of $Z$ maps to unit circle of $G(z)$: $|G(e^{j\omega})| = 1$.
• ② Interior/boundary/exterior preserved: $|z| \gtreqless 1 \implies |G(z)| \gtreqless 1$ (Wait: for $Z = g(z)$, $|z| < 1 \implies |G(z)| < 1$ so causal stable maps to causal stable).
• ③ Rational $H(Z) \implies$ Rational $H(G(z))$.

Therefore, $G(z)$ must be an all-pass function of order $N$ (where $N$ equals the number of passbands of the target filter as $z$ traverses the unit circle once):

$$G(z) = \pm \frac{D(z)}{z^N D(z^{-1})}$$

Target Filter	Mapping $Z \leftarrow G(z)$	DC ($z = 1$) Maps To	Cutoff $e^{j\theta_p}$ Maps To	Cutoff $e^{-j\theta_p}$ Maps To
Prototype	$z$	$z = 1$	$e^{j\theta_p}$	$e^{-j\theta_p}$
Lowpass (LP)	$\dfrac{z - \alpha}{1 - \alpha z}$	$1$	$e^{j\omega_p}$	$e^{-j\omega_p}$
Highpass (HP)	$-\dfrac{z + \alpha}{1 + \alpha z}$	$-1$	$e^{j\omega_p}$	$e^{-j\omega_p}$
Bandpass (BP)	$-\dfrac{z^2 + \alpha_1 z + \alpha_2}{1 + \alpha_1 z + \alpha_2 z^2}$	$e^{\pm j\omega_0}$	$e^{j\omega_2}$	$e^{j\omega_1}$
Bandstop (BS)	$\dfrac{z^2 + \alpha_1 z + \alpha_2}{1 + \alpha_1 z + \alpha_2 z^2}$	$\pm 1$	$e^{j\omega_1}$	$e^{j\omega_2}$

XIII. State-Space Analysis of Continuous and Discrete Systems

1. Formulation of Continuous-Time State-Space Equations

Choose integrator outputs as state variables $\bm{\lambda}(t)$ (input $\bm{e}(t)$, output $\bm{r}(t)$):

$$\begin{cases} \dfrac{d}{dt}\bm{\lambda}(t) = \bm{A}\bm{\lambda}(t) + \bm{B}\bm{e}(t) \\[1ex] \bm{r}(t) = \bm{C}\bm{\lambda}(t) + \bm{D}\bm{e}(t) \end{cases}$$

2. Solution of Continuous-Time State-Space Equations

① Time-Domain ($t$-Domain) Solution:

From the state equation, $\frac{d}{dt}[e^{-\bm{A}t}\bm{\lambda}(t)] = e^{-\bm{A}t}\bm{B}\bm{e}(t)$. Integrating from $0_-$ to $t$ yields:

$$\bm{\lambda}(t) = e^{\bm{A}t}\bm{\lambda}(0_-) + \int_{0_-}^t e^{\bm{A}(t - \tau)}\bm{B}\bm{e}(\tau) \, d\tau = e^{\bm{A}t}\bm{\lambda}(0_-) + e^{\bm{A}t}\bm{B} * \bm{e}(t)$$ $$\bm{r}(t) = \bm{C}e^{\bm{A}t}\bm{\lambda}(0_-) + \int_{0_-}^t \bm{C}e^{\bm{A}(t - \tau)}\bm{B}\bm{e}(\tau) \, d\tau + \bm{D}\bm{e}(t) = \bm{C}e^{\bm{A}t}\bm{\lambda}(0_-) + [\bm{C}e^{\bm{A}t}\bm{B} + \bm{D}\delta(t)] * \bm{e}(t)$$

② Laplace-Domain ($s$-Domain) Solution:

Taking the unilateral Laplace transform of the state equation gives $s\bm{\Lambda}(s) - \bm{\lambda}(0_-) = \bm{A}\bm{\Lambda}(s) + \bm{B}\bm{E}(s)$, so:

$$\bm{\Lambda}(s) = (s\bm{I} - \bm{A})^{-1}\bm{\lambda}(0_-) + (s\bm{I} - \bm{A})^{-1}\bm{B}\bm{E}(s)$$ $$\bm{\lambda}(t) = \mathcal{L}^{-1}[(s\bm{I} - \bm{A})^{-1}]\bm{\lambda}(0_-) + \mathcal{L}^{-1}[(s\bm{I} - \bm{A})^{-1}]\bm{B} * \bm{e}(t)$$ $$\bm{r}(t) = \bm{C}\mathcal{L}^{-1}[(s\bm{I} - \bm{A})^{-1}]\bm{\lambda}(0_-) + \left\{ \bm{C}\mathcal{L}^{-1}[(s\bm{I} - \bm{A})^{-1}]\bm{B} + \bm{D}\delta(t) \right\} * \bm{e}(t)$$

Under zero-state conditions ($\bm{\lambda}(0_-) = \bm{0}$), $\bm{\Lambda}(s) = (s\bm{I} - \bm{A})^{-1}\bm{B}\bm{E}(s)$ and $\bm{R}(s) = \bm{C}(s\bm{I} - \bm{A})^{-1}\bm{B}\bm{E}(s) + \bm{D}\bm{E}(s)$, yielding the transfer function matrix and impulse response:

$$\bm{H}(s) = \bm{C}(s\bm{I} - \bm{A})^{-1}\bm{B} + \bm{D} = \bm{C}\frac{\text{adj}(s\bm{I} - \bm{A})}{|s\bm{I} - \bm{A}|}\bm{B} + \bm{D}, \quad \bm{h}(t) = \bm{C}e^{\bm{A}t}\bm{B}u(t) + \bm{D}\delta(t)$$

• State Transition Matrix: $e^{\bm{A}t} = \mathcal{L}^{-1}[(s\bm{I} - \bm{A})^{-1}]$
• Resolvent (Characteristic) Matrix: $(s\bm{I} - \bm{A})^{-1}$

---

3. Formulation of Discrete-Time State-Space Equations

Choose delay element outputs as state variables $\bm{\lambda}(n)$ (input $\bm{x}(n)$, output $\bm{y}(n)$):

$$\begin{cases} \bm{\lambda}(n + 1) = \bm{A}\bm{\lambda}(n) + \bm{B}\bm{x}(n) \\[1ex] \bm{y}(n) = \bm{C}\bm{\lambda}(n) + \bm{D}\bm{x}(n) \end{cases}$$

4. Solution of Discrete-Time State-Space Equations

① Discrete Time-Domain ($n$-Domain) Solution:

From the state equation, $\bm{\lambda}(n) = \bm{K}\bm{A}^n u(n) + \bm{x}(n) * \bm{h}_0(n)$, solving for $\bm{K} = \bm{\lambda}(0)$ and $\bm{h}_0(n) = \bm{A}^{n-1}u(n - 1)\bm{B}$ gives:

$$\bm{\lambda}(n) = \bm{A}^n \bm{\lambda}(0) u(n) + \bm{A}^{n-1}\bm{B} u(n - 1) * \bm{x}(n) u(n)$$ $$\bm{y}(n) = \bm{C}\bm{A}^n \bm{\lambda}(0) u(n) + \bm{C}\bm{A}^{n-1}\bm{B} u(n - 1) * \bm{x}(n) u(n) + \bm{D}\bm{x}(n) u(n)$$

② $z$-Domain Solution:

Taking the unilateral $z$-transform gives $z\bm{\Lambda}(z) - z\bm{\lambda}(0) = \bm{A}\bm{\Lambda}(z) + \bm{B}\bm{X}(z)$, so:

$$\bm{\Lambda}(z) = (z\bm{I} - \bm{A})^{-1} z \bm{\lambda}(0) + (z\bm{I} - \bm{A})^{-1}\bm{B}\bm{X}(z)$$ $$\bm{\lambda}(n) = \mathcal{Z}^{-1}[(z\bm{I} - \bm{A})^{-1} z]\bm{\lambda}(0) + \mathcal{Z}^{-1}[(z\bm{I} - \bm{A})^{-1}\bm{B}] * \bm{x}(n)$$ $$\bm{y}(n) = \bm{C}\mathcal{Z}^{-1}[(z\bm{I} - \bm{A})^{-1} z]\bm{\lambda}(0) + \left\{ \bm{C}\mathcal{Z}^{-1}[(z\bm{I} - \bm{A})^{-1}\bm{B}] + \bm{D}\delta(n) \right\} * \bm{x}(n)$$

Under zero-state conditions ($\bm{\lambda}(0) = \bm{0}$), $\bm{\Lambda}(z) = (z\bm{I} - \bm{A})^{-1}\bm{B}\bm{X}(z)$ and $\bm{Y}(z) = \bm{C}(z\bm{I} - \bm{A})^{-1}\bm{B}\bm{X}(z) + \bm{D}\bm{X}(z)$, yielding:

$$\bm{H}(z) = \bm{C}(z\bm{I} - \bm{A})^{-1}\bm{B} + \bm{D} = \bm{C}\frac{\text{adj}(z\bm{I} - \bm{A})}{|z\bm{I} - \bm{A}|}\bm{B} + \bm{D}, \quad \bm{h}(n) = \bm{C}\bm{A}^{n-1}\bm{B}u(n - 1) + \bm{D}\delta(n)$$

• State Transition Matrix: $\bm{A}^n = \mathcal{Z}^{-1}[(z\bm{I} - \bm{A})^{-1} z]$, and $\bm{A}^{n-1} u(n - 1) = \mathcal{Z}^{-1}[(z\bm{I} - \bm{A})^{-1}]$.
• Resolvent Matrix: $(z\bm{I} - \bm{A})^{-1} z$.

Appendix A19: Linear Similarity Transformations of State Variables, Controllability, and Observability

1. Similarity Transformation & Transfer Function Invariance

Let $\bm{\gamma} = \bm{P}\bm{\lambda}$ be a non-singular linear transformation of the state vector ($\det \bm{P} \neq 0$). The transformed state-space matrices are:

$$\bm{A}' = \bm{P}\bm{A}\bm{P}^{-1}, \quad \bm{B}' = \bm{P}\bm{B}, \quad \bm{C}' = \bm{C}\bm{P}^{-1}, \quad \bm{D}' = \bm{D}$$

The transfer function matrix is invariant under similarity transformations:

$$\bm{H}'(s / z) = \bm{C}'(s\bm{I} - \bm{A}')^{-1}\bm{B}' + \bm{D}' = \bm{H}(s / z)$$

2. Jordan Canonical Form & Kalman Controllability / Observability Criteria

Using the modal matrix $\bm{P}$ formed by the eigenvectors (and generalized eigenvectors) of $\bm{A}$ from its characteristic equation, transform $\bm{A}$ into its Jordan Canonical Form $\bm{A}' = \bm{J}$:

• Controllability Criterion:
  • Gilbert / Jordan Form Criterion: For each distinct eigenvalue's Jordan blocks, the rows of $\bm{B}' = \bm{P}\bm{B}$ corresponding to the last row of each Jordan sub-block contain no all-zero row (and for distinct blocks sharing the same eigenvalue, those last rows are linearly independent).
  • Kalman Rank Criterion: Equivalent to the controllability matrix $\bm{M}$ having full row rank $N$: $$\text{rank}\,\bm{M} = \text{rank}\begin{bmatrix} \bm{B} & \bm{A}\bm{B} & \cdots & \bm{A}^{N-1}\bm{B} \end{bmatrix} = N \Longleftrightarrow \text{System is State Controllable.}$$
• Observability Criterion:
  • Gilbert / Jordan Form Criterion: For each Jordan sub-block, the columns of $\bm{C}' = \bm{C}\bm{P}^{-1}$ corresponding to the first column of each Jordan sub-block contain no all-zero column (and for blocks sharing the same eigenvalue, those first columns are linearly independent).
  • Kalman Rank Criterion: Equivalent to the observability matrix $\bm{N}$ having full column rank $N$: $$\text{rank}\,\bm{N} = \text{rank}\begin{bmatrix} \bm{C} \\ \bm{C}\bm{A} \\ \vdots \\ \bm{C}\bm{A}^{N-1} \end{bmatrix} = N \Longleftrightarrow \text{System is State Observable.}$$