Signals and Systems

I. Signals

1. Signal Classification
   • Time: Continuous vs. Discrete
     • Continuous: Analog (e.g., continuous sampling)
     • Discrete: Digital (quantized sequence)
2. Signal Decomposition
   • ① Convergence Property:
     • Continuous-time non-periodic (Fourier Transform): $$ f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{j\omega t} d\omega $$ Example: $ t \leftrightarrow \delta(t) $, Laplace Transform kernel $ e^{-st} $, Fourier Transform kernel $ e^{-j\omega t} $
     • Continuous-time periodic (Fourier Series): $$ f(t) = \sum_{n=-\infty}^{\infty} F_n e^{j n \omega_0 t} $$ Example: Fourier coefficients, Laplace Transform, Fourier Transform
     • Discrete-time non-periodic (DTFT): $$ f[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} F(e^{j\omega}) e^{j\omega n} d\omega $$ Example: $ n \leftrightarrow \delta[n] $, z-Transform kernel $ z^{-n} $, DTFT kernel $ e^{-j\omega n} $
     • Discrete-time periodic (DFS): $$ f[n] = \sum_{k=0}^{N-1} \tilde{X}[k] e^{j \frac{2\pi}{N} k n} $$ Example: DFS coefficients, z-Transform, DFT
   • ② Orthogonality:
     • Inner product of signals: $ \langle \phi_n(t), f(t) \rangle = \int_{-\infty}^{\infty} f(t) \phi_n^*(t) dt $
     • Discrete signals inner product: $ \langle f[n], g[n] \rangle = \sum_{n=-\infty}^{\infty} f[n] g^*[n] $
     • Orthogonality relation: $ \sum_{n} \phi_m[n] \phi_n^*[n] = \lambda_n \delta_{mn} $

Energy Signals:

• Cross-correlation: $ R_{21}(t) = \langle f_2(\tau), f_1(\tau - t) \rangle = R_{12}^*(-t) $
• Total energy: $ E = R_{11}(0) = \int_{-\infty}^{\infty} |f(t)|^2 dt $

Power Signals:

• Average power: $ P = R_{11}(0) = \lim_{T\to\infty} \frac{1}{T} \int_{-T/2}^{T/2} |f(t)|^2 dt $
• Autocorrelation: $ R_{ff}(\tau) = \lim_{T\to\infty} \frac{1}{T} \int_{-T/2}^{T/2} f(t) f^*(t-\tau) dt $

II. Systems

1. System Properties
   • Superposition (Linearity): $ \mathcal{H}\{a x_1(t) + b x_2(t)\} = a \mathcal{H}\{x_1(t)\} + b \mathcal{H}\{x_2(t)\} $
   • Time-invariance: If $ x(t) \rightarrow y(t) $, then $ x(t - t_0) \rightarrow y(t - t_0) $
   • Causality: Output depends only on present and past inputs.
   • Stability: BIBO (Bounded-Input Bounded-Output) stable.
2. Impulse Response of Combined LTI Systems
   • Parallel Connection: $ h(t) = h_1(t) + h_2(t) $
   • Cascade Connection: $ h(t) = h_1(t) * h_2(t) $

III. Time-Domain Analysis of LTI Systems

1. Differential & Difference Equation Formulations
   • Continuous LTI systems: $$ \sum_{k=0}^N a_k \frac{d^k y(t)}{dt^k} = \sum_{m=0}^M b_m \frac{d^m x(t)}{dt^m} $$
   • Discrete LTI systems: $$ \sum_{k=0}^N a_k y[n-k] = \sum_{m=0}^M b_m x[n-m] $$
2. Convolution Integrals & Sums
   • Convolution Integral: $$ y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) d\tau $$
   • Convolution Sum: $$ y[n] = x[n] * h[n] = \sum_{k=-\infty}^{\infty} x[k] h[n - k] $$
3. Impulse Response matching method

   Used to find initial conditions at $ t = 0^+ $ from $ t = 0^- $ when an impulse input $ \delta(t) $ is applied.

IV. Laplace Transform and s-Domain Analysis

1. Definition of Bilateral Laplace Transform $$ X(s) = \int_{-\infty}^{\infty} x(t) e^{-st} dt, \quad s = \sigma + j\omega $$
2. Region of Convergence (ROC) properties
   • Determined by $ \text{Re}(s) = \sigma $.
   • For right-sided signals, ROC is $ \text{Re}(s) > \sigma_{\text{max}} $.
   • For left-sided signals, ROC is $ \text{Re}(s) < \sigma_{\text{min}} $.
   • For two-sided signals, ROC is a strip $ \sigma_1 < \text{Re}(s) < \sigma_2 $.
3. Partial Fraction Expansion (PFE)

   Real poles: $ H(s) = \sum_{i=1}^N \frac{r_i}{s - p_i} \leftrightarrow h(t) = \sum_{i=1}^N r_i e^{p_i t} u(t) $

   Conjugate complex poles: $ H(s) = \frac{A s + B}{(s + \alpha)^2 + \beta^2} \leftrightarrow h(t) = C e^{-\alpha t} \cos(\beta t + \theta) u(t) $

Laplace Transform Properties

V. Fourier Analysis of Continuous-Time Signals

1. Energy and Power Signal Spectrum
   • Energy spectral density: $ S_e(j\omega) = |X(j\omega)|^2 $, with total energy: $$ E_y = \frac{1}{2\pi} \int_{-\infty}^{\infty} |X(j\omega)|^2 d\omega $$
   • Power spectral density: $ S_p(j\omega) = \lim_{T \to \infty} \frac{|X_T(j\omega)|^2}{T} $, with average power: $$ P_y = \frac{1}{2\pi} \int_{-\infty}^{\infty} S_p(j\omega) d\omega $$
2. LTI System Transfer Function
   • Reciprocity: $ H(j\omega) H(-j\omega) = 1 $
   • Causality: ROC of rational $H(s)$ lies in the right-half plane $\text{Re}(s) > \sigma_0$.
   • Stability: BIBO stable if ROC contains the $j\omega$ axis ($\text{Re}(s) = 0$).

Fourier Transform of Periodic Signals

If $f(t)$ is periodic with period $T$, then we can write it using the Fourier series representation:

$$ f(t) = \sum_{n=-\infty}^{\infty} F_n e^{j n \omega_0 t}, \quad \omega_0 = \frac{2\pi}{T} $$

Evaluating its Fourier Transform yields:

$$ F(\omega) = 2\pi \sum_{n=-\infty}^{\infty} F_n \delta(\omega - n \omega_0) $$

where the coefficients $F_n$ are given by:

$$ F_n = \frac{1}{T} \int_{-T/2}^{T/2} f(t) e^{-j n \omega_0 t} dt $$

For periodic power signals, the autocorrelation spectral density satisfies:

$$ R_f(\tau) = \sum_{n=-\infty}^{\infty} |F_n|^2 e^{j n \omega_0 \tau} \leftrightarrow S_f(\omega) = 2\pi \sum_{n=-\infty}^{\infty} |F_n|^2 \delta(\omega - n \omega_0) $$

VI. Discrete-Time LTI System Response

1. Time Domain Representation $$ x[n] = \sum_{m=-\infty}^{\infty} x[m] \delta[n-m] $$
2. Response Decomposition

   Total response is decomposed into the homogeneous solution (impulse response) and the particular solution (forced response):

   $$ y[n] = y_h[n] + y_p[n] $$

VII. The Hilbert Transform

1. Continuous-Time Definition $$ \hat{x}(t) = \mathcal{H}\{x(t)\} = x(t) * \frac{1}{\pi t} = \frac{1}{\pi} \int_{-\infty}^{\infty} \frac{x(\tau)}{t-\tau} d\tau $$

   In the frequency domain, the transfer function is:

   $$ H(j\omega) = -j \text{sgn}(\omega) $$
2. Properties

   An analytic signal is defined as: $ z(t) = x(t) + j\hat{x}(t) $. Its spectrum is double-sided, containing only positive frequencies:

   $$ Z(j\omega) = \begin{cases} 2 X(j\omega), & \omega > 0 \\ X(j\omega), & \omega = 0 \\ 0, & \omega < 0 \end{cases} $$

VIII. The z-Transform

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

   ROC: the set of values $z$ for which $ \sum_{n=-\infty}^{\infty} |x[n] z^{-n}| < \infty $ converges.

   • Right-sided signal: ROC is $|z| > r_0$
   • Left-sided signal: ROC is $|z| < r_0$
   • Two-sided signal: ROC is an annular region $r_i < |z| < r_o$
2. Rational Transfer Functions & Pole locations

   For causal real-pole and complex-pole systems:

   $$ H(z) = \sum_{i=1}^P \frac{A_i}{1 - p_i z^{-1}} \leftrightarrow h[n] = \sum_{i=1}^P A_i p_i^n u[n] $$
3. System Stability
   • Stable: ROC of $H(z)$ contains the unit circle $|z| = 1$.
   • Causal: ROC is the exterior of a circle $|z| > r_{\text{max}}$.

z-Transform Properties

1. Line multiplication: $ n^m x[n] \leftrightarrow \left(-z \frac{d}{dz}\right)^m X(z) $
2. Time Shifting: $ x[n - n_0] \leftrightarrow z^{-n_0} X(z) $
3. Frequency Scaling (Rotation): $ a^n x[n] \leftrightarrow X(a^{-1} z) $
4. Time Reversal: $ x[-n] \leftrightarrow X(z^{-1}) $
5. Conjugation: $ x^*[n] \leftrightarrow X^*(z^*) $
6. Convolution: $ x_1[n] * x_2[n] \leftrightarrow X_1(z) X_2(z) $
7. Initial Value Theorem: If $x[n]$ is causal, $ x[0] = \lim_{z\to\infty} X(z) $
8. Final Value Theorem: If $x[n]$ is causal and stable, $ x[\infty] = \lim_{z\to 1} (z-1) X(z) $

Single-sided z-Transform properties:

• Time Shifting (Advance): $ x[n+m] u[n] \leftrightarrow z^m \left[X(z) - \sum_{k=0}^{m-1} x[k] z^{-k}\right] $
• Time Shifting (Delay): $ x[n-m] u[n] \leftrightarrow z^{-m} \left[X(z) + \sum_{k=-m}^{-1} x[k] z^{-k}\right] $

IX. Discrete-Time Fourier Transform (DTFT)

1. Definition $$ X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n] e^{-j n \omega} $$
2. Relation to z-Transform

   If the ROC of $X(z)$ contains the unit circle ($|z|=1$), the DTFT is equivalent to evaluating the z-transform on the unit circle:

   $$ X(e^{j\omega}) = X(z)\Big|_{z=e^{j\omega}} $$
3. LTI System output DTFT $$ Y(e^{j\omega}) = X(e^{j\omega}) H(e^{j\omega}) $$

DTFT of Periodic Sequences

Given periodic sequence $x[n]$ with period $N$, its Discrete Fourier Series (DFS) representation is:

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

Its DTFT is represented as a train of impulse deltas in the frequency domain:

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

where the DFS/DFT coefficients $X[k]$ are:

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

For a finite length sequence of length $N_0$, if we extend it periodically to $x_N[n]$ with period $N \ge N_0$, the DTFT samples are directly related to the DFT coefficients:

$$ X_N[k] = X(e^{j\omega})\Big|_{\omega = \frac{2\pi}{N} k} $$

X. Sampling in Time and Frequency

1. Time-Domain Sampling of Continuous Signals

   Continuous signal $f(t)$ sampled with interval $T$ gives $x[n] = f(nT)$. In the frequency domain:

   $$ X(e^{j\omega}) = \frac{1}{T} \sum_{m=-\infty}^{\infty} F_a\left(j\left(\frac{\omega}{T} - m\frac{2\pi}{T}\right)\right) $$

   If the continuous spectrum is bandlimited to $\omega_m$ and the sampling frequency satisfies the Nyquist rate $\omega_s = \frac{2\pi}{T} > 2\omega_m$, reconstruction is possible using ideal interpolation:

   $$ f(t) = \sum_{n=-\infty}^{\infty} f(nT) \text{sinc}\left(\frac{t - nT}{T}\right) $$

XI. The Fast Fourier Transform (FFT)

1. Decimation-in-Time Base-2 FFT (DIT-FFT)

   Splitting the DFT sum into even and odd indices:

   $$ X[k] = \sum_{r=0}^{N/2-1} x[2r] W_{N/2}^{kr} + W_N^k \sum_{r=0}^{N/2-1} x[2r+1] W_{N/2}^{kr} $$ $$ X[k + N/2] = \sum_{r=0}^{N/2-1} x[2r] W_{N/2}^{kr} - W_N^k \sum_{r=0}^{N/2-1} x[2r+1] W_{N/2}^{kr} $$
2. Decimation-in-Frequency Base-2 FFT (DIF-FFT)

   Splitting the output sequence into even and odd bins:

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

XII. Decimation and Interpolation

1. Decimation (Downsampling by $M$)

   Anti-aliasing low-pass filtering followed by downsampling:

   $$ y[n] = v[nM] \leftrightarrow Y(e^{j\omega}) = \frac{1}{M} \sum_{i=0}^{M-1} V\left(e^{j\frac{\omega - 2\pi i}{M}}\right) $$
2. Interpolation (Upsampling by $L$)

   Upsampling by inserting $L-1$ zeros followed by interpolation low-pass filtering:

   $$ w[n] = \begin{cases} x[n/L], & n = 0, \pm L, \pm 2L, \dots \\ 0, & \text{otherwise} \end{cases} \leftrightarrow W(e^{j\omega}) = X(e^{j\omega L}) $$

XIII. Filter Structures and Lossless Network Synthesis

1. Lossless Double-Terminated Networks

   For a continuous network with source resistance $R_s$ and load resistance $R_L$, the rated power of the source is $P_{\text{max}} = \frac{|E_s|^2}{4 R_s}$. The power transmission coefficient is:

   $$ |H(j\Omega)|^2 = \frac{P_L}{P_{\text{max}}} = 4 \frac{R_s}{R_L} \left| \frac{V_o(j\Omega)}{E_s} \right|^2 $$

   The reflection coefficient function satisfies:

   $$ |\rho(j\Omega)|^2 = 1 - |H(j\Omega)|^2 $$
2. Digital Filter Realization (Difference Equations)

   Direct realization of IIR systems:

   $$ H(z) = \frac{\sum_{k=0}^M b_k z^{-k}}{1 + \sum_{k=1}^N a_k z^{-k}} \leftrightarrow y[n] = \sum_{k=0}^M b_k x[n-k] - \sum_{k=1}^N a_k y[n-k] $$

Butterworth Filter Design

1. Magnitude squared frequency response $$ |H_a(j\Omega)|^2 = \frac{1}{1 + \left(\frac{\Omega}{\Omega_c}\right)^{2N}} $$
2. Pole distributions $$ s_k = \Omega_c e^{j\left[\frac{\pi}{2} + \frac{2k-1}{2N}\pi\right]}, \quad k = 1, 2, \dots, 2N $$
3. Stable lowpass transfer function

   Selecting only LHP (left-half plane) poles to ensure stability:

   $$ H_a(s) = \frac{\Omega_c^N}{\prod_{k \in \text{LHP}} (s - s_k)} = \frac{\Omega_c^N}{B_N(s')} $$ where $s' = s/\Omega_c$ is the normalized Laplace variable, and $B_N(s')$ is the normalized Butterworth polynomial.

XIV. Infinite Impulse Response (IIR) Digital Filter Design

1. Impulse Invariance Method

   The continuous impulse response is sampled to form the discrete system: $ h[n] = T_d h_a(n T_d) $.

   $$ \frac{A}{s - s_0} \Rightarrow \frac{A T_d}{1 - e^{s_0 T_d} z^{-1}} $$

   Conclusion: Suffers from aliasing at high frequencies.

2. Bilinear Transform Method

   A mapping that maps the continuous s-plane to the discrete z-plane without aliasing:

   $$ s = \frac{2}{T_d} \left( \frac{1 - z^{-1}}{1 + z^{-1}} \right) $$

   Frequency warping relation:

   $$ \Omega = \frac{2}{T_d} \tan\left(\frac{\omega}{2}\right) $$

   Conclusion: Free of aliasing, but exhibits non-linear frequency warping distortion.

XV. Finite Word Length Effects in Digital Filters

1. Quantization Error Models
   • Rounding (uniform on $[-q/2, q/2]$): Mean $\mu_e = 0$, Variance $\sigma_e^2 = \frac{q^2}{12}$
   • Truncation (uniform on $[-q, 0]$): Mean $\mu_e = -q/2$, Variance $\sigma_e^2 = \frac{q^2}{12}$
2. Output Noise Power

   For quantization noise passing through an LTI system with impulse response $h[n]$:

   $$ \sigma_{yo}^2 = \sigma_e^2 \sum_{n=-\infty}^{\infty} |h[n]|^2 = \sigma_e^2 \frac{1}{2\pi} \int_{-\pi}^{\pi} |H(e^{j\omega})|^2 d\omega $$
3. Scaling to Avoid Overflow

   To ensure node variables $w[n]$ do not exceed $1$, we apply a scaling factor $\beta$ to the input signal:

   $$ \beta \le \frac{1}{\sum_{n=-\infty}^{\infty} |f[n]|} \quad \text{($L_1$ bound)} $$

XVI. Linear Phase FIR Filter Design

1. Symmetry and Anti-symmetry Conditions

   Symmetric (Type 1 & 2): $ h[n] = h[N - 1 - n] $

   Anti-symmetric (Type 3 & 4): $ h[n] = -h[N - 1 - n] $

2. Frequency Response $H(e^{j\omega})$ Types
   • Type 1 (Symmetric, $N$ odd): $ H(e^{j\omega}) = e^{-j\omega(N-1)/2} \sum_{n=0}^{(N-1)/2} a[n] \cos(\omega n) $
   • Type 2 (Symmetric, $N$ even): $ H(e^{j\omega}) = e^{-j\omega(N-1)/2} \sum_{n=1}^{N/2} b[n] \cos\left(\omega\left(n - \frac{1}{2}\right)\right) $
3. Frequency Transformation of Lowpass Digital Filters

   Lowpass mapping function for the unit circle:

   $$ Z^{-1} = G(z^{-1}) = \pm \frac{z^{-1} - \alpha}{1 - \alpha z^{-1}} $$

XVII. State-Space Representation of Systems

1. Continuous-Time State-Space Equations $$ \dot{\bm{x}}(t) = \bm{A} \bm{x}(t) + \bm{B} \bm{u}(t) $$ $$ \bm{y}(t) = \bm{C} \bm{x}(t) + \bm{D} \bm{u}(t) $$
   • Time domain solution: $$ \bm{x}(t) = e^{\bm{A}t} \bm{x}(0) + \int_{0}^t e^{\bm{A}(t-\tau)} \bm{B} \bm{u}(\tau) d\tau $$
   • s-domain solution: $$ \bm{X}(s) = (s\bm{I} - \bm{A})^{-1} \bm{x}(0) + (s\bm{I} - \bm{A})^{-1} \bm{B} \bm{U}(s) $$ $$ \bm{Y}(s) = \bm{C}(s\bm{I} - \bm{A})^{-1} \bm{x}(0) + [\bm{C}(s\bm{I} - \bm{A})^{-1} \bm{B} + \bm{D}] \bm{U}(s) $$
2. Discrete-Time State-Space Equations $$ \bm{x}[n+1] = \bm{A} \bm{x}[n] + \bm{B} \bm{u}[n] $$ $$ \bm{y}[n] = \bm{C} \bm{x}[n] + \bm{D} \bm{u}[n] $$
   • Time domain solution: $$ \bm{x}[n] = \bm{A}^n \bm{x}[0] + \sum_{k=0}^{n-1} \bm{A}^{n-1-k} \bm{B} \bm{u}[k] $$
   • z-domain solution: $$ \bm{X}(z) = z(z\bm{I} - \bm{A})^{-1} \bm{x}[0] + (z\bm{I} - \bm{A})^{-1} \bm{B} \bm{U}(z) $$ $$ \bm{Y}(z) = z\bm{C}(z\bm{I} - \bm{A})^{-1} \bm{x}[0] + [\bm{C}(z\bm{I} - \bm{A})^{-1} \bm{B} + \bm{D}] \bm{U}(z) $$

State-Space Linear Coordinate Transformation

1. Coordinate Mapping

   Let $ \bm{x}' = \bm{P} \bm{x} $, where $ \bm{P} $ is a non-singular transformation matrix. The transformed matrices are:

   $$ \bm{A}' = \bm{P}\bm{A}\bm{P}^{-1}, \quad \bm{B}' = \bm{P}\bm{B}, \quad \bm{C}' = \bm{C}\bm{P}^{-1}, \quad \bm{D}' = \bm{D} $$
2. Controllability and Observability Criteria
   • An $N$-th order system is controllable if the controllability matrix $ \bm{Q}_c $ has rank $N$: $$ \bm{Q}_c = [\bm{B}, \bm{A}\bm{B}, \bm{A}^2\bm{B}, \dots, \bm{A}^{N-1}\bm{B}] $$
   • An $N$-th order system is observable if the observability matrix $ \bm{Q}_o $ has rank $N$: $$ \bm{Q}_o = \begin{bmatrix} \bm{C} \\ \bm{C}\bm{A} \\ \bm{C}\bm{A}^2 \\ \vdots \\ \bm{C}\bm{A}^{N-1} \end{bmatrix} $$