Signals & Systems (EC-402) - Unit 5 Short Notes

1.0 SIGNAL CLASSIFICATION & FUNDAMENTAL PROPERTIES

1.1 Continuous-Time (CT) vs. Discrete-Time (DT) Signals

• CT Signal: Defined for every instant of time, represented as $x(t)$. Example: Speech waveform.

• DT Signal: Defined only at discrete instants $nT$ (sampling interval $T$), represented as $x[n]$. Example: Daily stock average.

• Key Difference: CT signals exist in continuous time domain, DT signals are sequences (functions of integer index $n$).

1.2 Basic Signal Operations & Standard Signals

• Unit Step: $$\displaystyle u(t) = \begin{cases} 1, & t \geq 0 \\ 0, & t < 0 \end{cases} $$; $$\displaystyle u[n] = \begin{cases} 1, & n \geq 0 \\ 0, & n < 0 \end{cases} $$

• Unit Ramp: $$\displaystyle r(t) = t\,u(t) $$; $$\displaystyle r[n] = n\,u[n] $$

• Unit Impulse: $\delta(t)$ (Dirac delta, sifting property); $$\displaystyle \delta[n] = \begin{cases} 1, & n=0 \\ 0, & n \neq 0 \end{cases} $$ (Kronecker delta).

• Exponential: $$\displaystyle x(t)=e^{at} $$ (CT), $$\displaystyle x[n]=a^n $$ (DT).

• Sinusoidal: $$\displaystyle x(t)=A\cos(\omega_0 t + \phi) $$, $$\displaystyle x[n]=A\cos(\Omega_0 n + \phi) $$.

[!TIP] Unit Step vs. Unit Ramp: Ramp is the integral of step: $$\displaystyle r(t)=\int_{-\infty}^{t} u(\tau)d\tau $$. Step is the derivative of ramp: $$\displaystyle u(t)=\frac{d}{dt}r(t) $$ (in distribution sense for CT).

1.3 Periodic vs. Aperiodic Signals

• CT Periodic: $$\displaystyle x(t) = x(t + mT_0) $$ for all $t$, $m \in \mathbb{Z}$. Fundamental period $$\displaystyle T_0 $$ is smallest positive $T$.

• DT Periodic: $$\displaystyle x[n] = x[n + mN_0] $$ for all $n$, $m \in \mathbb{Z}$. Fundamental period $$\displaystyle N_0 $$ is smallest positive integer $N$.

• Periodicity Condition:

  • CT: $$\displaystyle x(t)=A\cos(\omega_0 t + \theta) $$ is periodic iff $$\displaystyle \omega_0/2\pi $$ is rational.

  • DT: $$\displaystyle x[n]=A\cos(\Omega_0 n + \theta) $$ is periodic iff $$\displaystyle \Omega_0/2\pi $$ is rational.

  • Sum of periodic signals: Periodic if ratio of individual periods is rational.

[!TIP] Common Pitfall: $$\displaystyle x(t)=\cos(t)+\cos(\pi t) $$ is aperiodic because $1/\pi$ is irrational. $$\displaystyle x[n]=\cos(0.1\pi n) $$ is periodic with $$\displaystyle N_0=20 $$ since $$\displaystyle 0.1\pi / 2\pi = 1/20 $$ is rational.

1.4 Deterministic vs. Random Signals

• Deterministic: Completely specified for all $t$ (or $n$). Can be described by an explicit mathematical expression. Example: $$\displaystyle x(t)=5\cos(2\pi t) $$.

• Random: Described only probabilistically (e.g., mean, variance). Cannot be predicted exactly. Example: Thermal noise voltage.

[!TIP] Key Difference: Deterministic signals are predictable and repeatable; random signals are unpredictable and require statistical description.

1.5 Even, Odd, and Neither Symmetry

• Even: $$\displaystyle x(t) = x(-t) $$ (symmetric about y-axis). Example: $$\displaystyle \cos(\omega_0 t) $$.

• Odd: $$\displaystyle x(t) = -x(-t) $$ (symmetric about origin). Example: $$\displaystyle \sin(\omega_0 t) $$.

• Any signal: $$\displaystyle x(t) = x_e(t) + x_o(t) $$, where $$\displaystyle x_e(t)=\frac{1}{2}[x(t)+x(-t)] $$, $$\displaystyle x_o(t)=\frac{1}{2}[x(t)-x(-t)] $$.

1.6 Energy & Power Signals

• CT Energy: $$\displaystyle E = \int_{-\infty}^{\infty} |x(t)|^2 dt $$

• CT Power: $$\displaystyle P = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} |x(t)|^2 dt $$

• DT Energy: $$\displaystyle E = \sum_{n=-\infty}^{\infty} |x[n]|^2 $$

• DT Power: $$\displaystyle P = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^{N} |x[n]|^2 $$

• Classification:

  • Energy Signal: $$\displaystyle 0 < E < \infty $$, $$\displaystyle P = 0 $$.

  • Power Signal: $$\displaystyle 0 < P < \infty $$, $$\displaystyle E = \infty $$.

  • Neither: $$\displaystyle E = \infty $$, $$\displaystyle P = \infty $$ or $$\displaystyle E = 0 $$, $$\displaystyle P = 0 $$ (null signal).

[!TIP] How to Determine: Compute $E$ and $P$. If $E$ finite → energy. If $E$ infinite but $P$ finite → power. If both infinite → neither.

2.0 SYSTEM ANALYSIS & CLASSIFICATION

2.1 System Definition

A system is a transformation that maps an input signal $x(t)$ (or $x[n]$) to an output signal $y(t)$ (or $y[n]$), denoted $$\displaystyle y = T\{x\} $$.

2.2 Linearity (Additivity & Homogeneity)

• Definition: System is linear if it satisfies superposition principle:

  $$\displaystyle T\{a_1x_1(t) + a_2x_2(t)\} = a_1T\{x_1(t)\} + a_2T\{x_2(t)\} $$ for any scalars $$\displaystyle a_1,a_2 $$.

• Test: Check if output is a linear combination of inputs. No powers, products, or nonlinear functions of input.

• Example: $$\displaystyle y(t)=3x(t) $$ is linear; $$\displaystyle y(t)=x^2(t) $$ is non-linear.

2.3 Time-Invariance (TI) vs. Time-Variance (TV)

• TI: A time shift in input causes identical time shift in output.

  $$\displaystyle x(t-t_0) \longrightarrow y(t-t_0) $$.

• Test: Replace $t$ by $$\displaystyle t-t_0 $$ in system equation. If output becomes $$\displaystyle y(t-t_0) $$, system is TI.

• Example: $$\displaystyle y(t)=x(2t) $$ is TV (time scaling); $$\displaystyle y(t)=t\,x(t) $$ is TV (explicit time dependence).

2.4 Causality

• Definition: Output at any time depends only on present and past values of input.

  • CT: $$\displaystyle y(t_0) $$ depends on $x(\tau)$ for $$\displaystyle \tau \leq t_0 $$.

  • DT: $$\displaystyle y[n_0] $$ depends on $x[k]$ for $$\displaystyle k \leq n_0 $$.

• Causal System: $$\displaystyle h(t)=0 $$ for $$\displaystyle t<0 $$ (CT), $$\displaystyle h[n]=0 $$ for $$\displaystyle n<0 $$ (DT).

• Non-Causal: Output depends on future inputs (e.g., $$\displaystyle y(t)=x(t+1) $$).

• Anti-Causal: Output depends only on future inputs (e.g., $$\displaystyle y(t)=x(t+1) $$ for all $t$).

2.5 Stability (BIBO Stability)

• Definition: Bounded input $\Rightarrow$ bounded output.

  • CT: $$\displaystyle |x(t)| \leq M_x < \infty $$ for all $t$ $\Rightarrow$ $$\displaystyle |y(t)| \leq M_y < \infty $$.

  • DT: $$\displaystyle |x[n]| \leq M_x < \infty $$ for all $n$ $\Rightarrow$ $$\displaystyle |y[n]| \leq M_y < \infty $$.

• Test for LTI Systems:

  • CT: $$\displaystyle \int_{-\infty}^{\infty} |h(t)| dt < \infty $$ (absolute integrability).

  • DT: $$\displaystyle \sum_{n=-\infty}^{\infty} |h[n]| < \infty $$ (absolute summability).

2.6 Memoryless vs. Dynamic Systems

• Memoryless: Output at any time depends only on input at that same time. $$\displaystyle y(t)=g(x(t)) $$.

• Dynamic (With Memory): Output depends on past/future inputs (e.g., systems with derivatives, integrals, or delays).

2.7 Invertibility

• A system is invertible if its output uniquely determines its input.

• For LTI systems, invertible iff $H(z) \neq 0$ for all $z$ in ROC (DT) or $H(j\omega) \neq 0$ for all $\omega$ (CT).

2.8 Comprehensive System Analysis (From Equation)

Given a system described by a differential (CT) or difference (DT) equation:

1. Linearity: Check if equation is linear in $x(t)$ and $y(t)$ (constant coefficients, no powers/products of $x$ or $y$).

2. Time-Invariance: Check for explicit time dependence (e.g., coefficients that are functions of $t$ or $n$).

3. Causality: Check if $$\displaystyle y(t_0) $$ or $$\displaystyle y[n_0] $$ depends on $x(\tau)$ for $$\displaystyle \tau > t_0 $$ or $$\displaystyle k > n_0 $$.

4. Stability: For LTI, find impulse response $h(t)$ or $h[n]$ and test absolute integrability/summability.

[!TIP] Example Analysis: $$\displaystyle y(t) = \frac{d}{dt}x(t) $$

• Linear: Yes (derivative is linear operation).

• TI: Yes (no explicit $t$).

• Causal: Yes (derivative at $t$ depends on $x$ in an infinitesimal neighborhood around $t$, effectively present/past).

• Stable: No. For $$\displaystyle x(t)=u(t) $$, $$\displaystyle y(t)=\delta(t) $$ (bounded? $\delta(t)$ is unbounded, but BIBO test: input $$\displaystyle x(t)=u(t) $$ is bounded, output $$\displaystyle y(t)=\delta(t) $$ is unbounded in magnitude? Actually, $\delta(t)$ is not a bounded function; it's a distribution. More rigorously, use impulse response: $$\displaystyle h(t)=\frac{d}{dt}\delta(t) $$, and $$\displaystyle \int |h(t)| dt $$ does not exist. So unstable.

3.0 LTI SYSTEMS: IMPULSE RESPONSE & CONVOLUTION

3.1 LTI System Definition & Importance

• LTI: Linear and Time-Invariant.

• Impulse Response $h(t)$ (CT) / $h[n]$ (DT): Output when input is $\delta(t)$ / $\delta[n]$. Completely characterizes an LTI system.

• Importance: Any input $x(t)$ can be expressed as superposition of scaled/delayed impulses: $$\displaystyle x(t) = \int_{-\infty}^{\infty} x(\tau)\delta(t-\tau) d\tau $$. By linearity and TI, output $$\displaystyle y(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau) d\tau = x(t) * h(t) $$.

3.2 Convolution Integral (CT) & Convolution Sum (DT)

• CT Convolution: $$\displaystyle y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau) h(t-\tau) d\tau $$

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

• Graphical Method: Flip $h(\tau)$ to $h(-\tau)$, shift by $t$ (or $n$), multiply with $x(\tau)$, integrate/sum.

• Example: $$\displaystyle x(t)=u(t-1)-u(t+1) $$ (rect from -1 to 1), $$\displaystyle h(t)=e^{-at}u(t) $$.

  $$\displaystyle y(t) = \int_{-\infty}^{\infty} [u(\tau-1)-u(\tau+1)] e^{-a(t-\tau)}u(t-\tau) d\tau $$.

  For $$\displaystyle t < -1 $$: $$\displaystyle y(t)=0 $$.

  For $$\displaystyle -1 \leq t < 1 $$: $$\displaystyle y(t)= \int_{-1}^{t} e^{-a(t-\tau)} d\tau = \frac{1}{a}(1-e^{-a(t+1)}) $$.

  For $t \geq 1$: $$\displaystyle y(t)= \int_{-1}^{1} e^{-a(t-\tau)} d\tau = \frac{1}{a}(e^{a}-e^{-a})e^{-at} $$.

3.3 Properties of Convolution

• Commutative: $$\displaystyle x(t)*h(t) = h(t)*x(t) $$.

• Associative: $$\displaystyle (x*h_1)*h_2 = x*(h_1*h_2) $$.

• Distributive: $$\displaystyle x*(h_1+h_2) = x*h_1 + x*h_2 $$.

• With Delta: $$\displaystyle x(t)*\delta(t) = x(t) $$.

• With Shifted Delta: $$\displaystyle x(t)*\delta(t-t_0) = x(t-t_0) $$.

• Scaling: $$\displaystyle x(at)*h(at) = \frac{1}{|a|} [x*h](at) $$ (CT).

3.4 System Response using Convolution

Given $x(t)$ and $h(t)$, compute $$\displaystyle y(t)=x(t)*h(t) $$ via integration (or summation for DT).

3.5 Stability & Causality from Impulse Response

• Causality: $$\displaystyle h(t)=0 $$ for $$\displaystyle t<0 $$ (CT), $$\displaystyle h[n]=0 $$ for $$\displaystyle n<0 $$ (DT).

• Stability (BIBO): $$\displaystyle \int_{-\infty}^{\infty} |h(t)| dt < \infty $$ (CT), $$\displaystyle \sum_{n=-\infty}^{\infty} |h[n]| < \infty $$ (DT).

[!TIP] Quick Check: For $$\displaystyle h(t)=e^{-at}u(t) $$ ($$\displaystyle a>0 $$): Causal (yes, $$\displaystyle h(t)=0 $$ for $$\displaystyle t<0 $$), Stable: $$\displaystyle \int_0^\infty e^{-at} dt = 1/a < \infty $$ → stable.

4.0 Z-TRANSFORM & ANALYSIS

4.1 Definition & Region of Convergence (ROC)

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

• ROC: Set of $z$ in complex plane for which the sum converges. Never includes poles.

• Properties of ROC:

  1. ROC is a ring/annulus (or exterior/interior of circle) centered at origin.

  2. For right-sided sequences ($$\displaystyle n \geq n_1 $$), ROC is exterior of outermost pole: $$\displaystyle |z| > r_{\max} $$.

  3. For left-sided sequences ($$\displaystyle n \leq n_2 $$), ROC is interior of innermost pole: $$\displaystyle |z| < r_{\min} $$.

  4. For two-sided sequences, ROC is annular region between poles: $$\displaystyle r_{\min} < |z| < r_{\max} $$.

  5. For finite-duration sequences, ROC is entire $z$-plane except possibly $$\displaystyle z=0 $$ and/or $$\displaystyle z=\infty $$.

  6. ROC must be a connected region.

  7. ROC of $x[n]$ and $x[-n]$ are reciprocals.

4.2 Z-Transform Properties (State & Prove Frequently)

[!TIP] FVT Condition: Poles of $(z-1)X(z)$ must lie strictly inside unit circle $$\displaystyle |z|<1 $$ for limit to exist.

4.3 Inverse Z-Transform

• Methods:

  1. Partial Fraction Expansion (PFE): For rational $X(z)$. Expand into simpler terms, then use standard pairs.

  2. Power Series Expansion: Expand $X(z)$ as $$\displaystyle \sum_{n=-\infty}^{\infty} x[n] z^{-n} $$ (coefficients are $x[n]$).

  3. Contour Integration (Residue): $$\displaystyle x[n] = \frac{1}{2\pi j} \oint_C X(z) z^{n-1} dz $$.

• Example: $$\displaystyle X(z) = \frac{1}{(1-az^{-1})^2} $$, ROC $$\displaystyle |z|>|a| $$.

  PFE: $$\displaystyle X(z) = \frac{a}{(1-az^{-1})^2} \cdot \frac{1}{a} $$? Better: $$\displaystyle X(z) = \frac{z^2}{(z-a)^2} $$. Inverse: $$\displaystyle x[n] = n a^{n-1} u[n-1] $$? Actually, standard: $$\displaystyle \frac{1}{(1-az^{-1})^2} \leftrightarrow n a^{n} u[n] $$ for ROC $$\displaystyle |z|>|a| $$. So $$\displaystyle x[n] = n a^{n} u[n] $$.

4.4 System Function & Difference Equations

• System Function: $$\displaystyle H(z) = \frac{Y(z)}{X(z)} $$ for LTI DT system (assuming zero initial conditions).

• From Difference Equation: Take Z-transform (using time-shift property), solve for $H(z)$.

  • Example: $$\displaystyle y[n] - \frac{1}{4}y[n-1] - \frac{3}{8}y[n-2] = -x[n] + 3x[n-1] $$

  $$\displaystyle \Rightarrow Y(z) - \frac{1}{4}z^{-1}Y(z) - \frac{3}{8}z^{-2}Y(z) = -X(z) + 3z^{-1}X(z) $$

  $$\displaystyle \Rightarrow H(z) = \frac{Y(z)}{X(z)} = \frac{-1 + 3z^{-1}}{1 - \frac{1}{4}z^{-1} - \frac{3}{8}z^{-2}} $$.

• From $h[n]$: $$\displaystyle H(z) = \sum_{n=-\infty}^{\infty} h[n] z^{-n} $$.

4.5 Unilateral Z-Transform

• Definition: $$\displaystyle X_u(z) = \sum_{n=0}^{\infty} x[n] z^{-n} $$ (sum from $$\displaystyle n=0 $$ to $\infty$).

• Key Difference from Bilateral: Ignores $$\displaystyle n<0 $$ part. Used for solving difference equations with non-zero initial conditions.

• Property: For $n \geq 0$, $$\displaystyle x[n] \leftrightarrow X_u(z) $$ with ROC $$\displaystyle |z|>r_{\max} $$ (for causal part).

• Application: Take unilateral Z-transform of difference equation, include initial conditions as extra terms, solve for $$\displaystyle Y_u(z) $$, then inverse.

[!TIP] Unilateral vs. Bilateral: Bilateral uses full sequence and ROC depends on both sides. Unilateral is like bilateral but with $$\displaystyle x[n]=0 $$ for $$\displaystyle n<0 $$, so ROC is always exterior of outermost pole (if causal).

5.0 FOURIER ANALYSIS: SERIES & TRANSFORM

5.1 Continuous-Time Fourier Series (CTFS)

• For periodic $x(t)$ with period $$\displaystyle T_0 $$:

  $$\displaystyle x(t) = \sum_{k=-\infty}^{\infty} X_k e^{jk\omega_0 t} $$, where $$\displaystyle \omega_0 = 2\pi/T_0 $$.

• Fourier Coefficients (Exponential Form):

  $$\displaystyle X_k = \frac{1}{T_0} \int_{T_0} x(t) e^{-jk\omega_0 t} dt $$.

• Example: $$\displaystyle x(t)=\cos^2 t = \frac{1}{2} + \frac{1}{2}\cos 2t = \frac{1}{2} + \frac{1}{4}e^{j2t} + \frac{1}{4}e^{-j2t} $$.

  So $$\displaystyle X_0 = 1/2 $$, $$\displaystyle X_{\pm1} = 1/4 $$, all other $$\displaystyle X_k=0 $$.

• Magnitude & Phase Spectra: Plot $$\displaystyle |X_k| $$ vs $$\displaystyle k\omega_0 $$ and $$\displaystyle \angle X_k $$ vs $$\displaystyle k\omega_0 $$.

5.2 Continuous-Time Fourier Transform (CTFT)

• Definition (Aperiodic): $$\displaystyle X(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt $$.

• Inverse: $$\displaystyle x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(j\omega) e^{j\omega t} d\omega $$.

• Standard Transforms:

  • $$\displaystyle u(t) \leftrightarrow \pi\delta(\omega) + \frac{1}{j\omega} $$

  • $$\displaystyle e^{-at}u(t), a>0 \leftrightarrow \frac{1}{a+j\omega} $$

  • $$\displaystyle \cos(\omega_0 t) \leftrightarrow \pi[\delta(\omega-\omega_0)+\delta(\omega+\omega_0)] $$

  • $$\displaystyle \text{rect}(t/T) \leftrightarrow T\text{sinc}(\omega T/2) $$

• Key Properties (State & Prove Frequently):

  • Linearity: $$\displaystyle a_1x_1(t)+a_2x_2(t) \leftrightarrow a_1X_1(j\omega)+a_2X_2(j\omega) $$.

  • Time Shifting: $$\displaystyle x(t-t_0) \leftrightarrow e^{-j\omega t_0}X(j\omega) $$.

  • Frequency Shifting: $$\displaystyle e^{j\omega_0 t}x(t) \leftrightarrow X(j(\omega-\omega_0)) $$.

  • Time Scaling: $$\displaystyle x(at) \leftrightarrow \frac{1}{|a|}X(j\omega/a) $$.

  • Convolution: $$\displaystyle x_1(t)*x_2(t) \leftrightarrow X_1(j\omega)X_2(j\omega) $$.

  • Multiplication: $$\displaystyle x_1(t)x_2(t) \leftrightarrow \frac{1}{2\pi}[X_1(j\omega)*X_2(j\omega)] $$.

  • Duality: If $$\displaystyle x(t) \leftrightarrow X(j\omega) $$, then $$\displaystyle X(t) \leftrightarrow 2\pi x(-\omega) $$.

  • Differentiation in Time: $$\displaystyle \frac{d^n x(t)}{dt^n} \leftrightarrow (j\omega)^n X(j\omega) $$.

  • Integration in Time: $$\displaystyle \int_{-\infty}^{t} x(\tau) d\tau \leftrightarrow \frac{X(j\omega)}{j\omega} + \pi X(0)\delta(\omega) $$.

[!TIP] Applying Differentiation Property: To find FT of triangular pulse (which is integral of rectangular), start with FT of rect, then use integration property (inverse of differentiation). For triangular pulse $x(t)$ of width $T$ and height 1, $$\displaystyle x(t) = \int_{-\infty}^{t} \text{rect}(\tau/T) d\tau $$? Actually, triangular is convolution of two rects, so FT is square of sinc. Alternatively, differentiate triangular to get rectangular pulses, use differentiation property, then divide by $$\displaystyle (j\omega)^2 $$.

5.3 Discrete-Time Fourier Series (DTFS)

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

  $$\displaystyle x[n] = \sum_{k=0}^{N-1} X_k e^{jk(2\pi/N)n} $$.

• Coefficients: $$\displaystyle X_k = \frac{1}{N} \sum_{n=0}^{N-1} x[n] e^{-jk(2\pi/N)n} $$.

• Properties: Similar to CTFS but discrete frequencies $$\displaystyle \Omega_k = 2\pi k/N $$. $$\displaystyle X_k $$ are periodic with period $N$.

5.4 Discrete-Time Fourier Transform (DTFT)

• Definition: $$\displaystyle X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n} $$.

• Inverse: $$\displaystyle x[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(e^{j\omega}) e^{j\omega n} d\omega $$ (integration over one period $2\pi$).

• Periodicity: $$\displaystyle X(e^{j(\omega+2\pi)}) = X(e^{j\omega}) $$.

• Convergence: Exists if $x[n]$ is absolutely summable: $$\displaystyle \sum_{n=-\infty}^{\infty} |x[n]| < \infty $$. Can also exist for some signals not absolutely summable (e.g., $$\displaystyle x[n]=u[n] $$) in the sense of mean-square convergence.

• Properties: Similar to CTFT but with periodicity in frequency.

  • Linearity, Time Shifting: $$\displaystyle x[n-n_0] \leftrightarrow e^{-j\omega n_0}X(e^{j\omega}) $$.

  • Frequency Shifting: $$\displaystyle e^{j\omega_0 n}x[n] \leftrightarrow X(e^{j(\omega-\omega_0)}) $$.

  • Time Reversal: $$\displaystyle x[-n] \leftrightarrow X(e^{-j\omega}) $$.

  • Convolution: $$\displaystyle x_1[n]*x_2[n] \leftrightarrow X_1(e^{j\omega})X_2(e^{j\omega}) $$.

  • Multiplication: $$\displaystyle x_1[n]x_2[n] \leftrightarrow \frac{1}{2\pi} \int_{-\pi}^{\pi} X_1(e^{j\theta}) X_2(e^{j(\omega-\theta)}) d\theta $$.

• Applications:

  1. Frequency Response of LTI Systems: $$\displaystyle H(e^{j\omega}) $$ is DTFT of $h[n]$.

  2. Spectral Analysis: Analyze frequency content of DT signals.

  3. Filter Design: Design filters by specifying $$\displaystyle H(e^{j\omega}) $$.

  4. Solving Difference Equations: Use DTFT to find steady-state response.

[!TIP] Convergence Note: For $$\displaystyle x[n]=a^n u[n] $$, $$\displaystyle |a|<1 $$, absolutely summable → DTFT exists: $$\displaystyle X(e^{j\omega}) = \frac{1}{1-ae^{-j\omega}} $$. For $$\displaystyle x[n]=u[n] $$, not absolutely summable, but DTFT exists as $$\displaystyle \frac{1}{1-e^{-j\omega}} + \pi\delta(\omega) $$ (in distribution sense).

6.0 STATE-SPACE ANALYSIS

6.1 State & State Variables

• State: Minimal set of variables $$\displaystyle x_1(t),...,x_n(t) $$ (or $$\displaystyle x_1[n],...,x_n[n] $$) that, together with input $u(t)$ (or $u[n]$), uniquely determine future output for all $$\displaystyle t \geq t_0 $$ (or $$\displaystyle n \geq n_0 $$).

• State Variables: The chosen variables that constitute the state.

6.2 State-Space Representation (SSR)

• State Equation: Describes evolution of state.

  • CT: $$\displaystyle \dot{\mathbf{x}}(t) = \mathbf{A}\mathbf{x}(t) + \mathbf{B}u(t) $$

  • DT: $$\displaystyle \mathbf{x}[n+1] = \mathbf{A}\mathbf{x}[n] + \mathbf{B}u[n] $$

• Output Equation: Relates output to state and input.

  • CT: $$\displaystyle y(t) = \mathbf{C}\mathbf{x}(t) + Du(t) $$

  • DT: $$\displaystyle y[n] = \mathbf{C}\mathbf{x}[n] + Du[n] $$

• Matrix Dimensions (SISO):

  • $\mathbf{x}$: $n \times 1$ state vector.

  • $\mathbf{A}$: $n \times n$ system matrix.

  • $\mathbf{B}$: $n \times 1$ input matrix.

  • $\mathbf{C}$: $1 \times n$ output matrix.

  • $D$: scalar (feedthrough).

• MIMO Systems: $u$ and $y$ become vectors; $\mathbf{B}$, $\mathbf{C}$, $D$ become matrices of appropriate size.

[!TIP] Example MIMO (3-input, 2-output): $$\displaystyle u = [u_1, u_2, u_3]^T $$ (3×1), $$\displaystyle y = [y_1, y_2]^T $$ (2×1). Then $\mathbf{B}$ is $n\times3$, $\mathbf{C}$ is $2\times n$, $D$ is $2\times3$.

6.3 State Transition Matrix $\Phi(t)$ / $\Phi[n]$

• Definition: Matrix that propagates state from initial time.

  • CT: $$\displaystyle \Phi(t) = e^{\mathbf{A}t} $$.

  • DT: $$\displaystyle \Phi[n] = \mathbf{A}^n $$.

• Properties:

  1. $$\displaystyle \Phi(0) = \mathbf{I} $$ (identity).

  2. $$\displaystyle \Phi(t_1 + t_2) = \Phi(t_1)\Phi(t_2) $$ (CT), $$\displaystyle \Phi[n_1+n_2] = \Phi[n_1]\Phi[n_2] $$ (DT).

  3. $$\displaystyle \Phi(-t) = \Phi^{-1}(t) $$ (CT), $$\displaystyle \Phi[-n] = \Phi^{-1}[n] $$ (DT) if invertible.

  4. $$\displaystyle \frac{d}{dt}\Phi(t) = \mathbf{A}\Phi(t) $$ (CT), $$\displaystyle \Phi[n+1] = \mathbf{A}\Phi[n] $$ (DT).

• Role in Solution:

  • CT: $$\displaystyle \mathbf{x}(t) = \Phi(t)\mathbf{x}(0) + \int_0^t \Phi(t-\tau)\mathbf{B}u(\tau) d\tau $$.

  • DT: $$\displaystyle \mathbf{x}[n] = \Phi[n]\mathbf{x}[0] + \sum_{k=0}^{n-1} \Phi[n-k-1]\mathbf{B}u[k] $$.

6.4 Methods to Determine State Transition Matrix

1. Laplace Transform (CT): $$\displaystyle \Phi(t) = \mathcal{L}^{-1}\{(s\mathbf{I} - \mathbf{A})^{-1}\} $$.

2. Cayley-Hamilton Theorem: Since $\mathbf{A}$ satisfies its own characteristic equation $$\displaystyle p(\mathbf{A})=0 $$, express $$\displaystyle e^{\mathbf{A}t} $$ as polynomial in $\mathbf{A}$: $$\displaystyle e^{\mathbf{A}t} = \alpha_0(t)\mathbf{I} + \alpha_1(t)\mathbf{A} + ... + \alpha_{n-1}(t)\mathbf{A}^{n-1} $$. Solve for $$\displaystyle \alpha_i(t) $$ using eigenvalues.

3. Series Expansion: $$\displaystyle e^{\mathbf{A}t} = \mathbf{I} + \mathbf{A}t + \frac{\mathbf{A}^2 t^2}{2!} + \frac{\mathbf{A}^3 t^3}{3!} + ... $$ (useful for diagonalizable $\mathbf{A}$).

[!TIP] Example: For $$\displaystyle \mathbf{A} = \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix} $$, find $\Phi(t)$ via Laplace: $$\displaystyle (s\mathbf{I}-\mathbf{A})^{-1} = \frac{1}{s^2+3s+2}\begin{bmatrix} s+3 & 1 \\ -2 & s \end{bmatrix} $$. Inverse Laplace gives $\Phi(t)$.

7.0 SAMPLING THEORY

7.1 Sampling Theorem (Shannon-Nyquist)

• Statement: A bandlimited signal $x(t)$ with no frequency components above $$\displaystyle f_m $$ Hz can be reconstructed perfectly from its samples taken at a rate $$\displaystyle f_s \geq 2f_m $$ samples/second.

• Nyquist Rate: $$\displaystyle f_N = 2f_m $$.

• Proof Sketch:

  1. Sampling: $$\displaystyle x_s(t) = x(t) \cdot \sum_{n=-\infty}^{\infty} \delta(t-nT) $$, $$\displaystyle T=1/f_s $$.

  2. Spectrum: $$\displaystyle X_s(j\Omega) = \frac{1}{T} \sum_{k=-\infty}^{\infty} X(j(\Omega - k\Omega_s)) $$, $$\displaystyle \Omega_s=2\pi f_s $$.

  3. If $$\displaystyle f_s \geq 2f_m $$, shifted spectra do not overlap.

  4. Ideal reconstruction: Pass $$\displaystyle x_s(t) $$ through ideal LPF with gain $T$ and cutoff $$\displaystyle \Omega_c = \pi/T $$ (i.e., $$\displaystyle f_c = f_s/2 $$). Output: $$\displaystyle x(t) = \sum_{n=-\infty}^{\infty} x(nT) \text{sinc}(t/T - n) $$.

7.2 Implications & Aliasing

• Aliasing: Occurs when $$\displaystyle f_s < 2f_m $$. High-frequency components fold back into lower frequencies, causing irreversible distortion.

• Effect: Original signal cannot be recovered from samples. High frequencies appear as low frequencies.

7.3 Practical Considerations

• Minimum Sampling Rate: $$\displaystyle f_{s,\min} = 2f_m $$ (for baseband). For bandpass signals, can be lower than $$\displaystyle 2f_m $$ under certain conditions.

• Maximum Sampling Interval: $$\displaystyle T_{\max} = 1/f_{s,\min} = 1/(2f_m) $$.

• Example: Spectral range 5.6 MHz to 6.8 MHz.

  • Highest frequency $$\displaystyle f_m = 6.8 $$ MHz.

  • Minimum sampling rate $$\displaystyle f_{s,\min} = 2 \times 6.8 = 13.6 $$ MHz.

  • Maximum sampling interval $$\displaystyle T_{\max} = 1/13.6 \approx 73.53 $$ ns.

[!TIP] Bandpass Note: If signal is strictly bandpass (no DC component), minimum $$\displaystyle f_s $$ can be as low as $2B$, where $B$ is bandwidth (6.8-5.6=1.2 MHz), so $$\displaystyle f_s \geq 2.4 $$ MHz, provided $$\displaystyle f_s > 2f_l $$? Actually, condition: $$\displaystyle f_s \geq 2B $$ and $$\displaystyle f_s \geq 2f_l $$? But standard Nyquist for bandpass: $$\displaystyle f_s \geq 2B $$ if the band is properly positioned. However, in many exam contexts, they assume baseband interpretation unless specified. Given the question says "spectral range extends from 5.6 MHz to 6.8 MHz", it's bandpass. But to be safe, state both: conservative $$\displaystyle f_s \geq 2f_{\max}=13.6 $$ MHz; if bandpass sampling applies, $$\displaystyle f_s \geq 2B = 2.4 $$ MHz, but must avoid aliasing of bands. Usually, exams expect $$\displaystyle f_s \geq 2f_{\max} $$.

7.4 Reconstruction (Interpolation)

• Ideal Reconstruction: Use ideal low-pass filter (sinc interpolation).

  $$\displaystyle x(t) = \sum_{n=-\infty}^{\infty} x(nT) \text{sinc}\left(\frac{t}{T} - n\right) $$, where $$\displaystyle \text{sinc}(x) = \frac{\sin(\pi x)}{\pi x} $$.

• Ideal Filter Impulse Response: $$\displaystyle h_r(t) = \frac{\sin(\pi t/T)}{\pi t/T} = \text{sinc}(t/T) $$ (scaled).

• Process: $$\displaystyle x(t) = x_s(t) * h_r(t) $$.

8.0 ADDITIONAL / INTEGRATIVE TOPICS

8.1 Block Diagram Representations of LTI DT Systems

• Direct Form I: Realizes $$\displaystyle H(z) = \frac{b_0 + b_1 z^{-1} + ... + b_M z^{-M}}{1 + a_1 z^{-1} + ... + a_N z^{-N}} $$ as cascade of feedforward (numerator) and feedback (denominator) sections. Uses $M+N$ delays.

• Direct Form II: Combines feedforward and feedback to share delays. Uses $\max(M,N)$ delays. More efficient.

• Cascade Form: Factor $H(z)$ into second-order (or first-order) sections: $$\displaystyle H(z) = H_1(z) H_2(z) ... $$. Each section implemented separately and cascaded. Good for modularity and sensitivity reduction.

• Parallel Form: Partial fraction expansion: $$\displaystyle H(z) = C + \sum_{k} \frac{r_k}{1 - p_k z^{-1}} $$. Each term is a first-order system in parallel.

[!TIP] Realization Example: For $$\displaystyle y[n] - 3y[n-1] + 2y[n-2] = x[n] $$, $$\displaystyle H(z) = \frac{1}{1-3z^{-1}+2z^{-2}} = \frac{1}{(1-z^{-1})(1-2z^{-1})} $$.

• Direct Form I: Two feedback delays, two feedforward paths.

• Direct Form II: Two shared delays.

• Parallel: $$\displaystyle H(z) = \frac{1}{1-z^{-1}} - \frac{1}{1-2z^{-1}} $$, so two parallel first-order sections.

8.2 Solving Differential/Difference Equations for LTI Systems

• CT (Differential): Take Laplace transform (zero initial conditions), solve for $Y(s)$, then $$\displaystyle y(t)=\mathcal{L}^{-1}\{Y(s)\} $$. Impulse response $$\displaystyle h(t)=\mathcal{L}^{-1}\{H(s)\} $$.

  • Example: $$\displaystyle \frac{d^2y}{dt^2}+6\frac{dy}{dt}+8y=2x(t) $$.

  $$\displaystyle s^2Y(s)+6sY(s)+8Y(s)=2X(s) \Rightarrow H(s)=\frac{2}{s^2+6s+8} = \frac{2}{(s+2)(s+4)} $$.

  Partial fractions: $$\displaystyle H(s)=\frac{1}{s+2} - \frac{1}{s+4} \Rightarrow h(t)=(e^{-2t}-e^{-4t})u(t) $$.

• DT (Difference): Take Z-transform (zero initial conditions), solve for $H(z)$, then inverse Z-transform.

  • Example: $$\displaystyle y[n]-\frac{1}{4}y[n-1]-\frac{3}{8}y[n-2]=-x[n]+3x[n-1] $$.

  $$\displaystyle Y(z)-\frac{1}{4}z^{-1}Y(z)-\frac{3}{8}z^{-2}Y(z) = -X(z)+3z^{-1}X(z) $$.

  $$\displaystyle H(z)=\frac{-1+3z^{-1}}{1-\frac{1}{4}z^{-1}-\frac{3}{8}z^{-2}} = \frac{-z+3}{z^2-\frac{1}{4}z-\frac{3}{8}} $$.

  Multiply numerator and denominator by $$\displaystyle z^2 $$: $$\displaystyle H(z)=\frac{-z^2+3z}{z^2-\frac{1}{4}z-\frac{3}{8}} $$. Partial fractions or long division to find $h[n]$.

8.3 Basic Operations on DT Signals

• Shifting: $$\displaystyle x[n-n_0] $$ (delay if $$\displaystyle n_0>0 $$, advance if $$\displaystyle n_0<0 $$).

• Folding: $x[-n]$ (reflection about $$\displaystyle n=0 $$).

• Scaling: $x[kn]$ (compression if $$\displaystyle |k|>1 $$, expansion if $$\displaystyle |k|<1 $$).

• Addition: $$\displaystyle x_1[n]+x_2[n] $$.

• Multiplication: $$\displaystyle x_1[n]x_2[n] $$.

[!TIP] Order Matters: For combination, apply folding first, then shifting, then scaling? Actually, standard: $x[an+b]$: first shift by $b$, then scale by $a$? But careful: $x[2n-1]$: shift by 1 then scale by 2? Better: $x[2(n-1/2)]$. Usually, scaling and shifting do not commute. Define clearly.

8.4 Short Note Topics (Frequent 4m/3m)

Convolution:

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

• Properties: Commutative, associative, distributive, with delta, with shifted delta.

• Significance: Characterizes LTI system response. Output is weighted sum of past inputs with weights given by $h[n]$.

Properties of LTI Systems:

• Commutative: $$\displaystyle x*h = h*x $$.

• Associative: $$\displaystyle (x*h_1)*h_2 = x*(h_1*h_2) $$.

• Distributive: $$\displaystyle x*(h_1+h_2) = x*h_1 + x*h_2 $$.

• Memoryless: $$\displaystyle h[n]=\delta[n] $$.

• Invertible: $H(z) \neq 0$ for all $z$ in ROC.

• Causal: $$\displaystyle h[n]=0 $$ for $$\displaystyle n<0 $$.

• Stable: $$\displaystyle \sum |h[n]| < \infty $$.

Convergence of DTFT:

• Sufficient Condition: $$\displaystyle \sum_{n=-\infty}^{\infty} |x[n]| < \infty $$ (absolute summability) → DTFT converges for all $\omega$.

• Necessary Condition for Convergence at a Particular $\omega$: $x[n]$ must be of exponential order: $$\displaystyle |x[n]| \leq B r^{|n|} $$ for some $B,r$.

• Mean-Square Convergence: For finite energy signals ($$\displaystyle \sum |x[n]|^2 < \infty $$), DTFT converges in mean square sense, even if not absolutely summable.

Unilateral Z-Transform:

• Definition: $$\displaystyle X_u(z) = \sum_{n=0}^{\infty} x[n] z^{-n} $$.

• Difference from Bilateral: Considers only $n \geq 0$. ROC is always exterior of outermost pole (for causal part).

• Use: Solving difference equations with non-zero initial conditions. When taking Z-transform of a difference equation, use unilateral property: $$\displaystyle \mathcal{Z}\{x[n-k]\} = z^{-k}X_u(z) + \sum_{m=0}^{k-1} x[m] z^{-m} $$ for $$\displaystyle k>0 $$.

Applications of DTFT:

1. Frequency Response: $$\displaystyle H(e^{j\omega}) $$ is DTFT of impulse response $h[n]$.

2. Spectral Analysis: Determine frequency content of DT signals.

3. Filter Design: Specify desired $$\displaystyle H(e^{j\omega}) $$, then find $h[n]$ via inverse DTFT.

4. Solving Difference Equations: For steady-state sinusoidal response, replace $$\displaystyle e^{j\omega n} $$ by $$\displaystyle e^{j\omega n} $$ and use $$\displaystyle H(e^{j\omega}) $$.

5. Sampling Theorem for DT Signals: Relates DTFT to CTFT after sampling.

State Space Analysis:

• Definition: Representation of system using state variables instead of input-output differential/difference equations.

• Advantages:

  • Unified framework for CT and DT systems.

  • Handles MIMO systems naturally.

  • Convenient for computer-aided analysis (matrix methods).

  • Allows analysis of internal behavior (not just input-output).

  • Useful for optimal control and estimation.

• Disadvantages: More abstract, requires matrix algebra, may be higher dimensional than transfer function.

Block Diagram Representation:

• Direct Form I: Separate feedforward and feedback paths. Easy to understand but uses more delays.

• Direct Form II: Shared delays, more efficient. Sensitive to coefficient quantization.

• Cascade/Parallel: Better for implementation and sensitivity reduction.

• Properties: All realizations are equivalent (same transfer function) but differ in numerical properties and hardware complexity.

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