Formula Sheet: Signals And Systems

Table of Contents
1. Signal Classification and Properties
2. Basic Signal Operations
3. Elementary Signals
4. System Properties
5. Convolution
6. Fourier Series
7. Fourier Transform
8. Laplace Transform
9. Z-Transform
10. Frequency Response and Filters
11. Sampling Theorem
12. Transfer Functions and Block Diagrams
13. State-Space Representation
14. Bode Plots
15. Correlation and Spectral Density
16. Discrete Fourier Transform (DFT)
17. Digital Filter Structures
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Signal Classification and Properties

Continuous-Time vs. Discrete-Time Signals

• Continuous-time signal: \(x(t)\), defined for all \(t\) in the continuous domain
• Discrete-time signal: \(x[n]\), defined only at integer values of \(n\)
• Sampling relationship: \(x[n] = x(nT_s)\), where \(T_s\) is the sampling period
• Sampling frequency: \(f_s = \frac{1}{T_s}\) (Hz)

Energy and Power Signals

• Energy of continuous-time signal:
\[E = \int_{-\infty}^{\infty} |x(t)|^2 \, dt\]
• Energy of discrete-time signal:
\[E = \sum_{n=-\infty}^{\infty} |x[n]|^2\]
• Average power of continuous-time signal:
\[P = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} |x(t)|^2 \, dt\]
• Average power of discrete-time signal:
\[P = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^{N} |x[n]|^2\]
• Energy signal: \(0 < e="">< \infty\)="" and="" \(p="">
• Power signal: \(E = \infty\) and \(0 < p=""><>

Periodic Signals

• Continuous-time periodic signal: \(x(t) = x(t + T_0)\) for all \(t\)
• Fundamental period \(T_0\): smallest positive value satisfying periodicity condition
• Fundamental frequency: \(\omega_0 = \frac{2\pi}{T_0}\) (rad/s) or \(f_0 = \frac{1}{T_0}\) (Hz)
• Discrete-time periodic signal: \(x[n] = x[n + N]\) for all \(n\)
• Fundamental period \(N\): smallest positive integer satisfying periodicity condition

Even and Odd Signals

• Even signal: \(x(-t) = x(t)\) or \(x[-n] = x[n]\)
• Odd signal: \(x(-t) = -x(t)\) or \(x[-n] = -x[n]\)
• Even part decomposition:
\[x_e(t) = \frac{1}{2}[x(t) + x(-t)]\]
• Odd part decomposition:
\[x_o(t) = \frac{1}{2}[x(t) - x(-t)]\]
• Signal decomposition: \(x(t) = x_e(t) + x_o(t)\)

Basic Signal Operations

Time Shifting

• Continuous-time: \(y(t) = x(t - t_0)\)
• Discrete-time: \(y[n] = x[n - n_0]\)
• Right shift: \(t_0 > 0\) or \(n_0 > 0\) (delay)
• Left shift: \(t_0 < 0\)="" or="" \(n_0="">< 0\)="">

Time Scaling

• Continuous-time: \(y(t) = x(at)\), where \(a\) is scaling factor
• Compression: \(a > 1\)
• Expansion: \(0 < a=""><>
• Time reversal: \(a = -1\), giving \(y(t) = x(-t)\)

Amplitude Scaling and Addition

• Amplitude scaling: \(y(t) = A \cdot x(t)\)
• Signal addition: \(y(t) = x_1(t) + x_2(t)\)
• Signal multiplication: \(y(t) = x_1(t) \cdot x_2(t)\)

Elementary Signals

Unit Step Function

• Continuous-time unit step:
\[u(t) = \begin{cases} 1, & t \geq 0 \\ 0, & t < 0="" \end{cases}\]="">
• Discrete-time unit step:
\[u[n] = \begin{cases} 1, & n \geq 0 \\ 0, & n < 0="" \end{cases}\]="">

Unit Impulse Function

• Continuous-time unit impulse (Dirac delta): \(\delta(t)\)
• Properties of \(\delta(t)\):
\[\int_{-\infty}^{\infty} \delta(t) \, dt = 1\] \[\delta(t) = 0 \text{ for all } t \neq 0\]
• Sifting property:
\[\int_{-\infty}^{\infty} x(t) \delta(t - t_0) \, dt = x(t_0)\]
• Relationship to unit step:
\[\delta(t) = \frac{du(t)}{dt}\] \[u(t) = \int_{-\infty}^{t} \delta(\tau) \, d\tau\]
• Discrete-time unit impulse:
\[\delta[n] = \begin{cases} 1, & n = 0 \\ 0, & n \neq 0 \end{cases}\]
• Discrete sifting property:
\[\sum_{k=-\infty}^{\infty} x[k] \delta[n - k] = x[n]\]
• Discrete step relationship:
\[u[n] = \sum_{k=-\infty}^{n} \delta[k]\] \[\delta[n] = u[n] - u[n-1]\]

Exponential Signals

• Continuous-time real exponential: \(x(t) = Ae^{at}\)
• Growing exponential: \(a > 0\)
• Decaying exponential: \(a <>
• Complex exponential: \(x(t) = Ae^{st}\), where \(s = \sigma + j\omega\)
• Discrete-time exponential: \(x[n] = Ar^n\) or \(x[n] = Ae^{\alpha n}\)

Sinusoidal Signals

• Continuous-time sinusoid:
\[x(t) = A\cos(\omega_0 t + \phi)\] where \(A\) is amplitude, \(\omega_0\) is angular frequency (rad/s), and \(\phi\) is phase (rad)
• Discrete-time sinusoid:
\[x[n] = A\cos(\Omega_0 n + \phi)\] where \(\Omega_0\) is normalized angular frequency (rad/sample)
• Euler's formula:
\[e^{j\theta} = \cos\theta + j\sin\theta\]
• Cosine from exponentials:
\[\cos(\omega t) = \frac{e^{j\omega t} + e^{-j\omega t}}{2}\]
• Sine from exponentials:
\[\sin(\omega t) = \frac{e^{j\omega t} - e^{-j\omega t}}{2j}\]

System Properties

Linearity

• Linear system definition: A system is linear if it satisfies both homogeneity and additivity
• Homogeneity (scaling): If \(x(t) \to y(t)\), then \(ax(t) \to ay(t)\)
• Additivity (superposition): If \(x_1(t) \to y_1(t)\) and \(x_2(t) \to y_2(t)\), then \(x_1(t) + x_2(t) \to y_1(t) + y_2(t)\)
• Combined linearity test:
\[a_1x_1(t) + a_2x_2(t) \to a_1y_1(t) + a_2y_2(t)\]

Time Invariance

• Time-invariant system: If \(x(t) \to y(t)\), then \(x(t - t_0) \to y(t - t_0)\) for all \(t_0\)
• Discrete-time invariance: If \(x[n] \to y[n]\), then \(x[n - n_0] \to y[n - n_0]\) for all \(n_0\)

Causality

• Causal system: Output at any time depends only on present and past inputs, not future inputs
• Mathematical condition: \(y(t_0)\) depends only on \(x(t)\) for \(t \leq t_0\)
• Impulse response condition: \(h(t) = 0\) for \(t <>

Stability

• BIBO stability (Bounded-Input Bounded-Output): Every bounded input produces a bounded output
• Continuous-time stability condition:
\[\int_{-\infty}^{\infty} |h(t)| \, dt < \infty\]="" where="" \(h(t)\)="" is="" the="" impulse="" response="">
• Discrete-time stability condition:
\[\sum_{n=-\infty}^{\infty} |h[n]| < \infty\]="">

Memory

• Memoryless system: Output at any time depends only on input at that same time
• System with memory: Output depends on past or future input values

Invertibility

• Invertible system: Distinct inputs produce distinct outputs; an inverse system exists such that cascading the system with its inverse yields the identity system

Convolution

Continuous-Time Convolution

• Convolution integral:
\[y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau)h(t - \tau) \, d\tau\] where \(x(t)\) is input, \(h(t)\) is impulse response, and \(y(t)\) is output
• Alternative form:
\[y(t) = \int_{-\infty}^{\infty} h(\tau)x(t - \tau) \, d\tau\]
• For causal systems with causal inputs:
\[y(t) = \int_{0}^{t} x(\tau)h(t - \tau) \, d\tau\]

Discrete-Time Convolution

• Convolution sum:
\[y[n] = x[n] * h[n] = \sum_{k=-\infty}^{\infty} x[k]h[n - k]\]
• Alternative form:
\[y[n] = \sum_{k=-\infty}^{\infty} h[k]x[n - k]\]
• For causal systems with causal inputs:
\[y[n] = \sum_{k=0}^{n} x[k]h[n - k]\]

Convolution Properties

• Commutativity: \(x(t) * h(t) = h(t) * x(t)\)
• Associativity: \([x(t) * h_1(t)] * h_2(t) = x(t) * [h_1(t) * h_2(t)]\)
• Distributivity: \(x(t) * [h_1(t) + h_2(t)] = x(t) * h_1(t) + x(t) * h_2(t)\)
• Identity: \(x(t) * \delta(t) = x(t)\)
• Shift property: \(x(t) * \delta(t - t_0) = x(t - t_0)\)
• Derivative property:
\[\frac{d}{dt}[x(t) * h(t)] = \frac{dx(t)}{dt} * h(t) = x(t) * \frac{dh(t)}{dt}\]
• Width property: If \(x(t)\) has duration \(T_1\) and \(h(t)\) has duration \(T_2\), then \(y(t) = x(t) * h(t)\) has duration \(T_1 + T_2\)

Fourier Series

Continuous-Time Fourier Series (CTFS)

• Trigonometric Fourier series:
\[x(t) = a_0 + \sum_{n=1}^{\infty} \left[a_n\cos(n\omega_0 t) + b_n\sin(n\omega_0 t)\right]\] where \(\omega_0 = \frac{2\pi}{T_0}\) is the fundamental frequency
• DC coefficient:
\[a_0 = \frac{1}{T_0}\int_{T_0} x(t) \, dt\]
• Cosine coefficients:
\[a_n = \frac{2}{T_0}\int_{T_0} x(t)\cos(n\omega_0 t) \, dt\]
• Sine coefficients:
\[b_n = \frac{2}{T_0}\int_{T_0} x(t)\sin(n\omega_0 t) \, dt\]
• Exponential Fourier series:
\[x(t) = \sum_{n=-\infty}^{\infty} c_n e^{jn\omega_0 t}\]
• Exponential coefficients:
\[c_n = \frac{1}{T_0}\int_{T_0} x(t)e^{-jn\omega_0 t} \, dt\]
• Relationship between coefficients:
\[c_0 = a_0\] \[c_n = \frac{a_n - jb_n}{2} \text{ for } n > 0\] \[c_{-n} = c_n^* = \frac{a_n + jb_n}{2}\]
• Magnitude and phase form:
\[x(t) = A_0 + \sum_{n=1}^{\infty} A_n\cos(n\omega_0 t + \phi_n)\] where \(A_n = \sqrt{a_n^2 + b_n^2}\) and \(\phi_n = -\tan^{-1}\left(\frac{b_n}{a_n}\right)\)

Discrete-Time Fourier Series (DTFS)

• Discrete-time Fourier series:
\[x[n] = \sum_{k=\langle N \rangle} c_k e^{jk\Omega_0 n}\] where \(\Omega_0 = \frac{2\pi}{N}\) and \(\langle N \rangle\) indicates sum over one period
• DTFS coefficients:
\[c_k = \frac{1}{N}\sum_{n=\langle N \rangle} x[n]e^{-jk\Omega_0 n}\]

Fourier Series Properties

• Linearity: If \(x(t) \leftrightarrow c_n\) and \(y(t) \leftrightarrow d_n\), then \(ax(t) + by(t) \leftrightarrow ac_n + bd_n\)
• Time shifting: \(x(t - t_0) \leftrightarrow c_n e^{-jn\omega_0 t_0}\)
• Time reversal: \(x(-t) \leftrightarrow c_{-n}\)
• Time scaling: \(x(at) \leftrightarrow c_n\) with fundamental frequency \(a\omega_0\)
• Multiplication: \(x(t)y(t) \leftrightarrow \sum_{k=-\infty}^{\infty} c_k d_{n-k}\)
• Differentiation: \(\frac{dx(t)}{dt} \leftrightarrow jn\omega_0 c_n\)
• Integration: \(\int_{-\infty}^{t} x(\tau) \, d\tau \leftrightarrow \frac{c_n}{jn\omega_0}\) for \(n \neq 0\), assuming \(c_0 = 0\)
• Parseval's theorem:
\[\frac{1}{T_0}\int_{T_0} |x(t)|^2 \, dt = \sum_{n=-\infty}^{\infty} |c_n|^2\]

Symmetry Properties

• For real \(x(t)\): \(c_{-n} = c_n^*\), \(a_n\) and \(b_n\) are real
• For real and even \(x(t)\): \(b_n = 0\), \(c_n\) is real and even
• For real and odd \(x(t)\): \(a_n = 0\), \(c_n\) is imaginary and odd

Fourier Transform

Continuous-Time Fourier Transform (CTFT)

• Fourier transform (analysis equation):
\[X(j\omega) = \mathcal{F}\{x(t)\} = \int_{-\infty}^{\infty} x(t)e^{-j\omega t} \, dt\]
• Inverse Fourier transform (synthesis equation):
\[x(t) = \mathcal{F}^{-1}\{X(j\omega)\} = \frac{1}{2\pi}\int_{-\infty}^{\infty} X(j\omega)e^{j\omega t} \, d\omega\]
• Magnitude spectrum: \(|X(j\omega)|\)
• Phase spectrum: \(\angle X(j\omega)\)
• Existence condition:
\[\int_{-\infty}^{\infty} |x(t)| \, dt < \infty\]="">

Discrete-Time Fourier Transform (DTFT)

• DTFT (analysis equation):
\[X(e^{j\Omega}) = \sum_{n=-\infty}^{\infty} x[n]e^{-j\Omega n}\]
• Inverse DTFT (synthesis equation):
\[x[n] = \frac{1}{2\pi}\int_{2\pi} X(e^{j\Omega})e^{j\Omega n} \, d\Omega\]
• Periodicity: \(X(e^{j\Omega})\) is periodic with period \(2\pi\)
• Existence condition:
\[\sum_{n=-\infty}^{\infty} |x[n]| < \infty\]="">

Common Fourier Transform Pairs

• Unit impulse: \(\delta(t) \leftrightarrow 1\)
• Constant: \(1 \leftrightarrow 2\pi\delta(\omega)\)
• Unit step: \(u(t) \leftrightarrow \pi\delta(\omega) + \frac{1}{j\omega}\)
• Signum function: \(\text{sgn}(t) \leftrightarrow \frac{2}{j\omega}\)
• Exponential decay: \(e^{-at}u(t) \leftrightarrow \frac{1}{a + j\omega}\) (for \(a > 0\))
• Two-sided exponential: \(e^{-a|t|} \leftrightarrow \frac{2a}{a^2 + \omega^2}\) (for \(a > 0\))
• Rectangular pulse:
\[\text{rect}(t/\tau) = \begin{cases} 1, & |t| < \tau/2="" \\="" 0,="" &="" |t|=""> \tau/2 \end{cases} \leftrightarrow \tau\text{sinc}\left(\frac{\omega\tau}{2\pi}\right)\] where \(\text{sinc}(x) = \frac{\sin(\pi x)}{\pi x}\)
• Sinc function: \(\text{sinc}(t/\tau) \leftrightarrow \tau\text{rect}(\omega\tau/2\pi)\)
• Cosine: \(\cos(\omega_0 t) \leftrightarrow \pi[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)]\)
• Sine: \(\sin(\omega_0 t) \leftrightarrow j\pi[\delta(\omega + \omega_0) - \delta(\omega - \omega_0)]\)
• Complex exponential: \(e^{j\omega_0 t} \leftrightarrow 2\pi\delta(\omega - \omega_0)\)
• Gaussian: \(e^{-at^2} \leftrightarrow \sqrt{\frac{\pi}{a}}e^{-\omega^2/(4a)}\)

Fourier Transform Properties

• Linearity: \(ax_1(t) + bx_2(t) \leftrightarrow aX_1(j\omega) + bX_2(j\omega)\)
• Time shifting: \(x(t - t_0) \leftrightarrow X(j\omega)e^{-j\omega t_0}\)
• Frequency shifting: \(e^{j\omega_0 t}x(t) \leftrightarrow X(j(\omega - \omega_0))\)
• Time scaling: \(x(at) \leftrightarrow \frac{1}{|a|}X\left(\frac{j\omega}{a}\right)\)
• Time reversal: \(x(-t) \leftrightarrow X(-j\omega)\)
• Duality: If \(x(t) \leftrightarrow X(j\omega)\), then \(X(jt) \leftrightarrow 2\pi x(-\omega)\)
• Convolution in time: \(x(t) * h(t) \leftrightarrow X(j\omega)H(j\omega)\)
• Multiplication in time: \(x(t)y(t) \leftrightarrow \frac{1}{2\pi}[X(j\omega) * Y(j\omega)]\)
• Differentiation in time: \(\frac{dx(t)}{dt} \leftrightarrow j\omega X(j\omega)\)
• Integration in time: \(\int_{-\infty}^{t} x(\tau) \, d\tau \leftrightarrow \frac{X(j\omega)}{j\omega} + \pi X(0)\delta(\omega)\)
• Differentiation in frequency: \(tx(t) \leftrightarrow j\frac{dX(j\omega)}{d\omega}\)
• Parseval's theorem:
\[\int_{-\infty}^{\infty} |x(t)|^2 \, dt = \frac{1}{2\pi}\int_{-\infty}^{\infty} |X(j\omega)|^2 \, d\omega\]

Symmetry Properties for Real Signals

• For real \(x(t)\): \(X(-j\omega) = X^*(j\omega)\)
• Magnitude symmetry: \(|X(j\omega)| = |X(-j\omega)|\) (even function)
• Phase symmetry: \(\angle X(j\omega) = -\angle X(-j\omega)\) (odd function)
• Real part: \(\text{Re}\{X(j\omega)\}\) is even
• Imaginary part: \(\text{Im}\{X(j\omega)\}\) is odd
• For real and even \(x(t)\): \(X(j\omega)\) is real and even
• For real and odd \(x(t)\): \(X(j\omega)\) is imaginary and odd

Laplace Transform

Bilateral (Two-Sided) Laplace Transform

• Laplace transform:
\[X(s) = \mathcal{L}\{x(t)\} = \int_{-\infty}^{\infty} x(t)e^{-st} \, dt\] where \(s = \sigma + j\omega\) is the complex frequency
• Inverse Laplace transform:
\[x(t) = \mathcal{L}^{-1}\{X(s)\} = \frac{1}{2\pi j}\int_{\sigma - j\infty}^{\sigma + j\infty} X(s)e^{st} \, ds\]
• Region of convergence (ROC): Set of values of \(s\) for which \(X(s)\) converges

Unilateral (One-Sided) Laplace Transform

• Unilateral Laplace transform:
\[X(s) = \int_{0^-}^{\infty} x(t)e^{-st} \, dt\]
• Used for causal signals and initial condition problems

Common Laplace Transform Pairs

• Unit impulse: \(\delta(t) \leftrightarrow 1\), ROC: all \(s\)
• Unit step: \(u(t) \leftrightarrow \frac{1}{s}\), ROC: \(\text{Re}\{s\} > 0\)
• Ramp: \(tu(t) \leftrightarrow \frac{1}{s^2}\), ROC: \(\text{Re}\{s\} > 0\)
• Exponential: \(e^{-at}u(t) \leftrightarrow \frac{1}{s + a}\), ROC: \(\text{Re}\{s\} > -a\)
• Sine: \(\sin(\omega_0 t)u(t) \leftrightarrow \frac{\omega_0}{s^2 + \omega_0^2}\), ROC: \(\text{Re}\{s\} > 0\)
• Cosine: \(\cos(\omega_0 t)u(t) \leftrightarrow \frac{s}{s^2 + \omega_0^2}\), ROC: \(\text{Re}\{s\} > 0\)
• Damped sine: \(e^{-at}\sin(\omega_0 t)u(t) \leftrightarrow \frac{\omega_0}{(s + a)^2 + \omega_0^2}\), ROC: \(\text{Re}\{s\} > -a\)
• Damped cosine: \(e^{-at}\cos(\omega_0 t)u(t) \leftrightarrow \frac{s + a}{(s + a)^2 + \omega_0^2}\), ROC: \(\text{Re}\{s\} > -a\)
• Power of t: \(t^n u(t) \leftrightarrow \frac{n!}{s^{n+1}}\), ROC: \(\text{Re}\{s\} > 0\)

Laplace Transform Properties

• Linearity: \(ax_1(t) + bx_2(t) \leftrightarrow aX_1(s) + bX_2(s)\)
• Time shifting: \(x(t - t_0)u(t - t_0) \leftrightarrow e^{-st_0}X(s)\)
• Frequency shifting: \(e^{s_0 t}x(t) \leftrightarrow X(s - s_0)\)
• Time scaling: \(x(at) \leftrightarrow \frac{1}{|a|}X\left(\frac{s}{a}\right)\)
• Time reversal: \(x(-t) \leftrightarrow X(-s)\)
• Convolution: \(x(t) * h(t) \leftrightarrow X(s)H(s)\)
• Differentiation in time: \(\frac{dx(t)}{dt} \leftrightarrow sX(s) - x(0^-)\) (unilateral)
• Second derivative: \(\frac{d^2x(t)}{dt^2} \leftrightarrow s^2X(s) - sx(0^-) - x'(0^-)\)
• nth derivative:
\[\frac{d^n x(t)}{dt^n} \leftrightarrow s^n X(s) - s^{n-1}x(0^-) - s^{n-2}x'(0^-) - \cdots - x^{(n-1)}(0^-)\]
• Integration in time: \(\int_{0^-}^{t} x(\tau) \, d\tau \leftrightarrow \frac{X(s)}{s}\) (unilateral)
• Multiplication by t: \(tx(t) \leftrightarrow -\frac{dX(s)}{ds}\)
• Multiplication by \(t^n\): \(t^n x(t) \leftrightarrow (-1)^n \frac{d^n X(s)}{ds^n}\)
• Division by t: \(\frac{x(t)}{t} \leftrightarrow \int_{s}^{\infty} X(\sigma) \, d\sigma\)
• Initial value theorem: \(x(0^+) = \lim_{s \to \infty} sX(s)\)
• Final value theorem: \(x(\infty) = \lim_{s \to 0} sX(s)\) (if poles of \(sX(s)\) in left half-plane)

ROC Properties

• ROC consists of strips parallel to \(j\omega\)-axis in the s-plane
• For rational \(X(s)\), ROC does not contain poles
• Finite-duration signal: ROC is entire s-plane (possibly except \(s = 0\) or \(s = \infty\))
• Right-sided signal: ROC is \(\text{Re}\{s\} > \sigma_{\text{max}}\)
• Left-sided signal: ROC is \(\text{Re}\{s\} <>
• Two-sided signal: ROC is \(\sigma_1 < \text{re}\{s\}=""><>
• Causal signal: ROC is to the right of rightmost pole
• Stable signal: ROC includes \(j\omega\)-axis

Inverse Laplace Transform Techniques

• Partial fraction expansion for proper rational functions:
\[X(s) = \frac{N(s)}{D(s)} = \sum_{i=1}^{n} \frac{A_i}{s - p_i}\] for distinct poles \(p_i\)
• Residue for simple pole:
\[A_i = \lim_{s \to p_i} [(s - p_i)X(s)]\]
• For repeated pole of order \(r\):
\[A_k = \frac{1}{(r-k)!} \lim_{s \to p_i} \frac{d^{r-k}}{ds^{r-k}}[(s - p_i)^r X(s)]\] where \(k = 1, 2, \ldots, r\)

Z-Transform

Bilateral Z-Transform

• Z-transform:
\[X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n]z^{-n}\] where \(z\) is a complex variable
• Inverse Z-transform:
\[x[n] = \mathcal{Z}^{-1}\{X(z)\} = \frac{1}{2\pi j}\oint_C X(z)z^{n-1} \, dz\] where \(C\) is a counterclockwise contour in the ROC
• Region of convergence (ROC): Set of values of \(z\) for which \(X(z)\) converges

Unilateral Z-Transform

• Unilateral Z-transform:
\[X(z) = \sum_{n=0}^{\infty} x[n]z^{-n}\]
• Used for causal sequences and initial condition problems

Common Z-Transform Pairs

• Unit impulse: \(\delta[n] \leftrightarrow 1\), ROC: all \(z\)
• Unit step: \(u[n] \leftrightarrow \frac{z}{z - 1} = \frac{1}{1 - z^{-1}}\), ROC: \(|z| > 1\)
• Exponential: \(a^n u[n] \leftrightarrow \frac{z}{z - a} = \frac{1}{1 - az^{-1}}\), ROC: \(|z| > |a|\)
• Negative exponential: \(-a^n u[-n-1] \leftrightarrow \frac{z}{z - a}\), ROC: \(|z| <>
• Ramp: \(nu[n] \leftrightarrow \frac{z}{(z-1)^2}\), ROC: \(|z| > 1\)
• Sinusoid: \(\sin(\Omega_0 n)u[n] \leftrightarrow \frac{z\sin\Omega_0}{z^2 - 2z\cos\Omega_0 + 1}\), ROC: \(|z| > 1\)
• Cosinusoid: \(\cos(\Omega_0 n)u[n] \leftrightarrow \frac{z(z - \cos\Omega_0)}{z^2 - 2z\cos\Omega_0 + 1}\), ROC: \(|z| > 1\)
• Damped exponential: \(r^n\cos(\Omega_0 n)u[n] \leftrightarrow \frac{z(z - r\cos\Omega_0)}{z^2 - 2rz\cos\Omega_0 + r^2}\), ROC: \(|z| > r\)

Z-Transform Properties

• Linearity: \(ax_1[n] + bx_2[n] \leftrightarrow aX_1(z) + bX_2(z)\)
• Time shifting: \(x[n - n_0] \leftrightarrow z^{-n_0}X(z)\)
• Time shifting (unilateral):
\[x[n - k]u[n] \leftrightarrow z^{-k}X(z) + z^{-k}\sum_{n=0}^{k-1} x[n-k]z^{-n}\]
• Frequency scaling (multiplication by exponential): \(a^n x[n] \leftrightarrow X(a^{-1}z)\)
• Time reversal: \(x[-n] \leftrightarrow X(z^{-1})\)
• Convolution: \(x[n] * h[n] \leftrightarrow X(z)H(z)\)
• Multiplication: \(x[n]y[n] \leftrightarrow \frac{1}{2\pi j}\oint X(v)Y(z/v)v^{-1} \, dv\)
• First difference: \(x[n] - x[n-1] \leftrightarrow (1 - z^{-1})X(z)\)
• Accumulation: \(\sum_{k=-\infty}^{n} x[k] \leftrightarrow \frac{z}{z-1}X(z)\)
• Multiplication by n: \(nx[n] \leftrightarrow -z\frac{dX(z)}{dz}\)
• Multiplication by \(n^k\): \(n^k x[n] \leftrightarrow \left(-z\frac{d}{dz}\right)^k X(z)\)
• Initial value theorem: \(x[0] = \lim_{z \to \infty} X(z)\) (for causal \(x[n]\))
• Final value theorem: \(x[\infty] = \lim_{z \to 1} (z-1)X(z)\) (if poles of \((z-1)X(z)\) inside unit circle)

ROC Properties

• ROC consists of rings centered at origin in the z-plane
• For rational \(X(z)\), ROC does not contain poles
• Finite-duration sequence: ROC is entire z-plane (possibly except \(z = 0\) or \(z = \infty\))
• Right-sided sequence: ROC is \(|z| > r_{\text{max}}\) (exterior of circle)
• Left-sided sequence: ROC is \(|z| < r_{\text{min}}\)="" (interior="" of="">
• Two-sided sequence: ROC is \(r_1 < |z|="">< r_2\)="" (annular="">
• Causal sequence: ROC is exterior of circle beyond outermost pole
• Stable sequence: ROC includes unit circle \(|z| = 1\)

Relationship Between Transforms

• Z-transform and DTFT: \(X(e^{j\Omega}) = X(z)\big|_{z = e^{j\Omega}}\) when ROC includes unit circle
• Z-transform and Laplace transform (via sampling): \(z = e^{sT_s}\) or \(s = \frac{1}{T_s}\ln(z)\)

Inverse Z-Transform Methods

• Partial fraction expansion for proper rational functions:
\[X(z) = \sum_{i=1}^{n} \frac{A_i}{1 - p_i z^{-1}}\]
• Residue for simple pole at \(z = p_i\):
\[A_i = \lim_{z \to p_i} [(1 - p_i z^{-1})X(z)]\]
• Long division method: Expand \(X(z)\) as power series in \(z^{-1}\)
• Inversion integral (contour integration): Use residue theorem

Frequency Response and Filters

Frequency Response

• Frequency response of continuous-time LTI system:
\[H(j\omega) = \mathcal{F}\{h(t)\}\] where \(h(t)\) is the impulse response
• Frequency response from Laplace transform: \(H(j\omega) = H(s)\big|_{s = j\omega}\)
• Output for sinusoidal input: If \(x(t) = A\cos(\omega_0 t + \phi)\), then
\[y(t) = A|H(j\omega_0)|\cos(\omega_0 t + \phi + \angle H(j\omega_0))\]
• Magnitude response: \(|H(j\omega)|\)
• Phase response: \(\angle H(j\omega)\)
• Frequency response of discrete-time LTI system:
\[H(e^{j\Omega}) = \sum_{n=-\infty}^{\infty} h[n]e^{-j\Omega n}\]
• From Z-transform: \(H(e^{j\Omega}) = H(z)\big|_{z = e^{j\Omega}}\)

Ideal Filter Characteristics

• Ideal lowpass filter:
\[H(j\omega) = \begin{cases} 1, & |\omega| \leq \omega_c \\ 0, & |\omega| > \omega_c \end{cases}\] where \(\omega_c\) is the cutoff frequency
• Ideal highpass filter:
\[H(j\omega) = \begin{cases} 0, & |\omega| < \omega_c="" \\="" 1,="" &="" |\omega|="" \geq="" \omega_c="" \end{cases}\]="">
• Ideal bandpass filter:
\[H(j\omega) = \begin{cases} 1, & \omega_1 \leq |\omega| \leq \omega_2 \\ 0, & \text{otherwise} \end{cases}\]
• Ideal bandstop (notch) filter:
\[H(j\omega) = \begin{cases} 0, & \omega_1 \leq |\omega| \leq \omega_2 \\ 1, & \text{otherwise} \end{cases}\]
• Bandwidth of bandpass filter: \(B = \omega_2 - \omega_1\)
• Center frequency: \(\omega_0 = \sqrt{\omega_1 \omega_2}\) (geometric mean)

Filter Specifications

• Passband: Frequency range where signal passes with minimal attenuation
• Stopband: Frequency range where signal is significantly attenuated
• Transition band: Frequency range between passband and stopband
• Passband ripple \(\delta_p\): Maximum deviation from unity gain in passband
• Stopband attenuation \(\delta_s\): Maximum gain in stopband
• Cutoff frequency: Frequency where \(|H(j\omega)| = \frac{1}{\sqrt{2}}\) (-3 dB point)
• Gain in dB: \(G_{dB} = 20\log_{10}|H(j\omega)|\)

First-Order Filters

• First-order lowpass RC filter transfer function:
\[H(s) = \frac{1}{1 + sRC} = \frac{1/RC}{s + 1/RC}\]
• Cutoff frequency: \(\omega_c = \frac{1}{RC}\) (rad/s)
• First-order highpass RC filter transfer function:
\[H(s) = \frac{sRC}{1 + sRC} = \frac{s}{s + 1/RC}\]

Second-Order Filters

• Standard second-order lowpass transfer function:
\[H(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}\] where \(\omega_n\) is natural frequency and \(\zeta\) is damping ratio
• Standard second-order highpass transfer function:
\[H(s) = \frac{s^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}\]
• Standard second-order bandpass transfer function:
\[H(s) = \frac{2\zeta\omega_n s}{s^2 + 2\zeta\omega_n s + \omega_n^2}\]
• Quality factor: \(Q = \frac{1}{2\zeta}\)
• Bandwidth for bandpass filter: \(B = \frac{\omega_n}{Q}\)

Sampling Theorem

Nyquist Sampling Theorem

• Sampling theorem: A bandlimited signal with maximum frequency \(f_{\text{max}}\) can be perfectly reconstructed from its samples if the sampling frequency \(f_s\) satisfies:
\[f_s \geq 2f_{\text{max}}\]
• Nyquist rate: \(f_{\text{Nyquist}} = 2f_{\text{max}}\) (minimum sampling rate)
• Nyquist frequency: \(f_N = \frac{f_s}{2}\) (maximum frequency that can be represented)
• Angular frequency version: \(\omega_s \geq 2\omega_{\text{max}}\)
• Aliasing: Occurs when \(f_s < 2f_{\text{max}}\),="" causing="" high-frequency="" components="" to="" appear="" as="" lower="">
• Aliased frequency: For frequency \(f\) sampled at \(f_s\):
\[f_{\text{alias}} = |f - kf_s|\] where \(k\) is chosen such that \(f_{\text{alias}} \leq \frac{f_s}{2}\)

Reconstruction

• Ideal reconstruction formula (Whittaker-Shannon):
\[x(t) = \sum_{n=-\infty}^{\infty} x[n]\text{sinc}\left(\frac{t - nT_s}{T_s}\right)\] where \(T_s = 1/f_s\) is the sampling period
• Ideal reconstruction filter: Ideal lowpass filter with cutoff at \(f_s/2\)

Transfer Functions and Block Diagrams

Transfer Function

• Continuous-time transfer function:
\[H(s) = \frac{Y(s)}{X(s)}\] where \(Y(s)\) is the Laplace transform of output and \(X(s)\) is the Laplace transform of input
• Discrete-time transfer function:
\[H(z) = \frac{Y(z)}{X(z)}\]
• Rational transfer function:
\[H(s) = \frac{N(s)}{D(s)} = \frac{b_m s^m + b_{m-1}s^{m-1} + \cdots + b_1 s + b_0}{a_n s^n + a_{n-1}s^{n-1} + \cdots + a_1 s + a_0}\]
• Zeros: Values of \(s\) (or \(z\)) where \(N(s) = 0\) (numerator roots)
• Poles: Values of \(s\) (or \(z\)) where \(D(s) = 0\) (denominator roots)
• Pole-zero form:
\[H(s) = K\frac{(s - z_1)(s - z_2)\cdots(s - z_m)}{(s - p_1)(s - p_2)\cdots(s - p_n)}\] where \(K\) is the gain constant

System Stability from Poles

• Continuous-time stability: System is stable if all poles have \(\text{Re}\{p_i\} < 0\)="" (left="" half="" of="">
• Discrete-time stability: System is stable if all poles have \(|p_i| < 1\)="" (inside="" unit="" circle="" in="">
• Marginally stable: Poles on \(j\omega\)-axis (continuous) or unit circle (discrete)
• Unstable: At least one pole in right half-plane (continuous) or outside unit circle (discrete)

Block Diagram Operations

• Series (cascade) connection: \(H(s) = H_1(s) \cdot H_2(s)\)
• Parallel connection: \(H(s) = H_1(s) + H_2(s)\)
• Feedback connection:
\[H(s) = \frac{G(s)}{1 \pm G(s)H_f(s)}\] where \(G(s)\) is forward path, \(H_f(s)\) is feedback path, and - is for negative feedback
• Closed-loop transfer function (negative feedback):
\[T(s) = \frac{G(s)}{1 + G(s)H(s)}\]
• Error transfer function:
\[E(s) = \frac{1}{1 + G(s)H(s)}\]

State-Space Representation

Continuous-Time State-Space

• State equation:
\[\dot{\mathbf{x}}(t) = \mathbf{A}\mathbf{x}(t) + \mathbf{B}\mathbf{u}(t)\]
• Output equation:
\[\mathbf{y}(t) = \mathbf{C}\mathbf{x}(t) + \mathbf{D}\mathbf{u}(t)\]
• \(\mathbf{A}\): \(n \times n\) state matrix
• \(\mathbf{B}\): \(n \times p\) input matrix
• \(\mathbf{C}\): \(q \times n\) output matrix
• \(\mathbf{D}\): \(q \times p\) feedthrough matrix
• \(\mathbf{x}(t)\): \(n \times 1\) state vector
• \(\mathbf{u}(t)\): \(p \times 1\) input vector
• \(\mathbf{y}(t)\): \(q \times 1\) output vector

Discrete-Time State-Space

• State equation:
\[\mathbf{x}[n+1] = \mathbf{A}\mathbf{x}[n] + \mathbf{B}\mathbf{u}[n]\]
• Output equation:
\[\mathbf{y}[n] = \mathbf{C}\mathbf{x}[n] + \mathbf{D}\mathbf{u}[n]\]

Transfer Function from State-Space

• Continuous-time transfer function:
\[H(s) = \mathbf{C}(s\mathbf{I} - \mathbf{A})^{-1}\mathbf{B} + \mathbf{D}\]
• Discrete-time transfer function:
\[H(z) = \mathbf{C}(z\mathbf{I} - \mathbf{A})^{-1}\mathbf{B} + \mathbf{D}\]

Solution of State Equations

• Continuous-time solution:
\[\mathbf{x}(t) = e^{\mathbf{A}t}\mathbf{x}(0) + \int_0^t e^{\mathbf{A}(t-\tau)}\mathbf{B}\mathbf{u}(\tau) \, d\tau\]
• State transition matrix: \(\Phi(t) = e^{\mathbf{A}t}\)
• Discrete-time solution:
\[\mathbf{x}[n] = \mathbf{A}^n\mathbf{x}[0] + \sum_{k=0}^{n-1} \mathbf{A}^{n-1-k}\mathbf{B}\mathbf{u}[k]\]

Bode Plots

Bode Plot Fundamentals

• Magnitude plot: \(20\log_{10}|H(j\omega)|\) vs. \(\log_{10}\omega\) (dB vs. log frequency)
• Phase plot: \(\angle H(j\omega)\) vs. \(\log_{10}\omega\) (degrees or radians vs. log frequency)
• Decade: Frequency range from \(\omega\) to \(10\omega\)
• Octave: Frequency range from \(\omega\) to \(2\omega\)

Bode Plot Rules for Basic Factors

• Constant gain \(K\):
• Magnitude: \(20\log_{10}K\) dB (constant)
• Phase: 0° if \(K > 0\), -180° if \(K <>
• Pole at origin \(1/s\):
• Magnitude: -20 dB/decade slope
• Phase: -90° (constant)
• Zero at origin \(s\):
• Magnitude: +20 dB/decade slope
• Phase: +90° (constant)
• Simple pole \(1/(1 + s/\omega_0)\):
• Magnitude: 0 dB for \(\omega \ll \omega_0\); -20 dB/decade for \(\omega \gg \omega_0\)
• Phase: 0° for \(\omega \ll \omega_0\); -90° for \(\omega \gg \omega_0\); -45° at \(\omega = \omega_0\)
• Corner frequency: \(\omega_0\)
• Simple zero \((1 + s/\omega_0)\):
• Magnitude: 0 dB for \(\omega \ll \omega_0\); +20 dB/decade for \(\omega \gg \omega_0\)
• Phase: 0° for \(\omega \ll \omega_0\); +90° for \(\omega \gg \omega_0\); +45° at \(\omega = \omega_0\)
• Complex conjugate poles:
\[\frac{1}{1 + 2\zeta(s/\omega_n) + (s/\omega_n)^2}\]
• Magnitude: 0 dB for \(\omega \ll \omega_n\); -40 dB/decade for \(\omega \gg \omega_n\)
• Resonant peak at \(\omega = \omega_n\) if \(\zeta <>
• Phase: 0° for \(\omega \ll \omega_n\); -180° for \(\omega \gg \omega_n\); -90° at \(\omega = \omega_n\)

Gain and Phase Margins

• Gain margin (GM): Amount of gain increase at phase crossover frequency before instability
\[GM = -20\log_{10}|H(j\omega_{pc})| \text{ (dB)}\] where \(\omega_{pc}\) is frequency where \(\angle H(j\omega) = -180°\)
• Phase margin (PM): Amount of phase lag at gain crossover frequency before instability
\[PM = 180° + \angle H(j\omega_{gc})\] where \(\omega_{gc}\) is frequency where \(|H(j\omega)| = 1\) (0 dB)
• Stability requirement: GM > 0 dB and PM > 0° for stable closed-loop system
• Typical design targets: GM ≥ 6 dB, PM ≥ 30°

Correlation and Spectral Density

Autocorrelation

• Continuous-time autocorrelation:
\[R_x(\tau) = \int_{-\infty}^{\infty} x(t)x(t + \tau) \, dt\]
• For periodic signals:
\[R_x(\tau) = \frac{1}{T_0}\int_{T_0} x(t)x(t + \tau) \, dt\]
• Discrete-time autocorrelation:
\[R_x[m] = \sum_{n=-\infty}^{\infty} x[n]x[n + m]\]
• Properties:
• \(R_x(\tau)\) is even: \(R_x(\tau) = R_x(-\tau)\)
• Maximum at origin: \(R_x(0) \geq |R_x(\tau)|\) for all \(\tau\)
• \(R_x(0)\) equals signal energy or average power

Cross-Correlation

• Continuous-time cross-correlation:
\[R_{xy}(\tau) = \int_{-\infty}^{\infty} x(t)y(t + \tau) \, dt\]
• Discrete-time cross-correlation:
\[R_{xy}[m] = \sum_{n=-\infty}^{\infty} x[n]y[n + m]\]
• Property: \(R_{xy}(\tau) = R_{yx}(-\tau)\)

Power Spectral Density

• Wiener-Khinchin theorem: Power spectral density is the Fourier transform of autocorrelation
\[S_x(\omega) = \mathcal{F}\{R_x(\tau)\}\]
• Inverse relationship:
\[R_x(\tau) = \mathcal{F}^{-1}\{S_x(\omega)\}\]
• Average power:
\[P = R_x(0) = \frac{1}{2\pi}\int_{-\infty}^{\infty} S_x(\omega) \, d\omega\]
• Properties:
• \(S_x(\omega)\) is real and non-negative
• \(S_x(\omega)\) is even for real signals

Input-Output Relationships

• Output autocorrelation:
\[R_y(\tau) = R_x(\tau) * R_h(\tau)\] where \(R_h(\tau) = h(\tau) * h(-\tau)\)
• Output power spectral density:
\[S_y(\omega) = |H(j\omega)|^2 S_x(\omega)\]

Discrete Fourier Transform (DFT)

DFT Definition

• N-point DFT:
\[X[k] = \sum_{n=0}^{N-1} x[n]e^{-j2\pi kn/N}, \quad k = 0, 1, \ldots, N-1\]
• Inverse DFT (IDFT):
\[x[n] = \frac{1}{N}\sum_{k=0}^{N-1} X[k]e^{j2\pi kn/N}, \quad n = 0, 1, \ldots, N-1\]
• Twiddle factor: \(W_N = e^{-j2\pi/N}\)
• DFT in twiddle notation:
\[X[k] = \sum_{n=0}^{N-1} x[n]W_N^{kn}\]

DFT Properties

• Linearity: \(ax_1[n] + bx_2[n] \xrightarrow{DFT} aX_1[k] + bX_2[k]\)
• Circular time shift: \(x[(n - m) \bmod N] \xrightarrow{DFT} X[k]e^{-j2\pi km/N}\)
• Circular frequency shift: \(x[n]e^{j2\pi ln/N} \xrightarrow{DFT} X[(k - l) \bmod N]\)
• Time reversal: \(x[(-n) \bmod N] \xrightarrow{DFT} X[(-k) \bmod N]\)
• Circular convolution: \(x[n] \circledast y[n] \xrightarrow{DFT} X[k] \cdot Y[k]\)
• Multiplication: \(x[n] \cdot y[n] \xrightarrow{DFT} \frac{1}{N}X[k] \circledast Y[k]\)
• Parseval's theorem:
\[\sum_{n=0}^{N-1} |x[n]|^2 = \frac{1}{N}\sum_{k=0}^{N-1} |X[k]|^2\]

Relationship to Other Transforms

• DFT as sampled DTFT: \(X[k] = X(e^{j\Omega})\big|_{\Omega = 2\pi k/N}\)
• Frequency resolution: \(\Delta f = \frac{f_s}{N}\) or \(\Delta\Omega = \frac{2\pi}{N}\)

Fast Fourier Transform (FFT)

• FFT: Efficient algorithm for computing DFT
• Computational complexity:
• Direct DFT: \(O(N^2)\) operations
• FFT: \(O(N\log_2 N)\) operations
• Radix-2 FFT requirement: \(N = 2^m\) for integer \(m\)
• Zero-padding: Append zeros to signal to achieve desired \(N\) value

Digital Filter Structures

FIR Filter Structure

• FIR (Finite Impulse Response) difference equation:
\[y[n] = \sum_{k=0}^{M} b_k x[n-k]\]
• FIR transfer function:
\[H(z) = \sum_{k=0}^{M} b_k z^{-k}\]
• Properties:
• Always stable (all poles at origin)
• Can have linear phase
• No feedback
• Linear phase condition: Symmetric or antisymmetric coefficients

IIR Filter Structure

• IIR (Infinite Impulse Response) difference equation:
\[y[n] = \sum_{k=0}^{M} b_k x[n-k] - \sum_{k=1}^{N} a_k y[n-k]\]
• IIR transfer function:
\[H(z) = \frac{\sum_{k=0}^{M} b_k z^{-k}}{1 + \sum_{k=1}^{N} a_k z^{-k}}\]
• Properties:
• Can be unstable (poles not necessarily at origin)
• Uses feedback
• More efficient than FIR for similar specifications

Filter Realizations

• Direct Form I: Implements numerator and denominator separately
• Direct Form II: Canonical form with minimum number of delays
• Cascade form: Product of second-order sections
• Parallel form: Sum of first and second-order sections