Signals and Systems Cheat Sheet

Cheatsheet Content

1. Z-Transform (ZT) Bilateral Z-Transform: $$ X(Z) = \sum_{n=-\infty}^{\infty} x(n)z^{-n} $$ Unilateral Z-Transform: $$ X(Z) = \sum_{n=0}^{\infty} x(n)z^{-n} $$ Inverse Z-Transform: $$ x(n) = \frac{1}{2\pi j} \oint_C X(Z)Z^{n-1} dZ $$ 2. Properties of Z-Transform Linearity: If $Z\{x_1(n)\} = X_1(z)$ and $Z\{x_2(n)\} = X_2(z)$, then $Z\{a_1 x_1(n) + a_2 x_2(n)\} = a_1 X_1(z) + a_2 X_2(z)$. Time Shifting: If $Z\{x(n)\} = X(z)$, then $Z\{x(n-m)\} = z^{-m}X(z)$. Time Reversal: If $Z\{x(n)\} = X(z)$, then $Z\{x(-n)\} = X(z^{-1})$. Multiplication by an Exponential Sequence: If $Z\{x(n)\} = X(z)$, then $Z\{a^n x(n)\} = X(a^{-1}z)$. Differentiation Property: If $Z\{x(n)\} = X(z)$, then $Z\{nx(n)\} = -z \frac{dX(z)}{dz}$. Convolution Property: If $Z\{x_1(n)\} = X_1(z)$ and $Z\{x_2(n)\} = X_2(z)$, then $Z\{x_1(n) * x_2(n)\} = X_1(z)X_2(z)$. Conjugate Property: If $Z\{x(n)\} = X(z)$, then $Z\{x^*(n)\} = X^*(z^*)$. Initial Value Theorem: If $Z\{x(n)\} = X(z)$, then $x(0) = \lim_{z \to \infty} X(z)$. Final Value Theorem: If $Z\{x(n)\} = X(z)$, then $\lim_{n \to \infty} x(n) = \lim_{z \to 1} (z-1)X(z)$, if $(z-1)X(z)$ has no poles on or outside the unit circle. 3. Region of Convergence (ROC) of Z-Transform ROC is indicated by circles in the z-plane. ROC does not contain any poles. For causal sequences, ROC is the exterior of a circle. For anti-causal sequences, ROC is the interior of a circle. For finite duration causal sequences, ROC is the entire z-plane except $z=0$. For finite duration anti-causal sequences, ROC is the entire z-plane except $z=\infty$. 4. Z-Transform of Basic Signals $x(n)$ $X(Z)$ $\delta(n)$ $1$ $u(n)$ $\frac{Z}{Z-1}$ $a^n u(n)$ $\frac{Z}{Z-a}$ $nu(n)$ $\frac{Z}{(Z-1)^2}$ $n a^n u(n)$ $\frac{aZ}{(Z-a)^2}$ $\cos(\omega_0 n) u(n)$ $\frac{Z(Z-\cos \omega_0)}{Z^2 - 2Z\cos \omega_0 + 1}$ $\sin(\omega_0 n) u(n)$ $\frac{Z\sin \omega_0}{Z^2 - 2Z\cos \omega_0 + 1}$ 5. Solution to Linear Difference Equations using ZT Apply Z-transform to both sides of the difference equation. Use initial conditions. Solve for $Y(z)$ or $X(z)$ (algebraic equation). Apply inverse Z-transform to find the time-domain solution. 6. Relationship between Laplace Transform and Z-Transform Laplace transform for continuous-time systems (s-plane: $s = \sigma + j\omega$). Z-transform for discrete-time systems (z-plane: $z = r e^{j\omega}$). The relationship is $z = e^{sT}$ (Impulse Invariant method) or $s = \frac{2}{T} \frac{Z-1}{Z+1}$ (Bilinear Transform method). Mapping of s-plane to z-plane: LHS of s-plane maps to inside unit circle of z-plane. RHS of s-plane maps to outside unit circle of z-plane. $j\omega$-axis of s-plane maps to unit circle of z-plane. 7. Discrete Time Fourier Transform (DTFT) Definition: $$ X(\omega) = \sum_{n=-\infty}^{\infty} x(n)e^{-j\omega n} $$ Inverse DTFT: $$ x(n) = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(\omega)e^{j\omega n} d\omega $$ Periodicity: $X(\omega)$ is periodic with $2\pi$. 8. Properties of DTFT Linearity: $a x_1(n) + b x_2(n) \leftrightarrow a X_1(\omega) + b X_2(\omega)$. Time Shifting: $x(n-n_0) \leftrightarrow e^{-j\omega n_0} X(\omega)$. Frequency Shifting: $e^{j\omega_0 n} x(n) \leftrightarrow X(\omega - \omega_0)$. Time Reversal: $x(-n) \leftrightarrow X(-\omega)$. Convolution: $x_1(n) * x_2(n) \leftrightarrow X_1(\omega)X_2(\omega)$. Multiplication: $x_1(n)x_2(n) \leftrightarrow \frac{1}{2\pi} (X_1(\omega) * X_2(\omega))$. Parseval's Relation: $$ \sum_{n=-\infty}^{\infty} |x(n)|^2 = \frac{1}{2\pi} \int_{-\pi}^{\pi} |X(\omega)|^2 d\omega $$ 9. Comparison: Continuous-time Signal Analysis vs. Discrete-time Signal Analysis Feature Continuous-time Signal Discrete-time Signal Representation Analog, natural signal Digital, sampled signal Conversion CT to DT Euler's method Sampling Conversion DT to CT Complicated (sample & hold) Zero-order hold, first-order hold Domain Continuous (any point $t$) Discrete (sampling instants $n$) Notation $x(t)$ $x[n]$ Analysis Tool Laplace Transform, Fourier Transform Z-Transform, DTFT