SS Notes
Essential formulas reference for Signals and Systems — transform pairs, properties, system analysis formulas, and key relationships.
Introduction
This page collects the most important formulas in Signals and Systems into a single reference. These are the equations you need to have at your fingertips during examinations, homework, and professional work. They are organized by topic — transforms, properties, system analysis, and sampling — with brief notes on when each formula applies. Use this as a quick-reference sheet, but make sure you understand the derivation and significance of each formula, not just its form.
The formulas are grouped by the order you typically encounter them in a course: starting with basic signal operations, progressing through continuous-time and discrete-time transforms, and concluding with sampling theory and system stability conditions. Each formula includes its conditions of validity and common applications.
Signal Operations
Time reversal: $y(t) = x(-t)$ — reflects signal about the origin
Time scaling: $y(t) = x(at)$ — compresses if $|a|>1$, expands if $|a|<1$
Combined transformation: $y(t) = x(at - b) = x(a(t - b/a))$ — scale first, then shift by $b/a$
Even/Odd decomposition: $$x_e(t) = \frac{x(t) + x(-t)}{2}, \quad x_o(t) = \frac{x(t) - x(-t)}{2}$$
Fourier Transform Pairs
| Signal $x(t)$ | Transform $X(j\omega)$ | ||
|---|---|---|---|
| $\delta(t)$ | $1$ | ||
| $1$ | $2\pi\delta(\omega)$ | ||
| $u(t)$ | $\pi\delta(\omega) + \frac{1}{j\omega}$ | ||
| $e^{-at}u(t), a>0$ | $\frac{1}{a+j\omega}$ | ||
| $te^{-at}u(t)$ | $\frac{1}{(a+j\omega)^2}$ | ||
| $t^n e^{-at}u(t)$ | $\frac{n!}{(a+j\omega)^{n+1}}$ | ||
| $e^{-a | t | }$ | $\frac{2a}{a^2+\omega^2}$ |
| $\cos(\omega_0 t)$ | $\pi[\delta(\omega-\omega_0)+\delta(\omega+\omega_0)]$ | ||
| $\sin(\omega_0 t)$ | $\frac{\pi}{j}[\delta(\omega-\omega_0)-\delta(\omega+\omega_0)]$ | ||
| $\text{rect}(t/T)$ | $T\text{sinc}(\omega T/2\pi)$ | ||
| $\text{sinc}(Wt)$ | $\frac{\pi}{W}\text{rect}(\omega/2\pi W)$ | ||
| $e^{j\omega_0 t}$ | $2\pi\delta(\omega-\omega_0)$ | ||
| $\text{sgn}(t)$ | $\frac{2}{j\omega}$ | ||
| $e^{-at}\sin(\omega_0 t)u(t)$ | $\frac{\omega_0}{(a+j\omega)^2+\omega_0^2}$ | ||
| $e^{-at}\cos(\omega_0 t)u(t)$ | $\frac{a+j\omega}{(a+j\omega)^2+\omega_0^2}$ |
Fourier Transform Properties
| Property | Time Domain | Frequency Domain | ||||
|---|---|---|---|---|---|---|
| Linearity | $ax_1+bx_2$ | $aX_1+bX_2$ | ||||
| Time shift | $x(t-t_0)$ | $e^{-j\omega t_0}X(j\omega)$ | ||||
| Freq shift | $e^{j\omega_0 t}x(t)$ | $X(j(\omega-\omega_0))$ | ||||
| Scaling | $x(at)$ | $\frac{1}{ | a | }X(j\omega/a)$ | ||
| Convolution | $x_1*x_2$ | $X_1 \cdot X_2$ | ||||
| Multiplication | $x_1 \cdot x_2$ | $\frac{1}{2\pi}X_1*X_2$ | ||||
| Differentiation | $\frac{dx}{dt}$ | $j\omega X(j\omega)$ | ||||
| $n$-th derivative | $\frac{d^n x}{dt^n}$ | $(j\omega)^n X(j\omega)$ | ||||
| Integration | $\int_{-\infty}^t x(\tau)d\tau$ | $\frac{X(j\omega)}{j\omega} + \pi X(0)\delta(\omega)$ | ||||
| Time multiplication | $tx(t)$ | $j\frac{dX}{d\omega}$ | ||||
| Conjugation | $x^*(t)$ | $X^*(-j\omega)$ | ||||
| Parseval | $\int | x | ^2 dt$ | $\frac{1}{2\pi}\int | X | ^2 d\omega$ |
Laplace Transform Pairs
| $x(t)$ | $X(s)$ | ROC |
|---|---|---|
| $\delta(t)$ | $1$ | All $s$ |
| $u(t)$ | $1/s$ | $\text{Re}(s)>0$ |
| $e^{-at}u(t)$ | $1/(s+a)$ | $\text{Re}(s)>-a$ |
| $t^n e^{-at}u(t)$ | $n!/(s+a)^{n+1}$ | $\text{Re}(s)>-a$ |
| $\cos(\omega_0 t)u(t)$ | $s/(s^2+\omega_0^2)$ | $\text{Re}(s)>0$ |
| $\sin(\omega_0 t)u(t)$ | $\omega_0/(s^2+\omega_0^2)$ | $\text{Re}(s)>0$ |
| $e^{-at}\cos(\omega t)u(t)$ | $(s+a)/((s+a)^2+\omega^2)$ | $\text{Re}(s)>-a$ |
| $e^{-at}\sin(\omega t)u(t)$ | $\omega/((s+a)^2+\omega^2)$ | $\text{Re}(s)>-a$ |
| $t\cos(\omega_0 t)u(t)$ | $(s^2-\omega_0^2)/(s^2+\omega_0^2)^2$ | $\text{Re}(s)>0$ |
| $-e^{-at}u(-t)$ | $1/(s+a)$ | $\text{Re}(s)<-a$ |
Laplace Transform Properties
| Property | Time Domain | $s$-Domain |
|---|---|---|
| Time shift | $x(t-t_0)u(t-t_0)$ | $e^{-st_0}X(s)$ |
| $s$-shift | $e^{-at}x(t)$ | $X(s+a)$ |
| Differentiation | $dx/dt$ | $sX(s) - x(0^-)$ |
| 2nd derivative | $d^2x/dt^2$ | $s^2X(s) - sx(0^-) - x'(0^-)$ |
| Integration | $\int_0^t x(\tau)d\tau$ | $X(s)/s$ |
| Convolution | $x_1 * x_2$ | $X_1(s) \cdot X_2(s)$ |
| Initial value | $x(0^+)$ | $\lim_{s\to\infty} sX(s)$ |
| Final value | $x(\infty)$ | $\lim_{s\to 0} sX(s)$ (if stable) |
Z-Transform Pairs
| $x[n]$ | $X(z)$ | ROC | ||||
|---|---|---|---|---|---|---|
| $\delta[n]$ | $1$ | All $z$ | ||||
| $u[n]$ | $z/(z-1)$ | $ | z | >1$ | ||
| $a^n u[n]$ | $z/(z-a)$ | $ | z | > | a | $ |
| $na^n u[n]$ | $az/(z-a)^2$ | $ | z | > | a | $ |
| $-a^n u[-n-1]$ | $z/(z-a)$ | $ | z | < | a | $ |
| $n^2 a^n u[n]$ | $az(z+a)/(z-a)^3$ | $ | z | > | a | $ |
| $\cos(\omega_0 n)u[n]$ | $z(z-\cos\omega_0)/(z^2-2z\cos\omega_0+1)$ | $ | z | >1$ | ||
| $a^n\cos(\omega_0 n)u[n]$ | $z(z-a\cos\omega_0)/(z^2-2az\cos\omega_0+a^2)$ | $ | z | > | a | $ |
Convolution Formulas
Continuous: $y(t) = \int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau$
Discrete: $y[n] = \sum_{k=-\infty}^{\infty}x[k]h[n-k]$
Output length (discrete, finite sequences): $L_y = L_x + L_h - 1$
Circular convolution (length $N$): $y[n] = \sum_{k=0}^{N-1}x[k]h[(n-k) \mod N]$
Convolution with delta: $x(t)*\delta(t-t_0) = x(t-t_0)$
Sampling and Reconstruction
Nyquist rate: $f_s > 2f_{max}$ (minimum sampling rate for alias-free reconstruction)
Nyquist interval: $T_s < 1/(2f_{max})$
Reconstruction (ideal): $x(t) = \sum_n x[n]\text{sinc}\left(\frac{t-nT_s}{T_s}\right)$
Aliased frequency: $f_{alias} = |f_0 - kf_s|$ (folded into $[0, f_s/2]$)
Oversampling ratio: $\text{OSR} = f_s / (2f_{max})$
Quantization noise power (uniform PCM): $\sigma_q^2 = \Delta^2/12$ where $\Delta = V_{FSR}/2^B$
SQNR: $6.02B + 1.76$ dB (for B-bit uniform PCM of full-range sinusoid)
Signal Energy and Power
Energy: $E = \int_{-\infty}^{\infty}|x(t)|^2 dt$
Power: $P = \lim_{T\to\infty}\frac{1}{2T}\int_{-T}^T |x(t)|^2 dt$
Periodic signal power: $P = \frac{1}{T_0}\int_0^{T_0}|x(t)|^2 dt$
Parseval (energy): $E = \frac{1}{2\pi}\int|X(j\omega)|^2 d\omega$
Parseval (power, Fourier Series): $P = \sum_{n=-\infty}^{\infty}|c_n|^2$
Cross-energy: $\int x(t)y^*(t)dt = \frac{1}{2\pi}\int X(j\omega)Y^*(j\omega)d\omega$
System Stability
Continuous LTI: Stable iff $\int|h(t)|dt < \infty$ iff all poles have $\text{Re}(s)<0$
Discrete LTI: Stable iff $\sum|h[n]| < \infty$ iff all poles have $|z|<1$
Marginal stability: Poles on the boundary ($j\omega$-axis or unit circle) with multiplicity 1
Routh-Hurwitz: For continuous systems, tests stability without finding pole locations
Jury's test: For discrete systems, analogous to Routh-Hurwitz
Fourier Series
Exponential: $x(t) = \sum_{n=-\infty}^{\infty}c_n e^{jn\omega_0 t}$, $c_n = \frac{1}{T}\int_T x(t)e^{-jn\omega_0 t}dt$
Trigonometric: $x(t) = a_0 + \sum_{n=1}^{\infty}[a_n\cos(n\omega_0 t) + b_n\sin(n\omega_0 t)]$
Coefficient relations: $a_0 = c_0$, $a_n = 2\text{Re}(c_n)$, $b_n = -2\text{Im}(c_n)$
Parseval (series): $P = \sum|c_n|^2 = a_0^2 + \frac{1}{2}\sum(a_n^2+b_n^2)$
Bandwidth approximation: Most energy in first $N$ harmonics where $|c_n|$ becomes negligible
Correlation Functions
Autocorrelation: $R_{xx}(\tau) = \int x(t)x(t+\tau)dt$
Cross-correlation: $R_{xy}(\tau) = \int x(t)y(t+\tau)dt$
Wiener-Khinchin: $S_{xx}(\omega) = \mathcal{F}\{R_{xx}(\tau)\}$
Filtered PSD: $S_{yy}(\omega) = |H(j\omega)|^2 S_{xx}(\omega)$
Key Takeaways
- Memorize the basic transform pairs — all others derive from properties
- The convolution theorem ($x*h \leftrightarrow X\cdot H$) is the single most important property
- Stability = absolute integrability/summability = poles in stable region
- Sampling theorem: $f_s > 2f_{max}$ for perfect reconstruction
- Parseval's theorem connects time-domain energy to frequency-domain energy
- Always specify ROC for Laplace and Z-transforms — the expression alone is ambiguous
- Initial and final value theorems provide quick checks on transform computations
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