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Quick reference cheatsheet — all key formulas, transform pairs, properties, and important results for signals and systems in one place.
Introduction
This cheatsheet compiles all the essential formulas, transform pairs, and key results from the entire Signals and Systems course into a single quick-reference page. Use it for last-minute exam revision, during problem-solving practice, or as a handy reference while working on projects. Everything here assumes standard notation: $x(t)$ for continuous-time, $x[n]$ for discrete-time, $\omega$ for angular frequency, $f$ for cyclic frequency.
Basic Signals
| Signal | Definition | Laplace Transform |
|---|---|---|
| Unit impulse $\delta(t)$ | Zero everywhere except origin, area = 1 | 1 |
| Unit step $u(t)$ | 1 for $t>0$, 0 for $t<0$ | $1/s$ |
| Unit ramp $r(t)$ | $t \cdot u(t)$ | $1/s^2$ |
| Exponential $e^{-at}u(t)$ | Decay for $a>0$ | $1/(s+a)$ |
| Cosine $\cos(\omega_0 t)$ | Periodic, frequency $\omega_0$ | $s/(s^2+\omega_0^2)$ |
| Sine $\sin(\omega_0 t)$ | Periodic, frequency $\omega_0$ | $\omega_0/(s^2+\omega_0^2)$ |
System Properties Quick Check
| Property | Test | Example failing | ||
|---|---|---|---|---|
| Linear | $T\{ax_1+bx_2\} = aT\{x_1\}+bT\{x_2\}$ | $y=x^2$ | ||
| Time-invariant | Delay input → delay output | $y(t)=tx(t)$ | ||
| Causal | Output doesn't depend on future | $y(t)=x(t+1)$ | ||
| Stable (BIBO) | $\int | h(t) | dt < \infty$ | $h(t)=u(t)$ |
| Memoryless | Output depends only on current input | $y=x*h$ |
Fourier Transform Pairs
| $x(t)$ | $X(\omega)$ | ||
|---|---|---|---|
| $\delta(t)$ | $1$ | ||
| $1$ | $2\pi\delta(\omega)$ | ||
| $u(t)$ | $\pi\delta(\omega) + 1/(j\omega)$ | ||
| $e^{-at}u(t)$ | $1/(a+j\omega)$ | ||
| $e^{-a | t | }$ | $2a/(a^2+\omega^2)$ |
| $\text{rect}(t/T)$ | $T\text{sinc}(\omega T/2\pi)$ | ||
| $\text{sinc}(Wt)$ | $(\pi/W)\text{rect}(\omega/2W)$ | ||
| $e^{j\omega_0 t}$ | $2\pi\delta(\omega-\omega_0)$ | ||
| $\cos(\omega_0 t)$ | $\pi[\delta(\omega-\omega_0)+\delta(\omega+\omega_0)]$ | ||
| $te^{-at}u(t)$ | $1/(a+j\omega)^2$ |
Fourier Transform Properties
| Property | Time Domain | Frequency Domain | ||
|---|---|---|---|---|
| Linearity | $ax_1+bx_2$ | $aX_1+bX_2$ | ||
| Time shift | $x(t-t_0)$ | $X(\omega)e^{-j\omega t_0}$ | ||
| Freq shift | $x(t)e^{j\omega_0 t}$ | $X(\omega-\omega_0)$ | ||
| Scaling | $x(at)$ | $(1/ | a | )X(\omega/a)$ |
| Differentiation | $dx/dt$ | $j\omega X(\omega)$ | ||
| Convolution | $x*h$ | $X \cdot H$ | ||
| Multiplication | $x \cdot h$ | $(1/2\pi)X*H$ |
Laplace Transform Properties
| Property | $x(t)$ | $X(s)$ |
|---|---|---|
| Linearity | $ax_1+bx_2$ | $aX_1+bX_2$ |
| Time shift | $x(t-a)u(t-a)$ | $e^{-as}X(s)$ |
| s-shift | $e^{at}x(t)$ | $X(s-a)$ |
| Differentiation | $dx/dt$ | $sX(s)-x(0^-)$ |
| Integration | $\int_0^t x d\tau$ | $X(s)/s$ |
| Convolution | $x*h$ | $X(s)H(s)$ |
| Initial value | $x(0^+)$ | $\lim_{s\to\infty} sX(s)$ |
| Final value | $x(\infty)$ | $\lim_{s\to 0} sX(s)$ |
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 | $ |
Z-Transform Properties
| Property | $x[n]$ | $X(z)$ |
|---|---|---|
| Time shift | $x[n-k]$ | $z^{-k}X(z)$ |
| z-scaling | $a^n x[n]$ | $X(z/a)$ |
| Differentiation | $nx[n]$ | $-z\frac{dX}{dz}$ |
| Convolution | $x*h$ | $X(z)H(z)$ |
| Initial value | $x[0]$ | $\lim_{z\to\infty}X(z)$ |
Sampling Theorem
- Nyquist rate: $f_s > 2f_{max}$
- Aliased frequency: $f_{alias} = |f_{signal} - kf_s|$ (nearest fold)
- Quantization SNR: $\text{SNR}_q = 6.02N + 1.76$ dB ($N$ = bits)
- Reconstruction: $x(t) = \sum x[n]\text{sinc}((t-nT_s)/T_s)$
Convolution Formulas
- CT: $y(t) = \int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau$
- DT: $y[n] = \sum_{k=-\infty}^{\infty}x[k]h[n-k]$
- Duration: If $x$ has duration $M$ and $h$ has duration $N$, output has duration $M+N-1$
Key Relationships
- $\delta(t) = du(t)/dt$, $u(t) = dr(t)/dt$
- $h(t) = ds(t)/dt$ (impulse response = derivative of step response)
- Stability: poles in LHP (s-domain) or inside unit circle (z-domain)
- Parseval: $\int|x|^2 dt = (1/2\pi)\int|X|^2 d\omega$
- For periodic signals: $P = \sum|c_n|^2$
Filter Design Quick Reference
- FIR length for Hamming window: $M \approx 6.6\pi/\Delta\omega$
- Butterworth order: $N = \frac{\log(A_s/A_p)}{\log(\omega_s/\omega_p)}$
- Bilinear transform: $s = (2/T)(1-z^{-1})/(1+z^{-1})$
- Pre-warp: $\Omega = (2/T)\tan(\omega/2)$
Common Exam Problem Patterns
Pattern 1: System Classification Given a system equation, determine if it's linear, time-invariant, causal, stable, and memoryless. Check each property systematically:
- Linear: superposition holds (T{ax₁ + bx₂} = aT{x₁} + bT{x₂})
- Time-invariant: delay input → delay output equally
- Causal: output depends only on present and past inputs
- Stable: bounded input → bounded output (BIBO)
- Memoryless: output depends only on current input value
Pattern 2: Fourier Transform Computation Use properties to avoid direct integration whenever possible:
- Time shift → multiply by exponential in frequency
- Frequency shift → multiply by exponential in time
- Differentiation in time → multiply by jω in frequency
- Convolution in time → multiplication in frequency
Pattern 3: Filter Analysis Given H(s) or H(z), determine filter type and characteristics:
- Identify poles and zeros
- Determine stability (poles in LHP for CT, inside unit circle for DT)
- Find magnitude response |H(jω)| at key frequencies (0, cutoff, ∞)
- Classify as lowpass, highpass, bandpass, or bandstop
Quick Tricks for Exams
- For convolution of two rectangular pulses of widths T₁ and T₂, the result is a triangle with base T₁+T₂
- Energy of x(t) = Energy of X(ω)/2π (Parseval's theorem)
- Final value: lim(t→∞) y(t) = lim(s→0) sY(s) — but verify all poles of sY(s) are in LHP first
- For periodic signals, use Fourier Series (discrete spectrum), not Fourier Transform
- ROC of causal signal is always to the right of the rightmost pole
- Minimum phase system: all poles AND zeros in LHP (s-domain) or inside unit circle (z-domain)
Important Constants and Formulas to Memorize
- sinc(0) = 1, sinc(n) = 0 for integer n ≠ 0
- rect(t) ↔ sinc(f), sinc(t) ↔ rect(f) — duality
- Gaussian remains Gaussian under Fourier Transform
- δ(at) = δ(t)/|a| — scaling property of impulse
- For causal LTI: h(t) = 0 for t < 0
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