Signals and System Cheat Sheet Table

Z-Transform

The Z-transform of a discrete-time signal $x[n]$ is defined as:

$$X(z)=\mathcal{Z}\{x[n]\}=\sum _{n=-\infty }^{\infty }x[n]z^{-n}$$

Fourier Transform

The Fourier transform of a continuous-time signal $x(t)$ is defined as:

$$X(f)=\mathcal{F}\{x(t)\}={\int }_{-\infty }^{\infty }x(t)e^{-j2\pi ft}dt$$

Laplace Transform

The Laplace transform of a continuous-time signal $x(t)$ is defined as:

$$X(s)=\mathcal{L}\{x(t)\}={\int }_0^{\infty }x(t)e^{-st}dt$$

Initial and Final Value Theorems

Initial Value Theorem

For a Laplace transform $X(s)$, the initial value of $x(t)$ at $t=0^{+}$ is given by:

$$x(0^{+})=\underset{s\to \infty }{\lim }sX(s)$$

Final Value Theorem

For a Laplace transform $X(s)$, the final value of $x(t)$ as $t\to \infty$ is given by:

$$\underset{t\to \infty }{\lim }x(t)=\underset{s\to 0}{\lim }sX(s)$$

Impulse Response from Transfer Function

The impulse response $h(t)$ of a system with transfer function $H(s)$ is given by the inverse Laplace transform:

$$h(t)={\mathcal{L}}^{-1}\{H(s)\}$$

Properties of Fourier Transform

Property	Time Domain	Frequency Domain
Linearity	$a{x}_1(t)+b{x}_2(t)$	$a{X}_1(f)+b{X}_2(f)$
Time Shifting	$x(t-t_0)$	$X(f)e^{-j2\pi f{t}_0}$
Frequency Shifting (Modulation)	$x(t)e^{j2\pi f_{0}t}$	$X(f-f_0)$
Time Scaling	$x(at)$	$\frac{1}{
Convolution	$(x_1\ast x_2)(t)={\int }_{-\infty }^{\infty }{x}_1(\tau )x_2(t-\tau )d\tau$	$X_1(f)X_2(f)$
Duality	$X(t)$	$x(-f)$
Differentiation in Time	$\frac{d}{dt}x(t)$	$j2\pi fX(f)$
Differentiation in Frequency	$-jtx(t)$	$\frac{d}{df}X(f)$
Parseval’s Theorem	$\int_{-\infty}^{\infty}	x(t)

---

These formulae and properties are fundamental in signal processing and systems analysis, providing the tools necessary to analyze and design systems in both the time and frequency domains.

Common Laplace Transforms

$f(t)$	$F(s)$
$e^{at}$	$\frac{1}{s-a}$, where $\Re (s)>a$
$e^{-at}$	$\frac{1}{s+a}$, where $\Re (s)>-a$
$u(t)$	$\frac{1}{s}$, where $\Re (s)>0$
$u(t-t_0)$	$\frac{e^{-t_{0}s}}{s}$, where $\Re (s)>0$
$\sin (at)$	$\frac{a}{s^2+a^2}$
$\cos (at)$	$\frac{s}{s^2+a^2}$
$\sin (at)u(t)$	$\frac{a}{s^2+a^2}$, where $\Re (s)>0$
$\cos (at)u(t)$	$\frac{s}{s^2+a^2}$, where $\Re (s)>0$
$t^n$	$\frac{n!}{s^{n+1}}$, where $\Re (s)>0$
$\delta (t)$	$1$
$tu(t)$	$\frac{1}{s^2}$
$e^{at}u(t)$	$\frac{1}{s-a}$, where $\Re (s)>a$

---

Common Fourier Transforms

$f(t)$	$F(\omega )$
$e^{j{\omega }_{0}t}$	$2\pi \delta (\omega -{\omega }_0)$
$e^{-j{\omega }_{0}t}$	$2\pi \delta (\omega +{\omega }_0)$
$u(t)$	$\frac{1}{j\omega }+\pi \delta (\omega )$
$\sin ({\omega }_{0}t)$	$\pi \left[\delta (\omega -{\omega }_0)-\delta (\omega +{\omega }_0)\right]$
$\cos ({\omega }_{0}t)$	$\pi \left[\delta (\omega -{\omega }_0)+\delta (\omega +{\omega }_0)\right]$
$\text{rect}(t/T)$	$T\cdot \text{sinc}(\omega T)$
$e^{-a{t}^2}$	$\sqrt{\frac{\pi }{a}}{e}^{-\frac{{\omega }^2}{4a}}$
$\delta (t)$	$1$

---

Common Z-Transforms

$x[n]$	$X(z)$
$a^{n}u[n]$	$\frac{1}{1-a{z}^{-1}}$, for $
$u[n]$	$\frac{1}{1-z^{-1}}$, for $
$n^{k}u[n]$	$\frac{k!}{(1-z^{-1}{)}^{k+1}}$, for $
$\delta [n]$	$1$
$\sin ({\omega }_{0}n)u[n]$	$\frac{\sin ({\omega }_0)}{1-2z^{-1}\cos ({\omega }_0)+z^{-2}}$, for $
$\cos ({\omega }_{0}n)u[n]$	$\frac{1-z^{-1}\cos ({\omega }_0)}{1-2z^{-1}\cos ({\omega }_0)+z^{-2}}$, for $
$r^{n}u[n]$	$\frac{1}{1-r{z}^{-1}}$, for $
$e^{-an}u[n]$	$\frac{1}{1-a{e}^{-1}{z}^{-1}}$, for $
$nu[n]$	$\frac{z}{(z-1{)}^2}$, for $

---

Conclusion

Here are the tables for common transforms:

1. Laplace Transforms: These are used primarily for analyzing continuous-time signals and systems, and they have many common forms like exponential functions, sinusoids, and step functions.

2. Fourier Transforms: Used for converting signals from the time domain to the frequency domain, particularly helpful in analyzing frequency content. The transforms of sinusoidal signals, step functions, and impulses are frequently used.

3. Z-Transforms: Primarily used in discrete-time signal processing, Z-transforms are useful for analyzing discrete-time signals, especially when dealing with difference equations.

These tables and methods are the foundation for analyzing and solving problems in signal processing and systems theory.

Signals and System List of Signals

Signals & System List of Signals

Signals and systems can be classified into several categories based on various properties. Here’s a breakdown of the most common types:

1. Causal vs Non-Causal Systems:

• Causal System: The output of the system at any time depends only on present and past inputs, not future inputs.
  Example: $y(t)=x(t)+x(t-1)$
• Non-Causal System: The output of the system depends on future inputs as well.
  Example: $y(t)=x(t+1)+x(t)$

2. Static vs Dynamic Systems:

• Static (Memoryless) System: The output at any time depends only on the input at that exact time.
  Example: $y(t)=2x(t)$
• Dynamic System: The output at any time depends on past or future inputs, i.e., the system has memory.
  Example: $y(t)=x(t)+x(t-1)$

3. Linear vs Non-Linear Systems:

• Linear System: The system obeys the principles of superposition (additivity and homogeneity). Mathematically, a system is linear if: $y(a{x}_1+b{x}_2)=ay(x_1)+by(x_2)$
• Non-Linear System: The system does not satisfy superposition. Non-linear systems may exhibit complex behaviors like chaos or harmonics.
  Example: $y(t)=x^2(t)$

4. Time-Invariant vs Time-Variant Systems:

• Time-Invariant System: The behavior of the system does not change over time. If you shift the input, the output shifts accordingly.
  Example: $y(t)=x(t-1)$
• Time-Variant System: The system’s properties change over time. Shifting the input may result in a different output.
  Example: $y(t)=t\cdot x(t)$

5. Deterministic vs Stochastic (Random) Signals:

• Deterministic Signal: The signal can be exactly described by a mathematical function, and its future values are completely predictable.
  Example: $x(t)=\cos (2\pi ft)$
• Stochastic (Random) Signal: The signal cannot be predicted exactly due to inherent randomness. It is often described using statistical properties like mean and variance.
  Example: White noise

6. Continuous-Time vs Discrete-Time Signals:

• Continuous-Time Signal: The signal is defined for every time instant, and time is considered a continuous variable.
  Example: $x(t)=e^{-t}$
• Discrete-Time Signal: The signal is defined only at specific time instants (usually integer values), meaning time is considered a discrete variable.
  Example: $x[n]=\sin (2\pi fn)$

7. Periodic vs Aperiodic Signals:

• Periodic Signal: The signal repeats itself after a fixed interval of time (period $T$).
  Example: $x(t)=\sin (2\pi ft)$, where $T=\frac{1}{f}$
• Aperiodic Signal: The signal does not exhibit periodic behavior.
  Example: A single pulse or decaying exponential $x(t)=e^{-t}$

8. Even vs Odd Signals:

• Even Signal: A signal is even if it is symmetric about the vertical axis, meaning $x(t)=x(-t)$.
  Example: $x(t)=\cos (t)$
• Odd Signal: A signal is odd if it is anti-symmetric about the vertical axis, meaning $x(t)=-x(-t)$.
  Example: $x(t)=\sin (t)$

9. Energy vs Power Signals:

• Energy Signal: A signal is an energy signal if it has finite energy, meaning: ${\int }_{-\infty }^{\infty }|x(t){|}^{2}dt<\infty$ Example: A decaying exponential signal
• Power Signal: A signal is a power signal if it has finite average power but infinite energy. $P={\lim }_{T\to \infty }\frac{1}{2T}{\int }_{-T}^T|x(t){|}^{2}dt$ Example: A periodic signal like $x(t)=\sin (2\pi ft)$

10. Stable vs Unstable Systems:

• Stable System: A system is stable if bounded inputs always result in bounded outputs (BIBO stability).
  Example: $y(t)=e^{-t}x(t)$
• Unstable System: A system is unstable if bounded inputs can lead to unbounded outputs.
  Example: $y(t)=e^{t}x(t)$

11. Causal vs Non-Causal Signals:

• Causal Signal: A signal is causal if it is zero for all $t<0$.
  Example: $x(t)=u(t)$, where $u(t)$ is the unit step function.
• Non-Causal Signal: A signal that has non-zero values for times before $t=0$.
  Example: $x(t)=e^t$

12. Analog vs Digital Signals:

• Analog Signal: A signal with continuously varying amplitude over time.
  Example: An audio signal from a microphone
• Digital Signal: A signal that takes discrete values in both time and amplitude.
  Example: A binary signal representing a sequence of 0s and 1s

13. Symmetric vs Anti-Symmetric Systems:

• Symmetric System: The system behaves the same when input is reversed in time.
• Anti-Symmetric System: The output reverses sign when the input is reversed in time.

14. Bounded vs Unbounded Signals:

• Bounded Signal: The amplitude of the signal does not exceed a certain finite value.
  Example: $x(t)=\cos (2\pi ft)$
• Unbounded Signal: The amplitude of the signal grows without bound.
  Example: $x(t)=t^2$

---

Yes, a signal can be classified as either causal or non-causal based on when it exists in time.

1. Causal Signal:

A signal is said to be causal if it is zero for all time before a specific point, typically before $t=0$. In other words, the signal only exists or starts after time $t=0$. This is relevant in systems where the future input doesn’t affect the current output.

$$x(t)=0\text{for all}t<0$$

Example:

The unit step function $u(t)$ is a causal signal because it is zero for $t<0$ and 1 for $t\ge 0$:

$$u(t)=\{\begin{array}{ll}0,& t<0\\ 1,& t\ge 0\end{array}$$

2. Non-Causal Signal:

A signal is non-causal if it has non-zero values for times before the present (before $t=0$), meaning it can exist in the past, present, or even future. This implies that the signal may be defined for negative time as well.

$$x(t)\ne 0\text{for some}t<0$$

Example:

A signal like $x(t)=e^t$ for all $t$ is non-causal because it is non-zero for $t<0$.

Summary:

• Causal Signal: Exists or has non-zero values only for $t\ge 0$.
• Non-Causal Signal: Can have non-zero values for $t<0$.

---

Link to original