Classification of signals in signal and system: An introduction

In "signals and systems," we often need to categorize these signals to understand them better. It's like sorting different types of fruits to know what each one is like. So, let's look at the main classification of signals in signal and system. We'll also touch upon the types of signals in signal and system with simple examples.

1. Continuous Time and Discrete Time Signals

Imagine a light whose brightness can change smoothly, at every single moment. That's like a continuous time signal.

• Continuous Time Signals: These signals are defined for all points in time. If you pick any moment, the signal will have a value. Think of the temperature of a room changing throughout the day – it's a continuous curve. We usually write them as x(t).
  
  
  

Now, imagine taking a photo every hour. You only have the brightness of the light at those specific hours. That's like a discrete time signal.

• Discrete Time Signals: These signals are only defined at specific, separate points in time. We often get them by taking samples of a continuous signal. Think of the daily stock market closing price – you only have a value for the end of each day. We usually write them as x(n).

So, one is like a flowing river (continuous), and the other is like stepping stones in the river (discrete).

2. Deterministic and Non-deterministic Signals

Sometimes, you can perfectly predict the future value of a signal based on its past. That's a deterministic signal.

• Deterministic Signals: Their values are completely predictable. For example, if you have a sine wave with a known frequency and amplitude, you know exactly what its value will be at any future time.
  
  
  

Other times, the signal might have some randomness to it, and you can't predict its exact future values. That's a non-deterministic signal (also called a random signal).

• Non-deterministic Signals: Their future values cannot be precisely predicted. Think of the noise you sometimes hear on the radio, or the fluctuations in the stock market.
  
  
  

So, one is predictable like the sunrise every day (deterministic in a way!), and the other is a bit like guessing the outcome of a cricket match (non-deterministic).

3. Even and Odd Signals

These classifications are about the symmetry of a signal.

• Even Signals: A signal x(t) (for continuous time) or x(n) (for discrete time) is even if it's the same when you flip it around the vertical axis. Mathematically, $x(t)=x(-t)$ or $x(n)=x(-n)$. Think of a parabola $y=t^2$; it looks the same on both sides of the y-axis.

• Odd Signals: A signal is odd if flipping it around the vertical axis and then flipping it upside down gives you back the original signal. Mathematically, $x(t)=-x(-t)$ or $x(n)=-x(-n)$. Think of the line $y=t$; if you flip it around both axes, you get it back.

Most signals are neither even nor odd, just like most people are neither perfectly mirror-imaged nor perfectly anti-mirrored!

4. Periodic and Aperiodic Signals

Have you seen a clock's second hand go around and around, repeating its movement? That's a periodic signal.

• Periodic Signals: A signal repeats itself after a fixed interval of time (or in discrete time, after a fixed number of samples). For continuous time, $x(t)=x(t+T)$ for some $T>0$ (the period). For discrete time, $x(n)=x(n+N)$ for some integer $N>0$ (the period).

If a signal does not repeat itself, it's called an aperiodic signal.

• Aperiodic Signals: These signals do not have a repeating pattern. Most real-world signals, like a person speaking, are aperiodic.

So, one repeats like the seasons (periodic), and the other is unique like your fingerprint (aperiodic).

T= fundamental time period

1/T = f = fundamental frequency

5. Energy and Power Signals

This classification depends on the total energy and average power of a signal.

• Energy Signals: These signals have finite total energy. Roughly speaking, they die out over time. A short pulse is an example of an energy signal.

                                          EnergyE=∫∞−∞x2(t)dt

• Power Signals: These signals have finite average power (but infinite total energy). Periodic signals that continue forever are usually power signals. A continuous sine wave is an example.

                                     PowerP=limT→∞12T∫T−Tx2(t)dt

A signal cannot be both an energy signal and a power signal (unless it's a zero signal).

6. Real and Imaginary Signals

This one is simpler and relates to the values the signal takes.

• Real Signals: The value of the signal at any time is a real number. Most of the signals we encounter in basic applications (like temperature, audio voltage) are real.

• Imaginary Signals: The value of the signal at any time is an imaginary number (a multiple of $j=\sqrt{-1}$).

• In many practical scenarios, we deal with real signals, but imaginary signals are important in mathematical analysis, especially when using complex exponentials.

A signal is said to be real when it satisfies the condition x(t) = x*(t)

A signal is said to be imaginary when it satisfies the condition x(t) = -x*(t)

Ex: if X(3), then X(t)=3*=3, then X(t) is real signal.

      if X(3j), then X(t)=3j*=3J, then X(t) is imaginary signal. 

Conclusion:

So, we've seen that signals can be categorized in many ways: by whether they are continuous or discrete, predictable or random, symmetric or not, repeating or not, based on their energy or power, and whether their values are real or imaginary. This classification of signals with examples helps us to analyze and work with them effectively in the field of signal and system.

Understanding these types of signals is the first step in understanding how systems process them. Just like knowing the different ingredients helps you cook a good dish, knowing the different classification of signals helps you understand how electronics and communication work.

Frequently Asked Questions (FAQs) 

1. What are the main types of signals in "signals and systems"?

The main types include continuous time, discrete time, deterministic, non-deterministic, even, odd, periodic, aperiodic, energy, power, real, and imaginary signals.

2. Can you explain the difference between continuous and discrete signals with examples?
Continuous signals are defined at all times (e.g., temperature reading), while discrete signals are defined at specific time points (e.g., daily stock prices).

3. What is signal classification with examples of even and odd signals?
Signal classification based on symmetry includes even signals (where $x(t)=x(-t)$ or $x(n)=x(-n)$) and odd signals (where $x(t)=-x(-t)$ or $x(n)=-x(-n)$). An example of an even function is t2, and an odd function is t.

4. How are periodic and aperiodic signals classified?
Periodic signals repeat over a fixed interval, while aperiodic signals do not have a repeating pattern.

5. What are energy and power signals?
Energy signals have finite total energy, whereas power signals have finite average power.

6. Why is understanding the types of signals important in signal and system analysis?
Classifying signals helps in choosing the right mathematical tools and techniques to analyze and process them effectively.