Signals and systems
A signal is a dependent variable that varies with respect to some other independent variables. The independent variable can be time, spaceetc. Forexample, current flow in a resistor with respect to time is a signal. If a signal depends on only one variable, it is a one-dimensional signal. For example, the variation of room temperature with time is a one-dimensional signal. If a signal depends on more than one variable, it is a multi-dimensional signal. For example, X-ray images are multi-dimensional signal. If a signal can be represented by a mathematical equation, then the signal is a deterministic signal. If the signal cannot be represented by a mathematical equation, then it is a random signal.
Signals can be broadly classified into continuous time signals and discrete time signals.
If a signal is defined for all values of time 't', then it is a continuous time signal. Examples are x(t)=sin(2t) and x(t)=e-5t. The ECG wave form is a continuous time signal. If a signal is defined at certain discrete instant of time, then it is a discrete time signal. For example, the marks obtained by a student in several internal examinations is a discrete time signal. Most of the signals we come across everyday are continuous in nature. Discrete time signals can be obtained by sampling continuous time signals. The sampler output is discrete in time but continuous in amplitude. If this signal is passed through a quantizer, the output is a digital signal. A digital signal is discrete in time and in amplitude. A digital signal has got several advantages over its analog counterpart in the context of processing. The major advantage is its immunity to noise.
A system can be defined as a process that performs an operation on an input signal and produces another signal as output. Examples are electronic circuits such as amplifiers and filters. An amplifier amplifies the strength of the signal present at its input. A filter removes unwanted frequencies present at its input.
The systems can be classified as follows.
1. Continuous time and discrete time systems.
2. Static and dynamic systems.
3. Causal and non-causal systems.
4. Linear and non-linear systems.
5. Time-invariant and time-variant systems.
6. Stable and unstable systems.
A continuous time system has its input continuous in time and the result is also continuous in time. A discrete time system has discrete time input signal and discrete time output signal.
A static system has its output depends only on the present input value and not on past values. For example, y(t)=2x(t) and y(n)=nx(n) are static systems. A dynamic system has its output depends on past values also. For example, voltage across a capacitor V(t) = 1/C∫(i(t)dt) is dynamic system. In discrete time y(n)=x(n)+x(n-1) is a dynamic system.
A system is causal if the output at any time depends only on its present and past inputs. If the output depends on future input values then it is non-causal. For example, y(n)=x(n)+2x(n-1) is causal and y(t)=x(t)+2x(t+1) is non-causal.
A system is said to be linear if it obeys the theory of super position. For inputs treated as additive and scaled, the outputs must also be additive and scaled version. If the property is not obeyed, then the system is non-linear. For example, y(n)=nx(n) is linear and y(t)=x2(t) is non-linear.
A system is said to be time-invariant if its behavior and characteristics are fixed over time. For a time-invariant system a time shift in the input signal results in an identical time shift in the output. If the characteristics are varied over time it is a time variant system. For example,y(n)=x2(n-2) is a time-invariant system and y(t)=2x(t2) is a time-variant system.
A system is said to be stable if it produces a bounded output for a bounded input. Otherwise the system is unstable. For example, y(n)=nx(n) is an unstable system and y(t)=x(t)+x(t-1) is a stable system.
What are the best resources to learn about signals and systems
Text books on signals and systems are good enough to learn signals and systems. The text book 'Signals and Systems' authored by Alan.V. Oppenheim is one of the best books. Online lectures by Oppenheim is a good source to learn signals and systems.
Technique on Signals and Systems
Various techniques on signals and systems are
1. Fourier Series
2. Fourier Transform
3. Laplace Transform
5. Discrete Fourier Transform
According to Fourier series a periodic signal can be resolved into its sine and cosine waves. The harmonics present in a signal and their magnitude and phase can be found using Fourier series.
To analyses a signal or system it is easy if the signal is transformed from time domain to frequency domain. Laplace transform is applied to aperiodic signals and the signal can be analyzed. The various frequency components present in the signal, their magnitude and phase can be obtained. A continuous time system can be analyzed using Laplace transform and can be classified if the system is stable, causal, or LTI. The Laplace transform maps a function x (t) to a function X(S) of the complex variable 'S' where S= σ + jω. If the real part of the variable 'S' is set to zero i.e. σ=0, the transformation is Fourier transform.
The Z-transform is the discrete time counter part of Laplace transform. A discrete time signal or system can be analyzed and classified using Z-transform.
The Discrete Fourier Transform is used to evaluate the Fourier transform on a digital computer or any other DSP chips. DFT is obtained by sampling one period of the Fourier transform at a finite number of frequency points.
All students need to know to get started on signals and systems
To get started on signals and systems students must be familiar with infinite sum formulas, finite sum formulas, complex variables, how to find the magnitude phase of a complex variable, trigonometric identities, integral calculus and differential calculus.
Few tips to solve signals and systems problems
It is easy to analyze a signal or system in frequency domain instead of analyzing in time domain. So, transform the signal or system to frequency domain using transformation techniques. The problems can be solved efficiently if the properties of the techniques are used.
Convolution of two signals requires evaluation of an integral. If the transforms of the signals are multiplied and taking inverse transforms will give convolution result in easier way.
Knowledge of fundamental signals like unit step function, impulse function, ramp function and their properties will help to solve the problem.
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