Linear Time-Invariant System:
Linear time-invariant systems (LTI systems) are a class of systems used in signals and systems that are both linear and time-invariant. Linear systems are systems whose outputs for a linear combination of inputs are the same as a linear combination of individual responses to those inputs. Time-invariant systems are systems where the output does not depend on when an input was applied. These properties make LTI systems easy to represent and understand graphically.
Linear systems have the property that the output is linearly related to the input. Changing the input in a linear way will change the output in the same linear way. So if the input x1(t) produces the output y1(t) and the input x2(t) produces the output y2(t), then linear combinations of those inputs will produce linear combinations of those outputs. The input {x1(t)+x2(t)} will produce the output {y1(t)+y2(t)}. Further, the input {a1x1(t)+a2x2(t)} will produce the output {a1y1(t)+a2y2(t)} for some constants a1 and a2.
In other words, for a system T over time t, composed of signals x1(t) and x2(t) with outputs y1(t) and y2(t)
Homogeneity Principle:
Superposition Principle:
Thus, the entirety of an LTI system can be described by a single function called its impulse response. This function exists in the time domain of the system. For an arbitrary input, the output of an LTI system is the convolution of the input signal with the system's impulse response.
Conversely, the LTI system can also be described by its transfer function. The transfer function is the Laplace transform of the impulse response. This transformation changes the function from the time domain to the frequency domain. This transformation is important because it turns differential equations into algebraic equations, and turns convolution into multiplication.
In the frequency domain, the output is the product of the transfer function with the transformed input. The shift from time to frequency is illustrated in the following image:
Homogeneity, additivity, and shift invariance may, at first, sound a bit abstract but they are very useful. To characterize a shift-invariant linear system, we need to measure only one thing: the way the system responds to a unit impulse. This response is called the impulse response function of the system. Once we’ve measured this function, we can (in principle) predict how the system will respond to any other possible stimulus.
Introduction to Convolution
Because here’s not a single answer to define what is ? In “Signals and Systems” probably we saw convolution in connection with Linear Time Invariant Systems and the impulse response for such a system. This multitude of interpretations and applications is somewhat like the situation with the definite integral.
To pursue the analogy with the integral, in pretty much all applications of the integral there is a general method at work:
Convolution Theorem
F(g∗f)(s) = Fg(s)Ff(s)
Convolving in the Frequency Domain
By applying Duality Formula
F(Ff)(s)=f(−s) or F(Ff)=f− without the variable.
Note: Here we are trying to prove F(gf)(s) = (Fg∗Ff)(s) rather than F(g∗f)=(Ff)(Fg) Because, it seems more “natural” to multiply signals in the time domain and see what effect this has in the frequency domain, so why not work with F(fg) directly? But write the integral for F(gf); there’s nothing you can do with it to get toward Fg∗Ff.
There is also often a general method of convolutions:
Some of you who have seen convolution in earlier courses,you’ve probably heard the expression “flip and drag”
Meaning of Flip & Drag: here’s the meaning of Flip & Drag is as follow
Averaging
Smoothing
Other identities of Convolution
It’s not hard to combine the various rules we have and develop an algebra of convolutions. Such identities can be of great use — it beats calculating integrals. Here’s an assortment. (Lower and uppercase letters are Fourier pairs.)
Properties of Convolution
Here we are explaining the properties of convolution in both continuous and discrete domain
We shall now discuss the important properties of convolution for LTI systems.
1) Commutative property:
2. Distributive Property
By this Property we will conclude that convolution is distributive over addition.
3. Associative Property
Let Substitute λ3 = λ2
λ4 = λ1 - λ2
then the jacobian for the above transformation is
4. Invertibility
A system is said to be invertible if there exist an inverse system which when connected in series with the original system produces an output identical to input .
(x*δ)[n]= x[n]
(x*h*h-1)[n]= x[n]
(h*h-1)[n]= (δ)[n]
5. Causality
6. Stability
Sampling Theorem
Aliasing & Anti-aliasing
Condition For Sampling
This maximum value of n is given by
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