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Hi friends, today we will learn What is Causal and non-causal system?. These two are very important system in control systems. These systems are distinguished from their input output relationship. Let us see these systems one by one. A)     Causal systems: Definition: A system is said to be causal system if its output depends on present and past inputs only and not on future inputs. Examples: The output of casual system depends on present and past inputs, it means y(n) is a function of x(n), x(n-1), x(n-2), x(n-3)…etc. Some examples of causal systems are given below: 1)      y(n) = x(n) + x(n-2) 2)      y(n) = x(n-1) – x(n-3) 3)      y(n) = 7x(n-5) Significance of causal systems: Since causal system does not include future input samples; such system is practically realizable. That mean such system can be implemented practically. Generally all real time systems ...

Linear or Non-linear Systems (Linearity Property): A linear system is a system which follows the superposition principle. Let us consider a system having its response as ‘T’, input as x(n) and it produces output y(n). This is shown in figure below: Let us consider two inputs. Input x1(n) produces output y1(n) and input x2(n) produces output y2(n). Now consider two arbitrary constants a1 and a2. Then simply multiply these constants with input x1(n) and x2(n) respectively. Thus a1x1(n) produces output a1y1(n) and a2x2(n) produces output a2y2(n). Theorem for linearity of the system: A system is said to be linear if the combined response of a1x1(n) and a2x2(n) is equal to the addition of the individual responses. That means, T[a1 x1(n) + a2 x2(n)] = a1 T[x1(n)] + a2 T[x2(n)]…………….1) The above theorem is also known as superposition theorem. Important Characteristic: Linear system has one important characteristic: If the input to the system is zero then it produces zero...

Time Variant or Time Invariant Systems Definition: A system is said to be Time Invariant if its input output characteristics do not change with time. Otherwise it is said to be Time Variant system. Explanation: As already mentioned time invariant systems are those systems whose input output characteristics do not change with time shifting. Let us consider x(n) be the input to the system which produces output y(n) as shown in figure below. Now delay input by k samples, it means our new input will become x(n-k). Now apply this delayed input x(n-k) to the same system as shown in figure below. Now if the output of this system also delayed by k samples (i.e. if output is equal to y(n-k)) then this system is said to be Time invariant (or shift invariant) system. If we observe carefully, x(n) is the initial input to the system which gives output y(n), if we delayed input by k samples output is also delayed by same (k) samples. Thus we can say that input output characteristic...

Definition of Signal In a communication system, the word ‘signal’ is commonly used. Therefore we must know its exact meaning. Mathematically, signal is described as a function of one or more independent variables. Basically it is a physical quantity. It varies with some dependent or independent variables. So the term signal is defined as “A physical quantity which contains some information and which is function of one or more independent variables.” Classification of Signals: There are various types of signals. Every signal has its own characteristics. The processing of signals mainly depends on the characteristic of that particular signal. So classification of signal is necessary. Broadly the signals are classified as below: Continuous and discrete time signals Continuous valued and discrete valued signals Periodic and non-periodic signals Even and odd signals Energy and power signals Deterministic and random signals Multichannel and multidimensional signals ...

a)      Static systems: Definition: It is a system in which output at any instant of time depends on input sample at the same time. Example: 1)      y(n) = 9x(n) In this example 9 is constant which multiplies input x(n). But output at nth instant that means y(n) depends on the input at the same (nth) time instant x(n). So this is static system. 2)      y(n) = x2(n) + 8x(n) + 17 Here also output at nth instant, y(n) depends on the input at nth instant. So this is static system. Why static systems are memory less systems? Answer: Observe the input output relations of static system. Output does not depend on delayed [x(n-k)] or advanced [x(n+k)] input signals. It only depends on present input (nth) input signal. If output depends upon delayed input signals then such signals should be stored in memory to calculate the output at nth instant. This is not required in static systems. Thus for s...

 Here we will see how to determine whether the system is stable or unstable i.e. stability property. To define stability of a system we will use the term ‘BIBO’. It stands for Bounded Input Bounded Output. The meaning of word ‘bounded’ is some finite value. So bounded input means input signal is having some finite value. i.e. input signal is not infinite. Similarly bounded output means, the output signal attains some finite value i.e. the output is not reaching to infinite level. Definition of stable system: An infinite system is BIBO stable if and only if every bounded input produces bounded output. Mathematical representation: Let us consider some finite number Mx whose value is less than infinite. That means Mx < 8, so it’s a finite value. Then if input is bounded, we can write, |x(n)| = Mx < 8 Similarly for C.T. system |x(t)| = Mx < 8 Similarly consider some finite number My whose value is less than infinity. That means My < 8, so it’s a fin...