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1st March, 2006 Wednesday Signals and Systems (EEEG313) System In broader sense system is an interconnection of components, devices, or subsystems. A system manipulates the signal, processes and gives certain output for any signal input. Every system has a defined behavior unique to its self. Defined Behavior Every system has its own properties. Depending on the input and circuit parameters every system gives out a definite output defined according to the behavior of the system. Depending on the system concrete quantitative output is obtained. i.e. for a given input definite output is obtained. For e.g.: An Op-amp having gain = 2 Input = 2V Then, output = 4V The above system has a definite output and shows linear behavior until saturation of Op-amp is reached. Systems with Single Input Single Output (SISO) In a SISO system only a single input acts on the system as a result single output is obtained from the system. The block diagrams below represent system having single input and a single output in Continuous time (CT) and Discrete time (DT), respectively. x (t) CT System y (t) The system above transforms input signal x (t) in and gives y (t) as its output in CT. y (t) = T{ x (t) } x [n] y [n ] DT System Whereas the system above transforms the given input x [n] and gives y [n] as its corresponding output in DT. y [n] = T{ x [n] Linearity of the System For a given system: y (t) = 2x (t) At +15 V = System is in positive saturation At -15 V = System is in negative saturation Non-linear Region Li ne ar R eg io n +15v Saturation -15v Fig: graph of y (t) = 2 x (t) The given system shows linear behavior until saturation (i.e. 15 V) is reached. The system shows non-linear behavior at saturation due to the limitations of its circuit parameters. Every system on reaching saturation exhibits non-linear behavior. Various approximations can be done to make the system linear. Linear approximations are done for those systems which show non-linear characteristics. First order linear differential equation in time domain is given as: dy (t)/dt + a y(t) = bx (t) where x (t) and y (t) are the input and output respectively, and a and b are constants. In particular, we have, dy (t)/dt + 2y(t) = x (t) where a = 2 and b = 1 Difference Equation in discrete time system is given as: y [n ] –y [n-1] = x [n] y [n ] = x [n] + y[n-1] where ‘n’ takes discrete values . The system given above is with memory, as it takes into account previous value of the input i.e. such a system with memory is called accumulator. y [n-1] = previous value of input x [n] = input given at current time y[n] = accumulator which adds previous value of input with the current input value. System Properties System with or without memory A system is said to have memory if it stores previous input values and sums the current input value. A system with memory is called an accumulator. A memory less system is that which does not retain or store information about input values other than the current time. Examples of few systems with or without memory. 1. The capacitor described below is an example of a continuous-time system with memory, then t V (t) = 1/C ∫ idt -∞ t V (t) = 1/C ∫ idt o + i (t) DC C v (t) - 2. Purely resistive system in continuous-time shown below is a memory less system as resistor only dissipates energy, then v (t) = R i (t) + i (t) R v (t) DC - 3. Purely inductive system in continuous-time shown below is with memory. + v (t) DC i (t) L - System’s equation is described by: t i (t) = 1/ L ∫ v (t) dt -∞ t i (t) = 1/L ∫ v (t) dt + i (0) o A system with or without memory depends on system’s components and configuration. Systems with memory Systems with memory have storing element. Storing element may be a capacitor, inductor ,etc. In many physical systems, memory is directly associated with the storage of energy. If a single element stores past values then the whole system becomes memory element. In particular, the accumulator (stores previous inputs) computes the running sum of all inputs up to the current time. At each instant of time, the accumulator must add the current input value with the preceding value of the input. The input and output relationship of an accumulator in discrete time is given y [n] – y[n-1] = x [n] n y [n] = ∑ x [n] -∞ To obtain the output at the current time n, the accumulator must remember all the previous input values stored or accumulated by the system and add it with the current input. The system given by the equation above is with memory. Example : Calculation of differential equation in DT. Difference of two inputs in discrete time when the difference between two inputs is very small can be written as: y [n] = x [n] –x [n-1] which is equivalent to y (t) = dx (t)/dt x [n ] x [n-1] T t-T t Fig: graph showing two discrete inputs taking a small difference in time Differential equation in continuous time domain is given as, y = x (t) – x (t-1) which is equivalent to As T → 0; y = dx (t)/dt Submitted by: Runa Dahal Roll No.32015