Ch. 2 Time-Domain Models of Systems Kamen and Heck

2.1 Input/Output Representation of Discrete-Time Systems N-Point Moving Average – y[n] = (1/N) {x[n] + x[n-1] + …+x[n-N +1] (2.1) Generalization (linear, time-invariant,causal) –

2.1.1 Exponentially Weighted Moving Average Let a = (1-b)/(1-b n )

2.1.2 General Class of Systems Upper index (N-1) can be replaced by n. Unit impulse response of the system can be obtained by letting x[n] =  [n].

Convolution Equation The input/output representation can be rewritten with the weighting function replaced with the input response function values, h[i]. The result is called convolution.

2.2 Convolution of Discrete-Time Signals The convolution equation can be defined for arbitrary discrete-time signals x[n] and v[n]. The result can also be generalized for the case where – x[n] =0 for n<Q – v[n ] = 0 for n<P In such a case the convolution result is 0 for n<P+Q.

Array Method for Computation An array method can be used for computation of the general discrete convolution. The columns are labeled x[Q], x[Q+1],… The rows are labeled v[P], v[P+1]… The array is filled in with the products of the values—then sum of elements of the backwords diagonals gives the values, y[n]. Example 2.4 illustrates the process.

MATLAB The matlab function conv can be used to compute discrete convolution. Examples on page 52 and 53 illustrate the results.

Difference Equation Models In some applications, a causal linear time- invariant discrete-time system is given by an input/output difference equation instead of an input/output convolution model. First order linear difference equation – y[n] –ay[n-1] = -x[n]

2.3.1 Nth-Order Input/Output Difference Equations

Examples Example 2.6 Second Order System – System can be solved recursively. – Sometimes a “complete” solution can be found recursively.

2.4 Differential Equation Models Example 2.8 Series RC Circuit Example 2.9 Mass-Spring-Damper System Example 2.10 Motor with Load

2.5 Solution Of Differential Equations Example 2.11 Series RC Circuit Example 2.12 Using MATLAB ODE Solver Example MATLAB Symbolic MATH Solver

2.6 Convolution Representation of Continuous-Time Systems Example 2.14 RC Circuit Graphical Approach to Convolution