Linear Constantcoefficient Difference Equations ∑ ∑ = = − = − M m m N k ak y n k b x n m 0 0 An important subclass of linear timeinvariant systems consist of those system for which the input xn and output yn satisfy an Nthorder linear constantcoefficient difference equation. A general form is shown above. for all nSignal Flow Graph of the Difference Equation xn TD TD TD xn2 xn1 xnM + + + + b b0 1 b2 bM + + + + yn TD TD TD − a 1 − a 2 − a N ynN yn2 yn1 Assume that a 0 = 1. Let TD denote onesample delay.Difference Equation: FIR system The assumption a0 = 1 can be always achieved by dividing all the coefficients by a0 if a0≠0. The difference equation characterizes a recursive way of obtaining the output yn from the input xn. When a k = 0 for k = 1 … N, the difference equation degenerates to a FIR system. The output consists of a linear combination of finite inputs. ∑ = = − M m y n bmx n m 0Difference equation: IIR System When b m are not all zeros for m = 1 … M, the difference equation degenerates to This causes an IIR system The effect of an impulse response sequence applied to the input keeps on circulating around the feedback loops indefinitely. ∑( ) = = + − − N k y n x ak y n k 1 0Example Accumulator 1 1 = + = + − = ∑ ∑ − =−∞ =−∞ x n x k x n y n y n x k n n k kExample (continue) ∑ = − + = 2 0 2 1 1 M k x n k M y n Moving average system when M1=0: The impulse response is hn = un − un−M2 −1 Also, note that The term yn − yn−1 suggests the implementation can be cascaded with an accumulator. ( ) 1 1 1 1 2 2 − − − + − − = x n x n M M y n y nMoving Average System Hence, there are at least two difference equation representations of the moving average system. First, xn TD TD TD xn2 xn1 xnM + + + + b b b b yn where b = 1 (M2+1) and TD denotes onesample delayMoving Average System (continue) Second, The first representation is FIR, and the second is IIR.Solution of Difference Equation Just as differential equations for continuoustime systems, a linear constantcoefficient difference equation for discretetime systems does not provide a unique solution if no additional constraints are provided. Solution: yn = ypn + yhn yhn: homogeneous solution obtained by setting all the inputs as zeros. yhn: a particular solution satisfying the difference equation. 0 1 ∑ − = N = k ak y n k Additional constraints: consider the N auxiliary conditions that y1, y2, …, yN are given. The other values of yn (n≥0) can be generated by when xn is available, y1, y2, … yn, … can be computed recursively. To generate values of yn for n