Signals and Systems Lecture 19: FIR Filters

2 Today's lecture −System Properties:  Linearity  Time-invariance −How to convolve the signals −LTI Systems characteristics −Cascade LTI Systems

3 Time Invariance

4 Testing Time-Invariance

5 Examples of Time-Invariance −Square Law system y[n] = {x[n]} 2 −Time Flip system y[n] = x[- n] −First Difference system y[n] = x[n] - x[n-1] −Practice: Prove the system given Exercise 5.9 is not time-invariant

6 Linear System

7 Testing Linearity

8 Practice Problems − y[n] = x[n - 2] – 2 x[n] + x[n + 2] − y[n] = x[n] cos(0.2  n) − y[n] = n x[n] − Are all FIR filters Time-invariant and Linear?

9 There are two methods to convolve the signals: − Graphical Method − Tabular Method

10 Convolution −Example

11 Convolution −Example

12 Convolution −Example

13 Convolution −Example

14 Convolution −Example

15 Convolution −Example

16 Convolution −Example

17 Convolution −Example

18 Convolution −Example

19 Convolution Example

20 Convolution and LTI systems −Derivation of the convolution sum x[n] = ∑ x [l] δ[n - l] for l= any integer = …... x [-2] δ[n + 2] + x [-1] δ[n + 1] + x [0] δ[n] + x [1] δ[n - 1] + x [2] δ[n - 2] +……. x [0] δ[n]  x[0] h[n] x [1] δ[n - 1]  x[1] h[n - 1] x [2] δ[n - 2]  x[2] h[n - 2] x [l] δ[n - l]  x[l] h[n - l] x[n] = ∑ x [l] δ[n - l]  y[n] = ∑ x [l] h[n - l] l ll

21 Properties of LTI Systems −Convolution with an impulse x[n] * δ [n - n 0 ] = x[n - n 0 ] −Commutative Property of convolution x[n] * h [n] = h [n] * x[n] −Associative Property of convolution ( x 1 [n] * x 2 [n] )* x 3 [n] = x 1 [n] * (x 2 [n] * x 3 [n])

22 Assignment # 4 −Problems at the end of chapter 5 −P-5.2 −P-5.3 −P-5.4 −P-5.6 −P-5.10 −P-5.12 −Not Decided about Deadline Date −Tell you later