1. Basics of Digital Filters & Sub-band Coding Gilad Lerman Math 5467 (stealing slides from Gonzalez & Woods)

2. Digital Filters The basic setting Assumptions: 1. Input and output signals in or (n-periodic) 2. The filter is linear → matrix representation 3. The filter is shift invariant, i.e. 2 & 3 ↔ representing matrix is Toeplitz In finite case H = A

3. Filters or in book notation We note that In particular

4. Notation

5. Filters Z-transform Frequency Response Additional factor 2  will make it FT of the l 1 signal h

6. FIR Impulse Response = Filter response to  0 FIR = Finite Impulse Response K coefficients → length K filter (convolution is K-periodic) Example

7. More on example

8. Example: Transformations of Filters

9. Sub-band (two-band) Filters Need to have h 0 h 1 g 0 g 1 of perfect reconstruction