1. Digital Linear Filters 張智星 (Roger Jang) jang@mirlab.org http://mirlab.org/jang 多媒體資訊檢索實驗室 清華大學 資訊工程系

2. -2--2- Digital Linear Filters  Goal  Sound effects: echo, reverberation  Frequency-dependent manipulation: high- pass, low-pass, band-pass, etc.  Implementation  Use a set of coefficients to transform the original signal into a new one.

3. -3--3- General Formula for a Linear Filter zThe specs of a linear filter are represented by two vectors: zThe output of the filter:

4. -4--4- Example 1: Low-pass Filter za=[1], b=[1/5, 1/5, 1/5, 1/5, 1/5] zy[n]=(x[n]+x[n-1]+x[n-2]+x[n-3]+x[n-4])/5 z  A low-pass filter

5. -5--5- Example 2: High-pass Filter za=[1], b=[1, -1] zy[n]=x[n]-x[n-1] z  A high-pass filter

6. -6--6- Example 3: One-fold Echo za=[1], b=[1, 0, …, 0, 0.8] (3199 zeros) zy[n]=x[n]+0.8*x[n-3200] z  One-fold echo, with time delay = 3200/fs = 0.2 sec

7. -7--7- Example 4: Multiple-fold Echo za=[1, 0, …, 0, -0.8] (3199 zeros), b=[1] zy[n]=x[n]+0.8*y[n-3200] z  Multiple-fold echo, with time delay = 3200/fs = 0.2 sec

8. -8--8- Frequency-domain Decomposition zSignal decomposition yIf we only keep the first 10 components, then we have a low-pass version of x[n]. Amplitude Phase Freq=

9. -9--9- Filters for Frequency Manipulation zProblems yComputationally intensive to find the decomposition (via FFT) and then to resynthesize to obtain the low-pass version. yOnline implementation not readily available. zAlternatively yUse the Butterworth filter that has designated parameters for defining passing frequency range

10. -10- MATLAB Command for Butterworth Filter zWe can use “butter” command to design a Butterworth filter: y[b, a]=butter(order, wn, function) xorder: order of the filter (which is equal to the length of a and b.) xwn: normalized frequency (wn=0~1 corresponds to f=0~fs/2) xfunction: ‘low’, ‘high’, ‘band’ yPlease refer to the online tutorial for examples.