Topic. Filters, Reverberation & Convolution THEY ARE ALL ONE

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Pierce Ross
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1 Topic Filters, Reverberation & Convolution THEY ARE ALL ONE

2 What is reverberation? Reverberation is made of echoes Echoes are delayed copies of the original sound In the physical world these are caused by reflections off of walls and other surfaces In the digital world, this is done with impulse response functions and convolution

3 Impulse response An impulse is this signal: δ () t 1 if t = 0 = 0 else The impulse response h(t) of a filter is what happens when you do convolution of the impulse signal with the filter signal. The impulse response h(t) of a filter is the IFFT of the transfer function H(w) of that filter.

4 Example:1 tap delay

5 They are all ONE Reverb is filtering Filtering is convolution Convolution is reverb Bryan Pardo, 2008, Northwestern University EECS 352: Machine Perception of Music and Audio

6 Example: 5 tap delay

7 Acting as a low-pass filter The 5-tap delay makes 5 copies of the signal and sums them. You re summing multiple, slightly-offset copies. This causes high-frequency events to average out in the wash (low-pass filtering). What would happen if the copies are offset by the exact period of some frequency in the audio?

8 Note: Delay and filters The impulse response function of your filter has some length. That length determines how many samples go by before the filter s response becomes steady. Therefore, filtering imposes delays

9 Convolution = Filtering (in the time domain) Output Signal y(t) Filter h(t) Source Signal x(t) y(t) = h(t)* x(t) Impulse response function

10 Multiplication = Filtering (in the frequency domain) Output Signal y(t) Filter h(t) Source Signal x(t) Y(ω ) = H(ω )X (ω ) Transfer function

11 A room is a filter Carnegie Hall

12 Reverberant Rooms are louder (why?) make speech harder to understand (why?) may emphasize certain frequencies (why)?

13 Getting the impulse response Walk into a QUIET room with a recorder Turn on the recorder Clap your hands ONCE (this is your impulse) The recording captures the room s impulse response and other noise (air conditioner, etc.) Bryan Pardo, 2008, Northwestern University EECS 352: Machine Perception of Music and Audio

14 Noise and Distortion Source Signal X(w) Filter H(w) Distortion D(w) + Output Signal Y(w) Noise N(w) Distortion and noise make it a lot harder to estimate H Bryan Pardo, 2008, Northwestern University EECS 352: Machine Perception of Music and Audio

15 Estimating the transfer function H Assume we know what X is (because we made it) and what Y is ( cause we recorded it). Hope noise and distortion are not correlated with X. Call noise + distortion N. Then XkHk [ ] [ ] + Nk [ ] = Yk [ ] Bryan Pardo, 2008, Northwestern University EECS 352: Machine Perception of Music and Audio

16 Estimating H(k) (in the frequency domain) XkHk [ ] [ ] = Yk [ ] so... Hk [ ] = Yk [ ]/ Xk [ ]...more or less. Bryan Pardo, 2008, Northwestern University EECS 352: Machine Perception of Music and Audio

17 Estimating H Estimate H a lot of times, in the hopes that the noise will wash out in the mix Hk ˆ [ ] YkXk [ ] [ ] / XkXk [ ] [ ] where 1 YkXk [ ] [ ] = Y [ kx ] [ k] n n N N n= 1 The average value Over N estimates Bryan Pardo, 2008, Northwestern University EECS 352: Machine Perception of Music and Audio

18 Caveats This only works on the frequencies where there was energy in the input signal X. If there wasn t energy at a frequency then we re out of luck. So, best to use a broadband sound for X. Note: An impulse is broadband.

19 Adding room reverb to a recording Record an impulse in the room Estimate the impulse response of the room using the method described previously Record something in another quiet room (with the microphone up close to the source) Convolution between your new recording and the impulse response function of the first room

20 FIR Filter FIR means Finite Impulse Response Means it will stop making noise once you stop putting noise through it There are also Infinite Impulse Response (IIR) filters. These have feedback To find out more about IIR filters, take a DSP class.

21 Building a low-pass FIR filter Pick width of your LOW pass band (O Hz to X Hz, where X can be up to the Nyquist rate) Create a desired transfer function (don t forget the mirror frequencies above the Nyquist rate). Take the IFFT of the transfer function This is your impulse response function.

An 8-point 250 Hz low-pass filter Matlab Index h(x) H (k) 1 2 3 4 5 6 7 8 0.71 0.26-0.18 0.

22 An 8-point 250 Hz low-pass filter Matlab Index h(x) H (k) DC offset Complex conjugates Nyquist frequency Better index (-4) Frequency of FFT 0 S/N 2S/N 3S/N 4S/N -3S/N -2S/N -S/N If S=1000 Hz and N = Hz 250Hz 375Hz 500Hz -375Hz -250Hz -125Hz Bryan Pardo, 2008, Northwestern University EECS 352: Machine Perception of Music and Audio

23 The same 8-point low-pass filter (another view)

24 A 32-point low-pass filter for 8kHz sample rate

25 Making a high-pass FIR filter Pick how wide you want your pass band to be Create a transfer function (don t forget the mirror frequencies) from 0 up to the desired width Take the IFFT of the transfer function to get the impulse response function Multiply this by a sine wave at the Nyquist frequency You now have the transfer function

26 Making a band-pass FIR filter Just like making a high-pass filter EXCEPT Instead of multiplying by a sine at fs/2 (the Nyquist rate), you pick a sine wave whose frequency is at the center of your pass band. There are a few more details, but that is the basic idea.

27 Design issues to worry about There are lots of design issues here Ringing Side lobes Sharpness of cutoff We re not going to get into it all here (Take that DSP class, if interested)

Topic. Spectrogram Chromagram Cesptrogram. Bryan Pardo, 2008, Northwestern University EECS 352: Machine Perception of Music and Audio

Topic. Spectrogram Chromagram Cesptrogram. Bryan Pardo, 2008, Northwestern University EECS 352: Machine Perception of Music and Audio Topic Spectrogram Chromagram Cesptrogram Short time Fourier Transform Break signal into windows Calculate DFT of each window The Spectrogram spectrogram(y,1024,512,1024,fs,'yaxis'); A series of short term