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# ECONOMETRICS ECON 240B

UCSB

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This 31 page Class Notes was uploaded by Arno Leuschke on Thursday October 22, 2015. The Class Notes belongs to ECON 240B at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 63 views. For similar materials see /class/227166/econ-240b-university-of-california-santa-barbara in Economcs at University of California Santa Barbara.

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Date Created: 10/22/15

MAT 240B The Digital Audio Programming Series MAT 2400 Spectral Transformations Stephen Travis Pope stephenmatucsbedu Winter Quarter 2009 MAT 240 Topics 6 quarters gl A File W and Streaming Media gl B Spectral Transformations gl C Spatial Processing of Sound 4I D Sound Synthesis Techniques gl E Control and Muftirate Processing gl F Analysis and Music Information Retrieval MAT 2400 3 Logistics Meeting time TuesdayThursday 400600 PM gl Meeting place Music 22I5 CREATE class room Grading IncIass participation presentations proiects Of ce Hours TW I200 300 PM South Hall 4340 4 Web site and mailing list httpmatucsbedu240 240matucsbedu MAT 2400 5 The MAT 240 Series 4 Handson programming courses using primarily C C and Java for digital audio application development gl Sixquarter twoyear course sequence gl Students use a variety of software development tools on MSWindows LinuxUNIX and the Macintosh MAT 2400 2 MAT 240B Course Topics gl Timedomain and Frequencydomain Signals 4 Transformations and Analysis Synthesis gl Fourier Analysis and the FFT gl FFT Software Libraries Vocoders amp Applications 4 Digital Filters Theory and Design gl FIR and IIR Filter Libraries 4 Linear Prediction and LPC vocoders gl Pitch Detection and Analysis gl Advanced Applications rI1HT MAT 2400 4 Textbooks amp References Curtis Roads The Computer Music Tutorial MIT Press F Richard Moore Elements of Computer Music PrenticeHal D Gareth Loy Musimathics MIT Press Ken Steiglitz A Digital Signal Processing Primer AddisonWesley P Lynn amp W Fuerst Introductory Digital Signal Processing J Wiley Julius 0 Smith CCRMA Music 420 Notes heavily quoted WWW links at httpmatucsbedu240 LLLLLLL MAT 2400 6 Software We39ll Use mostly C and C MAT 24GB Outline F R Moore quotElementsquot Book Examples Weekl Timedomain and frequencydomain signals transformations and UCSD CARL filters amp SR convertor quotquotquotlys39SSymhes39s SySlems Week 2 Fourier analysis windowing and the FFT FFT software libraries FFTW lrbrarres 39 39 Week 3 FFTbased vocoders amp compressron spectral descrIptIons and file formats CREATE Sl39 8 MIXVlews source Week 4 Digital filters theory and design sound PV and WC code Week5 FIR and HR filter libraries Various filter design programs Week6 Filter examples and applications Squeak PVOC and LPC plugins Peter Kabal39s LibTSP WAVE Signal Processing Programs Week 7 Linear prediction and LPC vocoders Week 8 Pitch detection and analysis Week 9 Integrated Applications Week TO Random topics review proiects MAT 2403 8 MAT 240B Topic 1 Time and Frequencydomain Signals Topic T Time and Frequencydomain Signals Waveforms and spectra r Analysissynthesis systems Analysis for compression l l a ssssss FFT amp Convolution Readings El c li Smith Lecture 1 rel r I i 3 I i f W 107 radians pclr39 sample 8 Amplitude Moore Chapter 2 LynnampFuerst Chapter 3 MAT 240B MAT 240B 9 10 Functions of Time wave and Functions of Frequency spectrum xaxis time x axis frequency Timedomain Sampled discrete audio signal Control signal lD nD geometry Impulse response Other TD semantics l Frequencydomain 7 finite if H I H 1 52 r r Spectrum inst frame or continuous 3D l P windowed F0 lmk J a LPC spectrum Jl srnewave 1 Hf f Waveletmodulus domain 39 A ll Signal Domains in n e sine wave 7 g 2 g T r g g MAT 2403 MAT 2403 11 12 Frequencydomain Transforms l How to derive a spectrum from a timedomain signal l Swept bandpassfilter l Swept oscillator heterodyne Filter hank l Fourier transform l Polynomial approximation linear prediction l Others I39I39IHT39T L G quot739tl 39quot M AT 2403 13 Fourier AnalysisSynthesis A Fourier theorem periodic signals as sum of related sines Fourier transform continuous discrete infiniteshorttime Tinle Duration Finite In nite Fourier Series FS Fourier Transform FT cont P 30 Xk rfcTJw l1t Xou 2 139cTJ quotlrit time J 30 A oc w E oc oci f Discrete FT DFT Discrete Time FT DTFT discr 39 1 90 Xk Z I39Ii quot Xw Z 39I39nc time 0 quot730 A01 quot 1 we n1r u discrete my 1 continuous freq to MAT 2403 15 39 1 rn M quot DFT In Pseudocode r39l ZO 1quot i l To get a spectrum Xk k is spectral bin l For every output frequency k from 0 to F5 2 Compute the vector product of the input signal xn with a complex sine of freq Wk ie e39iwkquot bin center freq That real complex val is the energy at freq wk or Xk l Process requires 0n2 or worse operations l Process is hard to accelerate even with huge buffers MAT 240B 1 Example Spectrogram DSP Notations from Smith Frequency and Time Signal Notation 1v 39 rt 0 w denotes continuous radian frequency radsec T E C means I 395 a length A c mplex sequence 0 f denotes continuous frequency in Hertz Hz quot l39i v o X DFTt 6 ct Xk DFTkm o X k e C o 1a IDFTMX o xn E R or C 0 k E Z or n k E ZN integers modulo N o For r 6 CD X DTFTW E 71 k E Z integers o T conjugate of I o 121 E R times are always real m phase of 1 o w 27rf o w denotes discrete frequency w 27TkNf5 0 once 6 R frequencies are always real 0 T sampling interval sec 0 f8 sampling rate fS o tn nT discrete time MAT 240B 16 DFT in Pictures from Loy Musimafhics fable 3 Frequency Analysis Process of the DFT l I 3 4 5 6 7 8 usin C Sine Sine Cosine Sine Tut s b P Probe Product Sum Sum k S S UMWMW 39mwwWow wmwwwmww SWWWWWUW 4mmmmWw smmmmmw mmmmmw vmemmw mm 18 us no ruclucl ignn i r Slow Fourier Transform in SuperCoIIider mapslzg 1m 1 some appxnx pulse 1n nvaxlmrzs snuxce SlqlialJnrgl lllimzpslze c1 1 1 1 1 1 1 1 1 1n map glialarzvlclealimzpslzelt 1 meat sum 5m knife 1 mum to meal 3 mm mm Weclmavel lllixpeczicn n 16H 1 m1 menuo 1s nvnlnnes FIHT 39 MAr zoos 19 FFT Code Examples 7f SuperCoIIider spectrum display 7 Basic spectrum display IFFTsignaI synthesis 7f FFT example in SuperCoIIider FT via signal multiplication 7f CSLJUCE Examples quotwidgetquot example 7i Basic FRM Code FFT and phase vocoder MAT 2m 21 Code examples 7I Basic FFI function 7 DSPPrimerc 7 First steps 7 IN HIM bank 7 Iivein vs le ID i SMIAquot Incl 7 Windowing or not rm 7 FFTopIFFT in a loop 7 sound pvan app 7 Platform 7 Using 7 PAcaIIIrack 7 CSI 7f Signal Windowing 7 VST In in 7 Seeahove packages P N FIHT MAr zoos 23 Interpretations of the DFT 7f Multiply the signal by a swept sine wave Xt 61quot for w in infinity 7f Matrix multiplication X Wx W is square matrix of complex spectral coef cients 7f Proiection coef cients of signal x onto N sinusoidal components equalspaced complex roots of I 7f Coordinate transform from RN to Xk 7f others MAT 2m 2m Getting started Coding 7f Platforms tools compilers libraries 7 See MAT 240A Web site 7I Platform IDE Snd Libs 7f MSWindowx VisuaIStudio gcc DirectX ASIU 7f LinuxUNIX gcc KDE USS ALSA JACK 7f Macintosh Xcode gcc CoreAudio JACK 7f AN Eclipse Smalltalk PortAudio CSL JUCE JavaMF libSndFile RTAudio M 2m 22 About the Codezip File AFsp7v7rI 7 Peter Kahal39s IiIrTSP audio I0 lters etc in C DAFD 7 Digital Audio Filter Design and Implementation Utility DSPPrimerc 7Digital Signal Processing Iioutinesi lter 7 LICSD CAIiI FIIiIIIi lter implementation FFTExampIessc 7 SuperCoIIider code for a simple FT JSyn 7 Phil Burke39s Java synthesis library includes filters Ioris7 Kelly Fit139s Ioris analysissynthesis package in C Iinux IPC Folder7 Perry Cook39s minimal IPC imp ementatian IynnFuerst7Code to quotIntroductory Digital Signal Processingquot FFTWGIZ 7 FFTW see fftworg pv 7 Dick Moore39s vocoder as in quotElements of Computer Musicquot signaII4 7 John Culling39s WAVE Signal Processing Programs for linux snackZI3 7A sound toolkit for TcITk and Python by Kare SioIander Sudan 7 LIILIC UNIX Sound Analysis Package iIIIIIIIIIIIIII MAT 2m 24 Also get 4f libsndfile portaudio portmidi rtmidi if if if MAT 240A if OpenGL if Csound SuperCollider MAT 240 B 25 Why Fourier Physical relevance how the ear works Link to filter banks and add synth parameter est for a model an represent any LTI as a FT system May be compact and maybe not an be implemented efficiently FFF Some processes complex filters long convolution are more efficient in the frequency domain Many applications analysis coders classifiers processing etc 4 4 4 4 4 4 4 MAT 240B 2 How do you make the FT Fast Insight into handling of interleaved DFT buffers Take the formulation of the DFT and split the input into buffers of even and odd samples resp each as l 2 Fs but with phase shift Vk Zn0N1 WNkn Vn Zn even WNkn Vn Zn odd WNkn Vn ZroN2 1 WN2kr V2r WNk Zr0N2 1 WN2kr V2r1 where Wis complex phasor 4 The two sums are nothing else but N2point transforms of respectively the even subset and the odd subset of samples MAT 240 B 29 Properties of the DFT Linearity Time reversal Symmetry phase magnitude conjugate Shift Convolution Correlation Stretch Repeat DownsamplingAliasing 4 4 4 4 4 4 4 4 MAT 2 40 B 26 Spectral Processing Applications 4f CSLFMAK peak findingtracking 4f SMS in LAM 4f MQAN in SNDAN 4f HPS in CSL 4f Other implementations and applications MAT 2 40 B 28 Sample Block Decimation 4f Process even and odd sample as 2 subbuffers 4f To recombine use quottiddlequot factor for phase shift 4f Apply recursively FFT tree I 000 out 010 on too to no It 1 x 000 010 100 no I00 on to It MAT 2 40 B 30 OKsoquot I We can keep on reducing and recombining to reduce any powerof2Iength FT to a set of length2 FTs that are then combined with the complex exponential factors I This Is easy but requrres a lot of shuffling of the data set for the W X W even odd subdivision Ieadin to 391 a X hf b d39 MWEZ te amousFFT utter y ragram j MAT 240B 31 Other Frequency Transformations l Higherlevel analysis based on FFT PartiaITrackers Signature classifiers Noise reduction Pitch detection l Linear Prediction and PC I Wavelet Transform MAT 240B 33 Windowing l Problem the real world uses finitelength time varying signals l Solution analyze signal in short quotwindowsquot of fixed size I Impact need to consider frequency response of the window function itself and the time frequency accuracy tradeoff MAT 240B 35 Advanced FFT Implementations l FFTW Fastest FFT in the West Portable C code Supports nonpowerof2 window lengths Very fast Large and complex GNUware l Spectral Interpolation by Zeropadding in the timedomain MAT 240B 32 Windowing 2 freq39 MAT 240B 34 Windowing Overview I Input signaI l Windowed SignaI Shifted zerophase window W FFT gives magnitude spectrum I o39 freq MAT 240B 36 Windows and Spectral Smear 4 Any finitelength window will degrade the accuracy of analysis 4 Window spectra have a central lobe and symmetrical side lobes 4 There are many kinds of window functions with varying properties 4 The important variables are a the width of the central lobe b the Tst sidelobe level and c the sidelobe rolloff rate Window Functions MAT 2403 3 WllldDW Translonns WllldOW length 20 1 1 e 3105 E lt 04 03 gt02 01 0 01 02 03 04 05 m 0 50 I I blorrnallzed Frequency cycles per samplG I I I 1 Tlme lsampleSl 12 0 0s E 04 703 702 701 0 01 02 03 04 05 50 0 5 Normalized Frequency cycles per sample 1 TIH18SampeS 03 0 05 E lt 04 03 02 01 0 01 02 03 04 05 50 0 50 Normalized Frequenayayales per sample Tlme samples 00139 radian per mm MAT 2403 39 Man lobe 39dths l For S fs srdelobe wrdth In Hz Window Type MaulLulu l idlh B Hz Rectangular 2539 Hamming lt13 Harm lt13 Generalized Hmnrning 15 Blackmail 0539 Llerrn BlactlcmanHnrris 2L 1939 Kaiser depends on 3 Chelzvshel depends on ripple MAT 2403 41 Windows and their Spectra 4 Rectangular boxcar null mm sssss s DFF or a HectangularWlnduwr M 20 Magnilude 4 dB U l J radians per sample 4 Triangle Bartlett wider main lobe and lower Tst sidelobe both by a factor of 2 as well as better rolloff MAT 240B 38 General Windowing Issues l It39s an underconstrained design problem l Both timedomain and frequencydomain as well as magnitude and phase characteristics are relevant l There are some new 2000 techniques that promise good time and frequency features freq MAT 240B 40 39 39 739 39 39JF as I start up n l u z n a o 1 Peaks turn off dL 4L r E n o e o o u e n a q 5B revive i rim MAT 240B 42 What do you do with a Spectrum Spectral Metrics amp Measures Peak nding l Spectrum reduction and noctave band spectra Formant nding 7 39 r j 39i Peaktracking l Spectral weightings centroid slope LF HF Fo analysis peakiness Mlcccoel uems l Spectral change amp dynamicitY Thresholding Freq shifiing a H H l MPEG7 standard amp MIR feature vectors Additive crosssynthesis im MAT 240B MAT 240B 43 44 Example MIR Feature Vector PartialTracking it Time domain features Spatial features Windowed RMS amplitude LR difference gl Spectral peak delectlon Max sample amplitude FrontSurround difference RMS ratio of LPHP filtered signal Center vs LR sum difference 39 39 39 Count of zero crossings Spatial variety I RMS dynamic range of sub windows y y RMS peak sub window index Pitch estimates 1 40 E i DFTbin Tempo estimates several Bass pitch guess in Hz pppp bin quot estimated Beat histograms amp weights Bass note MIDI key number guess 39 ea Tempo weight amp off by 2 confidence Bass note dynamicity size of histogram Time signature guess Multi pitch estimates h Chromakey data Peak matching amp tracking i e v ounu e4 Windowed FFT data stored LPC features iq 1 octave FFT data 10 12 points List of LPG formant peaks 35 P REE 2 5 octave FFT data 4 spectral bands List of tracked LPC formants a c List of spectral peak indices LPC res dual level noisiness 4 Fiji l turn off List of tracked peak frequencies LPC formant track birthsdeaths Spectral peak track birthsdeaths 39 Spectral measures centroid slope variety Fluctuation Pattern features IT i 39 39 39 39 39 39 39 39 391 quot 39 i 39 l Relative HF level amp spectral variety FP flux rev W l Corr between HF and audio band FP gravity g MFCC coefficients 4 12 FP weight time quot MAT 240B 39337533913735 quot MAT 240B 45 46 File Edit Query View Select Spectrum Pitch Intensity Formant Pulses Help 105 113 I rmll llll kl 2H0 O O O O O O O incorporate numerical smoothing heuristics and iterative h 003021 tec niq ues 414733 5500 Hz 39439 Examples boundary conditions ignore peakat 0 or fs2 up l Thresholds and retry look for 4l6 peaks over threshold x if too many raise x if too few lower x h I osoooooo Window2136236 seconds 2133236 Total duration 2135235 seconds Many of er examp es I I fem MAT 240B MAT 240B 4 48 Spectrum amp Tracked Peaks Pitchtracking with Harmonic Product Spectra 4f Decimation of FFT spectra summation and spectral peak location 4f Assumes overtones are signi cant not that fundamental is singletome multi tone BPS Fundamental Frequency MAT 2403 HPS in Code Getting more from a Spectrum 4f Take magnitude FFT 4f 1 Think of spectrum in signal correlation terms 4f Copy into summation buffer 4f 2 Harmonic tones gt regularly spaced spectral peaks 4f l00P i l 0 6 or S0l 4f Take autocorrelation of spectrum and look for peaks sum every 39l Spe lml value or max 0f 39l hm quotquot0 4f The peaks relate to fundamental pitches In a warped summation bUffer frequency space bin 7 summallon bUffer accumUlaleS relaled peaks 4f The are called Mel frequency spectral coefficients MAT 2403 MAT 2403 51 52 MFCC Analysis MelFreq epstral Coefficients Analogy I l 4f Steps Start with log spectrum of s m 15 20mg 35 35 AD 45 7 Signal mixed complex tones several 34 7 FT 7 lnsleml of Ac39 use FFT or DU sets of related partial peaks 5 2 w and W Ho mm m i L09 mugnllUdelPDsl OfPDS 7 Take eg the autocorr of the WWW 7 Phase unwrappmg 7 Leads to interesting statistics FFT PDS FT or DCT Name reversal of higherlevel spectral Warped frequencies of peals 39 m m Equepjyzgz 20W 25W Interpretations properties see next section correspond to fundamental g 7 Quefrency frequencies of overtone series Melscale 2 A 5 310121416182522 quetrzncy lms Melscale filters MAT 2403 MAT 2403 53 54 Comparison FFT LPC MFCC by AndrianakisampWhite 3 39Esr QO v oQEQ in ni i DDDDDDDD acid eeeeeeeeeeeeee criptorsd ssss ca DDDDDDDDD rs nnnnnnnnnnnnnnnnnnnnn de aaaaaaa pus 4305 4353 01 M43 4497 45151 11589 5737 5795 33 31 4929 109771 52 wmniiiinnwwmww MWwk r 5 JK M l a MW 0 4 i i i ii Train WW a 75 2 no 1 JO i5 7 i0 750 5 Ca r 396 Vwi 2 r70 75 10 r a MM rm 8 7 V V 500 1000 i500 2000 2500 3000 3500 AD 0 500 1000 1500 2000 2500 3000 3500 40 0 50a man 1500 2000 25 a n 3 n o n 3 5 n a 4 no Frequency Hz MAT 2403 QB 55 56 Spectra Peaks amp Chroma in CLAM Tonal Analysis FFT Applications amp Extensions DC 1 3 1b 1 I a 3 O Vocoders and sinusoidal modeling Partial Tracking MQAN Transient Modeling Deterministic Stochastic Modeling Bandwidthenhanced partials 4 7s 2 Q 3 m r a llAllllllllllllllldllllllllluiililiii it LI L2 i o 2 E i lllllllllllllilllllgtlllllllllinlliizl ll 24 SMS Freq Shi pr SMS Residual Gain n a A o W In J77 SMS Si nnnn oidat Gain i Play Sto 7 Audio File IO 7 eneracors i gig g Continuous Phase Analysis 3 w g 12 1223 5 mm 5 58 Oldfashioned Vocoder Vocoder Structure magnitude or Sh ifi in Read a window of the input signal F Apply vector window function gt rnt b In f aw Spectral analysrs Data Corn ression W V r A quot 1 Hggg39jgg fh 1310 iiigt FFTofwindow FFT length may notwindow size may f A I 03233 E on maintain phase information V Z Filterbank analysis o o do something with spectral data f 11mm r iNIJ Resynthesize signal from spectral data 39l Overlappadd or sumofsines L f tH F 395 J Proc ssmg 331895 l Shift input by step srze may not wrndow srze MAT 2 40 B quotquotquotAquotquot quotdquot39mquot w MAT 2 40 B 59 60 Windowing for COLA 4 Overlapadd OLA resynthesis places special requirements on the window shapes and hop sizes constant OLA constraint 4 Options Rectangular window with no overlap Hanning window raised cosine with 50 overlap Bartlett triangular with 50 overlap Blackman window with 66 overlap MATZAOB 61 dd h d l A rtrve Synt esrsMo e l sum Of39smes 41m 30 Ago r20 Asa M 4105 M W Ara swim mot MATZAOB 63 Vocoder Applications Compression of quottonalquot sounds via sinusoidal modeling quotMalleablequot audio representation 7 Pitchtime warping Transpositiontime compression Morphing and crosssynthesis Pitch detection Transcription IFFl as synthesis technique 391 HT MAT 2403 65 Analysis Data Remember each quotbinquot or quotchannelquot delivers functions of amplitude and centerfreq and possibly phase It1 t MAT 2403 62 ConstantOLA Model IFFT Succeslve Wlndowed Frames causal window 50 overlapadd quotWkWVT39Avaxll illvlrquotl quotl 0 50 100 150 200 250 1 T T ltI o fquotfquotlJquotkf4 ifquot 39lo 5b 100 ll 260 230 r L 39quotv u M J J W w 0 so 100 150 200 250 gr jV39fgtm quot 0 5390 too 150 zoo 230 Time samples MAT 240B 64 FRM s PV Command pv R N NW D I P Np synt i lt samples gt samples R sampling rate in Hz N FFT length in samples must be 2quotn Nw window size in samples D decimation factor in samples 0 for synthesis only interpolation factor in samples 0 for analysis only P oscillator bank pitch factor 0 for overlapadd resynth Np linear prediction order 0 for none synt synthesis channel threshold 0 for all channels i means rw integer samples floats by default Filter fundamental R N Nw N or i N D Nw 8 very conservative Duration scale of result l D MAT 2403 66 Vocoder Processing Problems with Vocoders Time warping vary output buffer size Pitch warping transpose magnitude spectrum Poor model of noise transients quotdensequot sounds inharmonic sounds A A A rosswmhesis apply one spectrum to another Hope for at most l partIal per bIn needs finegrained FFT A A A Noise reduction quotgatequot components given a threshold Convolution multiply spectrum with spectrum of eg l N l revers39ble llormOllYl impulse response gl Maybe computeintensive See Supercollider and FRM pv code an actually increase data bandwidth PV in SoundStudio SoundHack MAT 240B MAT 240B Multi stage Analysis Loops Filter Banks gl segmenTWlndOWlnpUT a zFIXEIJEUER gl Preliminary analysis silence threshold m y gl Perform FFT gl Spectral processing peaks tracks F s gl Track birthdeath statistics transients gl Pitch detection HPS orautocorrelation gl 0theranalysis processes see later gl Feature weighting data reduction 7 Filter Banks Decimation for Filter Banks gl Filter bankvs l A I A Routine spectrum Downsample signal byafactor of2 wh e n to use w w m m mm m m Interpolate samples Subtract from original signal l HOWlo Implemenl 39 Difference is energy in top octave 7 Analog Repeatfor next octave 7 Digital Filter bankswidely used in coding a m m and compression 200 590 anuency Hz MAT 240B MAT 240B 1 2 Working with Spectral Data gl Instrument signatures ngolo instrument identi cation gllocation of styleindicative instruments ngtill a challenge in mixedcompressed music 4 Pitch tracking glMelody instrument vs bass tracking glGenerally requires segmentation andor bandpass filtering gl Spectral statistics nglope centroid band weights variety etc MAT 240B 3 Hybrid Models 4 Problem not all useful sounds consist solely of slowly changing harmonic overtone spectra Noisy transients esp attack Model as sines noise Inharmonic or stretched spectra MAT 240B 5 Transient Modeling 4 Find transient regions based on some measure of change and model them separately gl Use samples wavelets or PC for transients 4 Do not timescale the transients original signal timescaled sines sines sines k Em noise quot noise quot MAT 2AOB Code Exercises gl FFT spectral extraction Realvs complex Magnitude phase spectra 4 Weighting of spectral data Numerical scaling Perceptual scaling 4 Peak location continuation gl MFCC and HPS MAT 2403 4 Example Breathy Flute When are the upper partials so chaotic and when is the FFT a trying to model noise as partials Time s Deterministic Stochastic Modeling Spectral Modeling Synthesis SMS gl FFT models noise as partials with chaotic phase easy to spot in peak continuation process gl FFT captures quotdeterministicquot parts of spectrum gl Original signal FFT resynthesis noise component gl Use some good noise modeling technique such as loworder PC for the stochastic component MAT 240B 8 SMS Analysis MAT 2403 9 BandwidthEnhanced Partials Loris Take FFl track spectral peaks Tweak peak frequencies and times gl Resynthesize and subtract to locate noise gl Model noise as a nonzero bandwidth energy distributed around the partials gl Each partial has a of its energy dedicated to noise gl Allows for interesting transformations amp morphs 3911 HT yjvu MAT 2403 81 Loris Example Bass Note gl Beforeafter bandwidth reassignment Was v lily er WW 2 llll ll lllllillll Hm SMS Synthesis Model tl l Y Arl ll uslw39rll Nilll In quotlll MAT 2403 80 Timefrequency Reassignment and Bandwidth Enhancement in Loris n MAT 2403 82 Usrng Lorrs python c llber import loris 05 time print 39unulyzing voice le39 C a lorisAnulyzer 100 urg isfreq resolution cl lorisAil39lFile 39lZuluil39l39 gl Python via SWIG f I 0 v c sump es Smalltalk Vl sumplerute cfsampleRute vox uunulyze v sumplerute lorischunnelize vox loriscreuteFreqReference vox 0 1000 t lorisdistill vox lorisexportAiff 39voxtestuiff39 lorissynthesize vox sumplerute sumplerute 16 MAT 2403 84 Loris in Siren on Smalltalk 8 C O Loris Analysis Con guration Resolutldn 70 aw5care W Freq mm W savalsan IT Freq Floor l 30 Savetompulm Mln Freq 30 Playompuc W MaxFreq W AmpI Env true Amp Floor m Fled Env IT Analge I Play MAT 240B 8 Review gl Timedomain and spectraldomain signals gl Fourier transforms and DTF gl FFTusage gl Windowing and signal massaging gl FFTApplications gl LPC gl Pitchtracking gl Applications mm 89 MAT 2 40 B 85 FFT Applications gl Phase Vocoder gl Loris Morphing gl SMS Morphing gl Noise Reduction gl Compression and Perceptual Coding gl Dynamic Filters gl Others MAT240B 86 Exercises I0 Frameworks amp plugin architectures PV based applications Higherlevel spectral processing Multifeature analysisresynthesis MAT 240 B 88 End of Topic 1 m MAT 240B 90 MAT 2403 Digital Audio Programming Spectral Transformations 7lTopic 2 Digital Filters Input j Output Secondorder lter section MAT 240B 91 Background St hl a 7 Filter spectral phase response M u M quot M M M quot 7 Timeand frequencydomain 7 Window spectra 7 Analog filters 7 R L C circuits and complex math 7 The difference between resistance R and impedance Z MAT 240B 93 Analog Electronics Resistors Resistor AC Response Voltage and t 7 Analogies for circuits plumbing acoustical systems Imsinwt R Resistance Vmsinwt R 4 7 I 1 v I v g 95 MAT 2405 95 Topic 2 Digital Filters 7fTheory and Concepts 7fFilter Design and Transformations 7lDigital Filter Implementations 7nynamic Filters 7fComplex amp Parametric Filters 7prplications m Twzw MAT 2405 92 Kinds of Filters mgqu Filter classes 7 Low39puss high39pm 7 Bandpass bandreiect 7 Composite continuous multiband resonant 7 Filter topologies 7 Matrix equation description 7 Description of filter bands response regions 7 Filter freq phase response MAT 2405 94 Capacitors Capacitor AC Response inltagelags 39 L Examine B i a Inductor quot 4 Resistor Out of Phase s y X E f M E PVA U I 39 t 90 a b 1 Sin D m CONNDblth to Phasm diagram Figure 816 I cum 116mm edance Motordnclacousticzllcapacimr J I 39i u l 41 0c AV l l g V lift goes 100m high freq V 96 MAT 2405 96 Inductors Inductor AC Response Voltage leads current In C 39 90 39 n l v m p 832 Examine a s 39 I 39l a Visitor I s I x L y X Resistor Out of Phase f s I Figure 817 Cnntributlun tn Motor and acoustical inductor cnmplex impedance I 0c AV Pressure Current f Phasnr diagram gt 399 ij I goesto utDE 97 MAT240B 9 Analog Filters C HPF gnr e P OLQ HPF m an ATT V s 1 lt l VinO 5VTO0Ut 339 I quot1 quot quotSalt C Low Pass Type High Pass Type MATZAOB 99 Forward Sample Averager Simplest Example i ForwardSampleAverager Xquot X a y Ynlgtltngtltn Il2 ynXngXnIll i quotFunction smoother Lowpasstllter Sampleaveraging player app MATZAOB 101 RC RL and RLC Circuits 4 httpwebphysicsphmsstateeduiclibraryZl5ircuitiEhtml BAND OF DESIRED ALI FREQUENCIES FREQUFNCIES E REJECTED r ourpur R FREQUENCIEo UEHHKU BAND OF FREQUENCIES v C 93 MAT 2405 98 Acoustical Circuit Model Transfer Function Jinrlgl t v I ram 1 linltl 7 tall l fling 12M r rquotm I I t ticcr lrlrrrm Htm lr39Hl in 1 quotquot y 1 ml t ltlpllt iutlngc m mumh curl quot I39i 1 t l u iulcl mllpnt pl clll1 ill ICIIlUlC ummturing poinl Input currc39nt in speech mil u irnll h c ll 3 liltccriw uirtuucl u 7 n ncrugccl mgr lruglh ul spru39h mil in r l n r y I H I I mm Ultccmc lltlt ul39 39luinl l 7 mm Mu ul39 L39illiL and i1 usxcmhly um I m trm Resimc rm ril nrxlmnxiun i rm 39l39olnl compliance nl uslmnsimr xpniur und Vnrn Iltm rtlrl t mum f m k r quotatticnu 7 In A Constant or pmpurcimmlir ruml MAT 240B 100 Summing Several Past Inputs Yn 90 an 91 Xnd1l gm Xn dm i Short delays filter i Long delays reverberator MAT 240B 102 Past Output Sample Averager x y 4i Feedback loop 9 Ynan9Ynti comb filter 4i Generalform feedback delay line mm MAT ma ma Filter Classes 4 IIR Filters Feedback systems Speedspace efficient and exible 4 FIR Filters Delay lines Impulse response is simply the coel39licients Phase is linear if coefficients are symmetrical ie b0 bn bl bnl etc MHT MAT me me 2nd0rder General Form input 39 Output Al B1 M V m Secondorder lter section VI 71 Z 17 1112 7 1122 W xH t 2 1 b0 1212 1722 mm Mm m Filter Classes 4i Feedforward delaylines Play an impulsethrough it output ends at most dm samples later i Finite impulse response FIR filter 4 Feedback delaylines Play an impulse through it output recirculates quotforeverquot Infinite impulse response IIR filter NET 39 MAT ma m4 General Form Combination of feedforward and feedback terms poles and zeros 4 4 Feedforward weighted averager i zero yn Aoxn Aixni 4 Feedback comb filter i pole yn Aoxn Biyni 4 General 2pole2zero M 90 xn gt xni 92 xn 2 ltt yn t hzyn2 MAT non me General Form to Zform 4 Yn polynomal 4 Substitute z for ni zZ for nZ et gl Solve polynomial for roots in z 4 Example IIR system Yn Xln k Ylnl gl Expressed in Z Yz Xz K Yz z Hz Yz Xz i ikzI z z k gl ihisis forz0 and infinite lorzk 4 these are the 1 zero and i pole of the function MAT ma PoleZero Diagrams PoleZero Diagram Examples 7 Zplane and the unit circle rubber sheet analogy 7 See DSP Skriptum p 80 or other reference 7 Poles and zeros as roots of xfer function 7 Other topics HJL Product of vectors to zeros 7 Frequency transformation see below Product of vectors to poles OHUL 2 zero angles Z pole angles mums 39 t i 739 u we sl Figure 517 Complex frequency response MAT 240B Example 6pole lowpass Butterworth Filter Realworld Frlters wmrnacuk 7 Fs 20 kHz F0 1500 Hz 7 See CARL Filter Program filterc 7 Designed using MATLAB 7 See cmix Filters 1 7 General Filter Design I 1 39 39 39 39 Transition I l Unity gain A Passband ripple 05 I39D l Ideal lter 39 A 39 II39III Realistic lter l quot39 quot I39x Stopband Ll 39 39 39 ripple 1 H p 1 1 H mm ennn Hl39II39IIl FtI39II39II39I tnnl If Cut Off F MAT 2403 Freq MAT 2403 Filter Topologies Complex Filters 7 Each has different characteristics Two Options 7 Butterworth at passband response monotonic stopband 7 Higherorder filters attenuation 7 TChebyshev equalsized errors on passband and stopband 7 S nesParallel omblnallons 0f Slmple ller equiripple above or below knee 7 Some design programs decompose some combine 7 Bessel linearphase in passband Parametric Filters 7 Elliptical flexible specifications equalsized errors in both bands I o d o o h d f 7 wen excellent 510th d 7 Some topo ogres a ow esrgn wrt Q separate rom F0 7 many others MAT 240B 113 MAT 240B 114 Problems with Digital Filters AllPass Filters Numerical stability poles on the unit circle are very 4i If the pz pairs are each on the same radius vefry bad lead to ringing symmetrical on the unit circle the frequency Roundoff errors are everywhere floats are limited response remains ail bur one can design an Pre lSlonl arbitrary phasedelay response Ef ciency higherorder filters maybe slower than FFiIFFT combination MAT 24103 1 15 Spe i m WU33 PoleZero Diagram for Remez Filter Example FIR Spec Example ws00b617norp001 0 s 0K State ECEN 377 Design of ConstanKoef cient LinearPhase FIR U Filter Using the Remez Exchange Technique in MatLab or quot nl000wremezord33 67i 00 0 bremeznl000w quot hwlreqzbi256 Quip E 39Z d gll 9ml bslhll n 10 10 0 33 67 1 00 1 1 00 w 1 1 39 Q p l 00248 00000 00767 00000 03074 1 o gmelzl 05000 03074 00006 76719 00046 24776 39 pl0llwpl hdl NOTE Coell hat appear as zero are small nonzero as O 05 Real Pan MAT not 118 MAT 24103 1 1 PoleZero Dia ram for Remez Filter Exam le g p Form a cascade of secondorder sectrons zrootsb b1polyz1z2 10000 33231 49915 n k1b11b12b13 kt21684 m b2polyz3z4 10000 19429 10000 M k2b2tb22b23 k239429 a b3polyz5z6 10000 15437 10000 a k3b31b32b33 k335437 m b4polyz7z3 10000 10955 10000 k4b4t b42b43 k4 30955 quot 01 M 03 04 05 09 97 99 09 bspolyz9z10 10000 07659 02003 k5b5t 552553 k5 04344 MAT not 120 MAT 24103 1 19 Plot of First Section 812 10000 38231z1 4991522 21884 mm NETquot MA 2400 Plot of Third Section 832 10000 15487z1 1000022 35487 i 1 witquh3k3ll b M1H hgl nbsl13 mus1188 501 1020 1122 mar 211404 MA 2408 Plot of Fifth Section 852 10000 078592 1 020032 2 04344 itaei mm s ttk FR uruuuuuuua 141805 MAT ZAOB 125 Plot of Second Section 822 10000 19429z 1 100002 2 39429 till 3512 1255 V v 10 3911 IlFiwhWIJSL h2dl hgl nbshl Irlmi vii111W nx1s01 40 20 gum 148802 MAT ZAOB 122 Plot of Fourth Section 842 10000 109521 1000022 30955 ltll Sa mm 3232 Kisi XX 14 94141384154041 251 Mfli39lWrgIMnMM M W 11th nxis l 4020 09182 wind MAT ZAOB 124 Cascade of First Section 112 btkt 4 t 1 1 MM 1 4 111 1m11kr1251 11182048910015508 1 11811818 nxis01 1020 0 121 911 IIIIHIIEMD MA 2408 126 Cascade of Second Section t2 convtib2 k2 4 m E u I K 02 Iquz LI 50 mmmgrmnbsunzn 0 I 0i axis0 0020 09022 HT zplnneul MAT 2000 2 Cascade of Fourth Section t4 convt3b4k4 1 g 2 0K 39nuuu uu agu quota hat IHraszJJSb MH ZDWIJIMHHIIM gnu0000 nxis0 0020 MW mm MAT ma 29 Coding Usrng CSL Filters iiutterw0r1hhnndpn lter Whiteiiaise whiteZ white liaise saurce Butter huneriwhitel BWBANDPASS 1000f 100f th el0gtsetii001hutteri make same 50qu sleepllsec imdur10000000 the07gtclenrii001 urn 0H 0 generic handjmss lter created with inef cients ant hrae s 005f 0f 004 nal nrae sU 7104 09 Whiteiiaise whiteD white liaise saurce Filter lterwhi190 hrae smrae s 3 1 th el0gtsetii001 ter make same 50qu sleepllsec imdur10000000 the07gtclenrii001 turrr 0H MArona 131 Cascade of Third Section t3 convt2b3 k3 703 ins quzoa 250 memangmmhqma ph lpid d nxis0 0020 09022 m1 whneoa mm Cascade of Fifth Section t5 convt4b5k5 3 ins 014001021150 05040091000500 punpuny nxis0 0020 09022 m1 minus mm Examples on the Web 4 httpwwwmusicdsporg lesAudioFQCookbookat 4 httpwwwdspguidecom 4 hitpwwwenscstucapeopIetaculiycavers ENSCSBUpzresponsehtm 4 httpwwwmathworkscomaccesshelpdeskhelp toolboxsignal 4 httpwwwimrescomcoursesintroiir5poeshtm FIHT MArona Filter Design Time Domain Frequency Domain man In Ex 2 MAT 2AOB LowPass Coef cients Calculate lowPass lter coef cients yn c0d0xn cld0xnl c2d0xn 2 dld0ynl d2d0yn2 void lowPass doublefreq double bandwi th calcCommon freq bandwidth see above double c0 l0cs 05 double cl l0cs double c2 l0cs 05 double d0 10 alpha double dl 20 cs double d2 l0 alpha double temp 05d0 A0Value c0 temp AlValue cl temp A2Value c2 temp BlValue dl temp BZValue d2 temp MAT 21105 135 Models of Filters 4f Timedomain analogies see above Simple don39t scale 4f Windows as lter see part I 4f Convolution for lters Process smears the response 4f Vocoder as lter i Compute intensive artifacts MAT 21105 13 llllllllilillll 3 Mil Ll Calculating Filter Coef cients Calculate common variables used in every lter void calcCommon double freq double bandwidth Variables are double inst Vars Relative freq to SR double ratio freq SynthgetFrameRate Radian frequency omega 20 MathPI ratio cs Mathcos omega sine and cosine sn Mathsin omega Q steepness Q sn Mathlog20 bandwidth omega Alpha generic coef cient alpha sn sinhl 05 Q l From JSyn MAT 2405 BandPass Coef cients Calculate bandPass lter coef cients void bandPass double freq double bandwidth calcCommon freq bandwidth double c0 alpha double cl 00 double c2 alpha double d0 10 alpha double dl 20 cs double d2 l0 alpha double temp 05 d0 A0Value c0 temp AlValuecl temp A2Value c2 temp BlValuedl temp BZValue d2 temp MAT 2405 136 Windows as Filters Window length determines transition bandwidth Passband and stopband ripple determined by window sidelobes window type Ripple largest near transition band Narrow main lobe means narrow transition band Tighter specs require longer lters Very long FIR lters can be designed easily thought not usually optimum MAT 21105 138 Window Filter Design 4 Ideal filter has sinc as impulse response 4 Window the infinite sinc with a finite window 4 Shift the zerophase filter to get a linearphase causal filter Phase slope is I 2 the window length 4 Apply parameter transformations to make other filter classes MAT 2403 139 Coding Examples 4 Basic 2ndorder filter functions SI szn sound mxv mix etc 4 Filter design packages SL FIR design Mills DAFD mix ellipse ARL FIR Signal fir MAT 2403 141 onvolution and Filters 4 onvolution Definition examples 4 Timedomain solution nested loop 4 FFIbased solution use vocoder then multiply spectral frames with FFI of FIR impulse response 4 Same artifacts problems as vocoder 4 onstraints and workarounds for long convolution 4 Tradeoffs mnT MAT2AOB 143 fz39rgtttfrtg1 71d Better Filter Design Techniques 4 A huge still active area 4 Many techniques numerical iterative 4 Many functions in Matlab etc 4 StateOfTheArt Remez for optimal FIR with arbitrary magnitude and phase characteristics Secondorder one Problems SOCP Hilbert transform MAT 2403 1 40 onvolution MAT 2403 1 42 Filter Performance 4 Direct convolution is 0N2 4 FFTbased is 0NIog2N 4 Number of N FFT Direct Convolution 4 176 16 32 2560 1024 64 5888 4096 128 13312 16384 256 29696 65536 2048 311296 4194304 MAT 2403 1 44 Timevarying filters l Harder Numerical stability Coefficient interpolation and quotzipperingquot l Convolution method need to take FFT of changing filter ie 2 FFTs running at runtime MAT 240B 145 Review 4f Analog and Digital filters 4f Basic FIR amp IIR structures l Canonical form l Coefficient computation 4f Convolution for FIR filters 4f Applications MAT 240B 14 MAT 2403 Digital Audio Programming Spectral Transformations lTopic 3 Linear Prediction MAT 240B 149 Sample Rate Conversion l Simple interpolationdecimation l Need good filter to handle artifacts l Several goodsounding packages exist but the filters need to be carefully designed BU Input Y Secondorder lter section Topic 3 Linear Prediction lVoice and Vocal Tract Models lLPC model and execution llmplementing LPC encoders lLPC vocoders lProgramming LPC Output MAT 240B 146 MAT 240B 148 MAT 240B 150 Human Voice Production lSourceFilter Model lGlo al pulses or noise and vocal tract lModels of the source lModels of the Filter MAT 2403 151 SourceFilter Model pitch period T gt Sln vocal tract model MAT 2403 153 The Vocal Tract Movie MAT 2403 155 Vowel Spectra uuuuuuuuuuuuuuuuuuuuuuuuuu MAT 2403 152 The Human Vocal Trad as a Tube MAT 2403 154 MAT 2403 156 DSP Vocal Tract Model KellyLuchbmun Vocal Tract Model m 46m D m Glollnl Pi Ise Train or Noise 1 Wm Speec Output vlnl where 74 crnssrsectmnal area uftube Re ectmn cnef cient at 1 h wimdemyimder Junctmn k g 1L71tH MAT 240B 15 The PC Idea 7 Model a sequence of samples with a polynomial 7 sn aisnT a2sn2 apsnp 7 error 7 pthorder polynomial 7 Expect some error and capture it 7 For periodic sounds the error itself is periodic 7 Polynomial can be turned into a filter HTmivuwn iv MAT 24GB 1 59 LPC Model un 1 sn Hz A z Shaping Synthesis Filter G 7 un excitation function derived from PC error may include pitch function 7 G gain 7 Hz timevarying filter derived from PC polynomials mmw MAT 2405 161 gl Vocal Traci Editor 5 Glut Raft Gain SPASM Lector N as at C avity 7 httpwwwcsprincetonedu quotprcSingingSynthhtml 158 Weights and Error Functions 7 By a simple transformation we can view the weights oi as the coefficients of an allpole filter 7 Fora large class of sounds the error is either a noise unvoiced speech or b an impulse train voiced speech 7 This leads to a very convenient physical model MAT 240B 160 So what 7 PC analysis provides is a nice sourcefilter model glottal pulses vocal tract 7 The PC filter captures the spectral envelope of the signal well 7 There is a small family of excitation functions that can be used to model a large class of signals see CELP 7 PC is epecially applicable to speech 7 Both analysisresynthesis can be implemented efficiently NETWHW MAT 2405 162 LPC Analysis l Two methods for deriving pole weights autocorrelation and covariance l Complexity generally scales with window size and number of poles l Error can be classifies into noise or periodic l Pitchtracking necessary to exactly determine pitch of excitation MAT 240B 168 LPC APIs LibTSP in CSLFMAK LPC analysis get autocorrelation void SPautoc const float x int Nx float cor int Nt SPautoc mSampleBuffer mWindowSize mCorrGetRawData mLPCOrder 1 Find predictor coefficients from autocorrelation values double SPcorch const float rxx float pc int Np featureTablemLPCResidual SPcorch mCornGetRawData mPredGetRawData mLPCOrder Convert predictor coefficients to filter coefficients void SPchec const float pc float ec int Np SPchec mPredGetRawData mLPCCoeffs mLPCOrder Convert predictor coefficients to cepstral coefficients void SPchcep const float pc int Np float cep int Ncep SPchcepmPredGetRawData mLPCOrder mLPCCoeffs mCepst Locate the spectral peaks locatepeaksFtVector fv FtVector res epsilon spacing mPeakExtractorLocatePeaks mPred featureTablemLPCFormants 4 MATZAOB 165 efault s audio thions nuts 1 J 593ms Pquot value 9330 rilename Leslwav 030 049 o55 050 o7o l080 o9 LPC vs Power DensHy Spectrum LPC The Math 5a Rd 111 1 l again 2 l apshl p l As summation SW 2min a Guru In 52 Zora quot32 GUz ein5n nsn a shit Ir l Error signal quot 3in 1 1 l Transfer function H m 2 m MAT24OB 164 Examples for Voiced and Unvoiced Speech Voiced waveform with Linear Prediction Error Unvoiced waveform with Linear Prediction Error I ll lllllllllfllili rr i 011 0112 0114 0116 0118 012 0122 0124 0125 0128 013 1 Ci 53 c M J a sec sec Linear Prediction Spectrum With Error Signal Spectrum Linear Prediction Spectrum With Error Signal Spectrum 1 Freq Hz Freq Hz MAT 240B 166 le thiuns Help n El ll gt gl TImE 13n5n msec Amplitude 42400 I Spectral lti 7 i Analysis L Wssg iquotquotissquotm quotIquotquot15 39quot lgsgquot39Iquot Comparison IQVJl39ll Mi 1a ngtlllg upunns 1 ag CIquot Pitch time tradeoff 39 39 39 4 Elll 3 ll Time 793111 msec Amplitude 56m of the l quot lil 7 VHS 1 l l l l7 lil r l l l7 l l l l l l I l l39l lilgsLil il l I lil l 17L J l l7 lil Feature extraction 5 i 7 I r Ehl l ll Restze I 39 e on e I 0 ll Tlme 21330 msec Amplitude 72400 Spectra a MFCC 39 triangle I Tlnle 2135 mser Amplide 441m Bark m g i r I o W r LlrllIlll lllllllxslllllll SillllllllrLugnlllllll i V E5 19mins Problems f Both methods have potential stability problems because we need to invert the signal covariance matrix which may be hard if it39s nearly singular f Resulting filters can have roundoff problems lots of pols very close to the unit circle f It is not clear how to interpolate between filters on resynthesis MAT 240B PC In Csound l Lpanal program l Lpread and lpreson opcodes f Lpanal lpanal flags infilename outfilename lpanal performs both lpc and pitchtracking analysis on a soundfile to produce a timeordered sequence of frames of control information suitable for sound resyn hesis Analysis is conditioned by the control flags MAT 2403 1 3 Cepstrum envelope order 30 Discrete tepstrum order 30 r Partial peaks LPC envelope order 30 Log magnitude spectrum Amplltude 18 I LPC Implementations lTSP LPAnaILPSynth leound lCmix lSqueak 2 versions lMxv with GUI LPAnal Flags a alternate storage asks lpanal to write a file with filter poles values rather than the usual filter coefficient files s srate sampling rate of the audio input file c channel channel number sought The default is l b begin beginning time in seconds of the audio segment d duration duration in seconds of the audio segment p npoles number of poles default is 34 max 50 h hopsize hop size in samples default is 200 max 500 string text for the comments field of the lpfile header P mincps lowest frequency in Hz of pitch tracking 0 maxcps highest frequency in Hz of pitch tracking The narrower the pitch llliiilll i range the more accurate the pitch estimate The defaults are P70 1200 quot MAT 2403 1 4 LPC Playback in csound 4i Read LPC le into a set of variables 4i krmsr krmso kerr kcps 4i lpread ktimpntifilcod inpoles ifrmrate 4i Filter 0 signal given the most recent lpread 4i or Ipreson osig FIHT 39 MArona t 5 LPC Playback in Cmix 4Hatasetquotsampledatasetlpcquot 24 gllpcstuf thresh 0007 randamp l 0 0 0 4Upcplaystartstartincrl incr transp 8 frame frame2 amp D cf bw NHTquot MAr 2mg 1 Where to go with this glUsing LPC glPorting LPC implementations ngtandalone LPC vocoders for cross synthesis F39IHTquot MArona t a LPC analysis in Cmix 4i lpc o lpcanalfile p poles f framesize i inskip d duration sound le 4i ptrack ags soundfile 4i stablize 4i merge FIHT 39 MArona t 6 Using LPC 4i Source preprocessing sample rate 4i Pitchtracking and associated analysis 4i LPC data storage 4i Compression 4i Crosssynthesis 4i Challenges NHTquot MAr 2mg 1 a Review 4i Vocal tract and sourcefilter models 4i LPC paradigm 4i Implementing LPC 4i Associated analysis 4i LPC for compression crosssynthesis 4i Applications FIRT MAr 2mg tEEI Other Spectral Transformation Techniques Pitchdetection and Tracking 7l Wavelets as spectra 7l Pitch detection for single notes 7 Granular processing and hybrid timefrequencydomain 7 Simple GUTO39Correlu on M representations 7 FFT8 HPS a S 7l Vintagefilterimplementations mammal Mi 1 7 roma ata r 7l Spatralrzatlon wrthconvolutlon o 3 3 ll 7l MultIpltch detection 0 am 450 o am we 7 Convolutionbasedsynthesrs 7 Busstmcking We WM 7 See MAT 2403 The Sequel 7 chords MAT 240B MAT 240B Autocorrelation again Speech Analysis quotat lie all Q quotQ7 tux l l y L m it lw llll ll AWN M l i 39 i 5 i i i ml t l l mm 1 A o ACF pitch polnt 7 5 lb ll 1 l Vj r V l MAT 240B MAT 240B 200 Hz MultIprtch Detection 1 448Hz t Other Examples 5 g x 361 Hz 10 lScatter l Techniques ll ll 7 AssumeclearAC peaks 1 7 4 5 lSPEAR 7 Assumeseparated spectra 7lSonicVisualizer 7 Pre filter spectrum Vurious Plugins 7 Assume commonmode vibrato 7 Follow transients MAT 240B MAT 240B 185 186

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