Stochastic signals

Described by the laws of probablity; mean, variance, probability distributions

Autocorrelation

$Z_xx[k]=\sum\limits_{n=0}^{n=N-1} x[n]x[n+k]$ $k-N+1,...,N-1$

Measure periodicity of a signal, or degree of repeating patterns

Can be used to measure stochasticity; the lower the value, the higher the stochasticity

Power spectral density

$Xp[k] = lim_{N->\infty }|X[k]|^2$

where

$X[k]=\sum\limits_{n=0}^{N-1}x[n]e^{-j2\pi kn/N}$ $k=0,..,N-1,N$

basically the DFT to the limit; square value of the absolute value of the DFT N-> infiity; converges to a function that is our spectral density

Stochastic model

There are many. We will use:

$yst[n]=\sum\limits_{k=0}^{N-1} u[n]h[n-k]$

$u[n]$: white noise

$h[n]$: impulse response of filter approximating input signal x[n]

Convolution of two signals; convolution of signal with white noise

Spectral view:

$Yst_l[k]=|H_l[k]||U[k]|e^{j(\sphericalangle H[k]+\sphericalangle U[k])} =|\hat{X}_l[k]|e^{j\sphericalangle U[k]}$

$\hat{X}[k]|$: approximation of magnitude spectrum of input single $x[n]$

$\sphericalangle U[k]:$ spectral phases of noise signal

$l$: frame number

Convolution in the spectral domain is the product of the two spectra

In polar coordinates, the product of two magnitude spectra multiplied by exponential e to the j and the sum of the two phase spectra

Magnitude spectra of white noise is a flat line; a constant, so we can take out of equation.

As phase of model, use phase of white noise.

Take approximation of magnitude spectrum of our signal $|\hat{X}_l[k]|$ and take random phases $e^{j\sphericalangle U[k]}$for the modeling of the phase spectrum

details of shape are not relevant; rather, approximation of the time varying magnitude spectrum of the input signal.

phase = sequence of random numbers

LPC approximation

Linear Predictive Coding

$\hat{x}[n] = \sum\limits_{k=1}^{K}a_kx[n-k]$ Linear combination of past samples = expression of IR filter — infinite impulse response filter—linear combination of previous samples; goal=find coefficients

$Error=\sum\limits_{n=-\infty }^{n=\infty}(x[n]+\sum\limits_{k=1}^{K} a_kx[n-k])^2$ — sum of original with approximated ; then narrowed to finite length signals; find a coefficient that minimizes the error signal

obtain a sample set of filter coefficients (sub k) and the frequency response of the resulting filter approximates the spectrum

voice sound commonly approximated with LPC; a way to approximate the resonances, the formants, of a signal;

Envelope approximation

$\tilde{a}[k] = IDFT(LP(DFT(a[k])))$ LP: low-pass filter [only accept lower part of spectrum]

$b[k]=IDFT(ZP(DFT(\tilde{a}[k])))$ ZP: zero-padding

Stochastic Synthesis using LPC

$yst[n] = \sum\limits_{k=1}^{K}a_ku[n-k]$ $a_k$: filter coefficients $u[n]$: white noise

filter of white noise

can be direct for or lattice structure

Stochastic Synthesis using envelopes

$yst[n]=IDFT(|\tilde{X}[k]|e^{j\sphericalangle U[k]})$

smooth approximation of the signal; random phases; inverse of DFT

stochastic compression (take 1 of every 10 samples of magnitude spectrum

7T2: Sinusoidal + Residual Modeling

Sinusoidal/Harmonic plus residiual model (SpR, HpR) Residual subtraction Harmonic plus residual system Sinusoidal/Harmonic plus stochastic model (SpS, HpS) Stochastic model of residual Harmonic plus stochastic system

Sinusoidal + Residual Model

$y[n]=\sum\limits_{r=1}^{R}A_r[n]cos(2\pi f_r[n]n)+xr[n]=ys[n]+xr[n]$

$R$: number of sinusoidal components

$A_r[n]$: instantaneous amplitude

$f_r[n]$: instantaneous frequency (Hz)

$xr[n]$=$x[n]-ys[n]$: residual component

$ys[n]$: sinusoidal component

Spectral View

We develop from here; sum of the sinusoid as the sum of the transform the windows shifted to a frequency and scaled to the amplitudes of these sinusoids, plus the spectrum of the residiual components.

$Y_l[k]=\sum\limits_{r=1}^{R_l}A_{(r,l)}W[k-\hat{f}_{(r,l)}]+Xr_l[k]=Ys_l[k]+Xr_l[k]$]

$W[k]$ = spectrum of analysis window

$R_l$: number of sinusoidal components

$A_{(r,l)}$: amplitude of sinusoid

$\hat{f}_{(r,l)}$: normalized frequency of sinusoid

$Xr_l[k]=X_l[k]-Ys_l[k]$: resiudent component spectrum

$Ys_l[k]$: sinusoidal component spectrum

$l$: frame number

subtract generated from spectrum of same size and window=residual spectrum (usual 512 samples); use blackman-harris so it can be easily subtracted

inverse of that shows residual signal (e.g. breath noise of flute)

To improve stocastic:

smaller hop size, increase approx size-- A FINER GRAIN

With speak sound (not completely stochastic): 256 Hop, .2 approx

get rid of phase spectrum smooth out magnitude spectrum — sounds like a whispered type of sound FFT:

Blackman window—stable sound; good for low sidelobes; good signal to noise ratio

M: blackman: 6 bins * fps / fundamental frequency—odd; to minimize rest of components, make bigger

analyze middle of sound

look for high stochastic in high frequencies; less harmonic; not clearly defined as partials

1. Apply same parameters to harmonic model

inf crease max freq deviation in harmonic track shows unstable harmonics in high (with flute)

Stochastic approximation factor: reduces the whole spectrum by 90%

For stochastic, no need for odd-sized window or zero-padding for the FFT

DV?

resample: FFT approach; to downsample a signal

xw=x[10000:10000+M] w mX = 20 np.log10(abs([X[:

6*44100/200=1323