Notes of Digital Audio Signal Processing, Lecture 3.

  • From now on, the hat, \(\hat{}\), represents the normalized version.

Notable Notes

  • von Coler et al. 2018, Parametric Synthesis of Glissando Note Transitions - A User Study in a Real-Time Application, DAFx-18.
  • signal: a mathematical function that carries information, could be pressure, control parameters and so on.
  • Lagrange polynomial used in spatial sampling, non-integer delay as an interpolating filter.
  • A delay system is a system.
  • ADSR is a synthesizer.
  • \(s[n] = \sum\limits_{k\in \mathbb{Z}}s[k]\delta[n-k]\)
  • Dirac delta function
    • There are many ways to define Dirac delta function, see Wolfram MathWorld
    • See more here
    • integral, kind of dot product.
  • Block by Block (buffer) in applications equals vectorization
  • Sampling makes the spectrum periodic
    • For two frequency, \(\hat{f_0}\) and \(\hat{f_0}+r\)
      • Discrete-time domain: \(\cos(2\pi\hat{f_0}n) = \cos(2\pi(\hat{f_0}+r)n)\), because \(2\pi r n\) is an integer multiple of \(2\pi\), so it is periodic inthe frequency domain.
      • Continuous-time domain: \(\cos(2\pi f_0 t) \neq \cos(2\pi (f_0+rf_0)t)\)
    • From \(x(t)=x(t+T_0)\) to \(x[n] = x[n+N_0]\), it only works when $N_0 = k $ and \(k=0,1,\dots, N_0\)
  • impulse (time-domain) \(\rightarrow\) 1 (frequency-domain) \(\rightarrow\) alias filter (works as a bandlimited filter) \(\rightarrow\) rectangular (frequency-domain) \(\rightarrow\) ADC \(\rightarrow\) sinc function (time-domain)
  • The basis of Fourier transform is simply rotating vectors in the 2D plane (\(e^{2\pi j\hat{f_0}n}\))

Discrete-time sequences

Impulse

\[\begin{equation} \delta[n] = \begin{cases} 1, \quad n=0,\\ 0, \quad n\neq0 \end{cases} \end{equation}\]

  • Delayed impulse: \(\delta_{n_0}[n]=\delta[n-n_0]\)
  • Impulse response

Unit step sequence

\[\begin{equation} u[n] = \begin{cases} 1, \quad n\leq0,\\ 0, \quad n<0 \end{cases} \end{equation}\]

  • \(u[n]=u[n-1]+\delta[n]=\sum\limits_{k=0}^\infty\delta[n-k]\)
  • works as a switch (control)
  • used to check the stability

Rectangular sequence

\[\begin{equation} r[n]=u[n]-u[n-N],\end{equation}\] where \(N\) is the length of the rectangular

  • \(r[n] = \delta[n]\) when \(N=1\) \(\rightarrow\) \[\delta[n]=u[n]-u[n-1],\] which is also explained as a finite-difference scheme, representing the slope the signal is we divide both sides by the sampling time $ T_s$
  • used to design waveforms like a square wave (a linear combination of rectangular sequence)

Damped Exponentials

\[\begin{equation} x[n] = \begin{cases} Aa^n, \quad n\leq0,\\ 0, \quad n<0 \end{cases} \end{equation}\]

  • \(0<a<1\), damped signal, \(-1<a<0\), damped osillating signal.
  • RC circuit and RC filter, working as a low pass filter, check here.
  • The frequency response of \(a^nu[n]\) is \(\dfrac{1}{1-ae^{-j\hat{\omega}}}\), where \(\hat{\omega}\) is the normalized radian frequency.
  • recursive

Sinusoids sequence

\[\begin{equation}x[n] = A_0\cos(2\pi \hat{f_0}n+\phi_0),\end{equation}\] where \(\hat{f_0}\) is the normalized frequency and \(\phi_0\) is the initial phase.

  • recursive computaion of \(sin\) functions

Complex exponential sequence

\[P_{f_0}[n]=e^{2\pi j \hat{f_0}n}\] and \[z[n] = A_0e^{j\phi_0}e^{j2\pi\hat{f_0}n}=z[n-1]e^{2\pi j \hat{f_0}},\] where the phasor \(\hat{A}=Ae^{j\phi_0}\) is the complex amplitude.

Damped sinusoids

\[x[n] = A_0e^{-\alpha n}\cos(2\pi\hat{f_0}n+\phi_0)\] and \[z[n] = \hat{A_0}e^{-\alpha+2\pi j\hat{f_0}n},\] where \(e^{-\alpha}\) represents the damping.

Comparing: \[x(t) \rightarrow x(t-d)\] and \[sin(2\pi f_0t+\phi_0) \rightarrow sin(2\pi f_0 (t-d)+\phi_0),\] where the phase shift \(-2\pi f_0 d\) is frequency dependent.

Others

spatial wave

\[p(t,r) = A_0\cos(2\pi f_0(t-\frac{r}{c}))\] - the link between the spatial domain and the phase domain - wavenumber is the spatial frequency of the wave \(k = \dfrac{2\pi f_0}{c}=\dfrac{2\pi}{\lambda}\) (radians or circle per unit distance), BTW, wavenumber is not dimensionless but Helmholtz number \(ka\) is. - Compared to frequency \(\omega = \dfrac{2\pi}{T}\), where \(T\) and \(\lambda\) are the length of the period in time and space, respectively.

Phasor

  • phasor \(\leftrightarrow\) vector

Linear chirp

\[\phi[n] = \phi_0 + 2\pi(\hat{f_0}+\frac{\beta(n+1)}{2})n\] instead of simply \(\phi[n] = \phi_0+2\pi f[n]n\), where \(f[n] = \hat{f_0}+\beta n\)

  • this is to maintain the phase continuity.