Notes of Digital Audio Signal Processing, Lecture 5.

Convolution and multiplication

  • \(Z(f)=X(f)Y(f)\leftrightarrow z=x*y\)
  • \(Z(f)=X(f)*Y(f)\leftrightarrow z=xy\)
  • Application in Source-Filter model

Parseval's theorem

\[\sum\limits_{n=-\infty}^\infty x^2[n]= \dfrac{1}{2\pi}\int_{-\pi}^{\pi}|X(e^{j\hat{\omega}})|^2\text{d}\hat{\omega}=\int_{-\frac{1}{2}}^{\frac{1}{2}}|X(f)|^2\text{d}f,\] where the \(|X(f)|^2\) is called the power spectral density which is with respect to frequency.

  • To prove:
    • Given signal and its DTFT \(x\leftrightarrow X\)
    • For the time reversed version of \(x\), \(y[n] = x[-n] \leftrightarrow Y(f)=\overline{X}(f)\)
    • For signal \(z\) whose DTFT is defined as \(Z(f)=X(f)Y(f) = X(f)\overline{X}(f)\). \(z=x*y\)
    • In time domain: \(z[n]=\sum\limits_{k\in\mathbb{Z}}x[k]y[n-k] = \sum\limits_{k\in\mathbb{Z}}x[k]x[k-n]\), where \(\sum\limits_{k\in\mathbb{Z}}x[k]x[k-n]\) is the autocorrelation. When \(n=0\), \[\begin{equation}z[0]=\sum_\limits{k\in\mathbb{Z}}x^2[k]\label{Parseval1}\end{equation}\].
    • Taking the inverse DTFT: \(z[n] = \int_{-\frac{1}{2}}^{\frac{1}{2}}Z(f)e^{2\pi jfn}\text{d}f = \int_{-\tfrac{1}{2}}^{\tfrac{1}{2}}X(f)Y(f)e^{2\pi jfn}\text{d}f\). When \(n=0\), \[\begin{equation}z[0]=\int_{-\frac{1}{2}}^{\frac{1}{2}}|X(f)|^2\text{d}f\label{Parseval2}\end{equation}\].
    • Eq. \(\eqref{Parseval1}\) = Eq. \(\eqref{Parseval2}\).
  • Even: \(x_e[n]=x_e[-n]\)
  • Odd: \(x_o[n]=-x_o[-n]\)
  • properties:
    • \(x_e\bot x_o\), (\(\sum x[n]y[n]=0\), dot product equals zero)
    • \(\text{odd}\times\text{even} = \text{even}\)
  • For any signal, it could be decomposed as into an even signal and an odd signal, meaning, \(x[n] = x_e[n]+x_o[n]\), where \(x_e[n] = \frac{x[n]+x[-n]}{2}\) and \(x_o[n] = \frac{x[n]-x[-n]}{2}\)
  • Apply it into Fourier transform \[ \begin{align} X(f) &= \sum\limits_{n\in\mathbb{Z}}x[n]e^{-2\pi jfn}\\ &= \sum\limits_{n\in\mathbb{Z}}(x_e[n]+x_o[n])(\cos 2\pi fn - j \sin2\pi fn)\\ &= \sum\limits_{n\in\mathbb{Z}}(x_e[n]\cos2\pi fn - jx_o[n]\sin2\pi fn),\\ \end{align}\] where the \(\Re{\{X(f)\}}\) is even and the \(\Im{\{X(f)\}}\) is odd.
    • So a real spectrum means the even signal and a pure imaginary spectrum corresponds to a odd signal.

Frequency shift and modulation

  • \(z=xp_{f_0}\) (\(z[n]=x[n]e^{2\pi jf_0n}\))
  • Its Fourier transform: \(Z(f) = X*P_{f_0}=X(f-f_0)\), where \(P_{f_0}(f)=\delta(f-f_0)=\delta_{f_0}(f)\)
  • Periodic in one domain means evenly spaced in the other domain.
  • Dirac comb

Derivative of a Spectrum

  • \[ \begin{align} \dfrac{\text{d} X(f)}{\text{d} f} &= \frac{\text{d} \sum\limits_{n\in\mathbb{Z}}x[n]e^{-2\pi jfn}}{\text{d} f}\\ &= -2\pi j \sum\limits_{n\in\mathbb{Z}}nx[n]e^{-2\pi jfn} \end{align}\]
  • Application: gain
    • \(x[n] \rightarrow y[n] = g[n]x[n]\), where \(g[n] = a+bn\) is a gain, linearly evolves over time.
    • \(y[n]=ax[n]+b(nx[n])\) \(\leftrightarrow\) \(Y(f) = aX(f)+\frac{bj}{2\pi}\frac{\text{d} X(f)}{\text{d}f}\).
    • gain in time \(\leftrightarrow\) derivative in spectrum
    • Further explanation?

Time scaling

  • \(y(t)=x(\alpha t)\) \(\leftrightarrow\) \(Y = \frac{1}{\alpha}X(\frac{f}{\alpha})\)

Discrete-time system

  • A system: \(y=\mathcal{T}\{x\}\) or \(y[n] = \mathcal{T}\{x\}[n]\)
  • Like delay (\(y[n]=x[n-n_0]\)), square (\(y[n]=x^2[n]\)), moving max, threshold and so on.
  • Distortion - Chebyshev polynomials?
  • Noise reduction need distortion?

classes

  • memoryless: only the current time (no past, no future samples);
  • linear: additivity and scalability
  • time invariance: the system propcessing doesn't depends on when you apply it (\(y[n-{n_0}]=\mathcal{T}\{x[n-{n_0}]\}\))
  • Stability: \(\lVert x\rVert < B_x\) \(\leftrightarrow\) \(\lVert y\rVert < B_y\).
    • delay: stable
    • amplifier: stable
    • accumulator: depends

Linear Time-Invariant system \(\leftrightarrow\) filter

  • A filter is a LTI system.
  • \(x[n] = \sum\limits_{k\in\mathbb{Z}}x[k]\delta[n-k]\) \[\begin{align} y[n] &= \mathcal{T}\{x\}[n] \\ &= \mathcal{T}\{\sum\limits_{k\in\mathbb{Z}}x[k]\delta_k[n]\}\\ &= \sum\limits_{k\in\mathbb{Z}}\mathcal{T}\{x[k]\delta_k[n]\}\\ &= \sum\limits_{k\in\mathbb{Z}}\{x[k]\mathcal{T}\{\delta_k[n]\}\} \quad \text{applying linearities}\\ &= \sum\limits_{k\in\mathbb{Z}}x[k]h_k[n], \end{align} \] where \(h_k=\mathcal{T}\{\delta_k\}\) is the impulse reponse.
  • \(y=x*h\)
  • \(Y=XH\), where \(H\) is the Fourier transform of \(h\) and is the frequency response.
  • Toeplitz matrix and convolution
  • For convoluion: \(N_y = N_x+N_h-1\)
  • Properties:
    • Stability: depends on \(h\), meaning the bound of \(\sum\limits_{k\in\mathbb{Z}}|h[n-k]|\)
    • causality: \(h[n-k]=0\) for \(k\leq n\).
    • memoryless: \(h[k]=0\) when \(k\neq 0\)

Notable notes

  • Energy: the accumulated version of the power \(x^2[n]\)