AC_03b_FilterBanks3.ipynb

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    "<center>\n",
    "    <img src=\"./images/ac_header.png\">\n",
    "</center>\n",
    "\n",
    "### <a target=\"_blank\" rel=\"noopener noreferrer\" href=\"https://www.tu-ilmenau.de/mt-ams/personen/schuller-gerald/\">Prof. Dr. -Ing. Gerald Schuller</a> <br> <a target=\"_blank\" rel=\"noopener noreferrer\" href=\"https://www.tu-ilmenau.de/mt-ams/lehre/msp-and-adsp-tutorials/\">Jupyter Notebook: Renato Profeta</a> \n",
    "\n",
    "[Applied Media Systems Group](https://www.tu-ilmenau.de/en/applied-media-systems-group/) <br>\n",
    "[Technische Universität Ilmenau](https://www.tu-ilmenau.de/)"
   ]
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   "source": [
    "# Filter Banks III"
   ]
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    "%%html\n",
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   "source": [
    "## Block Switching"
   ]
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     "text": [
      "https://onlinelibrary.wiley.com/doi/book/10.1002/0470041978\n"
     ]
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       "\n",
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   "source": [
    "from IPython.display import IFrame\n",
    "print('https://onlinelibrary.wiley.com/doi/book/10.1002/0470041978')\n",
    "IFrame('https://ieeexplore.ieee.org/book/8039676', width=900, height=400)\n"
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   "source": [
    " - **Problem:** In audio coding, Pre-echoes appear before transients.\n",
    "     - reason: blocks too long.\n",
    "\n",
    "<center>\n",
    "    <img src='./images/ac_46_blockSwit.png' width='900'>\n",
    "</center>\n",
    "\n",
    " - **Approach:** for fast changing signals use block switching to lower number of subbands.\n",
    " \n",
    "<center>\n",
    "    <img src='./images/ac_47_blockSwit2.png' width='900'>\n",
    "</center>"
   ]
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   "source": [
    "### Accomodate Overlap-Add for Block Switching"
   ]
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   "source": [
    "<center>\n",
    "    <img src='./images/ac_48_blockSwit3.png' width='700'>\n",
    "</center>\n",
    "<br>\n",
    "<center>\n",
    "    <img src='./images/ac_49_blockSwit4.png' width='700'>\n",
    "</center>\n"
   ]
  },
  {
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   },
   "source": [
    "### Block Switching"
   ]
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    " - Sequence of windows for switching the number of sub-bands.\n",
    " - Shorter windows $\\rightarrow$ better resolution.\n",
    " \n",
    "<center>\n",
    "    <img src='./images/ac_50_blockSwit5.png' width='700'>\n",
    "</center>\n"
   ]
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   "execution_count": 16,
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       "\n",
       "        <iframe\n",
       "            width=\"900\"\n",
       "            height=\"400\"\n",
       "            src=\"https://ccrma.stanford.edu/~pdelac/422/project/report.html\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
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    "IFrame(\"https://ccrma.stanford.edu/~pdelac/422/project/report.html\",width=900, height=400)"
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   "source": [
    "## Wavelets, QMF (Quadradutre Mirror Filter) Filter Banks"
   ]
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    "%%html\n",
    "<iframe width=\"560\" height=\"315\" src=\"https://www.youtube.com/embed/gRdIhI04wt8?rel=0\" frameborder=\"0\" allow=\"accelerometer; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen></iframe>"
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    " - Iterate 2-band system.\n",
    " - See also: Wavelet Packets (more general)\n",
    " - **Problem:** Aliasing propagation reduces frequency selectivity!\n",
    " - Important in image coding, but no big role in Audio Coding.\n",
    "\n",
    "<center>\n",
    "    <img src='./images/ac_51_qmf.png' width='700'>\n",
    "</center>\n"
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   "source": [
    "### How to Obtain a Two Band Filter Bank"
   ]
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    " - Application: QMF filter banks, Wavelets, ...\n",
    " - Analysis polyphase for a 2-band filter bank:\n",
    " \n",
    " $$\\large\n",
    " H(z)=\n",
    " \\begin{bmatrix}\n",
    " H_{0,0}(z) & H_{0,1}(z) \\\\\n",
    " H_{1,0}(z) & H_{1,1}(z)\n",
    " \\end{bmatrix}\n",
    " $$\n",
    " <br>\n",
    " - **Observe:** $H_{0,0}(z)$ contains the even coefficients of the low pass filter, and $H_{1,0}(z)$ its odd coefficients.\n",
    " - Accordingly for the high pass filter.\n",
    " - Given the analysis filters, the synthesis filters can be obtained by inverting the analysis polyphase matrix:\n",
    " \n",
    " $$\\large\n",
    " H^{-1}(z) = \\dfrac{1}{Det(H(z))}\n",
    " \\begin{bmatrix}\n",
    " H_{1,1}(z) & -H_{0,1}(z) \\\\\n",
    " -H_{1,0}(z) & H_{0,0}(z)\n",
    " \\end{bmatrix}\n",
    " $$\n",
    " <br>\n",
    " \n",
    " - **Observe:** If the analysis filters have a finite impulse response (FIR), and the synthesis is desired to also be FIR, the **determinant** of the polyphase matrix needs to be a **constant or a delay**!\n",
    " \n",
    " $$\\large\n",
    " det(H(z)) = H_{1,1}(z)H_{0,0}(z)-H_{0,1}(z)H_{1,0}(z)  \\\\ = \\text{const or delay}$$\n",
    " <br>\n",
    " - **Observe:** This is the output of the lower band of the filter bank if the input signal is:\n",
    " \n",
    " $$\\large\n",
    " x(z) = \\left \\lfloor H_{1,1}(z), -H_{0,1} \\right \\rfloor $$\n",
    " \n",
    " - Hence the determinant can be formulated as a **convolution**.\n",
    " - This input is the high band filter coefficients, with the sign of the even coefficients flipped and switched places with the odd coefficients.\n",
    " - Since this represents a critically sampled filter bank, the result represents **every second sample** of the convolution of the low band filter with the correspondingly modified high band filter.\n",
    " - This modified high band filter is a low band filter (every second sample sign flipped).\n",
    " - The desired output of this downsampled convolution is a single pulse (corresponding to a constant or a delay), hence flat in frequency.\n",
    " - Another interpretation: correlation of the 2 signals, where the even lags that appear after downsampling are zero, except for the one pulse."
   ]
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   "source": [
    "## QMF (Quadrature Mirror Filter)"
   ]
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    " - This suggests a simple design strategy:\n",
    "     - Design a low pass filter for analysis and synthesis.\n",
    "     - Obtain the high pass filters by flipping the low pass filters every second coefficient.\n",
    "     \n",
    "$$\\begin{array}\n",
    "\\text{analysis FB:} & h_1(n)=(-1)^nh_0(n) & n=0,1,...,N-1 \\\\\n",
    "\\text{synth. low pass:} & g_0(n)=h_0(n) & \\\\\n",
    "\\text{synth. FB high pass:} & g_1(n)=-h_1(n) & \n",
    "\\end{array}$$\n",
    "\n",
    " - This is an early two band filter bank: QMF, Quadrature Mirror Filter (Croisier, Esteban, Galand, 1976)\n",
    " - For more than 2 bands: GQMF (Cox, 1986), PQMF\n",
    " \n",
    " - Sign flipping to obtain the high band filter leads to the polyphase components:\n",
    " <br>\n",
    "$$\\large\n",
    "H_{0,1}(z) = H_{0,0}(z) \\\\\n",
    "\\large\n",
    "H_{1,1}(z) = -H_{1,0}(z)\n",
    "$$<br>\n",
    "\n",
    " - The resulting determinant is:\n",
    "\n",
    "$$\\large\n",
    "det(H(z))=H_{1,1}(z)H_{0,0}(z)-H_{0,1}(z)H_{1,0}(z) \\\\\n",
    "\\large\n",
    "=-2H_{0,1}(z)H_{0,0}(z)$$\n",
    "\n",
    " - **Observe:** This cannot be made a constant or delay for finite polynomials of order 1 or greater, hence no PR for finite length filters!\n",
    " - The QMF accounts for the sign flipping in the determinant equation.\n",
    " - But not for the trading places of even and odd coefficients.\n",
    " - Hence: **No Perfect Reconstruction** (only for simple Haar and IIR filters)\n",
    " - High stopband attenuation needed to keep reconstruction error small.\n",
    " - Numerical optimization to obtain:\n",
    " \n",
    "  $$\\large\n",
    " \\left |H_0\\left(e^{j\\omega}\\right)\\right |^2 + \\left |H_1\\left(e^{j\\omega}\\right)\\right |^2 \\approx 1$$\n",
    " <br>\n",
    " <center>\n",
    "    <img src='./images/ac_52_qmf2.png' width='600'>\n",
    "</center>\n",
    " \n",
    " <center>\n",
    "    <img src='./images/ac_53_qmf3.png' width='500'>\n",
    "</center>\n",
    "\n",
    " - QMF: Example with Impulse Response of Length 96\n",
    " <br>\n",
    " \n",
    " <center>\n",
    "    <img src='./images/ac_54_qmf4.png' width='800'>\n",
    "</center>"
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   "source": [
    "### CQMF: Conjugate QMF"
   ]
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    "%%html\n",
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    " - To also accommodate for the trading places of odd and even coefficients, a natural choice is to also reverse the temporal order of the synthesis filter.\n",
    " \n",
    " $$\\large\n",
    " h_1(n) = -n_0(L-1-n)(-1)^n$$\n",
    " <br>\n",
    "     - With L: filter length, and $g_0(n) = h_0(n) \\quad g_1(n) = - h_1(n)$\n",
    "     - Introduced e.g. by Smith, Barnwell, 1984\n",
    " <br>\n",
    " \n",
    " \n",
    " - For the polyphase components this means:\n",
    " \n",
    " $$\\large\n",
    " \\begin{array}{ccc}\n",
    " H_{0,1} & = & -z^{\\frac{L}{2}}H_{0,0}(z^{-1}) \\\\\n",
    " H_{1,1} & = & z^{\\frac{L}{2}}H_{1,0}(z^{-1}) \n",
    " \\end{array}$$\n",
    " \n",
    " - And the input for our determinant calculation is:\n",
    " \n",
    " $$\\large\n",
    " x(z) = z^{-\\frac{L}{2}} \\left \\lfloor H_{1,0}(z^{-1}), H_{0,0}(z^{-1}) \\right \\rfloor$$\n",
    " <br>\n",
    " - This corresponds exactly to the time reversed low band filter!\n",
    " \n",
    " - Let's define:\n",
    " \n",
    " $$\\large\n",
    " A(z) = H_{1,0}(z^{-1})H_{0,0}(z) $$\n",
    " \n",
    " - The determinant is now:\n",
    " \n",
    " $$\\large\n",
    " det(H(z)) = H_{1,1}(z)H_{0,0}(z)-H_{0,1}(z)H_{1,0}(z) \\\\\n",
    " \\large\n",
    " = z^{-\\frac{L}{2}} (A(z)+A(z^{-1}))$$<br>\n",
    "     - **Observe:** This can be a constant if all even coefficients of A(z) are zero, except for the center coefficient!\n",
    "     - **Remember:** the determinant was the output of the low band with this input.\n",
    "     - Hence: Every second sample of the convolution of the low band filter with its time reversed version.\n",
    "     - This is equal to **every second value** of the **autocorrelation** function of the l**ow band filter**!\n",
    "     - Determinant is a constant or a delay: only one sample of this downsampled autocorrelation function (all even coefficients) can be unequal zero (most even coefficients are zero).\n",
    "     - The Determinant is a constant means:\n",
    "         - The 0th autocorrelation coefficient is a constant (unequal 0), and all other even coefficients must be zero.\n",
    "         - Called Nyquist filter property.\n",
    " \n",
    " \n",
    " - In other terms: Define P(z) as the z-transform of this autocorrelation function, the **Power Spectrum**:\n",
    " \n",
    " $$\\large\n",
    " P(z) := H_0(z) \\cdot H_0(z^{-1})$$\n",
    " \n",
    " - Then all nonzero coefficients of P(z) are the 0th coefficient and the odd coefficients.\n",
    " - As a result:\n",
    " \n",
    " $$\\large\n",
    " P(z) + P(-z) = const$$\n",
    " \n",
    "     - The odd coefficients cancel.\n",
    "     - Frequency reversal $\\rightarrow$ (P(-z))\n",
    "     - This is also called the halfband filter property.\n",
    "     \n",
    "     \n",
    " - Design approach: Design P(z) accordingly, then H(z)"
   ]
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   "source": [
    "## Pseudo-QMF (PQMF)"
   ]
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    "%%html\n",
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    " - So far we only had 2 subband QMF filter banks.\n",
    " - Only for the 2-band case we get perfect reconstruction (in the CQMF case).\n",
    " - The PQMF extents the QMF approach to N>2 subbands.\n",
    " - But it has only \"Near Perfect Reconstruction\", meaning a reconstruction error by the filter bank.\n",
    " - It is modulated filter band (like the MDCT), using a baseband prototype filter h(n) (a lowpass).\n",
    " - Its analysis filters are given by the impulse responses (L being the length of the impulse response):\n",
    " \n",
    " $$\\large\n",
    " H_k(n) = h(n) \\cos \\left( \\dfrac{\\pi}{N} \\cdot (k+0.5)(n+0.5-\\dfrac{L}{2}+(-1)^k\\dfrac{\\pi}{4} \\right)$$\n",
    " \n",
    " - It is an (almost) **orthogonal filter bank**, which means that the synthesis filter impulse responses are simply the time inverses of the analysis impulse responses:\n",
    " \n",
    " $$\\large\n",
    " g_k(n)=h_k(n)(L-1-n)$$\n",
    " \n",
    " - Its baseband prototype filters h(n) are now designed such that aliasing cancels between adjacent neighbouring bands:\n",
    " \n",
    " $$\\large\n",
    " \\left| H(e^{j\\Omega}) \\right|^2 + \\left| H \\left(e^{j\\left(\\frac{\\pi}{N}-\\Omega\\right)}\\right) \\right|^2 = 1 \\\\\n",
    " \\large\n",
    " \\text{for } 0 < |\\Omega| < \\frac{\\pi}{2N}\n",
    " $$\n",
    " \n",
    " - Beyond the adjacent bands, the attenuation should go towards infinity:\n",
    " \n",
    " $$\\large\n",
    " \\left| H(e^{j\\Omega}) \\right|^2 = 0, \\text{for } |\\Omega| > \\frac{\\pi}{N}$$\n",
    " \n",
    " \n",
    " - This leads to **\"Near Perfect Reconstruction\"** (there is a **reconstruction error**).\n",
    " - The PQMF filter bank is used in MPEG1/2 Layer I,II and III. There it has N=32 subbands and filter length L=512.\n",
    " - Also used in MPEG 4 for so-called SBR (Spectral Band Replication) and for parametric sourround coding. There it has N=32 or N=64 subbands, and filter length L=320 or L=640."
   ]
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   "source": [
    "### PQMF used in MPEG4"
   ]
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    " - Impulse response of the baseband prototype (the window), with N=64 and L=640\n",
    " \n",
    "<br>\n",
    "<center>\n",
    "    <img src='./images/ac_55_pqmf1.png' width='600'>\n",
    "</center>\n",
    "<br>\n",
    "\n",
    " - Frequency response of the baseband prototype (the window)\n",
    " \n",
    "<br>\n",
    "<center>\n",
    "    <img src='./images/ac_56_pqmf2.png' width='600'>\n",
    "</center>\n",
    "<br>"
   ]
  }
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