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Commit faff43a6 authored by Eric Kooistra's avatar Eric Kooistra
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Add section on DCT.

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# * [JOS4] Spectral Audio Signal Processing, 2011
#
# * [WIKI] https://en.wikipedia.org/wiki/Bilinear_transform
# * [WIKI] https://en.wikipedia.org/wiki/Discrete_cosine_transform
# * [CHIPMUNK] https://github.com/chipmuenk : Python Frequency Design Analysis and DSP
# * [WHDLWHIZ] https://vhdlwhiz.com/articles/ : FIR filter design using DSP blocks
# * [BIQUAD]
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The DFT uses exp(-j w) and the IDFT uses exp(+j w), so applying IDFT on x(n)
will also result in a frequency domain representation.
- Matrix formulation, DFT as linear transformation [JOS1, PROAKIS 5.1.3]:
- Matrix formulation of DFT, is DFT as linear transformation [JOS1, PROAKIS 5.1.3]:
DFT:
XN = WN xN
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= [x_k * flip(w)](m), with x_k(n) = x(n) exp(-j w_k n)
9) Discrete Cosine Transform (DCT) [WIKI]
. The discrete transform of N samples implicitely assumes that the N samples
extend periodically, this causes discontinuities at the edges for the DFT.
. DCT type II has even symmetry on both sides, so block of N inputs extends
periodically like with DFT, but:
- for DCT it extends flipped to avoid a zero-th order discontinuity, the
slope (first order) is typically still discontinuous.
- and for type II it extends symmetrically half way between the end points
. The different types come from how the boudaries are defined.
. The DCT makes the transform converge faster than the DFT, because any
discontinuities in a function reduce the rate of convergence of the Fourier
series.
. DCT is used for image compression (like JPG), by keeping only few
coefficients of the transformed signal.
. DCT is equivalent to DFTs of roughly twice the length, operating on real
data with even symmetry.
. Discrete Sin Transform (DST) is equivalent to the imaginary parts of a
DFT of roughly twice the length, operating on real data with odd symmetry
(since the Fourier transform of a real and odd function is imaginary and
odd). The DCT is more common than the DST, because the with the DST the
boundaries typically still have discontinuities.
. The Modified DCT (MDCT) uses DCT type IV with overlap. With factor 2
overlap it is used for audio compression (like MP3). Factor 2 overlap
makes it computationally equivalent to DFT.
. The inverse of DCT-II is DCT-III multiplied by 2/N.
9) Multirate processing:
N-1
DCT-II: X(k) = sum x(n) cos(pi/n (n + 1/2) k), for k = 0, 1, ..., N-1
n=0
10) Multirate processing:
- Linear Time Variant (LTV) process, because it depends on when the
downsampling and upsampling start.
- Polyphase filtering ensures that only the values that remain are calculated,
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