Skip to content
Snippets Groups Projects
Commit 8f00ad45 authored by Eric Kooistra's avatar Eric Kooistra
Browse files

Improve multirate section.

parent 098c2af2
No related branches found
No related tags found
No related merge requests found
Pipeline #87585 passed
...@@ -31,6 +31,7 @@ ...@@ -31,6 +31,7 @@
# * [JOS2] Introduction to Digital Filters, 2007 # * [JOS2] Introduction to Digital Filters, 2007
# * [JOS3] Physical Audio Signal Processing, 2010 # * [JOS3] Physical Audio Signal Processing, 2010
# * [JOS4] Spectral Audio Signal Processing, 2011 # * [JOS4] Spectral Audio Signal Processing, 2011
# * [SP4COMM] Signal Processing for Communications, 2008, Paolo Prandoni and Martin Vetterli
# #
# * [WIKI] https://en.wikipedia.org/wiki/Bilinear_transform # * [WIKI] https://en.wikipedia.org/wiki/Bilinear_transform
# * [WIKI] https://en.wikipedia.org/wiki/Discrete_cosine_transform # * [WIKI] https://en.wikipedia.org/wiki/Discrete_cosine_transform
...@@ -61,6 +62,8 @@ ...@@ -61,6 +62,8 @@
# . "Realisering van Digitale Signaalbewerkende Systemen, Toepassingen", # . "Realisering van Digitale Signaalbewerkende Systemen, Toepassingen",
# 5N290, TUE, P.C.M. Sommen, --> DAC slide 19, 20, # 5N290, TUE, P.C.M. Sommen, --> DAC slide 19, 20,
# * [PM-REMEZ] https://pm-remez.readthedocs.io/en/latest/ # * [PM-REMEZ] https://pm-remez.readthedocs.io/en/latest/
# * [SELESNICK] Ivan Selesnick
# . https://eeweb.engineering.nyu.edu/iselesni/EL713/zoom/mrate.pdf
# #
# https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/ # https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/
# Youtube: Guitars 4RL # Youtube: Guitars 4RL
...@@ -1032,42 +1035,92 @@ c) s-plane and z-plane ...@@ -1032,42 +1035,92 @@ c) s-plane and z-plane
. H_i(z^Q) is upsampled-by-Q version of H(z), so with Q-1 zero coefficients . H_i(z^Q) is upsampled-by-Q version of H(z), so with Q-1 zero coefficients
in the H(z) power series, starting at phase i. in the H(z) power series, starting at phase i.
- LPF + downsampling = decimation [LYONS 10.9, PROAKIS 10, CROCHIERE Fig 3.2]: - LPF + downsampling = decimation [LYONS 10.9, PROAKIS 10, CROCHIERE Fig 3.2,
. Downsampling: Y(z) = X(z^(1/Q)) VAIDYANATHAN Fig 4.1.4, SP4COMM 11.1.2]:
. Downsampling:
y(n) = x(n Q), because y removes Q-1 values from x x_D[n] = x[nD], because x_D removes D-1 values from x
+inf +inf +inf Define x' at x time grid, but with x' = 0 for samples in x that will be
Y(z) = sum y(n) z^-n = sum x(Qn) z^-n = sum x(k) z^-k/Q = X(z^(1/Q)) discarded:
n=-inf n=-inf k=-inf
. Spectrum, evaluate Y(z) on unit circle [PROAKIS Eq 10.2.9]: x'[n] = d[n] x[n], with d[n] = 1, for n % D = 0
= 0, otherwise
Q-1 Use x' to first express z-transform X_D(z) in X'(z) and then z-transform
Y(w) = 1/Q sum H((w - 2 pi k) / Q) X((w - 2 pi k) / Q), w = w_y of X'(z) in X(z). The z-transform X_D(z) = X'(z^(1/D)), because x' is 0
when k is not a multiple of D:
+inf
X_D(z) = sum x_D[n] z^(-n)
n=-inf
+inf +inf
= sum x'[nD] z^(-n) = sum x'[k] z^(-k/D) = X'(z^(1/D))
n=-inf k=-inf
Make use of the identity that the selector d[n] is equal to the normalized
sum of the D roots of unity:
D-1
d[n] = 1/D sum W_D^(-kn), with W_D = exp(-j 2pi/D)
k=0
so:
+inf D-1 +inf
X'(z) = sum d[n] x[n] z^(-n) = 1/D sum sum x[n] (z W_D^k)^(-n)
n=-inf k=0 n=-inf
D-1
= 1/D sum X(z W_D^k)
k=0
Combining the results yields the z-transform of downsampled signal in
terms of z-transform of the original signal:
D-1
X_D(z) = 1/D sum X(z^(1/D) W_D^k)
k=0 k=0
. Discarding samples folds the spectrum around multiples of pi / Q = The Fourier transform of the downsampled signal is obtained by evaluating
fxNyquist / Q = fyNyquist. First the LPF has to remove all folds. X_D(z) on the unit circle with z = exp(jw) [PROAKIS Eq 10.2.9]:
D-1
X_D(e^jw) = 1/D sum X(exp(j (w - k 2pi) / D))
k=0
The resulting spectrum is the scaled sum of D superimposed copies of the
original spectrum X(e^jω), and each copy is shifted in frequency by a
multiple of 2pi/D and the result is stretched by a factor of D. This is
similar to sampling of an analogue signal that creates a periodization of
the analogue spectrum, but now the spectra are already inherently
2pi periodic, and downsampling creates D − 1 additional interleaved copies.
For the spectral copies not to overlap, the maximum (positive) frequency
the original spectrum must be less than pi / D. This is the non-aliasing
condition for the downsampling operator.
. Discarding samples folds the spectrum around multiples of pi / D =
fxNyquist / D. First the LPF has to remove all folds.
. Do not calculate samples that will be thrown away. . Do not calculate samples that will be thrown away.
- Upsampling + LPF = interpolation: - Upsampling + LPF = interpolation:
. Upsampling: Y(z) = X(z^Q), because y inserts Q-1 zeros in x . Upsampling:
y[m] = x[m / Q], when m is multiple of Q, else 0, so equivalently x_U[n] = x[n / U], for n % U = 0
y[m] = x[n], when m = Q n else 0 = 0, otherwise, because x_U inserts U-1 zeros in x
+inf +inf +inf +inf +inf
Y(z) = sum y(m) z^-m = sum y(Qn) z^-Qn = sum x(n) z^-Qn = X(z^Q) X_U(z) = sum x_U[n] z^-n = sum x[k] z^-kU = X(z^U), with n = kU
m=-inf n=-inf n=-inf n=-inf k=-inf
. Spectrum, evaluate Y(z) on unit circle [PROAKIS Eq 10.3.3]: . Spectrum, evaluate X_U(z) on unit circle [PROAKIS Eq 10.3.3]:
Y(w) = Q X(w Q), w = w_y X_U(e^jw) = X(exp(jw U))
. Inserting zeros replicates the spectrum around multiples of pi / Q = . Inserting zeros replicates the spectrum around multiples of pi / U. Then
fyNyquist / Q = fxNyquist. Then the LPF has to remove all replicas and the LPF has to remove all replicas and by that it interpolates to fill in
by that it interpolates to fill in the zeros. the zeros.
. Do not calculate samples that will be inserted as zeros. . Do not calculate samples that will be inserted as zeros.
. Using zero order hold would be a naive approach, because then all samples . Using zero order hold would be a naive approach, because then all samples
need to be calculated and the LPF then needs to compensate for the non-flat need to be calculated and the LPF then needs to compensate for the non-flat
......
0% Loading or .
You are about to add 0 people to the discussion. Proceed with caution.
Please register or to comment