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Commit 14c18009 authored by Eric Kooistra's avatar Eric Kooistra
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Clarify type 1 (delays) and type 2 (advances) polyphase.

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applications/lofar2/model/pfb_os/dsp_study_erko.txt
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......@@ -64,11 +64,15 @@
# * [PM-REMEZ] https://pm-remez.readthedocs.io/en/latest/
# * [SELESNICK] Ivan Selesnick
# . https://eeweb.engineering.nyu.edu/iselesni/EL713/zoom/mrate.pdf
# * [KIEL] Part 6: Multi-Rate Digital Signal Processing, Gerhard Schmidt
# https://dss-kiel.de/images/teaching/lectures/advanced_digital_signal_processing/slides/adsp_06_multirate_processing.pdf
#
# https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/
# Youtube: Guitars 4RL
# Youtube: MATHEMATICAL METHODS AND TECHNIQUES IN SIGNAL PROCESSING
# https://www.allaboutcircuits.com/technical-articles/pipelined-direct-form-fir-versus-the-transposed-structure/
#
# FBMC = filter bank-based multicarrier
1) Linear Time Invariant (LTI) system [LYONS 1.6]
......@@ -547,7 +551,7 @@ c) s-plane and z-plane
= sum bk x[n - k] - sum ak y[n - k]
k=0 k=1
- Implmentation structure [LYONS 6.6, JOS2 9.1]:
- Implmentation structure [LYONS 6.6, JOS2 9.1, VAIDYANATHAN 2.1.3]:
. Direct-Form I: B(z) * 1 / A(z)
x[n] ---v-----> b0 --> + --> + -------------v----> y[n]
......@@ -569,6 +573,22 @@ c) s-plane and z-plane
--> b1 --> z^-1 <-- a1 <--
--> b2 --> z^-1 <-- a2 <--
. With the Direct-Form FIR the input is passed through the delay line, and
then weighted at each tap and added all in parallel, so combinatorially.
With the Transposed Direct-Form FIR the input is first fanned out and
weighted in parallel for all taps and passed to the delay line for each
tap. The summation is done per tap and the (intermediate) sum is passed
through the delay line.
. Transposed Direct-Form is self pipelined
. Direct-Form implementation can be pipelined by adding registers in the
adder line (reg) and in the delay line (z^(-1) + reg).
. Transposed Direct-Form has input fanout to all multipliers.
. Direct-Form has b0, b1 at first delay,
Transposed Direct-Form has b_{N-1}, b_{N-2} at first delay, so the FIR
coefficients are flipped in order along the delay line. However, in
both cases b0 weights the current sample x[n] and connects
combinatorially to the output y[n], so the implementation differs, but
the formula is exactly the same.
- For FIR b = h. For IIR it is not possible to directly derive b, a from h
[LYONS 6.1]. Therefor use z-transform [LYONS 6.3]:
......@@ -1162,8 +1182,16 @@ c) s-plane and z-plane
. 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.
. If D and U are relatively prime, then they can be changed in order.
If D and U are relatively prime, then there are integers d, u such that
d*D - u*U = 1, so then a delay z^(-k) can be decomposed in [KIEL]:
z^(-k) = z^(-k(d*D - u*U))
= z^(-kdD) * z^(kuU)
- Polyphase decomposition of H(z) [VAIDYANATHAN 4.3, PROAKIS 10.5.2]:
which then can be pulled through a D and a U
- Polyphase decomposition of H(z) [VAIDYANATHAN 4.3, PROAKIS 10.5.2,
CROCHIERE 3.3]:
. Direct-Form FIR is first apply delay z^(-q) then apply coefficient. Fits
down sampling because coefficients are then applied at low rate.
......@@ -1178,19 +1206,26 @@ c) s-plane and z-plane
- up sampling output commutator rotates clockwise and yields U samples
every rotation
. Type I polyphase
. Type I polyphase representation, based on delays z^(-q), yielding counter
clockwise commutator
(Can also notate Hq = Eq, hq = eq)
H(z) = E0(z^Q) + E1(z^Q) z^-1 + E2(z^Q) z^-2 + ... + E_{Q-1}(z^Q) z^-(Q-1)
= E0(z^Q) +
z^-1 E1(z^Q) +
z^-2 E2(z^Q) +
... +
z^-(Q-1) E_{Q-1}(z^Q)
Q-1
= sum z^(-q) Eq(z^Q) [VAIDYANATHAN Eq 4.3.7]
q=0
where Eq(z^Q) is the z-transform of eq[n]:
eq[n] = h(nQ + q), +q for counter clockwise [PROAKIS
eq[n] = h(Qn + q), +q for counter clockwise with delays z^(-1)
+inf
Eq(z) = sum eq[n] z^(-n), 0 <= q <= Q - 1
......@@ -1198,16 +1233,44 @@ c) s-plane and z-plane
- Note: For Q = 1 the Eq(z) are the single FIR coefficients in b
. Type II polyphase, clockwise commutator:
. Type II polyphase, based on advances z^(q), yielding clockwise commutator:
H(z) = z^(-(Q-1)) z^(Q-1) H(z), multiply by delay * advance = 1
= z^(-(Q-1)) * [z^(Q-1) E0(z^Q) + # q = 0
z^(Q-1) z^-1 E1(z^Q) + # q = 1
z^(Q-1) z^-2 E2(z^Q) + # q = 2
... +
z^(Q-1) z^-(Q-2) E_{Q-2}(z^Q)] # q = Q-2
z^(Q-1) z^-(Q-1) E_{Q-1}(z^Q)] # q = Q-1
= z^(-(Q-1)) * [z^(Q-1) E0(z^Q) + # q = 0
z^(Q-1 - 1) E1(z^Q) + # q = 1
z^(Q-1 - 2) E2(z^Q) + # q = 2
... +
z^( 1) E_{Q-2}(z^Q) + # q = Q-2
E_{Q-1}(z^Q)] # q = Q-1
= z^(-(Q-1)) * [ E_{Q-1}(z^Q) + # q = Q-1 --> p = 0
z^( 1) E_{Q-2}(z^Q) + # q = Q-2 --> p = 1
... +
z^(Q-1 - 2) E2(z^Q) + # q = 2 --> p = Q-3
z^(Q-1 - 1) E1(z^Q) + # q = 1 --> p = Q-2
z^(Q-1) E0(z^Q) + # q = 0 --> p = Q-1
Q-1
H(z) = sum z^(-(Q-1-q) Rq(z^Q) [VAIDYANATHAN Eq 4.3.7]
q=0
H(z) = sum z^(-(Q-1-p)) Rp(z^Q) [VAIDYANATHAN Eq 4.3.9]
p=0
Q-1
= z^(-(Q-1)) sum z^p Rp(z^Q)
p=0
where:
rq[n] = h(nQ - q)
rp[n] = h(Qn - p), -p for clockwise with advances z^(+1)
Rq(z) = E_{M-1-q}(z)
Rq(z) = E_{Q-1-p}(z), so flipud phases, but keep coefficient order
per phase
. Phase q of h(n) with Q - 1 zeros, so [VAIDYANATHAN Fig 4.3.1]:
......
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