diff --git a/applications/lofar2/model/pfb_os/dsp_study_erko.txt b/applications/lofar2/model/pfb_os/dsp_study_erko.txt
index baf30dad7c15f9739ca2274d23b8540498c214db..72eea03d208d04678a3baa39e80c01ff10d28605 100644
--- a/applications/lofar2/model/pfb_os/dsp_study_erko.txt
+++ b/applications/lofar2/model/pfb_os/dsp_study_erko.txt
@@ -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]: