Resolve RTSD-265

Closes RTSD-265

applications/lofar2/model/pfb_os/dsp_study_erko.txt

@@ -1145,23 +1145,74 @@ c) s-plane and z-plane
       X_U(e^jw) = X(exp(jw U)), w in [0:2pi>
                                 X(jwL) traverses unit circle U times
 
-- Polyphase decomposition of H(z) [VAIDYANATHAN 4.3]:
-
-  H(z) = H0(z^N) + H1(z^N) z^-1 + H2(z^N) z^-2 + ... + Hi(z^N) z^-i
-
-  . Hi(z^N ) is the z-transform of h(mN + i)
-  . Phase i of h(n) with N - 1 zeros
-
 - Noble identities [LYONS Fig 10.20], HARRIS 2.2.1, VAIDYANATHAN Fig 4.2.3]
-
   . Down sampling : x[n] --> H(z^Q) --> Q:1  --> y[m], is equivalent to:
                     x[n] --> Q:1    --> H(z) --> y[m]
 
+    The output from a filter H(z^Q) followed by a Q:1 downsampler is identical
+    to a Q:1 downsampler followed by a filter H(z).
+
   . Up sampling   : x[n] --> 1:Q  --> H(z^Q) --> y[m], is equivalent to:
                              H(z) --> 1:Q
 
+  . The filter H() can be FIR or IIR.
+  . The H(z^Q) operates at the higher rate, but has zero weights at all Q-1
+    samples that will get discarded by a downsampler or inserted by an
+    upsampler. The filter H(z) operates at the lower rate.
   . 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.
+
+- Polyphase decomposition of H(z) [VAIDYANATHAN 4.3, PROAKIS 10.5.2]:
+
+  . Direct-Form FIR is first apply delay z^(-q) then apply coefficient. Fits
+    down sampling because coefficients are then applied at low rate.
+  . Transposed Direct-Form FIR is first apply coefficient, then delay z^(-q)
+    result. Fits up sampling because coefficients are then applied at low rate.
+  . Commutator direction from oldest phase (q = Q-1 at end of delay line) to
+    current phase (q = 0 direct path). Assume FIR delay lines are drawn from
+    top to bottom for phases q = 0 to Q-1 for both Direct-Form and Transposed
+    Direct-Form, then:
+    - down sampling input commutator rotates counter clockwise and yields 1
+      sample every rotation, because the summation stage is combinatorial
+    - up sampling output commutator rotates clockwise and yields U samples
+      every rotation
+
+  . Type I polyphase
+
+    (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)
+
+           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
+
+                  +inf
+           Eq(z) = sum eq[n] z^(-n),  0 <= q <= Q - 1
+                  n=-inf
+
+    - Note: For Q = 1 the Eq(z) are the single FIR coefficients in b
+
+  . Type II polyphase, clockwise commutator:
+
+           Q-1
+    H(z) = sum z^(-(Q-1-q) Rq(z^Q)    [VAIDYANATHAN Eq 4.3.7]
+           q=0
+
+       where:
+           rq[n] = h(nQ - q)
+
+           Rq(z) = E_{M-1-q}(z)
+
+  . Phase q of h(n) with Q - 1 zeros, so [VAIDYANATHAN Fig 4.3.1]:
+
+     h[n] --> z^(-q) --> Q:1 --> eq[n]
+
 
 - Fractional time delay [CROCHIERE 6.3]
   . Up sampling Q --> LPF --> z^(-d) --> down sampling Q yields semi allpass