diff --git a/applications/lofar2/model/pfb_os/dsp_study_erko.txt b/applications/lofar2/model/pfb_os/dsp_study_erko.txt
index d0ef8e813f66e3f1cfe165daee50be5a4299557b..aaea9f9210c5fd8823a27390b63fd97a8249433b 100644
--- a/applications/lofar2/model/pfb_os/dsp_study_erko.txt
+++ b/applications/lofar2/model/pfb_os/dsp_study_erko.txt
@@ -68,6 +68,7 @@
# https://dss-kiel.de/images/teaching/lectures/advanced_digital_signal_processing/
# slides/adsp_06_multirate_processing.pdf
# * [TUTHILL] Compensating for oversampling effects in polyphase channelizers, 2015
+# * [BUNTON] Multi-resolution FX Correlator, ALMA memo 447, 2003
#
# https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/
# Youtube: Guitars 4RL
@@ -92,8 +93,9 @@
[PROAKIS 3.5]. Only unique for causal signals, because these are 0 for
n < 0.
* DFT: Every signal x(n) can be expressed as a linear combination of complex
- sinusoids W_N^kn = exp(j w_k t_n). The coefficients of projecting x(n) on
- W_N^kn for n = 0,1,...,N-1 yield the DFT of x is X(k) for k = 0,1,...,N-1.
+ sinusoids W_N^(kn) = exp(j w_k t_n). The coefficients of projecting x(n)
+ on W_N^(kn) for n = 0,1,...,N-1 yield the DFT of x is X(k) for k =
+ 0,1,...,N-1.
* DTFT: For N --> inf, linear combination of exp(j w t_n) = exp(j w T)^n
[LYONS 3.14]
* z-transform: sum n=0 --> inf, linear combination of z^n. Generalization
@@ -760,9 +762,10 @@ c) s-plane and z-plane
9) Discrete Fourier Transform (DFT)
-- The N roots of unity [JOS1 3.12, 5.1, PROAKIS 5.1.3, LYONS 4.3]. Note JOS
- uses +j in W_N because inproduct is with conj(W_N), others use -j because
- then W_N can be used directly in equation and matrix:
+- The N roots of unity [JOS1 3.12, 5.1, PROAKIS 5.1.3, LYONS 4.3, CROCHIERE Eq
+ 7.8]. Note JOS uses +j in W_N because inproduct is with conj(W_N). CROCHIERE
+ also use +j. Wikipedia and others use -j because then W_N can be used directly
+ in equation and matrix:
W_N = exp(-j 2pi/N) is primitive Nth root of unity
W_N^k = exp(-j 2pi/N k)
@@ -815,7 +818,7 @@ c) s-plane and z-plane
6.6, 7.1, PROAKIS 5.1.2, 5.1.3]:
N-1
- X(w_k) = X(k) = sum x(n) W_N^kn
+ X(w_k) = X(k) = sum x(n) W_N^(kn)
n=0 exp(-j w_k t_n)
exp(-j 2pi/N k n)
with:
@@ -1554,8 +1557,7 @@ c) s-plane and z-plane
-
-14) Polyphase filterbank (PFB) [HARRIS Fig 6.21, 9.21]
+13c) Polyphase DFT filterbank (PFB) [HARRIS Fig 6.21, 9.21]
. The PFB implements M single channel down converters to output all k bins.
The output rate per branch p in the single channel down converter is a
factor M less, so all k = 0:M-1 bins of y[mM, k] can be calculated by using
@@ -1573,6 +1575,42 @@ c) s-plane and z-plane
at bin 0, so bin k = M-1 is the newest and output last.
+14) WOLA DFT filterbank (PFB) [CROCHIERE 7.2.5]
+
+ * Mixer local oscillator (LO) and LPF and downsampler D [CROCHIERE Eq 7.65 =
+ Eq 7.9, HARRIS Eq 6.1] yields the short-time spectrum of signal x at time
+ n = mM, so M is the block size or downsample rate:
+
+ +inf
+ Xk(m) = sum h[mM - n] x[n] W_K^(kn), W_K = exp(-jw_k) = exp(-j 2pi/K)
+ n=-inf
+
+ The filter h weigths x at n = nM.
+
+ * In Eq 7.9 = 7.65 the signal time frame is fixed to n = 0 and the window h
+ slides along at n = mM. For implementation is convenient to keep the filter
+ h invariant and slide the signal, therefor use r = n - mM:
+
+ +inf
+ Xk(m) = sum h[-r] x[r + mM] W_K^(k(r + mM)) = W_K^(kmM) Rk(m)
+ n=-inf
+
+ with: +inf
+ Rk(m) = sum h[-r] x[r + mM] W_K^(kr)
+ n=-inf
+
+ The term W_K^(kmM) converts the transform Rk(m) with sliding time frame r,
+ into Xk(m) with fixed time frame n.
+
+ * Define ym[r] = h[-r] x[r + mM]. Length h is Ncoef = Ntaps * K. The DFT of
+ ym[n] has Ncoef bins k', but for Rk only bins k' = Ntaps * k are needed.
+ This is equivalent to DFT of Ntaps blocks of stacked K time samples. Define
+ xm[r] = sum ym[r + lK], then:
+
+ K-1
+ Rk(m) = sum xm[r] W_K^(kr) = DFT(xm[r])
+ r = 0
+
15) Quadrature Mirror Filter (QMF) [CROCHIERE 7.7, PROAKIS 10.9.6]