diff --git a/applications/lofar2/model/pfb_os/dsp_study_erko.txt b/applications/lofar2/model/pfb_os/dsp_study_erko.txt
index d6141fb1fb13c8226f142c5752a2a84658a45499..34b5b1f82b347aa6fbab38f76bf31e08df4cf6e5 100644
--- a/applications/lofar2/model/pfb_os/dsp_study_erko.txt
+++ b/applications/lofar2/model/pfb_os/dsp_study_erko.txt
@@ -1004,7 +1004,8 @@ c) s-plane and z-plane
12) Multirate processing:
- Linear Time Variant (LTV) process, because it depends on when the
- downsampling and upsampling start.
+ downsampling and upsampling start. This causes that order of operations
+ matters [LYONS 10.3.1]
- Sampling and sampling rate conversion can be viewed as a modulation process
in which the spectrum of the digital signal contains periodic repetitions of
the baseband signal (images) spaced at harmonics of the sampling frequency.
@@ -1013,36 +1014,66 @@ c) s-plane and z-plane
baseband [CROCHIERE 2.4.2].
- Polyphase filtering ensures that only the values that remain are calculated,
so there are D or U phases [LYONS 10.7]. The LPF with all phases is called
- the prototype filter.
+ the prototype filter. Do not calculate samples that will be:
+ . discarded,
+ . inserted as zeros.
- For large D or U use two stage D = D1 * D2 or U = U1 * U2, where D1 > D2 and
U1 < U2 [LYONS 10.8.2]
-- 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], [VAIDYANATHAN Fig 4.2.3]
-
- . Up sampling : x[n] --> up Q --> H(z^Q) --> y[m], is equivalent to:
- H(z) --> up Q
-
- . Down sampling : x[n] --> H(z^Q) --> down Q --> y[m], is equivalent to:
- x[n] --> down Q --> H(z) --> y[m]
-
- . 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.
-
-- LPF + downsampling = decimation [LYONS 10.9, PROAKIS 10, CROCHIERE Fig 3.2,
- VAIDYANATHAN Fig 4.1.4, SP4COMM 11.1.2]:
- . Downsampling:
+- Sampling, downsampling and upsampling
+ . Sampling causes the analoge spectrum to alias around k 2pi, similar for
+ downsamping the the digital spectrum aliases around k 2pi / D, as if the
+ analogue signal was sampled directly at the downsampled rate [LYONS 10.1].
+ . Downsampled spectrum [LYONS 10.3.2]
+ 1. Draw original spectrum beyond -2pi to + 2pi, to show 0 and at least one
+ spectral replication (alias) for both negative (-2pi) and positive
+ (+2pi) frequency directions of the original sample frequency fs_old =
+ 2pi.
+ 2. Draw D - 1 copies of the original spectrum shifted by k 2pi / D, note
+ k = 0 is the original spectrum of step 1
+ 3. Scale up the frequency axis of the new spectrum by factor D to get the
+ frequency axis for the down sample frequency. The downsampled spectrum
+ now ranges from -pi to pi for fs_new = 2pi.
+ 4. Scale down magnitude of new spectrum by factor D. The time domain
+ amplitude of downsampled signal remains the same, but the frequency
+ domain magnitude decreases by factor D, because the DFT magnitude is
+ proportional to number of time-domain samples used in the
+ transformation [LYONS 10.3.1].
+ . Upsampled spectrum [LYONS 10.5.2]
+ 1. Draw original spectrum beyond -U 2pi to +U 2pi, to show at least U
+ spectral replications of the original spectrum in both negative and
+ positive frequency directions of the original sample frequency fs_old =
+ 2pi. That is all, because inserting U - 1 zeros merely increases the
+ effective sample frequency to fs_new = U fs_old. It does not change the
+ spectrum, but it does cause that U - 1 spectral replications (aliases)
+ are now also in the 2pi range of fs_new. Hence it looks like inserting
+ zeros replicates the spectrum around multiples of 2pi / U, but it is
+ easier to understand as that increasing fs_new now includes U - 1
+ replications of the original spectrum.
+ 2. Scale down the frequency axis of the new spectrum by factor U to get the
+ frequency axis for the up sample frequency. The upsampled spectrum now
+ ranges from -pi to pi for fs_new = 2pi.
+ 3. The magnitude of new spectrum remains the same.
+ . Decimation = LPF + Downsampling [LYONS 10.1]:
+ To avoid overlapping aliasing after downsampling to fs_new an LPF needs
+ to band limit the original spectrum to pi / D = fs_old/2 / D.
+ . Interpolation = Upsampling + LPF:
+ To interpolate the zero values for fs_new an LPF needs to band limit the
+ new spectrum to pi / U = fs_new/2 / D.
+ 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 pass band of sin(x)/x [LYONS 10.5.1].
+ . Decimation and interpolation can also use a BPF to select another part of
+ the band [HARRIS 2.2, VAIDYANATHAN 4.1.1].
+
+- Downsampling [LYONS 10.1, PROAKIS 10, CROCHIERE Fig 3.2, VAIDYANATHAN Fig
+ 4.1.4, JOS4 11.1, SP4COMM 11.1.2]:
x_D[n] = x[nD], because x_D removes D-1 values from x
Define x' at x time grid, but with x' = 0 for samples in x that will be
- discarded:
+ discarded. This operation has no name in literature, probably because it
+ is a conceptual step and not an implementation operation:
x'[n] = d[n] x[n], with d[n] = 1, for n % D = 0
= 0, otherwise
@@ -1087,25 +1118,20 @@ c) s-plane and z-plane
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))
+ X_D(e^jw) = 1/D sum X(exp(j (w / D - k 2pi / D)))
k=0
+ w in [0:2pi>
+ summation terms for k != 0 are aliasing terms
+
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 copies the spectrum around multiples of 2pi / D. First
- the LPF has to remove all copies.
- . Do not calculate samples that will be thrown away.
-
-- Upsampling + LPF = interpolation:
- . Upsampling:
+- Upsampling:
x_U[n] = x[n / U], for n % U = 0
= 0, otherwise, because x_U inserts U-1 zeros in x
@@ -1116,15 +1142,26 @@ c) s-plane and z-plane
. Spectrum, evaluate X_U(z) on unit circle [PROAKIS Eq 10.3.3]:
- X_U(e^jw) = X(exp(jw U))
+ 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]:
- . Inserting zeros replicates the spectrum around multiples of 2pi / U. Then
- the LPF has to remove all replicas and by that it interpolates to fill in
- the zeros. To remove the replicas, the LPF pass band is pi / L.
- . Do not calculate samples that will be inserted as zeros.
- . 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
- pass band of sin(x)/x [LYONS 10.5.1]
+ 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]
+
+ . Up sampling : x[n] --> 1:Q --> H(z^Q) --> y[m], is equivalent to:
+ H(z) --> 1:Q
+
+ . 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.
- Fractional time delay [CROCHIERE 6.3]
. Up sampling Q --> LPF --> z^(-d) --> down sampling Q yields semi allpass