diff --git a/applications/lofar2/model/pfb_os/dsp_study_erko.txt b/applications/lofar2/model/pfb_os/dsp_study_erko.txt
index e3756e2303ba6eca1a05778a6368ea6f37411af1..ea5cd95160d70c5fb73125d8dcfeae0f6bbfeea0 100644
--- a/applications/lofar2/model/pfb_os/dsp_study_erko.txt
+++ b/applications/lofar2/model/pfb_os/dsp_study_erko.txt
@@ -160,7 +160,7 @@ a) DTFT [LYONS 3.52, MATLAB]
- Discrete time to continuous frequency domain:
- +inf
+ +inf
X(w) = sum x[n] exp(-jw n)
n=-inf
@@ -168,7 +168,7 @@ a) DTFT [LYONS 3.52, MATLAB]
b) z-transform [LYONS 6.3, MATLAB]
- Decrete time to z-domain:
- +inf
+ +inf
X(z) = sum x[n] z^-n, z = r exp(jw) = r (cos(w) + j sin(w)),
n=-inf
@@ -563,7 +563,7 @@ c) s-plane and z-plane
- 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]:
- +inf +inf
+ +inf +inf
H(z) = sum h[n] z^-n = sum h[n] r^-n exp(-j w n)
n=-inf n=-inf
@@ -830,30 +830,47 @@ c) s-plane and z-plane
X(m) = sin(pi * m) / sin(pi * m / K)
~= K * sinc(m) for K = N >~ 10
-- Fourier transform theorems [JOS4 B]
+- DTFT properties [JOS4 B, PROAKIS 4.3]
+ . Linearity: a1 x1[n] + a2 x2[n] <==> a1 X1(w) + a2 X2(w)
. Scaling: x(t / a) <==> |a| X(a w)
- . Shift: x(t - T) <==> exp(-j w T) X(w)
- . Modulation: x(t) exp(j v t) <==> X(w - v), is dual of shift
+ . Time shift: x(t - T) <==> X(w) exp(-j w T)
+ x[n - k] <==> X(w) exp(-j w k), t = n Ts
+ . Frequency shift (complex modulation): x[n] exp(+j v n) <==> X(w - v), is
+ dual of time shift
+ . Real modulation: x[n] cos(v n) <==> 1/2 [X(w + v) + X(w - v)]
+ . Conjugation: x*[n] <==> X*(-w)
. Convolution:
x * y <==> X Y
x y <==> 1 / (2 pi) X * Y
- . flip(x) <==> flip(X)
- . d(t) <==> 1, dirac pulse with area 1 at t = 0
+ . flip(x) <==> flip(X), so when signal is folded (time reversed) about the
+ origin in time, then its magnitude spectrum remains unchanged, and the
+ phase spectrum changes sign (phase reversal).
+ . d[n] <==> 1, dirac pulse with area 1 at n = 0
+ d[n - k] <==> exp(-j w k), dirac pulse with area 1 at n = k
+inf
- . d_train_P(t) = sum d(t - m P), period P
- m=-inf
+ . d_train_P[n] = sum d(n - k P), period P
+ k=-inf
<==>
- +inf
- d_train_P(f) = 1 / P sum d(f - m / P)
- m=-inf
+ +inf
+ d_train_P(w) = 2 pi / P sum d(w - 2 pi k / P)
+ k=-inf
+
+ . Sampling: x_d(t) = x_a(t) d_train_Ts(t)
+ <==> +inf
+ X_d(f) = X_a * d_train_fs(f) = fs sum X_a(f - k fs)
+ k=-inf
- . sampling: x_d(t) = x(t) d_train_Ts(t)
- <==>
- X_d(f) = X * d_train_fs(f)
- +inf
- = fs sum X(f - k fs)
- k=-inf
+ - The sampling theorem [PROAKIS 4.2.9, CROCHIERE 2.1]:
+ The digital spectrum is a periodic repetition of the scaled analogue
+ spectrum with period fs. If spectrum X_a = 0 for |f| >= B, then for
+ fs >= 2 B there is no overlapping aliasing (= spectral folding) and
+ then it is possible to reconstruct x_a from x_d using an LPF.
+ - The sinc() is the ideal interpolation formula:
+
+ +inf sin(pi (t - nT) / T) +inf
+ x_a(t) = sum x_d[n] -------------------- = sum x_d[n] sinc((t - nT) / T)
+ n=-inf (pi (t - nT) / T) n=-inf
10) Short Term Fourier Transform (STFT) [JOS4 7, 8]
@@ -979,9 +996,15 @@ c) s-plane and z-plane
12) Multirate processing:
- Linear Time Variant (LTV) process, because it depends on when the
downsampling and upsampling start.
+- 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.
+ This property can be used to advantage when dealing with bandpass signals by
+ associating the bandpass signal with one of these images instead of with the
+ 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 protype filter.
+ the prototype filter.
- 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]
@@ -994,48 +1017,59 @@ c) s-plane and z-plane
- 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
+ . 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 --> y[m], is equivalent to:
- x[n] --> down Q --> H(z) --> y[m]
+ . Down sampling : x[n] --> H(z^Q) --> down --> y[m], is equivalent to:
+ x[n] --> down Q --> H(z) --> y[m]
- . Hi(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
+ . 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:
- . Do not calculate samples that will be thrown away.
- . Discarding samples folds the spectrum, first the LPF has to remove all
- folds.
- . Sequence w(m) is an upsampled-by-D version of sequence x(n), and sequence
- x(n) is a downsampled-by-D version of sequence w(m) [LYONS 10.9].
- . Downsampling: W(z) = X(z^D)
+- LPF + downsampling = decimation [LYONS 10.9, PROAKIS 10, CROCHIERE Fig 3.2]:
+ . Downsampling: Y(z) = X(z^(1/Q))
- w(m) = x(m / D), when m is multiple of D else 0
+ y(n) = x(n Q), because y removes Q-1 values from x
- +inf +inf +inf
- W(z) = sum w(m) z^-m = sum w(Dk) z^-Dk = sum x(k) z^-Dk = X(z^D)
- m=-inf k=-inf k=-inf
+ +inf +inf +inf
+ Y(z) = sum y(n) z^-n = sum x(Qn) z^-n = sum x(k) z^-k/Q = X(z^(1/Q))
+ n=-inf n=-inf k=-inf
- . Upsampling: W(z^1/D) = X(z)
+ . Spectrum, evaluate Y(z) on unit circle [PROAKIS Eq 10.2.9]:
- w(u) = x(m), when u is Dm else 0
+ Q-1
+ Y(w) = 1/Q sum H((w - 2 pi k) / Q) X((w - 2 pi k) / Q), w = w_y
+ k=0
- +inf +inf +inf
- W(z) = sum w(u) z^-u = sum w(Dm) z^-Dm = sum x(m) z^-Dm = X(z^D)
- u=-inf m=-inf m=-inf
+ . Discarding samples folds the spectrum around multiples of pi / Q =
+ fxNyquist / Q = fyNyquist. First the LPF has to remove all folds.
+ . Do not calculate samples that will be thrown away.
- Upsampling + LPF = interpolation:
+ . Upsampling: Y(z) = X(z^Q), because y inserts Q-1 zeros in x
+
+ y[m] = x[m / Q], when m is multiple of Q, else 0, so equivalently
+ y[m] = x[n], when m = Q n else 0
+
+ +inf +inf +inf
+ Y(z) = sum y(m) z^-m = sum y(Qn) z^-Qn = sum x(n) z^-Qn = X(z^Q)
+ m=-inf n=-inf n=-inf
+
+ . Spectrum, evaluate Y(z) on unit circle [PROAKIS Eq 10.3.3]:
+
+ Y(w) = Q X(w Q), w = w_y
+
+ . Inserting zeros replicates the spectrum around multiples of pi / Q =
+ fyNyquist / Q = fxNyquist. Then the LPF has to remove all replicas and
+ by that it interpolates to fill in the zeros.
. Do not calculate samples that will be inserted as zeros.
- . Inserting zeros replicates the spectrum, the LPF remove all replicas and by
- that it interpolates to fill in the 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]
- Fractional time delay [CROCHIERE 6.3]
- . Up sampling M --> LPF --> z^(-L) --> down sampling M yields semi
- allpass filter and delay of L / M samples
+ . Up sampling Q --> LPF --> z^(-d) --> down sampling Q yields semi allpass
+ filter and delay of d / Q samples.
- Oversampling ADC and DAC
. Every oversampling factor of 4 yields 1 extra bit, because then 1 / 4 of the
@@ -1074,6 +1108,54 @@ c) s-plane and z-plane
output.
+13) Quadrature Mirror Filter (QMF) [CROCHIERE 7.7, PROAKIS 10.9.6]
+
+ |-- h0[n] --> Down Q --> x0[m] --> Up Q --> f0[n] --|
+ x[n] --| +--> x^[n]
+ |-- h1[n] --> Down Q --> x1[m] --> Up Q --> f1[n] --|
+
+ Q = 2
+
+ X^(z) = T(z)X(z) + A(z)X(-z), with:
+ . T(z) = H0(z)F0(z) + H1(z)F1(z), transfer part
+ . A(z) = H0(-z)F0(z) + H1(-z)F1(z), aliasing part
+
+- Choose:
+ h0[n] = h[n] <==> H0(w) = H(w), prototype LPF
+ h1[n] = (-1)^n h[n] <==> H1(w) = H(w - pi), mirror image HPF
+ f0[n] = Q h[n] <==> F0(w) = Q H(w)
+ f1[n] = -Q (-1)^n h[n] <==> F1(w) = -Q H(w - pi), to eliminate aliasing,
+ so A(w) = 0
+
+ then: X^(w) = T(w) X(w), with T(w) = H^2(w) - H^2(w - pi)
+
+- Get HPF from LPF using frequency shift (complex modulation) by pi = fNyquist,
+ so: h1[n] = h[n] exp(+j pi n) <==> H(w - pi)
+ = h[n] cos(j pi n)
+ = h[n] (-1)^n
+- For perfect reconstruction T(w) = 1. This can only be achieved for a two tap
+ FIR filter, because Q = 2, so each phase then becomes a delay.
+- Choose linear phase (= symmetric) FIR filter:
+
+ h[n] = h[N - 1 - n), for n = 0,1,...,N-1
+
+ H(w) = Hr(w) exp(-j w (N - 1) / 2), where Hr is real function, so:
+
+ Hr^2(w) = |H(w)|^2
+
+ H^2(w) = |H(w) |^2 exp(-j w (N - 1))
+ H^2(w - pi) = (-1)^(N-1) |H(w - pi)|^2 exp(-j w (N - 1))
+
+ T(w) = |H(w)|^2 - (-1)^(N-1) |H(w - pi)|^2
+
+ For N is odd, T(pi / 2) = 0, therefore choose N is even:
+
+ T(w) = |H(w)|^2 + |H(w - pi)|^2
+
+ For approximate reconstruction optimize for both maximum attenuation in
+ stop band of H(w) and all pass for T(w).
+
+
Appendix A) Signal operators [JOS1 7.2]
- Operator(x) is element of C^N for all x element of C^N