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Commit 87722c35 authored by Eric Kooistra's avatar Eric Kooistra
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Add section on QMF.

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applications/lofar2/model/pfb_os/dsp_study_erko.txt
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......@@ -830,31 +830,48 @@ 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
d_train_P(w) = 2 pi / P sum d(w - 2 pi k / P)
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)
. 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
- 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]
- Purpose of the STFT is to display frequency spectra in time (spectrogram). A
......@@ -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:
. 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:
. 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
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
......
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