Equality between two equations
$begingroup$
at the moment I am reading the following paper
Benno, Steven A., and José MF Moura. "On translation invariant
subspaces and critically sampled wavelet transforms." Multidimensional
Systems and Signal Processing 8.1-2 (1997): 89-110.
The step between the equations (10) and (11) I can't comprehend. The step is as follows
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k e^{-j2pi(f-omega)k}Big)domega =
$$
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k delta(f-omega+k)Big) domega.
$$
I know that $e^{-i2pi k}$ for $kinmathbb{Z}$ is an orthnormal basis, but not over $mathbb{R}$ and I have no idea, why $k$ comes into the dirac function with an "+". At most I woud expect something like $delta(f-omega)$ since it is in a product with k.
Has someone an idea about this?
Thanks Matthias
exponential-function dirac-delta
asked Jan 5 at 17:37
Matthias LauberMatthias Lauber
283
$endgroup$
add a comment |
$begingroup$
at the moment I am reading the following paper
Benno, Steven A., and José MF Moura. "On translation invariant
subspaces and critically sampled wavelet transforms." Multidimensional
Systems and Signal Processing 8.1-2 (1997): 89-110.
The step between the equations (10) and (11) I can't comprehend. The step is as follows
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k e^{-j2pi(f-omega)k}Big)domega =
$$
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k delta(f-omega+k)Big) domega.
$$
I know that $e^{-i2pi k}$ for $kinmathbb{Z}$ is an orthnormal basis, but not over $mathbb{R}$ and I have no idea, why $k$ comes into the dirac function with an "+". At most I woud expect something like $delta(f-omega)$ since it is in a product with k.
Has someone an idea about this?
Thanks Matthias
exponential-function dirac-delta
asked Jan 5 at 17:37
Matthias LauberMatthias Lauber
283
$endgroup$
1
$begingroup$
good catch. interesting too.
$endgroup$
– Nick
Jan 5 at 20:56
add a comment |
$begingroup$
at the moment I am reading the following paper
Benno, Steven A., and José MF Moura. "On translation invariant
subspaces and critically sampled wavelet transforms." Multidimensional
Systems and Signal Processing 8.1-2 (1997): 89-110.
The step between the equations (10) and (11) I can't comprehend. The step is as follows
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k e^{-j2pi(f-omega)k}Big)domega =
$$
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k delta(f-omega+k)Big) domega.
$$
I know that $e^{-i2pi k}$ for $kinmathbb{Z}$ is an orthnormal basis, but not over $mathbb{R}$ and I have no idea, why $k$ comes into the dirac function with an "+". At most I woud expect something like $delta(f-omega)$ since it is in a product with k.
Has someone an idea about this?
Thanks Matthias
exponential-function dirac-delta
asked Jan 5 at 17:37
Matthias LauberMatthias Lauber
283
$endgroup$
at the moment I am reading the following paper
Benno, Steven A., and José MF Moura. "On translation invariant
subspaces and critically sampled wavelet transforms." Multidimensional
Systems and Signal Processing 8.1-2 (1997): 89-110.
The step between the equations (10) and (11) I can't comprehend. The step is as follows
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k e^{-j2pi(f-omega)k}Big)domega =
$$
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k delta(f-omega+k)Big) domega.
$$
I know that $e^{-i2pi k}$ for $kinmathbb{Z}$ is an orthnormal basis, but not over $mathbb{R}$ and I have no idea, why $k$ comes into the dirac function with an "+". At most I woud expect something like $delta(f-omega)$ since it is in a product with k.
Has someone an idea about this?
Thanks Matthias
exponential-function dirac-delta
exponential-function dirac-delta
asked Jan 5 at 17:37
Matthias LauberMatthias Lauber
283
asked Jan 5 at 17:37
Matthias LauberMatthias Lauber
283
asked Jan 5 at 17:37
Matthias LauberMatthias Lauber
283
asked Jan 5 at 17:37
Matthias LauberMatthias Lauber
283
asked Jan 5 at 17:37
Matthias LauberMatthias Lauber
283
283
1
$begingroup$
good catch. interesting too.
$endgroup$
– Nick
Jan 5 at 20:56
add a comment |
1
$begingroup$
good catch. interesting too.
$endgroup$
– Nick
Jan 5 at 20:56
1
1
$begingroup$
good catch. interesting too.
$endgroup$
– Nick
Jan 5 at 20:56
$begingroup$
good catch. interesting too.
$endgroup$
– Nick
Jan 5 at 20:56
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Clearly there is a typo in equation (10), the correct expression is
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k e^{-j2pi(f-omega+k)}Big)domega,
$$
which can be seen by substituting the definition of $a_k$ exactly as the authors describe.
answered Jan 5 at 20:52
pre-kidneypre-kidney
12.9k1849
$endgroup$
$begingroup$
Hi, thank you for your answer. I checked equation (10) if $a_k$ is inserted. The sum is reordered by k and (10) should be correct.
$endgroup$
– Matthias Lauber
Jan 6 at 11:19
add a comment |
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1 Answer
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1 Answer
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active
oldest
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$begingroup$
Clearly there is a typo in equation (10), the correct expression is
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k e^{-j2pi(f-omega+k)}Big)domega,
$$
which can be seen by substituting the definition of $a_k$ exactly as the authors describe.
answered Jan 5 at 20:52
pre-kidneypre-kidney
12.9k1849
$endgroup$
$begingroup$
Hi, thank you for your answer. I checked equation (10) if $a_k$ is inserted. The sum is reordered by k and (10) should be correct.
$endgroup$
– Matthias Lauber
Jan 6 at 11:19
add a comment |
$begingroup$
Clearly there is a typo in equation (10), the correct expression is
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k e^{-j2pi(f-omega+k)}Big)domega,
$$
which can be seen by substituting the definition of $a_k$ exactly as the authors describe.
answered Jan 5 at 20:52
pre-kidneypre-kidney
12.9k1849
$endgroup$
$begingroup$
Hi, thank you for your answer. I checked equation (10) if $a_k$ is inserted. The sum is reordered by k and (10) should be correct.
$endgroup$
– Matthias Lauber
Jan 6 at 11:19
add a comment |
$begingroup$
Clearly there is a typo in equation (10), the correct expression is
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k e^{-j2pi(f-omega+k)}Big)domega,
$$
which can be seen by substituting the definition of $a_k$ exactly as the authors describe.
answered Jan 5 at 20:52
pre-kidneypre-kidney
12.9k1849
$endgroup$
Clearly there is a typo in equation (10), the correct expression is
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k e^{-j2pi(f-omega+k)}Big)domega,
$$
which can be seen by substituting the definition of $a_k$ exactly as the authors describe.
answered Jan 5 at 20:52
pre-kidneypre-kidney
12.9k1849
answered Jan 5 at 20:52
pre-kidneypre-kidney
12.9k1849
answered Jan 5 at 20:52
pre-kidneypre-kidney
12.9k1849
answered Jan 5 at 20:52
pre-kidneypre-kidney
12.9k1849
12.9k1849
$begingroup$
Hi, thank you for your answer. I checked equation (10) if $a_k$ is inserted. The sum is reordered by k and (10) should be correct.
$endgroup$
– Matthias Lauber
Jan 6 at 11:19
add a comment |
$begingroup$
Hi, thank you for your answer. I checked equation (10) if $a_k$ is inserted. The sum is reordered by k and (10) should be correct.
$endgroup$
– Matthias Lauber
Jan 6 at 11:19
$begingroup$
Hi, thank you for your answer. I checked equation (10) if $a_k$ is inserted. The sum is reordered by k and (10) should be correct.
$endgroup$
– Matthias Lauber
Jan 6 at 11:19
$begingroup$
Hi, thank you for your answer. I checked equation (10) if $a_k$ is inserted. The sum is reordered by k and (10) should be correct.
$endgroup$
– Matthias Lauber
Jan 6 at 11:19
add a comment |
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1
$begingroup$
good catch. interesting too.
$endgroup$
– Nick
Jan 5 at 20:56