Equality between two equations












2














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







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  • 1




    good catch. interesting too.
    – Nick
    2 days ago
















2














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







share|cite|improve this question


















  • 1




    good catch. interesting too.
    – Nick
    2 days ago














2












2








2







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







share|cite|improve this question













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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 2 days ago









Matthias LauberMatthias Lauber

283




283








  • 1




    good catch. interesting too.
    – Nick
    2 days ago














  • 1




    good catch. interesting too.
    – Nick
    2 days ago








1




1




good catch. interesting too.
– Nick
2 days ago




good catch. interesting too.
– Nick
2 days ago










1 Answer
1






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oldest

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0














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.






share|cite|improve this answer





















  • 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.
    – Matthias Lauber
    yesterday











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1 Answer
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1 Answer
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active

oldest

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active

oldest

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active

oldest

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0














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.






share|cite|improve this answer





















  • 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.
    – Matthias Lauber
    yesterday
















0














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.






share|cite|improve this answer





















  • 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.
    – Matthias Lauber
    yesterday














0












0








0






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.






share|cite|improve this answer












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.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 2 days ago









pre-kidneypre-kidney

12.8k1748




12.8k1748












  • 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.
    – Matthias Lauber
    yesterday


















  • 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.
    – Matthias Lauber
    yesterday
















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.
– Matthias Lauber
yesterday




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.
– Matthias Lauber
yesterday


















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