Mixing
Mathematical terms for “bandlimited” and “timelimited”?
$begingroup$
I have read
"Signals that are bandlimited are not timelimited" and the reverse; "Signals that are timelimited are not bandlimited".
Q1: Is this because of the Fourier transform?
Q2: What are the mathematical terms for "bandlimited" and "timelimited"?
real-analysis fourier-analysis signal-processing
asked Feb 1 '18 at 9:39
JDoeDoeJDoeDoe
7701614
$endgroup$
add a comment |
$begingroup$
I have read
"Signals that are bandlimited are not timelimited" and the reverse; "Signals that are timelimited are not bandlimited".
Q1: Is this because of the Fourier transform?
Q2: What are the mathematical terms for "bandlimited" and "timelimited"?
real-analysis fourier-analysis signal-processing
asked Feb 1 '18 at 9:39
JDoeDoeJDoeDoe
7701614
$endgroup$
add a comment |
$begingroup$
I have read
"Signals that are bandlimited are not timelimited" and the reverse; "Signals that are timelimited are not bandlimited".
Q1: Is this because of the Fourier transform?
Q2: What are the mathematical terms for "bandlimited" and "timelimited"?
real-analysis fourier-analysis signal-processing
asked Feb 1 '18 at 9:39
JDoeDoeJDoeDoe
7701614
$endgroup$
I have read
"Signals that are bandlimited are not timelimited" and the reverse; "Signals that are timelimited are not bandlimited".
Q1: Is this because of the Fourier transform?
Q2: What are the mathematical terms for "bandlimited" and "timelimited"?
real-analysis fourier-analysis signal-processing
real-analysis fourier-analysis signal-processing
asked Feb 1 '18 at 9:39
JDoeDoeJDoeDoe
7701614
asked Feb 1 '18 at 9:39
JDoeDoeJDoeDoe
7701614
asked Feb 1 '18 at 9:39
JDoeDoeJDoeDoe
7701614
asked Feb 1 '18 at 9:39
JDoeDoeJDoeDoe
7701614
asked Feb 1 '18 at 9:39
JDoeDoeJDoeDoe
7701614
7701614
add a comment |
add a comment |
1 Answer
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A signal $x(t)$ is time limited if there exists a $T>0$ such that $$x(t)=0qquad,qquad |t|>T$$so we can define band limited (in frequency) signals where their FT is zero for $omega>omega_0$ and for some $omega_0$. Also a signal limited on one domain (time or frequency) cannot be limited in the other domain either, but there are signals neither limited in time nor in frequency for example consider $x(t)=e^{-|t|}$ with corresponding FT $X(omega)=dfrac{2}{1+omega^2}$. Neither $x(t)$ nor $X(omega)$ are limited in time and frequency respectively. To show that, let $x_T(t)$ be a time limited version of signal $x(t)$ in $|t|<T$ therefore$$x_T(t)=x(t)Pi(dfrac{t}{2T})$$by taking FT we have$$X_T(omega)=X(omega)*2Tsinc(dfrac{Tomega}{pi})=$$regardless of $X(omega)$ being band limited or not, $X_T(omega)$ is never band limited because of the convolution of $X(omega)$ with $sinc$ function and this is what we wanted to show. The same argument can be used in dual case for band limited signals.
answered Feb 1 '18 at 9:50
Mostafa AyazMostafa Ayaz
15.6k3939
$endgroup$
$begingroup$
I don't understand the point of the computations following "To show that...". Are you trying to prove that a time-limited signal cannot be band-limited? If so, then I fail to understand why "$X_T$ is never band-limited because of its convolution with $text{sinc}$".
$endgroup$
– Giuseppe Negro
Jan 14 at 18:16
$begingroup$
Yes that's right. A signal can't be limited in both time and frequency range. By "$X_T$ is never band-limited because of its convolution with sinc" I mean since sinc is spread throughout the frequency range, then even if $X(omega)$ is limited, the convolution takes it to presence in higher frequencies too.....
$endgroup$
– Mostafa Ayaz
Jan 14 at 18:22
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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active
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$begingroup$
A signal $x(t)$ is time limited if there exists a $T>0$ such that $$x(t)=0qquad,qquad |t|>T$$so we can define band limited (in frequency) signals where their FT is zero for $omega>omega_0$ and for some $omega_0$. Also a signal limited on one domain (time or frequency) cannot be limited in the other domain either, but there are signals neither limited in time nor in frequency for example consider $x(t)=e^{-|t|}$ with corresponding FT $X(omega)=dfrac{2}{1+omega^2}$. Neither $x(t)$ nor $X(omega)$ are limited in time and frequency respectively. To show that, let $x_T(t)$ be a time limited version of signal $x(t)$ in $|t|<T$ therefore$$x_T(t)=x(t)Pi(dfrac{t}{2T})$$by taking FT we have$$X_T(omega)=X(omega)*2Tsinc(dfrac{Tomega}{pi})=$$regardless of $X(omega)$ being band limited or not, $X_T(omega)$ is never band limited because of the convolution of $X(omega)$ with $sinc$ function and this is what we wanted to show. The same argument can be used in dual case for band limited signals.
answered Feb 1 '18 at 9:50
Mostafa AyazMostafa Ayaz
15.6k3939
$endgroup$
$begingroup$
I don't understand the point of the computations following "To show that...". Are you trying to prove that a time-limited signal cannot be band-limited? If so, then I fail to understand why "$X_T$ is never band-limited because of its convolution with $text{sinc}$".
$endgroup$
– Giuseppe Negro
Jan 14 at 18:16
$begingroup$
Yes that's right. A signal can't be limited in both time and frequency range. By "$X_T$ is never band-limited because of its convolution with sinc" I mean since sinc is spread throughout the frequency range, then even if $X(omega)$ is limited, the convolution takes it to presence in higher frequencies too.....
$endgroup$
– Mostafa Ayaz
Jan 14 at 18:22
add a comment |
$begingroup$
A signal $x(t)$ is time limited if there exists a $T>0$ such that $$x(t)=0qquad,qquad |t|>T$$so we can define band limited (in frequency) signals where their FT is zero for $omega>omega_0$ and for some $omega_0$. Also a signal limited on one domain (time or frequency) cannot be limited in the other domain either, but there are signals neither limited in time nor in frequency for example consider $x(t)=e^{-|t|}$ with corresponding FT $X(omega)=dfrac{2}{1+omega^2}$. Neither $x(t)$ nor $X(omega)$ are limited in time and frequency respectively. To show that, let $x_T(t)$ be a time limited version of signal $x(t)$ in $|t|<T$ therefore$$x_T(t)=x(t)Pi(dfrac{t}{2T})$$by taking FT we have$$X_T(omega)=X(omega)*2Tsinc(dfrac{Tomega}{pi})=$$regardless of $X(omega)$ being band limited or not, $X_T(omega)$ is never band limited because of the convolution of $X(omega)$ with $sinc$ function and this is what we wanted to show. The same argument can be used in dual case for band limited signals.
answered Feb 1 '18 at 9:50
Mostafa AyazMostafa Ayaz
15.6k3939
$endgroup$
$begingroup$
I don't understand the point of the computations following "To show that...". Are you trying to prove that a time-limited signal cannot be band-limited? If so, then I fail to understand why "$X_T$ is never band-limited because of its convolution with $text{sinc}$".
$endgroup$
– Giuseppe Negro
Jan 14 at 18:16
$begingroup$
Yes that's right. A signal can't be limited in both time and frequency range. By "$X_T$ is never band-limited because of its convolution with sinc" I mean since sinc is spread throughout the frequency range, then even if $X(omega)$ is limited, the convolution takes it to presence in higher frequencies too.....
$endgroup$
– Mostafa Ayaz
Jan 14 at 18:22
add a comment |
$begingroup$
A signal $x(t)$ is time limited if there exists a $T>0$ such that $$x(t)=0qquad,qquad |t|>T$$so we can define band limited (in frequency) signals where their FT is zero for $omega>omega_0$ and for some $omega_0$. Also a signal limited on one domain (time or frequency) cannot be limited in the other domain either, but there are signals neither limited in time nor in frequency for example consider $x(t)=e^{-|t|}$ with corresponding FT $X(omega)=dfrac{2}{1+omega^2}$. Neither $x(t)$ nor $X(omega)$ are limited in time and frequency respectively. To show that, let $x_T(t)$ be a time limited version of signal $x(t)$ in $|t|<T$ therefore$$x_T(t)=x(t)Pi(dfrac{t}{2T})$$by taking FT we have$$X_T(omega)=X(omega)*2Tsinc(dfrac{Tomega}{pi})=$$regardless of $X(omega)$ being band limited or not, $X_T(omega)$ is never band limited because of the convolution of $X(omega)$ with $sinc$ function and this is what we wanted to show. The same argument can be used in dual case for band limited signals.
answered Feb 1 '18 at 9:50
Mostafa AyazMostafa Ayaz
15.6k3939
$endgroup$
A signal $x(t)$ is time limited if there exists a $T>0$ such that $$x(t)=0qquad,qquad |t|>T$$so we can define band limited (in frequency) signals where their FT is zero for $omega>omega_0$ and for some $omega_0$. Also a signal limited on one domain (time or frequency) cannot be limited in the other domain either, but there are signals neither limited in time nor in frequency for example consider $x(t)=e^{-|t|}$ with corresponding FT $X(omega)=dfrac{2}{1+omega^2}$. Neither $x(t)$ nor $X(omega)$ are limited in time and frequency respectively. To show that, let $x_T(t)$ be a time limited version of signal $x(t)$ in $|t|<T$ therefore$$x_T(t)=x(t)Pi(dfrac{t}{2T})$$by taking FT we have$$X_T(omega)=X(omega)*2Tsinc(dfrac{Tomega}{pi})=$$regardless of $X(omega)$ being band limited or not, $X_T(omega)$ is never band limited because of the convolution of $X(omega)$ with $sinc$ function and this is what we wanted to show. The same argument can be used in dual case for band limited signals.
answered Feb 1 '18 at 9:50
Mostafa AyazMostafa Ayaz
15.6k3939
edited Jan 14 at 18:20
answered Feb 1 '18 at 9:50
Mostafa AyazMostafa Ayaz
15.6k3939
answered Feb 1 '18 at 9:50
Mostafa AyazMostafa Ayaz
15.6k3939
answered Feb 1 '18 at 9:50
Mostafa AyazMostafa Ayaz
15.6k3939
15.6k3939
$begingroup$
I don't understand the point of the computations following "To show that...". Are you trying to prove that a time-limited signal cannot be band-limited? If so, then I fail to understand why "$X_T$ is never band-limited because of its convolution with $text{sinc}$".
$endgroup$
– Giuseppe Negro
Jan 14 at 18:16
$begingroup$
Yes that's right. A signal can't be limited in both time and frequency range. By "$X_T$ is never band-limited because of its convolution with sinc" I mean since sinc is spread throughout the frequency range, then even if $X(omega)$ is limited, the convolution takes it to presence in higher frequencies too.....
$endgroup$
– Mostafa Ayaz
Jan 14 at 18:22
add a comment |
$begingroup$
I don't understand the point of the computations following "To show that...". Are you trying to prove that a time-limited signal cannot be band-limited? If so, then I fail to understand why "$X_T$ is never band-limited because of its convolution with $text{sinc}$".
$endgroup$
– Giuseppe Negro
Jan 14 at 18:16
$begingroup$
Yes that's right. A signal can't be limited in both time and frequency range. By "$X_T$ is never band-limited because of its convolution with sinc" I mean since sinc is spread throughout the frequency range, then even if $X(omega)$ is limited, the convolution takes it to presence in higher frequencies too.....
$endgroup$
– Mostafa Ayaz
Jan 14 at 18:22
$begingroup$
I don't understand the point of the computations following "To show that...". Are you trying to prove that a time-limited signal cannot be band-limited? If so, then I fail to understand why "$X_T$ is never band-limited because of its convolution with $text{sinc}$".
$endgroup$
– Giuseppe Negro
Jan 14 at 18:16
$begingroup$
I don't understand the point of the computations following "To show that...". Are you trying to prove that a time-limited signal cannot be band-limited? If so, then I fail to understand why "$X_T$ is never band-limited because of its convolution with $text{sinc}$".
$endgroup$
– Giuseppe Negro
Jan 14 at 18:16
$begingroup$
Yes that's right. A signal can't be limited in both time and frequency range. By "$X_T$ is never band-limited because of its convolution with sinc" I mean since sinc is spread throughout the frequency range, then even if $X(omega)$ is limited, the convolution takes it to presence in higher frequencies too.....
$endgroup$
– Mostafa Ayaz
Jan 14 at 18:22
$begingroup$
Yes that's right. A signal can't be limited in both time and frequency range. By "$X_T$ is never band-limited because of its convolution with sinc" I mean since sinc is spread throughout the frequency range, then even if $X(omega)$ is limited, the convolution takes it to presence in higher frequencies too.....
$endgroup$
– Mostafa Ayaz
Jan 14 at 18:22
add a comment |
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