ASN6 a,b- Prove given Fourier Transform property: Difference between revisions
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Problem Statement |
'''Problem Statement''' |
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6(a) Show <math> \mathcal{F}\left[ \int_{- \infty}^{t} s(\lambda ) \,d\lambda \right] = \frac{S(f)}{j2 \pi f} \mbox{ if } S( |
6(a) Show <math> \mathcal{F}\left[ \int_{- \infty}^{t} s(\lambda ) \,d\lambda \right] = \frac{S(f)}{j2 \pi f} \mbox{ if } S(f_0) = 0 </math>. |
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6(b) If <math> S( |
6(b) If <math> S(f_0) \neq 0 </math> can you find <math> \mathcal{F}\left[ \int_{- \infty}^{t} s(\lambda ) \,d\lambda \right] </math> in terms of <math> \displaystyle S(f_0) </math>? |
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'''Answer''' |
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Solution |
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a)Remember that dummy variable <math> \lambda \!</math> was used in substitution such that <math> \lambda= t-t_0 \! </math> |
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<math> |
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\mathcal{F} \left[ g(t-t_{0})e^{j2 \pi f_{0}t} \right] = \int_{- \infty}^{\infty} \left[ g(t-t_{0})e^{j2 \pi f_{0}t} \right] e^{-j2 \pi ft} \,dt |
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</math> |
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Then <math> s(\lambda)= s(t-t_0)= \mathcal{F}\left[ S (f)- S(f_0) \right] \!</math> |
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and <math> \int_{- \infty}^{t} s(\lambda) \,d\lambda = \int_{- \infty}^{t}\mathcal{F}\left[ S (f)- S(f_0) \right] \,d\lambda \! </math> |
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And |
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The problem statement says to make <math>S(f_0)=0 \!</math> that makes the above equation simplify to |
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<math> |
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\int_{- \infty}^{\infty} g(t-t_{0})e^{-j2 \pi (f-f_{0})t} \,dt = \int_{- \infty}^{\infty} g(\lambda )e^{-j2 \pi (f-f_{0})(\lambda + t_{0})} \,dt |
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</math> |
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⚫ | |||
And |
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Taking the inverse Fourier Transform and changing the order of intgration |
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<math> |
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⚫ | |||
⚫ | |||
And |
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<math> |
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⚫ | |||
</math> |
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Then |
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Result |
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<math>\int_{- \infty}^{t} s(\lambda ) \,d\lambda = \int_{\infty}^{\infty} S(f)\frac{ e^{j2 \pi f t}} {j2 \pi f }\,df = \mathcal{F }^{-1}\left[ \frac{S(f)}{j2 \pi f} \right] \! </math> |
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<math> |
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\mathcal{F} \left[ g(t)e^{j2 \pi f_{0}t} \right] = G(f-f_{0})e^{-j2 \pi (f-f_{0})t_{0}} |
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Therefore it is demonstrated that <math> \mathcal{F}\left[ \int_{- \infty}^{t} s(\lambda ) \,d\lambda \right] = \frac{S(f)}{j2 \pi f} \!</math> |
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</math> |
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b) If <math>S(f_0)\neq 0</math> |
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Then |
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<math> \int_{- \infty}^{t} s(\lambda) \,d\lambda = \int_{- \infty}^{t}\mathcal{F}^{-1}\left[ S (f)- S(f_0) \right] \,d\lambda = \int_{- \infty}^{t}\int_{- \infty}^{\infty} e^{j2 \pi f t} [S (f)- S(f_0)] \,d\lambda\,d\lambda \! </math> |
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<math> \int_{- \infty}^{t} s(\lambda) \,d\lambda = \int_{- \infty}^{\infty} e^{j2 \pi f t}S (f)\,d\lambda\,d\lambda - \int_{- \infty}^{t}\int_{- \infty}^{\infty} e^{j2 \pi f t} S(f_0) \,d\lambda\,d\lambda \! </math> |
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<math> \int_{- \infty}^{t} s(\lambda) \,d\lambda = \int_{- \infty}^{\infty}\frac{ e^{j2 \pi f t}} {j2 \pi f } S (f)\,d\lambda - \int_{- \infty}^{\infty}\frac{ e^{j2 \pi f t}} {j2 \pi f } S (f_0)\,d\lambda \! </math> |
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Using <math> dt=d\lambda \!</math> and taking the Fourier transform of the equation |
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answer is <math>\mathcal{F}\left[ \int_{- \infty}^{t} s(\lambda ) \,d\lambda \right] = \frac{ e^{j2 \pi f t}} {j2 \pi f }S(f) - \frac{ e^{j2 \pi f t}} {j2 \pi f }S(f_0) \! </math> |
Latest revision as of 13:21, 19 December 2009
Problem Statement
6(a) Show .
6(b) If can you find in terms of ?
Answer
a)Remember that dummy variable was used in substitution such that
Then
and
The problem statement says to make that makes the above equation simplify to
Taking the inverse Fourier Transform and changing the order of intgration
Then
Therefore it is demonstrated that
b) If
Then
Using and taking the Fourier transform of the equation
answer is