ASN6 a,b- Prove given Fourier Transform property: Difference between revisions
Jump to navigation
Jump to search
Jodi.Hodge (talk | contribs) (New page: Back to my home page) |
Jodi.Hodge (talk | contribs) No edit summary |
||
Line 1: | Line 1: | ||
[[Jodi Hodge|Back to my home page]] |
[[Jodi Hodge|Back to my home page]] |
||
(a) Show <math> \mathcal{F}\left[ \int_{- \infty}^{t} s(\lambda ) \,d\lambda \right] = \frac{S(f)}{j2 \pi f} \mbox{ if } S(0) = 0 </math>. HINT: <math> S(0) = S(f) \vert _{_{f=0}} = \int_{- \infty}^{\infty} s(t)e^{-j2 \pi (f \rightarrow 0)t} \,dt = \int_{- \infty}^{\infty} s(t) \,dt </math> |
|||
(b) If <math> S(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(0) </math>? |
|||
'''Find <math>\mathcal{F} \left[ g(t-t_{0})e^{j2 \pi f_{0}t} \right]</math><br/>''' |
|||
<math> |
|||
\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 |
|||
</math> |
|||
<math> |
|||
\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 |
|||
</math> |
|||
After some simplification and rearranging terms, we get: |
|||
<math> |
|||
\int_{- \infty}^{\infty} g(\lambda )e^{-j2 \pi (f-f_{0})(\lambda + t_{0})} \,dt = \int_{- \infty}^{\infty} g(\lambda )e^{-j2 \pi (f-f_{0})\lambda } e^{-j2 \pi (f-f_{0})t_{0}} \,dt = |
|||
\int_{- \infty}^{\infty} g(\lambda )e^{-j2 \pi (f-f_{0})\lambda } e^{-j2 \pi (f-f_{0})t_{0}} \,dt = e^{-j2 \pi (f-f_{0})t_{0}} \left[ \int_{- \infty}^{\infty} g(\lambda )e^{-j2 \pi (f-f_{0})\lambda } \,dt \right] |
|||
</math> |
|||
Result |
|||
<math> |
|||
\mathcal{F} \left[ g(t)e^{j2 \pi f_{0}t} \right] = G(f-f_{0})e^{-j2 \pi (f-f_{0})t_{0}} |
|||
</math> |
Revision as of 14:59, 3 December 2009
(a) Show . HINT:
(b) If can you find in terms of ?
Find
After some simplification and rearranging terms, we get:
Result