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(a) Show F [ ∫ − ∞ t s ( λ ) d λ ] = S ( f ) j 2 π f if S ( 0 ) = 0 {\displaystyle {\mathcal {F}}\left[\int _{-\infty }^{t}s(\lambda )\,d\lambda \right]={\frac {S(f)}{j2\pi f}}{\mbox{ if }}S(0)=0} . HINT: S ( 0 ) = S ( f ) | f = 0 = ∫ − ∞ ∞ s ( t ) e − j 2 π ( f → 0 ) t d t = ∫ − ∞ ∞ s ( t ) d t {\displaystyle 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}
(b) If S ( 0 ) ≠ 0 {\displaystyle S(0)\neq 0} can you find F [ ∫ − ∞ t s ( λ ) d λ ] {\displaystyle {\mathcal {F}}\left[\int _{-\infty }^{t}s(\lambda )\,d\lambda \right]} in terms of S ( 0 ) {\displaystyle \displaystyle S(0)} ?
Find F [ g ( t − t 0 ) e j 2 π f 0 t ] {\displaystyle {\mathcal {F}}\left[g(t-t_{0})e^{j2\pi f_{0}t}\right]}
F [ g ( t − t 0 ) e j 2 π f 0 t ] = ∫ − ∞ ∞ [ g ( t − t 0 ) e j 2 π f 0 t ] e − j 2 π f t d t {\displaystyle {\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}
∫ − ∞ ∞ g ( t − t 0 ) e − j 2 π ( f − f 0 ) t d t = ∫ − ∞ ∞ g ( λ ) e − j 2 π ( f − f 0 ) ( λ + t 0 ) d t {\displaystyle \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}
After some simplification and rearranging terms, we get:
∫ − ∞ ∞ g ( λ ) e − j 2 π ( f − f 0 ) ( λ + t 0 ) d t = ∫ − ∞ ∞ g ( λ ) e − j 2 π ( f − f 0 ) λ e − j 2 π ( f − f 0 ) t 0 d t = ∫ − ∞ ∞ g ( λ ) e − j 2 π ( f − f 0 ) λ e − j 2 π ( f − f 0 ) t 0 d t = e − j 2 π ( f − f 0 ) t 0 [ ∫ − ∞ ∞ g ( λ ) e − j 2 π ( f − f 0 ) λ d t ] {\displaystyle \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]}
Result
F [ g ( t ) e j 2 π f 0 t ] = G ( f − f 0 ) e − j 2 π ( f − f 0 ) t 0 {\displaystyle {\mathcal {F}}\left[g(t)e^{j2\pi f_{0}t}\right]=G(f-f_{0})e^{-j2\pi (f-f_{0})t_{0}}}
![{\displaystyle {\mathcal {F}}\left[\int _{-\infty }^{t}s(\lambda )\,d\lambda \right]={\frac {S(f)}{j2\pi f}}{\mbox{ if }}S(0)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e30ea2d81a3c0586359e282685bc252899dbaab)
. HINT: 
can you find ![{\displaystyle {\mathcal {F}}\left[\int _{-\infty }^{t}s(\lambda )\,d\lambda \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0af3b7173a594a013131920839a34842568e41f3)
in terms of 
?
![{\displaystyle {\mathcal {F}}\left[g(t-t_{0})e^{j2\pi f_{0}t}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/760cf8aebe40a93a519c795041b22948916fedfd)
![{\displaystyle {\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bac1763263678d6f66de5064a4a0b087717f58a)
![{\displaystyle \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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/491841badafc8c526d6347c9f601c2ba3425b259)
![{\displaystyle {\mathcal {F}}\left[g(t)e^{j2\pi f_{0}t}\right]=G(f-f_{0})e^{-j2\pi (f-f_{0})t_{0}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cfc8ad8f910e3f77cdc8622a23c620ce21717d0)