Explanation of Wednesday October 22nd In-Class Quiz
Today's Quiz was based on taking the Fourier Transform of a signal that had been time inverted and time shifted by using the definition of a Fourier Transform. The quiz consisted of two very similar questions. The first was to be answered working by oneself without notes. Following the completion of this problem, students papers were exchanged with a partner and then graded as they saw appropriate in comparison to the correct solution that was given on the overhead. After seeing how their paper was graded and having had a chance to study the solution given, the second problem was assigned and was to be graded by the TA for credit.
First Signal:
Show that $ \mathcal{F}(x(-t-1)) = e^{jw} \mathcal{X}(w) $
By the definition of Fourier Transform
$ = \int_{-\infty}^{\infty} x(-t-1)e^{-jwt}dt $
Let y = -t - 1, dy = -dt
Substituting into the above equation yields:
$ = \int_{\infty}^{-\infty} x(y)e^{-jw(-1-y)}(-dy) $
Split using properties of e^(t) and flip bounds by using the negative from (-dy)
$ = \int_{-\infty}^{\infty} x(y)e^{jw}e^{jwy}(dy) $
Factor out the constant term,
$ = e^{jw}\int_{-\infty}^{\infty} x(y)e^{jwy}(dy) $
This is very close to being the desired value, just insert a negative in front of the "w" and in front of the term "jwy"
$ = e^{jw}\int_{-\infty}^{\infty} x(y)e^{-j(-w)y}(dy) $
