Talked about DFT (Fourier transform of an infinite discrete sequence also called DTFT) \(X\left ( e^{j\omega }\right ) ={\sum \limits _{n=-\infty }^{\infty }}x\left [ n\right ] e^{-j\omega n}\) where \(x\left [ n\right ] =\frac{1}{2\pi }{\int \limits _{-\pi }^{\pi }}X\left ( e^{j\omega }\right ) e^{j\omega n}d\omega \). Notice that \(X\left ( e^{j\omega }\right ) \) is continuous in \(\omega \) which has units radians. and is periodic of period \(2\pi .\) Not every sequence \(x\left [ n\right ] \) has DFT. It must be absolutely summable on its own for it to have DFT, i.e. \({\sum \limits _{n=-\infty }^{\infty }}\left \vert x\left [ n\right ] \right \vert <\infty \). This means \(u\left ( n\right ) \), the unit step function do not have DFT.
If system is stable, then the \(h\left ( n\right ) \) will have DFT (called frequency response)
Next, \(H\left ( e^{j\omega }\right ) \) was shown and its inverse was calculated. For a low pass filter. The result is sinc function\(.\)Note, low pass filter is over \(-\pi ,\pi \), and if the cut off frequency \(\omega _{c}\) gets close to \(\pi \), filter becomes all pass filter. We only need to look at region \(-\pi ,\pi \)
Next talk about symmetry property of DTFT. Definition of even/odd for \(x\left [ n\right ] \) and for \(X\left ( e^{j\omega }\right ) \)
proofs given:
To remember these, note that when \(x\left [ n\right ] \) is even, then \(X\left ( e^{j\omega }\right ) \) follows \(x\left [ n\right ] \) , so not need to worry about this part. When \(x\left [ n\right ] \) is odd, then \(X\left ( e^{j\omega }\right ) \) is also odd, but it is opposite to what \(x\left [ n\right ] \) is. When \(x\left [ n\right ] \) real, \(X\left ( e^{j\omega }\right ) \) becomes imaginary, and when \(x\left [ n\right ] \) is imaginary, \(X\left ( e^{j\omega }\right ) \) becomes real. So just remember the \(x\left [ n\right ] \) odd part. Learn the proofs in notes.
Next talked about conjugate symmetric. sequence is CS, if \(x\left [ n\right ] =x^{\ast }\left [ -n\right ] \) this means the real part of \(x\left [ n\right ] \) is even and its imaginary part is odd. Next more properties about CS for \(x\left [ n\right ] \) and \(X\left ( e^{j\omega }\right ) \) are given. Not sure if these will come up in exam.
Learn how to find even and odd part of sequence.
Talked about sampling of \(x\left ( t\right ) \) and sample and hold, and sampling function (impulse train)
This was the end of the lecture.