A question about Gibbs phenomenon example in Richard Lyons book If anyone has a copy of this book could you please shed some light here: I was trying to reproduce the example shown in Chapter 5, section 5.3.1 Window Design Method's Figure 5-19 which illustrates the ripples in the frequency domain in octave. Just as the book suggested, when you have any dropped tap in time domain, you will see Gibbs phenomenon in the frequency domain. However, my question is that, why don't you use all 32 taps in this case (the book is dropping 1 tap from 32 taps to get 5-19(c) )?  When I use all 32 taps, I can see the un-distorted frequency domain here since the iDFT and DFT pretty much cancelled out each other here...I think I am missing some points here the book is trying to make...

Thanks,

Will
I don't have the book, but from what I understand from your question, you take a discrete desired frequency response with $32$ elements, apply an IDFT to get a $32$-tap filter, and then you remove taps to see the effects of truncation in the time domain. Well, this is not exactly what the window design method is about, and it is no surprise that if you don't remove filter taps in the time domain and apply a DFT that you get back your desired $32$-tap frequency response (because, as you've figured out yourself, IDFT and DFT are inverse operations of each other).

However, your actual frequency response does not have $32$ values in the frequency domain but it is continuous in frequency. It is given by the discrete-time Fourier transform (DTFT):

$$H(e^{j\omega})=\sum_{n=0}^{N-1}h[n]e^{-jn\omega}\tag{1}$$

where $N$ is the number of time domain taps (filter length), and $h[n]$ are the filter coefficients. You can compute a sampled version of $(1)$ by using the DFT, and for example, if $N=32$, you can (and should) compute more than $32$ frequency domain samples by using zero-padding. In Matlab/Octave, you can use the commands fft or freqz to compute sampled versions of $(1)$ with an arbitrary number of frequency domain samples.

To help you improve your understanding of the window method, and of DSP in general, I would like to recommend to you a really good (and free) book: Introduction to Signal Processing by S.J. Orfanidis.

Also take a look at this answer, this answer and this answer to related questions.