Hi, I needed some help understanding a derivation in Mr. Oppenheim's book Discrete Time Signal Processing. In Edition 3, Chapter 4 Sampling of Continuous Time Signals, 4.6.1 Sampling Rate Reduction, Page 180, Eq. 4.74, he replaced the variable of summation from r to i + kM. I didn't understand how he did it or why it is valid and I would appreciate it if someone could clarify it. Thanks and Best Regards Adil Ahsan
Help Needed Understanding Decimation from Discrete Time Signal Processing
Started by ●January 24, 2016
Reply by ●January 24, 20162016-01-24
On Sun, 24 Jan 2016 13:43:19 -0800, adilmahsan wrote:> Hi, > > I needed some help understanding a derivation in Mr. Oppenheim's book > Discrete Time Signal Processing. In Edition 3, Chapter 4 Sampling of > Continuous Time Signals, 4.6.1 Sampling Rate Reduction, Page 180, Eq. > 4.74, he replaced the variable of summation from r to i + kM. I didn't > understand how he did it or why it is valid and I would appreciate it if > someone could clarify it. > > Thanks and Best Regards Adil AhsanIf you post the relevant equations then those of us who don't have the book may be able to answer your question. OTOH, puzzling it out yourself will teach you a lot, one way or another. -- www.wescottdesign.com
Reply by ●January 25, 20162016-01-25
On Monday, January 25, 2016 at 4:55:26 AM UTC+1, Tim Wescott wrote:> On Sun, 24 Jan 2016 13:43:19 -0800, adilmahsan wrote: > > > Hi, > > > > I needed some help understanding a derivation in Mr. Oppenheim's book > > Discrete Time Signal Processing. In Edition 3, Chapter 4 Sampling of > > Continuous Time Signals, 4.6.1 Sampling Rate Reduction, Page 180, Eq. > > 4.74, he replaced the variable of summation from r to i + kM. I didn't > > understand how he did it or why it is valid and I would appreciate it if > > someone could clarify it. > > > > Thanks and Best Regards Adil Ahsan > > If you post the relevant equations then those of us who don't have the > book may be able to answer your question. > > OTOH, puzzling it out yourself will teach you a lot, one way or another. > > -- > www.wescottdesign.comThanks for your reply. No need. Thanks. I just needed to take a shower. Things tend to become clearer when one takes a shower. :)
Reply by ●January 25, 20162016-01-25
On Sun, 24 Jan 2016 13:43:19 -0800 (PST), adilmahsan@gmail.com wrote:>Hi, > >I needed some help understanding a derivation in Mr. Oppenheim's book Discr= >ete Time Signal Processing. In Edition 3, Chapter 4 Sampling of Continuous = >Time Signals, 4.6.1 Sampling Rate Reduction, Page 180, Eq. 4.74, he replace= >d the variable of summation from r to i + kM. I didn't understand how he di= >d it or why it is valid and I would appreciate it if someone could clarify = >it. > >Thanks and Best Regards >Adil AhsanI have the book, but it's a different edition and it took me a while to sort out what you were asking about. Since our books are different editions, I don't know exactly what's in yours, but I can describe a response from what's in mine. In my edition (the first, I think), he gives a definition of the DTFT of X from the continuous time spectrum Xc, sampled discretely with an index, k, ranging from minus infinity to plus infinity with a sample period of T. He then rewrites everything when the sample period is MT, and shows the downsampled DTFT expression, in terms of T' = MT shown in separate equations with both T' and M, this time with an index, r, that runs from minus infinity to plus infinity. He then describes that r is essentially r = i + kM and rewrites X in terms of two summations, an outer summation with i running from 0 to M-1 and and inner summation with k running from minus infinity to plus infinity. The trick of all of this is thinking about what's happening to the DFT frequency samples when you downsample. When T grows from T to MT, the spacing in the DFT output actually *decreases* from 1/T to 1/MT, so there are M-1 more samples along the same frequency spacing for each "bin" relative to sampling at T. Rewriting the equation with the outer summation with i running from 0 to M-1 fills in these additional samples on the frequency axis. Basically, as the time sampling spaces gets longer the frequency sampling spaces get narrower. In my book this is shown a little bit in a Figure that shows a triangular spectrum being resampled to eliminate the gaps in between. Essentially, the triangles are sampled at a higher rate in the frequency domain as the time domain sample rate is decreased, which is, again, where the additional index, i, comes from to fill in those samples. This lesson also shows the fundamtal idea that there are two ways to increase discrete frequency "resolution", or bin width: increase the observations time (typically N), or sample *more slowly*. Either decreases frequency bin width. Eric Jacobsen Anchor Hill Communications http://www.anchorhill.com
Reply by ●January 25, 20162016-01-25
On Mon, 25 Jan 2016 10:40:55 -0800, adilmahsan wrote:> On Monday, January 25, 2016 at 4:55:26 AM UTC+1, Tim Wescott wrote: >> On Sun, 24 Jan 2016 13:43:19 -0800, adilmahsan wrote: >> >> > Hi, >> > >> > I needed some help understanding a derivation in Mr. Oppenheim's book >> > Discrete Time Signal Processing. In Edition 3, Chapter 4 Sampling of >> > Continuous Time Signals, 4.6.1 Sampling Rate Reduction, Page 180, Eq. >> > 4.74, he replaced the variable of summation from r to i + kM. I >> > didn't understand how he did it or why it is valid and I would >> > appreciate it if someone could clarify it. >> > >> > Thanks and Best Regards Adil Ahsan >> >> If you post the relevant equations then those of us who don't have the >> book may be able to answer your question. >> >> OTOH, puzzling it out yourself will teach you a lot, one way or >> another. >> >> -- >> www.wescottdesign.com > > Thanks for your reply. No need. Thanks. I just needed to take a shower. > Things tend to become clearer when one takes a shower. :)That happens to me. I think that some months I generate more value for my customers in the shower or during long walks than I do sitting at my desk. It raises the question of how I should be compensated -- I've decided that it's OK to dream up the stuff for free, as long as I get paid to write it down and flesh it out. -- www.wescottdesign.com
