EECE 3 Sigals & Systems Prof. Mark Fowler Note Set #6 D-T Systems: DTFT Aalysis of DT Systems Readig Assigmet: Sectios 5.5 & 5.6 of Kame ad Heck /
Course Flow Diagram The arrows here show coceptual flow betwee ideas. Note the parallel structure betwee the pik blocks C-T Freq. Aalysis ad the blue blocks D-T Freq. Aalysis. New Sigal Models Ch. Itro C-T Sigal Model Fuctios o Real Lie System Properties LTI Causal Etc Ch. 3: CT Fourier Sigal Models Fourier Series Periodic Sigals Fourier Trasform CTFT No-Periodic Sigals Ch. Diff Eqs C-T System Model Differetial Equatios D-T Sigal Model Differece Equatios Zero-State Respose Ch. 5: CT Fourier System Models Frequecy Respose Based o Fourier Trasform New System Model Ch. Covolutio C-T System Model Covolutio Itegral Ch. 6 & 8: Laplace Models for CT Sigals & Systems Trasfer Fuctio New System Model New System Model D-T Sigal Model Fuctios o Itegers New Sigal Model Powerful Aalysis Tool Zero-Iput Respose Characteristic Eq. Ch. 4: DT Fourier Sigal Models DTFT for Had Aalysis DFT & FFT for Computer Aalysis D-T System Model Covolutio Sum Ch. 5: DT Fourier System Models Freq. Respose for DT Based o DTFT New System Model Ch. 7: Z Tras. Models for DT Sigals & Systems Trasfer Fuctio New System Model /
5.5: System aalysis via DTFT We ow retur to Ch. 5 for its DT coverage! Back i Ch., we saw that a D-T system i zero state has a output-iput relatio of: x h y h x i h i x i Recall that i Ch. 5 we saw how to use frequecy domai methods to aalyze the iput-output relatioship for the C-T case. We ow do a similar thig for D-T Defie the Frequecy Respose of the D-T system H h e j Perfectly parallel to the same idea for CT systems!!! DTFT of h 3/
4/ From Table of DTFT properties: H X h x So we have: H h X x x h y H X Y + H X Y H X Y So So i geeral we see that the system frequecy respose re-shapes the iput DTFT s magitude ad phase. System ca: -emphasize some frequecies -de-emphasize other frequecies Perfectly parallel to the same ideas for CT systems!!! The above shows how to use DTFT to do geeral DT system aalyses virtually all of your isight from the CT case carries over!
Now lets look at the special case: Respose to Siusoidal Iput x Acos + θ 3,,,,,,3,... From DTFT Table: X jθ jθ e δ + + e δ A < < periodic elsewhere X We oly eed to focus our attetio here Y H X y j H X e d 5/
So what does Y look like? Y jθ jθ H e δ + + H e δ A < < periodic elsewhere Now H H H H e j H e j H Used symmetry properties Y j θ + H + j θ + H e δ + + e δ A H < < periodic elsewhere From DTFT Table we see this is the DTFT of a cosie sigal with: Amplitude A H Phase θ + H 6/
So y H Acos + θ + H Acos + θ H H Acos + θ + H System chages amplitude ad phase of siusoidal iput Perfectly parallel to the same ideas for CT systems!!! 7/
Example 5.8 error i book Suppose you have a system described by H + e Ad you put the followig sigal ito it x + si cos + si Cosie with Fid the output. So we eed to kow the system s frequecy respose at oly frequecies. j H + j e Im H j + e j e j 4 Re j Im Re 8/
9/ + 4 si si 4 cos H H y H y Sice the system is liear we ca cosider each of the iput terms separately. Ad the add them to get the complete respose + 4 si 4 y
Note: I the above example we used Asi + θ H H Asi + θ + H which is the sie versio of our result above for a cosie iput Q: Why does that follow? A: It is a special case of the cosie result that is easy to see: - covert si + θ ito a cosie form - apply the cosie result - covert cosie output back ito sie form /
Aalysis of Ideal D-T lowpass Filter LPF Just as i the CT case we ca specify filters. We looked at the ideal lowpass filter for the CT case here we look at it for the DT case. H Ideal lowpass filter 4 3 -B B 3 4 As always with DT we oly eed to look here Cut-off frequecy B rad/sample /
This slide shows how a DT filter might be employed but ideal filters ca t be built i practice. We ll see later a few practical DT filters. xt x y yt ADC @ D-T Ideal DAC Fs /T LPF ~ ~ let B F cut - off B < ~ sample @ Fs F X ω Y ω B ~ B ~ ω B /T B /T ω X B Y B Whole System ADC D-T filter DAC acts like a equivalet C-T system /
Why ca t a ideal LPF exist i practice?? We kow the frequecy respose of the ideal LPF so fid its impulse respose: From DTFT Table : B B h sic h Key Poit: h is o-zero here starts before the impulse that makes it is eve applied! Ca t build a Ideal LPF Same thig is true i C-T 3/
Causal Lowpass Filter But ot Ideal I practice, the best we ca do is try to approximate the ideal LPF If you go o to study DSP you ll lear how to desig filters that do a good job at this approximatio Here we ll look at two seat of the pats approaches to get a good LPF Approach # Trucate & Shift Ideal h Trucate Shift h truc h, ideal, N N otherwise N eve h approx N h truc h truc N+ o-zero samples h approx 4/
Let s see how well these work H H H..8.6.4. - -.8 -.6 -.4 -...4.6.8 /..8.6.4. - -.8 -.6 -.4 -...4.6.8 /..8.6.4. N N 6 N - -.8 -.6 -.4 -...4.6.8 / Frequecy Respose of a filter trucated to samples Frequecy Respose of a filter trucated to 6 samples Frequecy Respose of a filter trucated to samples Some geeral isight: Loger legths for the trucated impulse respose Gives better approximatio to the ideal filter respose!! 5/
6/ Approach #: Movig Average Filters Here is a very simple, low quality LPF: otherwise h,,, h -3 - - 3 4 5 To see how well this works as a lowpass filter we fid its frequecy respose: + + j j j j e e e e h H By defiitio of the DTFT Oly o-zero terms i the sum
Now, to see what this looks like we fid its magitude. H j + e Euler! + cos j si It is ow i rect. form H + cos + si + cos + cos + si + cos + cos Trig. ID Now.. Plot this to see if it is a good LPF! 7/
Here s a plot of this filter s freq. resp. magitude: repeats periodically H repeats periodically / / High Frequecies Low Frequecies High Frequecies Well this does atteuate high frequecies but does t really stop them! It is a low pass filter but ot a very good oe! How do we make a better LPF??? We could try loger movig average filters: h N,,,,,, otherwise N 8/
Plots of various Movig Average Filters h N,,,,,, otherwise N N N 3 N 5 N N We see that icreasig the legth of the all-oes movig average filter causes the passbad to get arrower but the quality of the filter does t get better so we geerally eed other types of filters. 9/
Commets o these Filter Desig Approaches The two approaches to DT filters we ve see here are simplistic approaches There are ow very powerful methods for desigig REALLY good DT filters we ll look at some of these later i this course. A complete study of such issues must be left to a seior-level course i DSP!! /