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by: Eloy Ferry


Eloy Ferry
GPA 3.84

John Gowdy

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John Gowdy
Class Notes
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This 18 page Class Notes was uploaded by Eloy Ferry on Saturday September 26, 2015. The Class Notes belongs to E C E 467 at Clemson University taught by John Gowdy in Fall. Since its upload, it has received 23 views. For similar materials see /class/214305/e-c-e-467-clemson-university in ELECTRICAL AND COMPUTER ENGINEERING at Clemson University.

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Date Created: 09/26/15
ECE T467667 Introduction to Digital Signal Processing Lecture 24 L RealiZationsof Digital Filters 3 01 bkxm quot k Emsn 39 quotDirect Form I 1 b0 6 mu 5 G3 gt gtyltn V 39 v E1 31 rag t o x 32 a EU v 4be Corresponds to I H2Z H1 Z yn ECE quot1 467667 Introduction to Digital Signal Pi occsxing 39 M where H2 2 Zbkz k zeros kD quot39 and H1z 1 Mathematically equivalent implementation Since H2Z41Z H1 242 0a Difference equations pinxngt akpltn k k1 yin gnarl k ECE 39139467667 Introduction to Digital Signal Processing The above realization can be simplified to 5 pin Direct Form II A canonical form min no of delay elements Still another approach Write as yn b0pn ibkpm k v2 k1 The kO term split out from the 23 ECE 391 467667 Introduction to Digital Signal Processing Now substitute the expression of for pn in b0pn b yltngtb0xngtu akpltnkgtgbkpnk ECE T467667 Introduction to Digital Signal Processing Biguadratic Section ratio of two quadratics in zquot1 I Z boy bizquot1 bzz 2 Example H 1 2 1 2112 a2 Typically a digital filter is implemented as a cascade or parallel bank of biquadraticsections Hz H1z H2 zHRzlt R biquadratic sections xn yln HA2 v H2Z Hn2 V xnL H2z yin Reason for using second order sections The resulting filter typically is less sensitive to round off error ECE T4o7667 Introduction to Di gitnl Signal Processing Discrete Fourier Transfo rmDFT Applies to finitelength sequence39sh inite duration signals Recall X e39m 2xne jw Definition of Discrete Time Fourier Transform n DTFT Consider xn0 for nltO and nZN 39 N imam 0 For this case Xequot 23 Cohsider N Samples of Xej Xe Xk r m k k O1N1 N N 1 gik Xk xne N jquot k O1N 1 n0 132 Let WN e N N 1 z Xk ZxnWNk o lt k lt N 1 DFT 0 ECE 1 467667 Introduction to Digital Signal Processing 339 27 47 671 21rN 1 quot 711 7E N N N N I N1 kn Inverse DFT IDFT xn 139 2XkwN O S n S N1 v ko Verify the inverse relation l39i lfi xewn Nk0 p0 1 N1 N 1 H Roam N 2 Xp 39 2 WM p0 k0 2 1 wNquotquot N 1eWp nN For paen o 1le9 quot7 i1wN5 5 N1 For pn EWNO N k0 gt right hand side of expression isJ xn N xn ECE 391 467667 Introduction to Di gi 11 Signal Processing Examgie finite duration xa t X136 Xa 1 39 Q xn xanT Xequot l W liilHITIll IJJI 39 0 39 1 I I 2n39 21 Xk TTTTTTI O N1 k Summarize XkXequot nggk k01N1 l X 2 213k k 0 1 N 1 except for aliasing T a NT 2Xf f lfk k 01N 1 except for aliasing 27 27 1 1 39 Agzm Af mReSOIUt on Am N 39 NT NT Signal duration ECE 391 467667 Introduction to Digitnl39Signai Processing Zero paddinq to improve resolution make At smaller N 1 Consider Xk ExnWNk n0 Let x101 xn with N1 N 0 s appended Length of the signal isartificially extended from N to N1 N1 N xk 2x1nwmkn 2xnWN1kquot k 01N1 n0 nO N 43221 N Zxne 1 2xnequot Dn n0 n0 mggg N1 22gt A0 E lt 21 1 N A9 3i and N J N1T N1T k01 N1 ECE T467667 Introduction to Di giml Signal Processing Effects of Windowincuto get a restricted length signalto apply the DFT to L xt X053 2 menw wnuuwm WM Effects of Sampli ng to get a discrete time signal yn we Xjsz W052 quotlcaligcquot We aliasing wk TITfTTI 012 N l k 10 ECE T467667 Introduction to Di giml Signal Processing Symmetry of DFT when xn is gal XkXquotN kgtXkXN k k01N1 ReXk RexN k lmXk lmXN k arg Xk arg39XNk 11 ECE T467667 Introduction to Digital Signal Processing DFF of windoWed sinusoids Consider xt cos 21rf0t Xt X m V T f0 f0 f Pe od l To Use rectangular window to extract exactly one period 1 wt 1 f0 t yt cos 2m0t w l U 1 f0 12 ECE T467667 Introduction to Di gi ta Signal Processing Yn I quotquotNfo 9100 1 03 21tf T 2 2m 0 0 0 Nfo Use N samplesperiod 2n 2 Mk 3 Tu 1 1 3 T I T W 0 0 Aw 373 N Apply DFT to yn get 39 2 I T x o 1 2 N1 Now consider DFT of sampies of exactly 2 periods of cos 271 0 xt wt W0 2 0 t f0 f U U2 t H 46 3 f 13 ECE T4671667 Introduction to Digital Signal Pmcessing 2N 1 Use T 1 as before Nf0 Have total of 2N samples since 2 periods are sampled DFT A0 H MIN 21 2 2N 1 14 271 E N LZE ONfO N ECE 391 467 667 Introduction to Digital Signal Processing Generalize Windowed cos N samplvesperiodM periods MN poin t DFT gives T M MN M MN1 k 27 A m 03 MN DFT of samples of 005 over noninteger no of periods We 3Yk DFT amples 15 ECE T467667 Introduction to Digital Signal Processing DFT Wk Heakagequot Parameters of DFT TN 1 Choose gt 2 fmaxto avoid aliasing 2 Frequency resolution Choose N to give desired frequency resolution using 2 m AfAQN 1 16 ECE 391 467667 Introduction to Digital Signal Processing Recall We can append 039s to xn to artificially increase N met 1 xn 1 IEHXOINI O 12 3 n xn l 4polnt DFT jgt 39 0 l 2 3 n 0 39 I 2 3 k xn 39 1X00 I Add4 8 point zeros DPT 1 0 l 2 3 7 n xn I Add 12 16point zero DFT gt 02468101215 02468l012 15quot Figure 622 Illustration of zero padding Note Appending 039s does not generate any new frequency domain information It just gives an easy way of implementing an interpolation that could have been done wo adding zeros IO N1 a 4335 4201 sin xe k2 JJXk m N where 10 9 2 Summarize quotPitfallsquot of DFT 1 Aliasing if not gt 2fmax 2 quotLeakag lquot if window must be used 3 quotPicket Fence Effectquotactually leakage for the case where the input is a noninteger no ofperiods ofua pure sinusoid 17 ECE T467667 Introduction to Digital Signal Processing Averaging DFT S of Noisy Signals When using the DFT to obtain spedtral estimates of a sig nal having a large amount of additive random noisesignal embedded in noise we must use averaging of DFT39s to the true signal from the noise Assume xt quotstationaryquot X1k signal sinusoid noise OI llil lljl39 l 0 X2 0 No obvious H signal I I 39 characteristics I l l l K o 1 39 XNk l K o 155 K average DR of windowed sinusoid


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