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# SIGNALS & SYSTEMS LAB EE 422G

UK

GPA 3.81

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This 6 page Class Notes was uploaded by Adaline Pollich on Friday October 23, 2015. The Class Notes belongs to EE 422G at University of Kentucky taught by Staff in Fall. Since its upload, it has received 46 views. For similar materials see /class/228327/ee-422g-university-of-kentucky in Electrical Engineering at University of Kentucky.

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Date Created: 10/23/15

EE 422G Notes Chapter 8 Instructor Cheung 82B DigitaltoAnalog Conversion DA Passmg Xst through an analog lowpass filter Mathematically 0 Jxxnha ndr Z x T5T I1Thl Tdr Z xnThZ nT X xowm e The estimate t at timet is based on a linear combination of all samples xnwT Two factors 1 How accurate can the continuoustime signal be reconstructed 2 How complex is it to realize We will look at four methods brie y 1 Ideal lowpass 2 SampleandHold 3 Linear Interpolation 4 General lowpass filter e g RC filter Ideal Lowpass RC lter Linear Interpolation SampleandHold Most accurate 39 m 39 quot 3 Least accurate rl 4Jer DEVI SampleandHold Linear Interpolation RC lter Ideal Lowpass Simplest Unrealizable V Sacrament 1 Ideal LowPass filter Idea In frequency domain multiply by a brick wall filter T alt 05as H 160 O Otherw1se is the same as convolVing with a sine lter in the time domain sin a t2 a a ht 3 s1nc2 t s1nc2 2 cost 2 Page 59 EE 422G Notes Chapter 8 Instructor Cheung t ht s1nc j I 0 AHow T L0 5 3 E t TrT TE T 39 I T 2T 3T 4T 5T 6T Interpretation in timedomain w t quotT xn 1Tsinc quot 1 t 2 2H xnTsinc x TSinc HTT 550 Snmplr u ur Diagram on right shows the contributions from each term only four terms are shown Notice At integral number of T i e nlT nT only one term has nonzero contribution For all other t 20 has contributions N from all other samples Why is it impossible to implement realtime sinc lter It goes from ltgtltgt to ltgtltgt which means we need to know XnT from ltgtltgt to ltgtltgt Not possible unless the signal is nite duration and stored 2 Sample andHold Idea n2T n1T nT n1T It is easy to see that the interpolation lter must look like Page 510 EE 422G Notes Chapter 8 Instructor Cheung m l HUM F VVV ZTUT 4m As the frequency spectrum of the sampleandhold is a sinc function many high frequency components remains after interpolation Though very simple to implement this is not a very effect interpolation filter 3 Linear Interpolation Idea n2T n1T nT n1T The interpolation filter looks like ht H003 l 1 gt T i 2 i T u I T n 39 27rT 39 2m 39 It is a better filter than mpl 4 h 1quot as less hi h fr 1quot n pass through There is still some lowfrequency distortion Page 511 EE 422G Notes Chapter 8 Instructor Cheung Timedomain interpolation let s say t nlT k with 0ltkltT nlT t nT Contribution from xn 1T xn 1TTTk Red part Contribution from xnT 9mm Blue part and there are no contribution from any other samples Thus xn 1T k xn 1T T k T xnT a linear combination of the closest two samples weighted based on how close they are to the point of interpolation 4 General filter In practice we can design a better filter using proper analog filter design To ensure that the high frequency components are remove the passband frequency 03p must be smaller than half of the Nyquist rate or 1203S The text page 382 lists the first order Butterworth lter but you can use anyone depending on the desired fidelity and complexity Page 512 EE 422G Notes Chapter 8 Instructor Cheung 82C Quantization Patterns Levels 11010 12250 11011 12300 12330352562 D 11100 12350 gt 11100 closest level 11101 12400 5bit computer Quantization represents a continuous real number ex 123303 52562 using the discrete level among 2n ex n5 different levels closest to the input ex 12350 Unlike sampling quantization almost always induces loss in precision 1f the input value is very large we may not care about a loss in the 10th decimal place To quantify this relative loss we use the Signal toquantization Noise Ratio SNR defined as follows SNR Average s1gnal power Average power of quantlzatlon error in dB Obviously a large SNR indicates high fidelity Quantization Error e 12330352562 12350 019647438 This error must be between A2 and A2 where A is the interval between successive levels ex A05 A depends on two parameters 1 Number of levels ie 2n 2 The dynamic range max min of the input signal Let s denote as D Then A 2 DZ quot n Assume the quantization error e is unifome distributed between A2 and A2 Average power of quantization error Ee2 e2 iale A A2 D2 22Vl E 12 Page 513 EE 422G Notes Chapter 8 Instructor Cheung If the signal power is PS then P SNR lOlog22M12 lOlonglOlogPS 2010gD20nlog2 Thus SNR can be improved by gt raising the signal power PS gt reducing the signal dynamic range D gt increase the number of bits n 6dB rule it s convenient to remember that every additional bit will improve the SNR by 2010g2 3 6dB Page 514

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