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# ELECTRIC CURCUIT ANALYS III ECE 223

PSU

GPA 3.78

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This 29 page Class Notes was uploaded by Miss Chadrick Doyle on Tuesday September 1, 2015. The Class Notes belongs to ECE 223 at Portland State University taught by Staff in Fall. Since its upload, it has received 23 views. For similar materials see /class/168241/ece-223-portland-state-university in ELECTRICAL AND COMPUTER ENGINEERING at Portland State University.

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Date Created: 09/01/15

Fast Fourier Transform Discretetime windowing Discrete Fourier Transform Relationship to DTFT Relationship to DTFS Zero padding J McNames Portland State University ECE 223 FFT Ver 105 1 Fourier Series amp Transform Summary mm 2 Xlltejmon kltNgt 1 XkN Z awe Won nltNgt 1 3399 an 2 Xe e d9 Xej9 Z nk 9399 o What are the similarities and differences between the DTFS 84 DTFT J McNames Portland State University ECE 223 FFT Ver 105 2 Periodic Signals mm 2 Xkejk90 ltggt2w Z mum mo kltNgt 0 Recall that DT periodic signals can be represented by the DTFT o Requires the use of impulses why oT his makes the DTFT more general than the DTFS J McNames Portland State University ECE 223 FFT Ver 105 3 Example 1 Relationship to Fourier Series Suppose that we have a periodic signal mph with fundamental period N Define the truncated signal as follows mph n01 n n0N 0 otherwnse Determine how the Fourier transform of is related to the discretetime Fourier series coefficients of xp Recall that 1 Xpk N Z xpne9k27TNn nltNgt J McNames Portland State University ECE 223 FFT Ver 105 4 Example 1 Workspace J McNames Portland State University ECE 223 FFT Ver 105 5 DFT Estimate of DTFT OgragN l 0 Otherwise 1 1 Xwe99 Xem GB WEN 27TX Mm W W ejw dw 0 Recall that windowing in the time domain is equivalent to filtering convolution in the frequency domain 0 We found earlier that a windowed signal 9 could be thought of as one period of a periodic signal xp 0 This enables us to calculate the DTFT at discretefrequencies using the DTFS analysis equation 0 Advantages DTFS consists of a finite sum we can calculate it DTFS can be calculated very efficiently using the Fast Fourier Transform FFT J McNames Portland State University ECE 223 FFT Ver 105 6 FFT Estimate of DTFT Derived 00 Z 9 Me 99 TL OO N 1 Z xw e nn n0 N l I 27r Xwejglgik27r E xwlnle jkwn N n0 DFT xw Xwej9 0 Calculation of the DTFT of a finiteduration signal at discrete frequencies is called the Discrete Fourier Transform DFT o The FFT isjust a fast algorithm for calculating the DFT J McNames Portland State University ECE 223 FFT Ver 105 7 DFT FFT and DTFS DTFS Xlk i Z acme Won nltNgt N l DFTFFT Xwk Z Mme MW n0 0 Note abuse of notation Xlk used for DTFS CTFS 84 DFT o The DFT is a transform 0 The FFT is a fast algorithm to calculate the DFT If is a periodic signal with fundamental period N then the DFT is the same as the scaled DTFS The DFT can be applied to nonperiodic signals For nonperiodic signals this is modelled with windowing o The DTFS cannot J McNames Portland State University ECE 223 FFT Ver 105 8 Example 2 FFT Estimate of DTFT Solve for the Fourier transform of 1 1 g n g 4 mm 1 13 n 16 0 Otherwise J McNames Portland State University ECE 223 FFT Ver 105 9 Example 2 Workspace J McNames Portland State University ECE 223 FFT Ver 105 10 Example 2 Workspace J McNames Portland State University ECE 223 FFT Ver 105 11 Example 2 Signal 08 06 02 04 06 Time J McNames Portland State University ECE 223 FFT Ver 105 12 Real Xei Imag Xej co Example 2 DTFT Estimate 25 3 0 05 1 5 2 25 3 Frequency rads per sample J McNames Portland State University ECE 223 FFT Ver 105 13 Zero Padding DFT yew X 9399 wle l9k o The FFT has two apparent disadvantages It requires that N be an integer power of 2 N 2g It only generates estimates at N frequencies equally spaced between 0 and 27r radsample 0 Both of these problems can be circumvented by zeropadding Recall that ew c We can choose N to be larger than the length of our window J McNames Portland State University ECE 223 FFT Ver 105 14 Zero Padding Derived Suppose we add M N zeros to the finitelength signal such that M is an integer power of 2 and M Z N Then the zeropadded signal 963 has a length M The frequency resolution then improves to 2V radsample rather than 2 radsample N l DFTxwlnl walnle kth 1211 I 2 DFTxZn szlnle lk N l Z mum Wt Xwe939Q DFT xzw l 2k2 7r J McNames Portland State University ECE 223 FFT Ver 105 15 Example 3 FFT Estimate of DTFT Repeat the previous example but use zeropadding so that the estimate is evaluated at no less than 1000 frequencies between 0 and 7t J McNames Portland State University ECE 223 FFT Ver 105 16 Example 3 DTFT Estimate with Padding 5 0 gtlt 7 3 5 ad True DTFT DFT Estimate 10 u 2 25 3 Imag Xej co 0 05 1 15 2 25 3 Frequency rads per sample J McNames Portland State University ECE 223 FFT Ver 105 17 Example 3 MATLAB Code qunction FFTEstimate close all n 017 X 1n21 amp n34 1n218 amp n316 Plot the signal figure FigureSet14528 plotminn maxn0 O k hold on h stemnx b seth1 MarkerFaceColor b seth1 MarkerSize 4 hold off ylabe1 xn X1abe1 Time Xlimminn maxn ylim105 1051 box off AXisSet8 print depsc FFTESignal Plot the True Transform amp Estimate J McNames Portland State University ECE 223 FFT Ver 105 18 n 116 X 1n21 amp n34 1n218 amp n316 N 1engthx k OzNl we ON 12piN Frequency of estimates Xe exp jk2piN1fftX w 0000i2pi10002pi X eXpjw1 expjw5 expjw15 expjw171expjw Z True spectrum figure FigureSet1 LTX subplot211 h plotwrea1X r werea1Xe k seth2 Marker o seth2 MarkerFaceColor k seth2 MarkerSize S ylabe1 Rea1 Xe jomega xlimO p11 box off AXisLines 1egendh12 True DTFT DFT Estimate 4 subplot212 h plotwimagX r weimagXe k seth2 Marker o seth2 MarkerFaceColor k seth2 MarkerSize S ylabe1 Imag Xe jomega xlimO p11 box off AXisLines J McNames Portland State University ECE 223 FFT Ver 105 19 x1abe1 Frequency rads per sample AXisSet8 print depsc FFTEstimate Plot the True Transform amp Estimate n 116 X 1n21 amp n34 1n218 amp n316 N 2 Ceil1og22000 k OzNl we ON 12piN Frequency of estimates Xe eXpjk2piN1fftXN figure FigureSet1 LTX subplot211 h plotwrea1X r werea1Xe k seth1 LineWidth 20 seth2 LineWidth 10 ylabe1 Real Xe jomega x1imO p11 box off AXisLines 1egendh12 True DTFT DFT Estimate 4 subplot212 h plotwimagX7r weimagXe k seth1 LineWidth 20 seth2 LineWidth 10 ylabe1 Imag Xe jomega xlimO pi J McNames Portland State University ECE 223 FFT Ver 105 20 box off AXisLines x1abel Frequency rads per sample AXisSet8 print depsc FFTEstimatePadded J McNames Portland State University ECE 223 FFT Ver 105 21 Example 4 FFT Estimate of DTFT of ECG Plot the energy spectral density for a short segment of an ECG signal J McNames Portland State University ECE 223 FFT Ver 105 22 Example 4 Time Domain Plot 2 15 E 1 E 8 DJ 05 0 M 0395 I I I I I I I I I I 0 02 04 06 08 1 12 14 16 18 2 Time s J McNames Portland State University ECE 223 FFT Ver 105 23 Example 4 DTFT Estimate via FFT 1024 7000 6000 5000 4000 ESD 3000 2000 100 I U I I I I 0 5 10 15 20 25 30 35 40 Frequency Hz J McNames Portland State University ECE 223 FFT Ver 105 24 Example 4 DTFT Estimate via FFT 2048 7000 6000 5000 4000 ESD 3000 2000 1000 U 0 5 10 15 20 25 30 35 40 Frequency Hz J McNames Portland State University ECE 223 FFT Ver 105 25 Example 4 DTFT Estimate via FFT 4096 7000 6000 5000 4000 ESD 3000 2000 1000 0 MW 0 5 10 15 20 25 30 35 40 Frequency Hz J McNames Portland State University ECE 223 FFT Ver 105 26 Example 4 DTFT Estimate via FFT 8192 7000 6000 5000 4000 ESI 3000 2000 1000 Frequency Hz J McNames Portland State University ECE 223 FFT Ver 105 27 Example 4 DTFT Estimate via FFT 16384 7000 6000 5000 4000 ESI 3000 2000 1000 Frequency Hz J McNames Portland State University ECE 223 FFT Ver 105 28 Some Key Points on Windowing amp the FFT 0 In practice most DT signals are truncated to a finite duration prior to processing 0 Mathematically this is equivalent to multiplying an signal with infinite duration by another signal with finite duration 0 This process is called windowing Windowing in time is equivalent to convolution in frequency xln wln 3 Xej9 Wej9 This causes a blurring of the estimated DTFT o The FFT is a very efficient means of calculating the DTFT of a finite duration DT signal at discrete frequencies Zero padding can be used to obtain arbitrary resolution 0 The FFT can also be used to efficiently perform convolution in time details were not discussed in class J McNames Portland State University ECE 223 FFT Ver 105 29

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