Dig Sig Proc&Analys
Dig Sig Proc&Analys EECS 451
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Date Created: 10/29/15
Review Session III EECS 451 Winter 2009 April 15th 2009 OUTLINE 0 Review of Important concepts Concepts 1 Spectral Estimation and resolution techniques Let be L samples of the signal t and Xk be the N pt DFT of the signal formed by zero padding with N 7 L zeros a Increasing the number of samples L improves the resolution of the spectrum Xk b Increasing N ie the number of zeros padded smoothens the spectrum produced but does not increase the resolution of the spectrum c To resolve frequencies whwg the number of samples should be atleast L gt 2 Design of digital IIR lters Let hat be an analog IIR lter with Has as its s transform and Haslsj9 being its frequency response a Impulse invariance method The digital IIR lter is constructed by sampling the con tinuous time impulse response7 ie Mn Ts hat ltnTs where T9 is the sampling interval The frequency response of the digital IIR lter is related to the frequency response of the analog IIR lter by 1 27m HQ T Z Ham s k7oo s where the digital frequency and the analog frequency are related by w 0T5 b Some properties of Impulse invariance method i A stable and causal analog lter hat yields a stable and causal digital lter hn ii A real analog lter hat yields a real digital lter hn c Bilinear Transformation The s domain of the analog HR lter is mapped to the 2 domain as follows Ha5l 2 lizil 5 Ts 1z 1 and the inverse mapping is given by Has 72T5s 1 21175 d Some properties of the Bilinear Transformation BLT When analog lter Has has a rational form7 BLT yields a which has a rational form The entire left half of the s plane is mapped inside the unit circle in the z plane iii The entire right half of the s plane is mapped outside the unit circle in the z plane iv The imaginary axis in the s plane is mapped onto the unit circle in the z plane V A stable Has yields a stable by performing BLT since all poles in the left half s plane is mapped inside the unit circle in the z plane vi Hw depends only on HAD since the imaginary axis in the s plane is mapped onto the unit circle in the z plane vii The relationship between an and Q is highly nonlinear called frequency warping given by 2 w 9 i tan 5 viii Poleszeros at s oo map to poleszeros at z 71 by BLT ix Real poleszeros remain real and complex conjugate pairs remain complex conjugate pairs after applying BLT 3 Some prominent analog HR lters a Butterworth lter They have rational system function Has The frequency response is given by 2 lHamll W where N is the order of the lter iii The lter has N poles equally spaced on a circle of radius 90 in the left half plane iv Pro Maximally at in the pass band v Con Not a sharp cut off b Chebyshev lter Type I i They have a rational system function Has ii They have equi ripples in the passband and monotonically decreasing in stopband iii They are all pole analog lters and the frequency response is given by 11119 W where CN is the N th order Chebyshev polynomial iv Pro The cut off is sharper than in Butterworth lters V Con It has ripples in either the passband or the stopband c Elliptic lter i The frequency response is given by 1 H o 2 l lt gt1 HEZUMQC where UNz is the Jacobian elliptic function don t need not know ii Pro Sharpest transition for a given N iii Con Ripple in both passband and stopband 4 Design of Linear phase FIR lters Let Hdw be the desired frequency response a Window method i Step 1 Find the impulse response ofthe lter hd using the inverse DTFS equation 1 W hdn Hdw57wndw 77r ii Step 2 The obtained impulse response hdn can be in nite in length and hence should be truncated using a window function wn to get the nal impulse response Mn b Frequency Sampling method i Step 1 Sample the given frequency response Hdw to form H k HdMl 7 k w727rm Wherek01 7M71 ii Step 2 Find the impulse response hdn using the M pt inverse DFT equation M71 1 hdn M Z Hke mMn 01 m M7 1 k0 iii Step 3 If the obtained MM is not real7 take the real part of hdn to get the nal impulse response hn 5 Multirate Signal Processing a Decimation by a factor D Let be a signal with spectrum Xw7 where Xw is non zero in the frequency interval 0 lwl 7139 or Fm2 Review Session II EECS 451 Winter 2009 March 25th7 2009 OUTLINE 0 Review of Important concepts Concepts from Chapter 4 1 Discrete Time Fourier series DTFS a For a discrete time periodic signal with period N7 we have 1 N71 k N71 k n n Ck N E xne 727rW E 57sz n0 k0 where ck is also periodic with period N b If is real and periodic7 we have ck cik ie ck cjvik c Parseval s relations yields N 1 1 N71 Average Power N Z lcklz n0 k0 where lcklz is the Power spectrum density 2 Discrete Time Fourier Transform DTFT a For a discrete time aperiodic signal7 we have Xw Z 00577 E 2WXW57wndw n7oo where Xw is periodic with period 27139 b If is real and aperiodic we have Xw X7w c Parseval s relation yields 00 1 7r AveraeEner zn27 szdw g gy Ecol lt l 2 4 l where lXwl2 is the Energy density spectrum 3 Symmetry properties For a discrete signal with Fourier transform Xw These symmetry properties hold for DTFS and DFT as well 4 Properties of DTFT Many follow from the fact Xw XZ Zejm a Linearity a1z1n a2z2n BET a1X1w a2X2w b Time shifting n 7 k BET E WkXQu DTFT lt gt c Time Reversal Min X7w Frequency shifting ejwonzm BET Xw 7 we Convolution 101 201 DEFT X1wX2w d e f Multiplication 3510035201 BET ff X1wX2w 7 d g Differentiation in the frequency domain BET 34 h Con u ation f 71 BEST X 7w J g Concepts from Chapter 5 1 Linear phase response is de ned as w now where no is an integer The linear phase results in a pure delay that is often desired in lter design 2 Pole zero placement and magnitude response Assuming ROC includes the unit circle7 ie system is BIBO stable7 a when 57 is close to a zero7 is small b when 57 is close to a pole7 is large 3 Digital lter design a For a causal system Number of poles 2 Number of zeros For a high pass lter Place poles near high frequencies and zeros near low frequencies For a band pass lter Place pairs of conjugate poles near the unit circle at the frequen cies we wish to pass The peak of the magnitude response of the lter becomes sharper as the poles approach the unit circle g For a notch lter Place pairs of complex conjugate zeros on the unit circle at the frequencies we wish to stop h For an all pass lter Place a pole at 2176 if a zero is present at 20 and vice versa i For a linear phase lter Place a pole at i if a pole is present at 20 and vice versa j For a minimum phase lter All poles and zeros should be within the unit circle 4 lnverse systems a A system T is said to be invertible iff each possible output signal is the response to only one input signal b If T is LTl and invertible then T 1 is also LTl and invertible c If T is LTl causal and stable then T 1 is causal iff of poles of T of zeros of T d If T is LTl causal and stable then T 1 is stable iff all zeros of T are strictly inside the unit circle e A system with a minimum phase has a stable inverse that is also minimum phase Concepts from Chapter 7 1 For a time limited signal of length N the N point DFT is de ned by N71 392 k Xk Z zne7n n0 where k 0 N 7 1 and the inverse DFT formula is given by 1 N4 2 9071 N Xke7Tk 0 w H wheren0N71 a Note that the indices range from 0 to N 7 1 for and Xk b The DFT is ef ciently implemented by the Fast Fourier transform FFT 2 Relationship to other transformations a If zpn is the N periodic superposition of a time limited signal oflength N then the DTFS of zpn is obtained by 1 N71 71m cp zzme 7N n0 thus leading to Xk ch