Lecture 19 Outline: Block-by Block Convolution, … Block Convolution, FFT/IFFT, Digital Spectral...
Transcript of Lecture 19 Outline: Block-by Block Convolution, … Block Convolution, FFT/IFFT, Digital Spectral...
Lecture 19 Outline: Laplace Transforms
l Announcements:l Midterms graded (grades will be released after lecture); details next slidel HW 5 posted, due Friday 5pml TAs will get to Piazza questions today (were grading MT all weekend)l Substitute lecturer Wednesday is Milind Rao, no OHs for me on Wed
l Review of DFTs
l Motivation for Laplace Transforms
l Bilateral Laplace Transform
l S-Plane and Region of Convergencel Examples: Right-sided Exponential, Rectangle,
Left-Sided Exponential
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Rough “curve”
A+AA-B+B
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Review of DFTs
DFT Definitionl Discrete Fourier Series (DFS) Pair for Periodic Signals
l Discrete Fourier Transform (DFT) Pair
l and are one period of and , respectively
l DFT is DTFT sampled at N equally spaced frequencies between 0 and 2p:
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𝑥"[𝑛] 𝑋'[𝑘] ={𝑁𝑎,}= 𝑥"[𝑛]DFS/IDFS IDTFS/DTFS
𝑋 𝑘 = 𝑋 𝑒01 ×S𝑘𝛿 𝑛 − 2p𝑘/𝑁𝑥"[n]=𝑥 𝑛 ∗ ∑ 𝛿 𝑛 − 𝑘𝑁�,
Review of DFTs
DFT/IDFT as Matrix Operation
l DFT
l Inverse DFT
l Computational Complexityl Computation of an N-point DFT or inverse
DFT requires N 2 complex multiplications.
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Review of DFTsKey Properties, Circular Convolution
l Circular Time Shift ® DFT mutiplication with exponential
l Circular Frequency Shift ® multiplication in time by exponential
l Circular convolution in time is multiplication in frequency
l Computing circular convolution:l Linearly convolve and :
l Place N-point sequences on circle in opposite directions, sum up all pairs, rotate outer sequence clockwise each time increment
l Multiplication in time is circular convolution in frequency
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Review of DFTsLinear Convolution from Circular,
Block-by-Block Convolution, FFTs
l Linear Convolution from Circular with Zero Padding
l Block-by-block linear convolutionl Breaks x[n] into shorter blocks; computes y[n] block-by-block. l Overlap-add method: breaks x[n] into non-overlapping segments
l Segments computed by circular method:
l FFT/IFFT has complexity .5Nlog2N (vs. N2 for DFT/IDFT)
x1[n] * x2[n]=
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New Topic: Laplace TransformsMotivation
l Why do we need another transform?l We have the CTFT, DTFT, DFT, FFT
l Most signals don’t have a Fourier Transform( ) ( ) dtetxjX tjww -¥
¥-ò=Requires that Fourier integral converges: always true if ( ) ¥<ò
¥
¥-dttx ||
l Need a more general transform to study signals and systems whose Fourier transform doesn’t existl Laplace transform x(t)«X(s) has similar properties as CTFT
In general there is no Fourier Transform for signals with finite power; some power signals have CTFTs (e.g. sinusoids)
x(t)h(t)
x(t)*h(t)X(s)H(s)
Y(s)=X(s)H(s) Holds even when Fouriertransforms don’t exit
Bilateral Laplace Transforms(Continuous Time)
l Definition:
l Relation with Fourier Transform:
l If we set s=0 then L[x(t)]=F[x(t)]:
l The bilateral Laplace transform exists if
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s-Plane and Region of Convergencel Definition of Region of Convergence (ROC) for
Laplace transform L[x(t)]=X(s)=X(s+jw): l Defined as all values of s=s+jw such that L[x(t)] existsl Convergence depends only on s, not jw, as it requires:
l s-Plane: Plot of s+jw with s on real (x) axis, jw on imaginary (y) axis. l Show the ROC (shaded region) for L[x(t)] on this plane
s-plane
Values of
wj
s
X(s)definedin ROC
X(jw)
Real Axis
Imaginary Axis
( ) ¥<ò¥
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Smallest s:X(s) exists
ROC consists ofstrips along jw axis
Example: Right-Sided Real Exponential
l This converges if
l Under this condition
l Special cases:
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More Laplace Transform Examples
l Rect function:l Familiar friend; has a Fourier transforml Laplace Transform: l ROC: Finite everywhere except possibly s=0:
● So ROC is entire s-plane: ROC={all s}: true for any finite duration absolutely integrable function
l Fourier transform (jwÎROC):
l Left-Sided Real Exponential:l Laplace: , converges if Re(s)<-a
l Does not have a Fourier transform if –a<0, else
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Main Pointsl Laplace transform allows us to analyze signals and systems
for which their Fourier transforms do not converge
l Laplace transform X(s) has similar properties as the Fourier Transform X(jw) and equals X(jw) when s=jw
l Laplace transform is defined over a range of s=s+jw values for which the transform converges
l The set of s=s+jw values for which the Laplace transform exists is called its Region of Convergence (RoC)l RoC plotted on the s-plane l Real axis for s, imaginary axis for jw
l Laplace transform includes the ROC; different functions can have same Laplace transform with different ROCs