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

Midterm Grade Distribution2016 GradesMedian: 65.25Mean: 67.77Std. Dev: 24.46Low: 24High: 98

Median 82.83%

Maximum 97.33%

Mean 77.32%

StandardDev 16.66%

Rough “curve”

A+AA-B+B

l Regrade requests must be submitted in writing to mel Describe grading error that occurred

l Course grade from all coursework: A-/B+ class averagel Before extra credit; extra credit points added after curve (separate weighting)

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:

[ ] [ ] knN

N

k

WkXN

nx --

=å=1

0

~1~ [ ] [ ] knN

N

n

WnxkX å-

=

=1

0

~~

[ ] [ ] knN

N

k

WkXN

nx --

=å=1

0

1 [ ] [ ] knN

N

n

WnxkX å-

=

=1

0

[ ]kX [ ]kX~[ ]nx~[ ]nx

[ ] ( ) 10,2 -££==W

W NkeXkXN

k

jp

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.

[ ] [ ] knN

N

k

WkXN

nx --

=å=1

0

1

[ ] [ ] knN

N

n

WnxkX å-

=

=1

0

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

( )( )[ ] [ ] [ ]kXekXWmnxkm

Njkm

NN

p2-

=«-

[ ] [ ] ( )( )[ ]Nln

Njln

N lkXnxenxW -«=-p2

[ ] ( )( )[ ] [ ] [ ]kXkXmnxmxDFTN

mN 21

1

021 «-å

-

=

[ ] [ ] [ ] [ ]kXkXnxnx N 2121 !«×

[ ] [ ] [ ] [ ]ïî

ïíì

-££-= å-

=

otherwise0

10~~1

021

21Nnmnxmxnxnx

N

mN![ ]nx1~ [ ]nx2~

9:

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]=

n0 3

1

1 2 64 5

2

3

4

][2 nx

n0 3

1

1 2 4 5

n0 3

1

1 2

[ ]nx1

87

1

2

3

4 4L=4

P=6

M=L+P-1=9n

0 3

1

1 2

[ ]nx zp,1

64 5 7 8

][,2 nx zp

0 3

1

1 2 4 5n

6 7 8

[ ] [ ]nxnx zpzp ,2,1 9

[ ] [ ]å¥

=

=0r

r nxnx [ ] [ ] [ ] [ ] [ ]å¥

=

==0

**r

r nhnxnhnxny[ ] [ ] ( )îíì -+££

=otherwise0

11 LrnrLnxnxr

[ ] [ ] [ ]{ } [ ]{ }{ }nhFFTnxFFTFFTnhnx rpNr zpzp,1

zzp, ×= -!

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

( )[ ] ( ) ( ) dtetxsXtxL st-¥

¥-ò== ws jsjss +=+= )Im()Re(

( )[ ] ( ) ( )[ ] ( )[ ]ttjt etxFdteetxjXtxL swsws -¥

¥-

-- ==+= ò

( ) ¥<ò¥

¥-

- dtetx ts

( ) ( ) dtetxjX tjww -¥

¥-ò=

( ) ( ) ( )[ ]txFjXsX js ===

ww

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¥

¥-

- dtetx ts

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:

( ) ( )tuetx at-=

, a real

( ) ( ) ( ) ( ) ( )[ ]tueFdteedtesX tatjtatas +--+-¥

+-¥

=== òò sws

00

( )tx

( ) asasa ->Þ>+=+ )Re(0Res

, i.e., if

( ) as ->Re( ) ,1as

sX+

=

ROC

( )aj

jX+

=w

w 1 ( ) 0)Re(,1 >« ss

tu

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

( )tx2

t0 t

1

t-

( )îíì

>£

=÷øö

çèæP=

tt

t ttttx

01

22

( )seedtesXss

stttt

t

--

-

-== ò2

( ) ( )t

tt

2limlim020

=-

=-

®® see

sXdsd

ssdsd

ss

( ) ÷øö

çèæ==

-=

-

pwtt

wtwtt

ww

wtwt

sinc2sin22 jeejX

jj

( ) ( )tuetx at --= -4

( )tx4t

0

1-

( ) ( ) dtesX tas+-

¥-ò-=0

4

( ) asas

sX -<+

= )Re(,14

Same as for right-sided case but with a different ROC (Re(s)>-a)

( )aj

jX+

=w

w 14

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