Signals & Systems Chapter3

Transcription

1 Sigals & Systems Chapter3

2 1.2 Discrete-Time (D-T) Sigals Electroic systems do most of the processig of a sigal usig a computer. A computer ca t directly process a C-T sigal but istead eeds a stream of umbers which is a D-T sigal. Sesor Aalog Electroics ADC DSP Computer Physical C-T Sigal Electrical C-T Sigal Electrical C-T System Electrical C-T Sigal Electrical D-T Sigal Electrical D-T System Electrical D-T Sigal

3 What is a discrete-time (D-T) sigal? A discrete time sigal is a sequece of umbers idexed by itegers Example: x[] =, - 3, -2, -1, 0, 1, 2, 3, Brackets idicate discrete-time sigal. Recall we used paretheses to idicate a C-T sigal. A stem plot emphasizes that the sigal does ot exist i-betwee iteger values x[]

4 D-T systems allow us to process iformatio i much more amazig ways tha C-T systems! Sesor T 2T 3T x(t) C-T system (RLC, etc.) y(t) Aalog to Digital Coverter (ADC) y[] D-T system (computer) z[] samplig is how we typically get D-T sigals I this case the D-T sigal y[] is related to the C-T sigal y(t) by: y[] = y(t) t =T = y(t ) T = time spacig betwee samples (secods) 1/T = samplig rate (F s ) i samples/secod T is samplig iterval F s is samplig rate

5 Major Questio: How fast should we sample a specific sigal? (We ca t aswer that util samplig theory!!) Hit: You may kow that humas ca t hear frequecies above approximately 20kHz. Therefore, audio sigals typically are limited to have o frequecies above 20kHz.

6 Some Commo D-T Sigals Much of what we leared about C-T sigals carries over to D-T sigals The D-T Uit Step is defied i a obviously similar way that the C-T Uit Step was defied. The D-T uit step is just a sampled versio of the C-T uit step The same holds true for the D-T Uit Ramp. However there are a Few Exceptios

7 Uit Pulse: δ[] D-T Impulse or D-T Delta δ[] 1 δ [ ] 1, = 0 = 0, 0 Note: δ[] is ot a sampled versio of δ[t]

8 Siftig Property for D-T Delta Fuctio Note: δ[] works iside summatios the same way δ(t) works iside itegrals

9 D-T Uit step fuctio The DT uit impulse is the first differece of the DT uit-step

10 D-T Siusoids Use upper case omega for frequecy of D-T siusoids x[ ] = A cos( Ω + θ ) What is the uit for Ω? Ω i radias Ω is how may radias jump for each sample Ω is i radias/sample Ω = 2π π π π cos( + ) Ω= = 24 sample i oe cycle 24 cycle per sample

11 Size of Discrete-Time Sigal E x = = [ ] x 2 a ecessary coditio for thr eergy to be fiite is that the sigal amplitude must 0 as other wise the sum will ot coverge P x = 1 lim 2 N + 1 N N N [ ] x 2 E p x x fiite fiite eergy sigal power sigal

12 D-T Covolutio: The Tool for Fidig the Zero-State Respose

13 Covolutio Our Iterest: Fidig the output of LTI systems (D-T & C-T cases)

14 Covolutio for LTI D-T systems We are tryig to fid y[] so the ICs = 0 i.e. o stored eergy x[] LTI D-T system ICs = 0 y[] Before we ca fid the Z-S outptut we eed somethig first: Impulse Respose ( uit-impulse respose ) The impulse respose h[] is what comes out whe δ[] goes i w/ ICs=0 δ[] h[] δ[] LTI h[] D-T system Note: If system is causal, ICs = 0 the h[] = 0 for < 0

15 The impulse respose h[] uiquely describes the system so we ca idetify the system by specifyig its impulse respose h[]. Thus, we ofte show the system usig a block diagram with the system s impulse respose h[] iside the box represetig the system: x[] LTI D-T system with h[] y[] Because impulse respose is oly defied for LTI systems, if you see a box with the symbol h[] iside it you ca assume that the system is a LTI system. x[] h[] y[]

16 How do we kow/get the impulse respose h[]? May possible ways: 1. Give by the desiger of D-T systems 2. Measured experimetally -Put i sequece See what comes out 3. Mathematically aalyze the D-T system -Give differece equatio -Derive h[] I what form will we kow h[]? 1. h[] kow aalytically as a fuctio ] = 1 []

17 Example of aalytically fidig h[] Give a system described by a 1 st order differece equatio: y[] = ay[ 1] + bx[] Recall that h[] is what comes out whe δ [] goes i (with zero ICs). So we ca re-write the above differece equatio as follows: h[] = ah[ 1] + bδ [] (a ad b are arbitrary umbers) Here we solve for h[] recursively ad the examie the results to deduce a closedform solutio (ote: we ca t always use this deductive approach):

18 Q: How do we use h[] to fid the Zero-State Respose? A: Covolutio We ll go through three aalysis steps that will derive The Geeral Aswer that covolutio is what we eed to do to fid the zerostate respose After that we wo t eed to re-do these steps we ll just Do Covolutio Step 1: Usig time-ivariace we kow: δ[-i] h[] (w/ ICs = 0) h[-i] Shifted iput gives shifted output Step 2: Use homogeeity part of liearity: The iput is a fuctio of so we view x[i] as a fixed umber for a give i x[i]δ[-i] h[] (w/ ICs = 0) So we scale the output by the same fixed umber x[i]h[-i]

19 Step 3: Use additivity part of liearity I Step 2 we looked at iputs like this: x[i]δ[-i] h[] ICs = 0 x[i]h[-i] For each i, a differet iput For each i, a differet respose Now we use the additivity part of liearity: Put the Sum of Those Iputs I Get the Iput: x[i]δ [ i] i = Sum of Their Resposes Output: x[i]h[ i] i = Out But what is this?? O the ext slide we show that it is the desired iput sigal x[]!

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23 2.2 Computig D-T covolutio -We kow about the impulse respose h[] -We foud out that h[] iteracts with x[] through covolutio to give the zero-state respose Two cases, depedig o form of x[]: = y [ ] x [ m] h[ m] m = 1. x[] is kow Uaalytically 2. x[] is kow UumericallyU or Ugraphically UAalytical Covolutio (used for by-had aalysis):u Pretty straightforward coceptually: - put fuctios ito covolutio summatio - exploit math properties to evaluate/simplify

24 Example: x[] = a u[] h[] = b u[] Recall this form from 1 st - order differece equatio example a u[] b u[] y[] =? m m y [ ] = x [ m] h[ m] = a u[ m] b u[ m] m= 1, m 0 ow use u [ m ] = 0, m 0 m m 1, m = a b u[ m] ow use u[ m] = m = 0 0, m = a b m m m= a = b b m= 0 m= 0 m

25 [ ] 0 m m a y b b = = If a = b you are addig ( + 1) 1 s ad that gives + 1 If a b, the a stadard math relatioship give 1 Geometric Sum, r 1 1 a N N i i a r r r r + = =

26 Graphical Covolutio Steps Ca do covolutio this way whe sigals are kow umerically or by equatio - Covolutio ivolves the sum of a product of two sigals: x[m]h[ m] - At each output idex, the product chages Repeat for each UStep 1U: Write both as fuctios of m: x[m] &h[m] Step 2: Flip h[m] to get h[-m] Step 3: For each output idex value of iterest, shift by to get h[ -m] Step 4: Form product x[m]h[ m] ad sum its elemets to get the umber y[]

27 Example of Graphical Covolutio x[] h[]... Fid y[]=x[]*h[] for all iteger values of Solutio

28 Step 1: Write both as fuctios of m: x[m] & h[m] h[m] m x[m] m Step 2: Flip x[m] to get x[-m] h[m] x[-m] 2 1

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30 Steps 3&4 for all < 0 Step 3: For < 0, shift by to get x[ - m ]... 3 h[m] x[-1 - m] 2... m Negative gives a left-shift Show here for = Step 4: For < 0, Form the product x[m]h[ m] ad sum its elemets to give y[]... h[m]x[-1 - m] = 0... Sum over m y[] = 0 < m

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32 Steps 3&4 for = 1 Step 3: For = 1, shift by to get x[ - m] h[m] x[1 - m]... m Positive gives a Right-shift m Step 4: For = 1, Form the product x[m]h[ m] ad sum its elemets to give y[]... x[1 - m] h[m] Sum over m y[1] = = 12 m

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34 Complete for all So ow we kow the values of y[] for all values of We just eed to put it all together as a fuctio Here it is easiest to just plot it you could also list it as a table y[]

35 So what we have just doe is foud the zero-state output of a system havig a impulse respose give by this h[] whe the iput is give by this x[]: x[] x[]... 3 h[] h[] y[] = x[] * h[] y[]

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37 Additio Method of Discrete-Time Covolutio Produces the same output as the graphical method Effectively a short cut method Let x[] = 0 for all <N Let h[] = 0 for all <M (sample value N is the first o-zero value of x[] (sample value M is the first o-zero value of h[] y for M [ ] = x[ ] h[ ] = x[ i] h[ i] for M + N i= N 0 < M + N To compute the covolutio, use the followig array

38 Discrete-Time Covolutio Array Sum dow a colum x[n] x[n+1] x[n+2] x[n+3] h[m] h[m+1] h[m+2] h[m+3] x[n]h[m] x[n+1]h[m] x[n+2]h[m] x[n+3]h[m] x[n]h[m+1] x[n+1]h[m+1] x[n_2]h[m+1] x[n]h[m+2] x[n+1]h[m+2] + 1st row values of x[] 2d row values of h[] (1 st row) x (1 st elemet of 2 d row) (1 st row) x (2 d elemet of 2 d row) (1 st row) x (3 rd elemet of 2 d row) y[n+m] y[n+m+1] y[n+m+2] y[n+m+3] The resultig values i the output sequece

39 Discrete-Time Covolutio Example Fid the output of a system if the iput ad impulse respose are give as follows. [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] h x δ δ δ δ δ δ δ = = Solutio The, N = -1 Idex of the first o-zero value of x[] M = -2 Idex of the first o-zero value of h[] Next, write a array

40 Discrete-Time Covolutio Example Coefficiets of x[] Coefficiets of h[] First Row times (-1) First Row times (5) First Row times (3) Summatio of colums y[] = 0 for < N+M = -3 y [ ] = δ [ + 3] + 3δ [ + 3] + 10δ [ + 1] + 17δ [ ] + 29δ [ 1] + 12δ [ 2]

41 D-T Covolutio Examples x[] = ( ) 1 u[] 2 h[i] h[] = u[] u[ 5] x[i] i i Choose to flip ad slide h[] This shows h[-i] for = 0 h[ i] i

42 chagig poit : 0,-4, sth, 0 y [ ] = sth,0 4 sth,4

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44 Now for = 4, = 5, = 4 case x[i] i h[ i] = 5 case h[ i] = h[5 i] i = 4 i = i i

45 Notice that: for = 4, 5, 6, for 4 [ ] ( 1 ) y = = 2 i = 4 i ( ) 1 ( 1 ) ( ) So 0, = 2 1, ( ) 4 ( ) , [ ] ( 1 ) y

46 . Same Impulse Respose: h[] x[] = cos(π u[] 2 x[m] etc Agai y[] = 0 for < 0: h[ m] i m y [ 0] = 1 y [ 1 ] = = 1 y [ 2] = = 0 y [ 3 ] = = 0 y [4] = = 0 y [ ] 5 = = 0 Notice: y[] = 0 = 2, 3, 4, 5,!

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