ECE%20352%20Systems%20II

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ECE 352 Systems II

• Manish K. Gupta, PhD
• Office Caldwell Lab 278
• Email guptam _at_ ece. osu. edu
• Home Page http//www.ece.osu.edu/guptam
• TA Zengshi Chen Email chen.905 _at_ osu. edu
• Office Hours for TA in CL 391  Tu Th
  100-230 pm
• Home Page http//www.ece.osu.edu/chenz/

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Acknowledgements

• Various graphics used here has been taken from
  public resources instead of redrawing it. Thanks
  to those who have created it.
• Thanks to Brian L. Evans and Mr. Dogu Arifler
• Thanks to Randy Moses and Bradley Clymer

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• Slides edited from
• Prof. Brian L. Evans and Mr. Dogu Arifler Dept.
  of Electrical and Computer Engineering The
  University of Texas at Austin course
• EE 313 Linear Systems and Signals Fall
  2003

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Continuous-Time Domain Analysis

• Example Differential systems
• There are n derivatives of y(t) and m derivatives
  of f(t)
• Constants a0, a1, , an-1 and b0, b1, , bm
• Linear constant-coefficient differential equation
• Using short-hand notation,above equation becomes

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Continuous-Time Domain Analysis

• PolynomialsQ(D) and P(D)where an 1
• Recall n derivatives of y(t) and m derivatives of
  f(t)
• We will see that this differential system behaves
  as an (m-n)th-order differentiator of high
  frequency signals if m gt n
• Noise occupies both low and high frequencies
• We will see that a differentiator amplifies high
  frequencies. To avoid amplification of noise in
  high frequencies, we assume that m ? n

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Continuous-Time Domain Analysis

• Linearity for any complex constants c1 and c2,

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Continuous-Time Domain Analysis

• For a linear system,
• The two components are independent of each other
• Each component can be computed independently of
  the other

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Continuous-Time Domain Analysis

• Zero-input response
• Response when f(t)0
• Results from internal system conditions only
• Independent of f(t)
• For most filtering applications (e.g. your stereo
  system), we want no zero-input response.

• Zero-state response
• Response to non-zero f(t) when system is relaxed
• A system in zero state cannot generate any
  response for zero input.
• Zero state corresponds to initial conditions
  being zero.

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Zero-Input Response

• Simplest case
• Solution
• For arbitrary constant C
• Could C be complex?
• How is C determined?

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Zero-Input Response

• General case
• The linear combination of y0(t) and its n
  successive derivatives are zero.
• Assume that y0(t) C e l t

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Zero-Input Response

• Substituting into the differential equation
• y0(t) C el t is a solution provided that Q(l)
  0.
• Factor Q(l) to obtain n solutionsAssuming
  that no two li terms are equal

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Zero-Input Response

• Could li be complex? If complex,
• For conjugate symmetric roots, and conjugate
  symmetric constants,

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Zero-Input Response

• For repeated roots, the solution changes.
• Simplest case of a root l repeated twice
• With r repeated roots

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System Response

• Characteristic equation
• Q(D)y(t) 0
• The polynomial Q(l)
• Characteristic of system
• Independent of the input
• Q(l) roots l1, l2, , ln
• Characteristic roots a.k.a. characteristic
  values, eigenvalues, natural frequencies

• Characteristic modes (or natural modes) are the
  time-domain responses corresponding to the
  characteristic roots
• Determine zero-input response
• Influence zero-state response

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RLC Circuit

• Component values
• L 1 H, R 4 W, C 1/40 F
• Realistic breadboard components?
• Loop equations
• (D2 4 D 40) y0(t) 0
• Characteristic polynomial
• l2 4 l 40 (l 2 - j 6)(l 2 j 6)
• Initial conditions
• y(0) 2 A
• ý(0) 16.78 A/s

y0(t) 4 e-2t cos(6t - p/3) A
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Linear Time-Invariant System

• Any linear time-invariant system (LTI) system,
  continuous-time or discrete-time, can be uniquely
  characterized by its
• Impulse response response of system to an
  impulse
• Frequency response response of system to a
  complex exponential e j 2 p f for all possible
  frequencies f
• Transfer function Laplace transform of impulse
  response
• Given one of the three, we can find other two
  provided that they exist

May or may not exist
May or may not exist
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Impulse response

• Impulse response of a system is response of the
  system to an input that is a unit impulse (i.e.,
  a Dirac delta functional in continuous time)

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Example Frequency Response

• System response to complex exponential e j w for
  all possible frequencies w where w 2 p f
• Passes low frequencies, a.k.a. lowpass filter

H(w)
? H(w)
stopband
stopband
w
w
wp
ws
-ws
-wp
Phase Response
Magnitude Response
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Kronecker Impulse

• Let dk be a discrete-time impulse function,
  a.k.a. the Kronecker delta function
• Impulse response hk response of a
  discrete-time LTI system to a discrete impulse
  function

1
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Transfer Functions
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Zero-State Response

• Q(D) y(t) P(D) f(t)
• All initial conditions are 0 in zero-state
  response
• Laplace transform of differential equation,
  zero-state component

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Transfer Function

• H(s) is called the transfer function because it
  describes how input is transferred to the output
  in a transform domain (s-domain in this case)
• Y(s) H(s) F(s)
• y(t) L-1H(s) F(s) h(t) f(t) ? H(s)
  Lh(t)
• The transfer function is the Laplace transform of
  the impulse response

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Transfer Function

• Stability conditions for an LTIC system
• Asymptotically stable if and only if all the
  poles of H(s) are in left-hand plane (LHP). The
  poles may be repeated or non-repeated.
• Unstable if and only if either one or both of
  these conditions hold (i) at least one pole of
  H(s) is in right-hand plane (RHP) (ii) repeated
  poles of H(s) are on the imaginary axis.
• Marginally stable if and only if there are no
  poles of H(s) in RHP, and some non-repeated poles
  are on the imaginary axis.

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Examples

• Laplace transform
• Assume input f(t) output y(t) are causal
• Ideal delay of T seconds

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Examples

• Ideal integrator with
• y(0-) 0

• Ideal differentiator with f(0-) 0

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Cascaded Systems

• Assume input f(t) and output y(t) are causal
• Integrator first,then differentiator
• Differentiator first,then integrator
• Common transfer functions
• A constant (finite impulse response)
• A polynomial (finite impulse response)
• Ratio of two polynomials (infinite impulse
  response)

f(t)
f(t)
1/s
s
F(s)/s
F(s)
F(s)
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Frequency-Domain Interpretation

• y(t) H(s) e s tfor a particular value of s
• Recall definition offrequency response

h(t)
y(t)
ej 2p f t
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Frequency-Domain Interpretation

• s is generalized frequency s s j 2 p f
• We may convert transfer function into frequency
  response by if and only if region of convergence
  of H(s) includes the imaginary axis
• What about h(t) u(t)?
• We cannot convert this to a frequency response
• However, this system has a frequency response
• What about h(t) d(t)?

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Unilateral Laplace Transform

• Differentiation in time property
• f(t) u(t)
• What is f (0)? f(t) d(t).f (0) is
  undefined.
• By definition of differentiation
• Right-hand limit, h ? ? h 0, f (0) 0
• Left-hand limit, h -? ? h 0, f (0-) does not
  exist

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Block Diagrams
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Derivations

• Cascade
• W(s) H1(s)F(s) Y(s)
  H2(s)W(s)
• Y(s) H1(s)H2(s)F(s) ? Y(s)/F(s) H1(s)H2(s)
• One can switch the order of the cascade of two
  LTI systems if both LTI systems compute to exact
  precision
• Parallel Combination
• Y(s) H1(s)F(s) H2(s)F(s)
• Y(s)/F(s) H1(s) H2(s)

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Derivations

• Feedback System
• Combining these two equations

• What happens if H(s) is a constant K?
• Choice of K controls all poles in the transfer
  function
• This will be a common LTI system in Intro. to
  Automatic Control Class (required for EE majors)

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Stability
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Stability

• Many possible definitions
• Two key issues for practical systems
• System response to zero input
• System response to non-zero but finite amplitude
  (bounded) input

• For zero-input response
• If a system remains in a particular state (or
  condition) indefinitely, then state is an
  equilibrium state of system
• Systems output due to nonzero initial conditions
  should approach 0 as t??
• Systems output generated by initial conditions
  is made up of characteristic modes

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Stability

• Three cases for zero-input response
• A system is stable if and only if all
  characteristic modes ? 0 as t ? ?
• A system is unstable if and only if at least one
  of the characteristic modes grows without bound
  as t ? ?
• A system is marginally stable if and only if the
  zero-input response remains bounded (e.g.
  oscillates between lower and upper bounds) as t ?
  ?

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Characteristic Modes

• Distinct characteristic roots l1, l2, , ln

• Where l s j win Cartesian form
• Units of w are in radians/second

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Characteristic Modes

• Repeated roots
• For r repeated roots of value l.
• For positive k

• Decaying exponential decays faster thantk
  increases for any value of k
• One can see this by using the Taylor Series
  approximation for elt about t 0

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Stability Conditions

• An LTIC system is asymptotically stable if and
  only if all characteristic roots are in LHP. The
  roots may be simple (not repeated) or repeated.
• An LTIC system is unstable if and only if either
  one or both of the following conditions exist
• (i) at least one root is in the right-hand plane
  (RHP)
• (ii) there are repeated roots on the imaginary
  axis.
• An LTIC system is marginally stable if and only
  if there are no roots in the RHP, and there are
  no repeated roots on imaginary axis.

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Response to Bounded Inputs

• Stable system a bounded input (in amplitude)
  should give a bounded response (in amplitude)
• Linear-time-invariant (LTI) system
• Bounded-Input Bounded-Output (BIBO) stable

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Impact of Characteristic Modes

• Zero-input response consists of the systems
  characteristic modes
• Stable system ? characteristic modes decay
  exponentially and eventually vanish
• If input has the form of a characteristic mode,
  then the system will respond strongly
• If input is very different from the
  characteristic modes, then the response will be
  weak

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Impact of Characteristic Modes

• Example First-order system with characteristic
  mode e l t
• Three cases

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System Time Constant

• When an input is applied to a system, a certain
  amount of time elapses before the system fully
  responds to that input
• Time lag or response time is the system time
  constant
• No single mathematical definition for all cases
• Special case RC filter
• Time constant is t RC
• Instant of time at whichh(t) decays to e-1 ?
  0.367 of its maximum value

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System Time Constant

• General case
• Effective duration is th seconds where area under
  h(t)
• C is an arbitrary constant between 0 and 1
• Choose th to satisfy this inequality
• General case appliedto RC time constant

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Step Response

• y(t) h(t) u(t)
• Here, tr is the rise time of the system
• How does the rise time tr relate to the system
  time constant of the impulse response?
• A system generally does not respond to an input
  instantaneously

y(t)
A th
t
tr
tr
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Filtering

• A system cannot effectively respond to periodic
  signals with periods shorter than th
• This is equivalent to a filter that passes
  frequencies from 0 to 1/th Hz and attenuates
  frequencies greater than 1/th Hz (lowpass filter)
• 1/th is called the cutoff frequency
• 1/tr is called the systems bandwidth (tr th)
• Bandwidth is the width of the band of positive
  frequencies that are passed unchanged from
  input to output

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Transmission of Pulses

• Transmission of pulses through a system (e.g.
  communication channel) increases the pulse
  duration (a.k.a. spreading or dispersion)
• If the impulse response of the system has
  duration th and pulse had duration tp seconds,
  then the output will have duration th tp

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System Realization
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Passive Circuit Elements

• Laplace transforms with zero-valued initial
  conditions
• Capacitor

• Inductor
• Resistor

Transfer Function
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First-Order RC Lowpass Filter
R


x(t)
y(t)
C
i(t)
Time domain
R


X(s)
Y(s)
I(s)
Laplace domain
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Passive Circuit Elements

• Laplace transforms with non-zero initial
  conditions
• Capacitor

• Inductor

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Operational Amplifier

• Ideal case model this nonlinear circuit as
  linear and time-invariant
• Input impedance is extremely high (considered
  infinite)
• vx(t) is very small (considered zero)

_


vx(t)
_

y(t)
_
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Operational Amplifier Circuit

• Assuming that Vx(s) 0,
• How to realize gain of 1?
• How to realize gain of 10?

I(s)
H(s)
Zf(s)
_

Z(s)
F(s)

Vx(s)
_

Y(s)

_
_
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Differentiator

• A differentiator amplifies high frequencies, e.g.
  high-frequency components of noise
• H(s) s where s s j 2p f
• Frequency response is H(f) j 2 p f ? H( f )
  2 p f
• Noise has equal amounts of low and high
  frequencies up to a physical limit
• A differentiator may amplify noise to drown out a
  signal of interest
• In analog circuit design, one would use
  integrators instead of differentiators

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Initial and Final Values

• Values of f(t) as t ? 0 and t ? ? may be computed
  from its Laplace transform F(s)
• Initial value theorem
• If f(t) and its derivative df/dt have Laplace
  transforms, then provided
  that the limit on the right-hand side of the
  equation exists.
• Final value theorem
• If both f(t) and df/dt have Laplace transforms,
  then provided that s
  F(s) has no poles in the RHP or on the imaginary
  axis.

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Final and Initial Values Example

• Transfer function
• Poles at s 0, s -1 ? j2
• Zero at s -3/2

Initial Value
Final Value