lecture21

Document Sample
lecture21 Powered By Docstoc
					EECS 20 N—March 9, 2001
     Lecture 21:
  Time Responses
   Laurent El Ghaoui
           1
                  announcements: recall
• reading assignment: Chapter 6 of Lee and Varaiya
• no lab next week (3/12–3/16)
• still, go to lab sessions for problem set discussions
• this week’s lab due in two weeks only
                                  2
                              outline
 • time-invariance
 • linearity
 • impulse response
 • response to sinusoids
look at discrete-time systems only
                                     3
                                              delaying signals
given a discrete-time signal, we can associate the same signal, delayed
by a time T :
                                                          a discrete−time sequence
                      5
                      4
           response




                      3
                      2
                      1
                      0
                          0   1           2           3          4         5         6    7   8   9
                                                               time (seconds)
                                                      a discrete−time sequence, delayed
                      5
                                  original sequence
                      4           delayed sequence
           response
                      3
                      2
                      1
                      0
                          0   1           2           3          4         5         6    7   8   9
                                                               time (seconds)
                                                                   4
                         delay operator
formally, if x is a function of time, then we define a new signal DN (x)
by
                 ∀ n ∈ Ints, DN (x)(n) = x(n − N )
here, N ∈ Ints is the delay
DN is called the delay operator
                                   5
                         time-invariance
if y is the zero-state response to input x of the system
            s(n + 1)   = As(n) + bx(n), n = 0, 1, 2, . . .
                y(n)   = cT s(n) + dx(n)
then, for any N ∈ Ints, the response to DN x is DN y
i.e., if we denote the zero-state response to input x by y = S(x), then
                       S(DN (x)) = DN (S(x))
this property is called time-invariance
                                   6
         time-invariance and output response
for the system (denoted S)
            s(n + 1)   = As(n) + bx(n), n = 0, 1, 2, . . .
                y(n)   = cT s(n) + dx(n)
recall the zero-state output response formula:
                              n−1
         (S(x))(n) = y(n) =         cT An−1−m bx(m) + dx(n)
                              m=0
                                    7
if we delay x by N , the output is delayed likewise:
                                 n−1−N
 DN (Sx)(n) = y(n − N )      =            cT An−1−m bx(m) + dx(n − N )
                                   m=0
                                  n−1
                             =          cT An−1−m bx(m − N ) + dx(n − N )
                                 m=0
                             = S(DN (x))(n)
we’ll see a more transparent proof next
                                    8
       example of a non-time-invariant system
                                              
                                               1 if n = 0,
                                              
                                              
          y(n) = d(n)x(n), where d(n) =           2 if n = 1
                                              
                                              
                                                  0 if n > 1
                                              
is not time-invariant, example:
          n    d x     D1 (x) S(x) D1 (S(x)) S(D1 (x))
           0   1   1     0        1       0             0
           1   2   2     1        4       1             2
           2   0   3     2        0       4             0
           3   0   4     3        0       0             0
           4   0   5     4        0       0             0
                                      9
                         a general result
the system with time-varying state-space matrices
         s(n + 1) =      A(n)s(n) + b(n)x(n), n = 0, 1, 2, . . .
             y(n)   =    c(n)T s(n) + d(n)x(n)
is in general not time-invariant
when A(n), b(n), c(n) and d(n) are independent of n, then the system
is time-invariant
                                   10
                        impulse response
impulse response is the zero-state response with impulse input:
                               
                                0 if n < 0
                               
                               
                        δ(n) =       1 if n = 0,
                                 
                                 
                                     0 if n > 0
                                 
we get
                                             
                                              0               if n < 0,
                                             
           n−1                               
  y(n) =         cT An−1−m bδ(m) + dδ(n) =         d           if n = 0,
                                             
           m=0                               
                                                   cT An−1 b   if n > 0
                                             
                                     11
                           convolution sum
the zero-state output response
                           n−1
                  y(n) =         cAn−1−m bx(m) + dx(n)
                           m=0
can be written as a convolution sum
                  n−1                      m=+∞
         y(n) =         h(n − m)x(m) =            h(n − m)x(n)
                  m=0                      m=−∞
where h is the sequence
                             
                              0           if n < 0,
                             
                             
                    h(n) =        d        if n = 0,
                             
                             
                                  cAn−1 b if n > 0
                             
h is the impulse response!
                                      12
                                result
for a linear, time-invariant system, the zero-state response to an
arbitrary input is the convolution of the input with the impulse
response
                  n−1                    m=+∞
         y(n) =         h(n − m)x(m) =          h(n − m)x(n)
                  m=0                    m=−∞
we denote this with the convolution operator:
                               y=h x
what does this mean?
                                   13
               LTI systems and convolution
assume we are given a discrete-time system y = S(x) that satisfies the
following properties:
 • it is linear
 • it is time-invariant
we say the system is LTI
given a signal x, what is the form of the output?
claim: the output signal can be written as
                              y = h x,
where h is the impulse response
                                  14
                          to get the idea
start with x(0), what is y(0)? by linearity, there exist a number h(0)
(independent of x(0)!) such that
                            y(0) = h(0)x(0)
now, what is the response at time n = 1? by linearity, there exist
numbers h(1) and h(0, 1) (independent of x(0), x(1)!) such that
                      y(1) = h(0, 1)x(1) + h(1)x(0)
by time-invariance, we must have h(0, 1) = h(0), so that
                       y(1) = h(0)x(1) + h(1)x(0)
similarly, at time n = 2 y(2) takes the form
                  y(2) = h(0)x(2) + h(1)x(1) + h(2)x(0)
and so on . . .
                                   15
                 signals as sums of impulses
let’s take an arbitrary signal x, can decompose it in a series of
impulses:
                               +∞
                     x(n) =          x(m)δ(n − m)
                              m=−∞
this is a weighted sum of delayed impulses
                                    16
              response to a delayed impulse
take the delayed impulse as input:
                                          
                                           0 if n < m,
                                          
                                          
             (Dm δ)(n) = δ(n − m) =           1 if n = m,
                                          
                                          
                                              0 if n > m
                                          
what is the output response?
                                     17
                 delayed impulse response
by time-invariance, the answer is: simply delay the impulse
response!
thus, the response to the delayed impulse (with delay m) is
                        (Dm h)(n) = h(n − m)
by linearity, response to arbitrary input x is weighted sum of delayed
impulse responses
                              +∞
                 y(n)   =     m=−∞     x(m)(Dm h)(n)
                              +∞
                        =     m=−∞     x(m)h(n − m)
the final result: y = h x, uses both time-invariance and linearity
                                  18
                       complex numbers
a complex number is one of the form
                                z = a + i · b,
                                √
where a, b are reals, and i =     −1 (so that i2 = −1)
a is called the real part of z, b the imaginary part
the magnitude of z is defined by
                            |z| =      a2 + b 2
it is the Euclidean distance of the 2-D vector
                                      
                                      a
                                          
                                      b
                                      19
        polar representation of complex numbers
            √
let |z| =       a2 + b 2 ,
                                                             a       b
                        z = a + i · b = |z| ·                   +i·         ,
                                                            |z|     |z|
since
                                    2               2
                              a               b             a2 + b 2
                                        +               =            = 1,
                             |z|             |z|              |z|2
any such number can be represented as
                               z = |z| · (cos θ + i · sin θ),
where θ is such that
                                              a             b
                                   cos θ =       , sin θ =
                                             |z|           |z|
                                                   20
                  complex exponentials
we can represent a complex number as
                               z = ρeiθ
where
 • ρ = |z| is the magnitude
 • the complex exponential is defined by
                         eiθ = cos θ + i · sin θ
where
                                     +∞
                                           un
                              eu =
                                     n=0
                                           n!
                                     21
                        sinusoidal inputs
a sinusoidal input is one of the form
                         x(n) = sin(nω + φ),
where φ ∈ [0 2π[ is the phase, and ω ∈ Reals is the frequency.
can also use a cos in the above (change φ)
                                   22
                     complex exponentials
it is more convenient to work with complex exponentials, for example
                               x(n) = einω
or, with a non-zero phase:
           x(n) = ei(nω+φ) = cos(nω + φ) + i · sin(nω + φ)
these signals model a lot of real-life signals (more later)
                                    23
          response to a complex exponential
consider the output response:
                  n−1                    m=+∞
         y(n) =         h(n − m)x(m) =          h(n − m)x(n)
                  m=0                    m=−∞
what is the response to a complex exponential
                             x(n) = einω ?
                                  24
           response to complex exponential
                         m=+∞           imω
          y(n)   =       m=−∞ h(n − m)e
                          m=+∞           i(m−n)ω
                 =        m=−∞ h(n − m)e            einω
                 = H(w)einω ,
where H(w) is the complex number
                            m=+∞
             H(w) =         m=−∞     h(n − m)ei(m−n)ω
                            m=+∞
                     =      m=−∞     h(m)eimω
H is called the frequency response
                                25

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:0
posted:5/4/2012
language:
pages:25