Signal and systems Linear Systems

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					Signal and systems
 Linear Systems
      Luigi Palopoli


   palopoli@dit.unitn.it




                           Signal and systems – p. 1/5
Wrap-Up




          Signal and systems – p. 2/5
    Fourier Series


•   We have see that is a signal is periodic, it can be conveniently expressed as a
    Fourier Series:
                                            X∞
                                   s(t) =          sn ejnωt
                                            n=−∞

    or, if s(t) is real,
                                    ∞
                                    X
                           s(t) =         2 |sn | cos(nωt + Φ(sn ))
                                    n=1

•   This is essentially a representation of the signal in an appropriate base made of
    exponential orthonormal functions
•   The convenience of this representation can easily be seen when we process a
    signal through a linear systems




                                                                                Signal and systems – p. 3/5
    Fourier Series


•   Because exponential functions are eigenfunctions, the response to a sinusoidal
    signal is:

                     st , H(s) = ∞ h(t)e−st dt →
      ˘ st ¯                    R
     S e     = H(s)e             −∞
     S {cos(ωt + φ0 )}                                    = |H(jω)| cos(ωt + φ0 + Φ(H(jω))
     S {1} = H(0)

    Where the last expression is true iff H(s) converges for s = jω.
•   Using the superimposition principle, for a generic periodic signal s(t), we get:

                              P∞
     S {s(t)}   = H(0)s0 +       n=1   2 |sn | |H(jnω)| cos(ωnt + Φ(sn ) + Φ(H(jnω)))

•   The expression above reveals the power of our formalism. By simply knowing
    H(jω), we are able to compute the response to any periodic signal without solving
    differential equations. We would like to apply this to non-periodic signals as well.



                                                                                  Signal and systems – p. 4/5
Linear systems: Frequency Domain




                                   Signal and systems – p. 5/5
Nonperiodic signals




                      Signal and systems – p. 6/5
    Fourier series of a non periodic signal


•   A non-periodic signal can be obtained as the limit of a periodic signal:
    s(t) = limT0 →∞ repT0 s(t) where repT0 s(t) = ∞
                                                   P
                                                       h=−∞ s(t + hT0 )
•   The coefficients of the periodic signal repT0 s(t) are given by:

                            1
                                 Z    T0 /2
                       sn =                   [repT0 s(t)]e−jnω0 t dt
                            T0       −T0 /2
                                      T0 /2     ∞
                            1
                                 Z              X
                          =                          s(t + hT0 )e−jnω0 t dt
                            T0       −T0 /2 h=−∞




                                                                               Signal and systems – p. 7/5
    Fourier series of a non periodic signal


•   Observing that e−j2πnh = 1, we can write

                                T0 /2    ∞
                     1
                            Z            X
                sn =                             s(t + hT0 )e−jnω0 t e−jω0 nhT0 dt
                     T0         −T0 /2 h=−∞

                                T0 /2    ∞
                     1
                            Z            X
                   =                             s(t + hT0 )e−jnω0 (t+hT0 ) dt
                     T0         −T0 /2 h=−∞

                                ∞        T0 /2
                     1          X Z
                   =                             s(t + hT0 )e−jnω0 (t+hT0 ) dt
                     T0     h=−∞        −T0 /2

                                ∞        T0 (h+1/2)
                     1          X Z                                    ′
                   =                                     s(t′ )e−jnω0 t dt′
                     T0     h=−∞        T0 (h−1/2)
                            Z ∞
                       1                             ′
                   =                 s(t′ )e−jnω0 t dt′
                       T0       −∞




                                                                                     Signal and systems – p. 8/5
    Fourier series of a nonperiodic signal


•   Define S(ω) =
                   R∞
                    −∞   s(t)e−jωt dt
•   We get: sn =    1
                      S(nω0 )
                   T0
•   Notice that two coefficients are spaced out by ∆ω = ω0 . Drawing the spectrum in
    the (ω − sn ) plan, the rows sn become closer and closer as T0 → ∞: the
    spectrum tends to be continuous. Therefore, sn = ∆ω S(n∆ω)
                                                       2π
•   We can write: repT0 s(t) = n=−∞ ∆ω S(n∆ω)ejnω0 t
                                P∞
                                         2π
•   As T → ∞ (∆ω → 0), we get the definition of the integral:

                                              1
                                                  Z   ∞
                     s(t) = lim repT0 s(t) =              S(ω)ejω dω
                           T0 →∞             2π    −∞




                                                                             Signal and systems – p. 9/5
    Fourier Transform


•   Starting from the definition of Fourier series (that applies to periodic signal) we
    have [informally] derived the fourier transform,
                           Z   ∞                 1
                                                           Z   ∞
                  S(ω) =    s(t)e−jωt dt s(t) =                     S(ω)ejωt dω
                         −∞                     2π             −∞

•   It can be seen that if the signal is absolutely integrable and bounded, (plus other
                                              R∞
    technical mathematical conditions) i.e., −∞ |s(t)| dt = L < ∞, then
     ◦   the fourier transform exists for almost all ω
     ◦   also the Fourier is squared -integrable: −∞ |S(ω)|2 dω < ∞
                                                   R∞

     ◦   its inverse transform converges to s(t) (in squared error sense)
•   Notice
     ◦   If a signal has finite energy, then it is absolutely integrable
     ◦   The above conditions are only sufficient (we will find the Fourier transform of
         several signals that do not meet these conditions).




                                                                                  Signal and systems – p. 10/5
    Example


•   Compute the FT of a rectangular impulse:
                                         8
                                         <A if τ /2 ≤ t ≤ τ /2
                        s(t) = AGτ (t) =
                                         :0 Otherwise

•   By applying the definition:
                                   Z   ∞
                         S(ω) =              s(t)e−jωt dt
                                       −∞
                                       Z    τ /2
                                 =A                e−jωt dt
                                           −τ /2
                                     1     −jωt ˛τ /2
                                                ˛
                                 =A      e       −τ /2
                                    −jω
                                     1                τ
                                 =A     (−2j sin(ω )
                                    −jω               2
                                      1        τ           τ
                                 = Aτ τ sin(ω ) = Aτ Sinc(ω )
                                     ω2        2           2

                                                                 Signal and systems – p. 11/5
    Example


•   Compute the FT of a rectangular impulse:

                                      s(t) = δ(t)

•   By applying the definition:
                                          Z   ∞
                                 S(ω) =           δ(t)e−jωt dt
                                          −∞
                                     =1




                                                                 Signal and systems – p. 12/5
    Properties of the Fourier transform


•   Linearity:
                                                    F(u1 (t)) = U1 (ω),
                                                     F(u2 (t)) = U2 (ω)
                                                                               ↔
                     F(a1 u1 (t) + a2 u2 (t)) = a1 U1 (ω) + a2 U2 (ω)

    This is a fairly obvious consequence of the linearity of the integral operator.
•   Time shifting:

                                   F(u1 (t)) = U1 (ω)
                                                          ↔
                         Z   ∞
     F(u1 (t − t0 )) =           u1 (t − t0 )e−jωt dt =
                         −∞
                                                              Z   ∞
                                                          =           u1 (t − t0 )e−jω(t−t0 ) e−jωt0 dt =
                                                              −∞

                                                          = e−jωt0 U1 (ω)


                                                                                            Signal and systems – p. 13/5
    Properties of the Fourier transform


•   Shifiting in the frequency domain

                  F(u1 (t)) = U1 (ω),
                                            ↔
                                                 Z   ∞
                             jω0 t
                       F(e           u1 (t)) =           ejω0 t u1 (t)e−jωt dt =
                                                 −∞
                                                 Z∞
                                            =            u1 (t)e−j(ω−ω0 )t dt =
                                                 −∞
                                            = U1 (ω − ω0 )




                                                                                   Signal and systems – p. 14/5
    Properties of the Fourier transform


•   Time scaling. Assume a positive

                   F(u1 (t)) = U1 (ω)
                                        ↔
                                            Z   ∞
                           F(u1 (at)) =             u1 (at)e−jωt dt =
                                            −∞
                                            Z∞1             ω
                                        =       u1 (t′ )e−j a t dt′ =
                                          −∞ a
                                         1    ω
                                        = U1 ( )
                                         a    a

•   For general a we can easily see

                                                F(u1 (t)) = U1 (ω)
                                                         ↔
                                             1      ω
                            F(u1 (at)) =        U1 ( )
                                            |a|     a

                                                                        Signal and systems – p. 15/5
    Properties of the Fourier transform


•   Differentiation in the time domain:

                   F(u1 (t)) = U1 (ω),
                                          ↔
                                              Z   ∞
                                 u1 (t) =               S(ω)ejωt dt   =
                                              −inf ty
                                          ↔
                                              Z ∞
                               d            d
                                  u1 (t) =            S(ω)ejωt dt =
                               dt          dt −inf ty
                                           Z ∞
                                                        d
                                         =          S(ω) ejωt dt      =
                                            −inf ty     dt
                                          = jωU1 (ω)

•   More generally
                                     F(u1 (t)) = U1 (ω),
                                                             ↔
                                dn u1
                              F( n ) = (jω)n U1 (ω)
                                 dt                                       Signal and systems – p. 16/5
    Properties of the Fourier transform


•   Duality
                                    F(f (t)) = F (ω)
                                                          ↔
                               F(F (t)) = 2πf (−ω)
•   Proof:
                       F(f (t)) = F (ω)
                                          ↔
                                             1
                                                 Z   ∞
                                    f (t) =              F (ω)ejωt dt
                                            2π   ∞

    If we simply swap the two variables t and ω, we find:
                                          Z ∞
                                        1
                               f (ω) =         F (t)ejωt dt
                                       2π ∞
                                        1
                                     =    F(F (t))|−ω
                                       2π



                                                                        Signal and systems – p. 17/5
    An example


•   Consider the signal s(t) = 1. This is not absolutely integrable (it does not
    converge to 0).
•   However, if we apply duality we get:

                                     F(δ(t)) = 1
                                           ↔
                                        F(1) = 2πδ(ω)

•   This is extremely important because it shows that if we consider generalised
    functions (δ(.)) we can find the Fourier Transform of function that are not
    absolutely integrable




                                                                                   Signal and systems – p. 18/5
    Another (important) example


•   Let us consider the signal (non absolutely integrable)
                                         8
                                         <1     t≥0
                                  1(t) =
                                         :0     Otherwise

•   We can see that:
                                           1
                                  1(t) =     (1 + sgn(t))
                                           2
                                           8
                                           <1     t≥0
                                sgn(t) =
                                           :−1 Otherwise

•   Function sgn(t) is not an absolutely integrable function, but we can manage it with
    some trick.....




                                                                               Signal and systems – p. 19/5
    Another (important) example


•   We can write:
                                     sgn(t) = lim Sα (t)
                                               α→0
                                               8
                                               <e−αt t ≥ 0
                                      Sα (t) =
                                               :−eαt t < 0

•   Sα (t) is absolutely integrable, hence we can deal with it:
                             Z   ∞
               F(Sα (t)) =           Sα (t)e−jωt dt
                                 −∞
                                  Z 0                        Z   ∞
                                            αt −jωt
                          =−            e    e        dt +           e−αt e−jωt dt
                                   −∞                        0
                               1                    1
                          =−        e(α−jω)t |0 −
                                              −∞         e−(α+jω)t |∞
                                                                    0
                             α − jω               α + jω
                               1         1
                          =−        +
                             α − jω    α + jω




                                                                                     Signal and systems – p. 20/5
    Another (important) example


•   Hence:
                                             1        1
                       F(sgn(t)) = lim −          +
                                    α→0    α − jω   α + jω
                                     2
                                =
                                    jω
•   We can conclude:                   „            «
                                    1    2
                          F(1(t)) =        + 2πδ(ω)
                                    2 jω
                                     1
                                  =     + πδ(ω)
                                    jω




                                                             Signal and systems – p. 21/5
    Properties of the Fourier transform


•   Convolution
                                         F(u1 (t)) = U1 (ω),
                                         F(u2 (t)) = U2 (ω),
                                                                     ↔
                         F(u1 (t) ∗ u2 (t)) = U1 (ω)U2 (ω)
•   Proof                            Z   ∞   Z   ∞
              F(u1 (t) ∗ u2 (t)) =                    u1 (τ )u2 (t − τ )dτ e−jωt dt
                                     −∞          −∞
                                     Z∞      Z    ∞
                                =                     u1 (τ )u2 (t − τ )dτ e−jωt dt
                                     −∞          −∞
                                     Z∞                »Z   ∞                    –
                                =            u1 (τ )            u2 (t − τ )e−jωt dt dτ
                                     −∞                  −∞
                                     Z∞
                                =            u1 (τ )U1 (ω)e−jωτ dτ
                                     −∞
                                = U1 (ω)U2 (ω)



                                                                                         Signal and systems – p. 22/5
    Properties of the Fourier transform


•   Product
                                         F(u1 (t)) = U1 (ω),
                                         F(u2 (t)) = U2 (ω),
                                                                 ↔
                                             1
                        F(u1 (t)u2 (t)) =      U1 (ω) ∗ U2 (ω)
                                            2π
•   Proof: It descends from duality + convolution




                                                                     Signal and systems – p. 23/5
    Properties of the Fourier transform


•   Integration

                                                                F(u1 (t)) = U1 (ω),
                                                                                      ↔
                       Z   t                                1
                  F(                       u1 (τ )dτ ) =      U1 (ω) + πU1 (0)δ(ω)
                       τ =−∞                               jω

•   Proof                      Z       t
                                                u1 (τ )dτ = u1 (t) ∗ 1(t)
                                   τ =−∞
                                                           ↔
                               Z   t
                       F(                      u1 (τ )dτ ) = F (u1 (t) ∗ 1(t))
                               τ =−∞
                                                               1
                                                           =(     + πδ(ω))U1 (ω)
                                                              jω
                                                             U1 (ω)
                                                           =        + πδ(ω))U1 (0)
                                                               jω

                                                                                          Signal and systems – p. 24/5
    Example


•   Consider the f (t) = B cos ω0 t

                                          B `` jω0 t
                                                     + e−jω0 t
                                                               ´´
                             F(f (t)) =     F e
                                          2

•   We have seen that F(δ(t)) = 1; applying the duality property:

                                      F (1) = 2πδ (−ω)

•   Now, we apply frequency shifting property:
                                   ` jω t ´
                                  F e 0 = 2πδ (ω − ω0 )

•   Therefore, we get:

                         F (B cos ω0 t) = Bπ (δ(ω − ω0 ) + δ(ω + ω0 ))




                                                                         Signal and systems – p. 25/5
 Example


• We have seen that

                                       τ
               F (AGτ (t)) = Aτ Sinc ω
                                       2
• Let us find: F AGτ /2

• We can apply time scaling rule:

                            Aτ      ωτ
               F AGτ /2   =    Sinc
                             2      22




                                           Signal and systems – p. 26/5
    Spectrum


•   Also for non periodic signals g(t) we can associate a frequency domain spectrum
    G(ω)
•   It is typically depicted by giving its norm |G(ω)| and its phase ∠(G(ω))
•   For real signals the following hold:
     ◦   |G(ω)| = |G(−ω)|, ∠(G(ω)) = −∠(G(−ω))
     ◦   If the signal is even, then G(ω) is real
     ◦   If the signal is odd, then G(ω) is imaginary
•   An interesting example is the following ideal filter.
•   Its spectrum is given by: |H(ω)| = 1GateB (ω) and ∠(H(ω) = −ωt0




                                                                               Signal and systems – p. 27/5
    Why is an ideal filter ideal?


•   Consider, for simplicity, t0 = 0
                                            “ ”
•   Applying duality, we get: h(t) = 2πBSinc t B
                                               2
•   As we can see this filter is not a causal system....
•   Therefore the system is not pysically implementable




                                                          Signal and systems – p. 28/5
    Fourier Transform of periodic signals


•   We have seen that the fourier transform of a cosine is the sum of two δ
•   The same applies also to other periodic signals
•   For a periodic signal, we have seen that it is possible to write them in terms of
    Fouries series:
                                          X∞
                                  s(t) =          sn ejnω0 t
                                           n=−∞

•   we have seen that F ejnω0 t = 2πδ(ω − nω0 )
                       `       ´

•   Therefore we get:
                                           ∞
                                           X
                              F (s(t)) =          sn δ(ω − nω0 )
                                           n=−∞




                                                                                  Signal and systems – p. 29/5
    Fourier Transform of periodic signals


•   What if we construct a periodic repeating a non periodic signal?

                                           ∞
                                           X
                                 s(t) =            sc (t + iT )
                                          i=−∞


•   The signal can be expressed as

                                                  ∞
                                                  X
                              s(t) = sc (t) ∗          δ(t + iT )
                                                i=−∞


•   We can compute the Fourier series of signal sr (t) =
                                                              P∞
                                                                  i=−∞   δ(t + iT ):

                                                      ∞
                                                      X
                                          sr (t) =          sn ejnω0 t
                                                     n=−∞

                               1
                                    Z   T0 /2                       1
                          sn =                  δ(t)e−jnω0 t dt =
                               T0    −T0 /2                         T0


                                                                                       Signal and systems – p. 30/5
    Fourier Transform of periodic signals


•   Therefore:
                                                 ∞
                                            1    X
                            s(t) = sc (t) ∗             ejnω0 t
                                            T0   n=−∞

    Which, corresponds, in the frequency domain, to:

                                             ∞
                                        1    X
                          S(ω) = Sc (ω)             δ(ω − nω0 )
                                        T0   n=−∞




                                                                  Signal and systems – p. 31/5
  Fourier Transform of periodic signals


Example:




                                          Signal and systems – p. 32/5
Mathematical Complements




                           Signal and systems – p. 33/5
    Discussion


•   We have seen that for signals compying with the following conditions:
     ◦   s(t) limited
     ◦   Finite number of minima and maxima and of singularities
     ◦   Absolutely integrable
    The Fourier transform exists and the inverxe transform converges to s(t) .
•   For these signals we have the Parseval equality:
                           Z    ∞               1
                                                    Z   ∞
                                  |s(t)|2 dt =              |S(ω)|2 dω
                               −∞              2π   −∞

•   We can compute the integral in the easier domain (for instance for a low pass filter
    it is much easier in the frequency domain)




                                                                                 Signal and systems – p. 34/5
    Discussion


•   If we consider signals with finite power (e.g. periodic signals) we can still compute
    the fourier transform using generalized functions (δ)
•   We have derived the Fourier transform from the Fourier series, but we also have
    seen that the Fourier series is a special case of the Fourier transform.




                                                                                Signal and systems – p. 35/5
Two interesting applications




                               Signal and systems – p. 36/5
 Amplitude Modulation


• We want to use the same medium (e.g., the air), to transmit
  multiple signals (e.g., different channels)
• Assume that each transmission can have a limited
  bandwidth
• One of the oldest ways for doing this was to modulate the
  amplitude of the signal by multiplying it by a sinusoidal
  oscillation:
                      xAM (t) = x(t) cos(5t)




                                                              Signal and systems – p. 37/5
     Amplitude Modulation


                                 x(t)

                       1.0


                       0.7

                       0.5

                       0.3

                       0.1


K             K
10            5               0
                              t
                                         5   10

                                                  *
                            cos(5t)

                       1.0


                       0.5




K             K              0

                       K
10            5                          5   10




                                                  =
                       0.5


                       K
                       1.0




                           x(t)cos(5t)



                       0.8

                       0.6

                       0.4

                       0.2



K
10            K
              5
                       K
                       0.2
                          0              5   10


                       K
                                         t
                       0.4

                       K
                       0.6




                                                      Signal and systems – p. 38/5
 Amplitude Modulation - Frequency domain


• It is important to see what happens in the frequency
  domain.
• Remember cos ω0 t =   ejω0 t +e−jω0 t
                                2
• Therefore

                                         ejω0 t + e−jω0 t
         xAM (t) = x(t) cos(ω0 t) = x(t)                  ↔
                                                 2
                                X(ω − ω0 ) + X(ω + ω0 )
                   XAM (ω) =
                                             2




                                                              Signal and systems – p. 39/5
  Fourier Transform of periodic signals


Example:




                                          Signal and systems – p. 40/5
 FDM


• The idea outlined above can be used to do a Frequency
  Division Demultiplexing
• In practice, the spectrum of each singnal is translated to a
  different frequency range: Xi (ω) → Xi (ω − ωi )
• In order for the idea to work, the frequency used to translate
  the signal must be sufficiently spaced out so as to avoid
  interference: ωi+1 − ω > B/2, where B is the bandwidth
• To demodulate the signal, we first isoltate the part of the
  spectrum we are interested in, translate the spectrum by ωi
  and then elimnate spurious component by a low pass filter.




                                                          Signal and systems – p. 41/5
FDM




      Signal and systems – p. 42/5
FDM - demodulation




                     Signal and systems – p. 43/5
Sampling




           Signal and systems – p. 44/5
 Ideal sampler


• Ideal sampling can intuitively be seen as generated by
  multiplying a signal by a sequence of dirac’s δ




                                                           Signal and systems – p. 45/5
    Properties of δ


•    +∞
     −∞ f (t)δ(t   − a)dt = f (a)
•    t
     −∞ δ(τ )dτ    = 1 → F (δ(t)) = 1
• f (t) ∗ δ(t − t0 ) = f (t − t0 )




                                        Signal and systems – p. 46/5
    F -trasform of r∗

•                                             to
    Using the above properties it is possible ” write:
                  “    P∞
    F (r∗ (t)) = F r(t) n=−∞ δ(t − nT )
•   We can express the sampling signal using the Fourier series:

                               ∞                    ∞
                               X                1   X            2π t
                                   δ(t − nT ) =            ejh   T

                              n=−∞
                                                T   h=−∞

                            “                       ”
•                                 1 P∞      jh 2π t
    Hence F   (r∗ (t))   = F r(t) T  h=−∞ e
                                               T


•   Applying the frequency shifting property we get:
                 1 P∞                 2π
    F (r∗ (t)) = T    h=−∞ R(ω − h T )




                                                                        Signal and systems – p. 47/5
      Example


            1                                                                            1


           0.9                                                                          0.9


           0.8                                                                          0.8


           0.7                                                                          0.7


           0.6                                                                          0.6
|R(j ω)|




                                                                             |R (jω)|
           0.5                                                                          0.5




                                                                             *
           0.4                                                                          0.4


           0.3                                                                          0.3

                                                                                                                           2 π /T
           0.2                                                                          0.2


           0.1                                                                          0.1


            0                                                                            0
            −1   −0.8   −0.6   −0.4   −0.2   0   0.2   0.4   0.6   0.8   1               −2   −1.5   −1   −0.5   0   0.5        1   1.5         2
                                             ω                                                                   ω




                                                                                                                                          Signal and systems – p. 48/5
   Aliasing


The spectrum might be altered (i.e., signal not attainable from
samples!)
                2


                                      sin(2 π t/3)
               1.5
                                                      samples collected with T = 3/2

                1



               0.5



                0



              −0.5



               −1


                         sin(2 π t)
              −1.5



               −2
                     0        0.5      1             1.5             2                 2.5   3




                                                                                                 Signal and systems – p. 49/5
 Shannon theorem


• If the signal has a finite badwidth then the signal can be reconstructed
                                                 1
  from samples collected with a period such that T   ≥ 2B
• Band-limited signals have infinite duration; many signals of
  interest have infinite bandwidth
• Typically a low-pass filter is used to de-emphasize higher
  frequencies




                                                                     Signal and systems – p. 50/5
    Data Extrapolation


•   If the following hypotheses hold
     ◦   the signal has limited bandwidth B
     ◦   the signal is sampled at frequency fs =        1
                                                            ≥ 2B
                                                        T
•   then the signal can be reconstructed using an ideal lowpass filter L(s) with
                                            8
                                            <T              π      π
                                                  if ω ∈ [− T ,    T
                                                                       ]
                                   |L(ω)| =
                                            :0    elsewhere.

•   Signal l(t) is given by:

                          1
                               Z   pi/T                 sin(πt/T )
                  l(t) =                  T ejωT dω =              = sinc(πt/T )
                         2π    −pi/T                       πt/T




                                                                                   Signal and systems – p. 51/5
    Data Extrapolation I


•   The reconstructed signal is computed as follows:

                                       R +∞    P              π(t−τ
                r(t) = r∗ (t) ∗ l(t) = −∞ r(τ ) δ(τ − kT )sinc T dτ =
                                      π(t−kT )
                = +∞ r(kT )sinc
                   P
                     −∞                  T

•   The function sinc is not causal and has infinite duration
•   In communication applications
      ◦   The duration problem can be solved truncating the signal
      ◦   The causality problem can be solved introducing a delay and collecting some
          sample before the reconstruction
•   Not viable in control applications since large delays jeopardise stability




                                                                                 Signal and systems – p. 52/5