Eigenfunction & eigenvalues of LTI systems Understanding...

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  • Overview of Complex Sinusoids Topics

    Eigenfunction & eigenvalues of LTI systems Understanding complex sinusoids Four classes of signals Periodic signals CT & DT Exponential harmonics

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 1

  • Complex Sinusoids

    A complex sinusoid is defined asx(t) = Aejt = A [cos(t) + j sin(t)]

    These are a special case of complex exponentialsx(t) = Aest = Aet [cos(t) + j sin(t)]

    when = 0

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 2

  • Review: Signals as Impulses

    h(t)x(t) y(t) h[n]x[n] y[n]

    Fourier series is used for signal decomposition In ECE 222 we decomposed signals into sums and of impulses

    x(t) =

    x() (t ) d x[n] =

    k=x[k] [n k]

    y(t) =

    x() h(t ) d y[n] =

    k=x[k] h[n k]

    We then used the LTI properties (linearity and time invariance) tosolve for the output as a sum of the impulse responses

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 3

  • Review of Energy and Power Signals

    E =

    |x(t)|2 dt P = limT

    12T

    TT

    |x(t)|2 dt

    E =

    n=|x[n]|2 P = lim

    N1

    2N + 1

    Nn=N

    |x[n]|2

    A signal is an energy signal if E < A signal is a power signal if 0 < P < Most signals are either energy signals or power signals A signal cannot be both

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 4

  • Orthogonalality

    A pair of real-valued energy signals are orthogonal if

    0 =

    x1(t)x2(t) dt

    0 =

    n=x1[n]x2[n]

    A pair of real-valued power signals are orthogonal if

    0 = limT

    12T

    TT

    x1(t)x2(t) dt

    0 = limN

    12N + 1

    Nn=N

    x1[n]x2[n]

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 5

  • Examples of Orthogonal Signals

    Like impulses, complex sinusoids are special

    Impulses are orthogonal to one another

    (t t1)(t t2) dt = 0 for t1 = t2

    n=[n n1][n n2] = 0 for n1 = n2

    Complex sinusoids are also orthogonal to one another

    limT

    12T

    TT

    ej1t ej2t dt = 0 for 1 = 2

    limN

    12N + 1

    Nn=N

    ej1n ej2n = 0 for 1 = 2

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 6

  • Importance of Orthogonality

    Orthogonal basis functions (e.g., complex sinusoids, shiftedimpulses) enable us to decompose signals into distinct(orthogonal) components

    More later

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 7

  • Eigenfunctions & Eigenvalues

    h(t)x(t) y(t) h[n]x[n] y[n]

    There are other basic signals that are also orthogonal But exponentials have another special property: You may be familiar with eigenvectors & eigenvalues for matrices There is a related concept for LTI systems Any signal x(t) or x[n] that is only scaled when passed through a

    system is called an eigenfunction of the system

    y(t) = x(t) h(t) = c x(t) y[n] = x[n] h[n] = c x[n]

    The scaling constant c is called the systems eigenvalue Complex exponentials are eigenfunctions of LTI systems Complex sinusoids are the only eigenfunctions of LTI systems that

    have finite power!

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 8

  • CT LTI System Response to Complex Exponentials

    Let x(t) = ejt. Then

    y(t) =

    h() x(t ) d

    =

    h() ej(t) d

    =

    h() ejtej d

    = ejt

    h()ej d

    = ejtH(j)

    where

    H(j) =

    h()ej d

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 9

  • DT LTI System Response to Complex Exponentials

    Let x[n] = ejny[n] =

    k=

    h[k] x[n k]

    =

    k=h[k] ej(nk)

    =

    k=h[k] ejnejk

    = ejn

    k=h[k] ejk

    = ejnH(ej)

    where

    H(ej) =

    k=h[k] ejk

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 10

  • Eigenfunctions & Eigenvalues

    h(t)x(t) y(t) h[n]x[n] y[n]

    Thus, complex sinusoids are eigenfunctions of LTI systemsejt H(j)ejtejn H(ejn)ejn

    The Fourier transform of the impulse response are the eigenvalues

    H(j) = F {h(t)} =

    h(t)ejt dt

    H(ej) = F {h[n]} =

    n=h[n] ejn

    H(j) and H(ej) are functions of frequency Called the frequency response of the system

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 11

  • Example 1: Real and Complex Sinusoids

    Are real sinusoids eigenfunctions of LTI systems?

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 12

  • Magnitude and Phase Response

    x(t) y(t) x[n] y[n]H(j) H(ej)

    H(j) = F {h(t)} =

    h(t)ejt dt

    H(ej) = F {h[n]} =

    n=h[n] ejn

    Both the eigenfunctions and eigenvalues of LTI systems arecomplex-valued

    |H(j)| and |H(ej)| are called the magnitude response The complex phase angle of H(j) and H(ej)

    Called the phase response Denoted as arg H(j) in the text

    This is not the same as arctan{

    Im{H(j)}Re{H(j)}

    }, in general

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 13

  • Magnitude and Phase Response Continued

    x(t) y(t) x[n] y[n]H(j) H(ej)

    ejt |H(j)|ej arg H(j)ejt = |H(j)|ej(t+arg H(j))

    ejn |H(ej)|ej arg H(ej)ejn = |H(ej)|ej(t+arg H(ej))

    Thus, if a complex sinusoid is applied at the input to an LTI system

    The system scales the amplitude by |H()| The system changes the phase by arg H()

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 14

  • Sums of Complex Exponentials

    h(t)x(t) y(t) h[n]x[n] y[n]

    Fourier series represent signals as sums (or integrals) of complexsinusoids

    x(t) =

    k=akejk0t x[n] =

    N01k=0

    akejk0n

    Since the signals are real, the imaginary portions of the complexexponentials must cancel out to zero

    This decomposition makes it particularly easy to solve for andanalyze the system output

    Sums of complex sinusoids can only represent periodic and nearlyperiodic signals

    Integrals of complex sinusoids are more general

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 15

  • Complex Exponential Sums

    x(t) y(t) x[n] y[n]H(j) H(ej)

    By linearity and time-invariance (LTI),

    x(t) =

    k

    akejkt y(t) =

    k

    akH(jk)ejkt

    x[n] =

    k

    akejkn y[n] =

    k

    akH(ejk)ejkn

    Thus if the input signal can be expressed as a sum of complexsinusoids, so can the output of the LTI system

    But what types of signals can be represented in this form? Virtually all of the (periodic) signals that we are interested in! Important and interesting idea

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 16

  • Continuous-Time Signal Intuition

    x(t) =

    k

    akejkt y(t) =

    k

    akH(jk)ejkt

    Fourier transforms represent signals as sums (or integrals) ofcomplex sinusoids

    It is therefore worthwhile to understand complex sinusoids asthoroughly as possible

    x(t) = ests=j

    = ejt = cos(t) + j sin(t)

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 17

  • Example 2: x(t) = ejt

    10 8 6 4 2 0 2 4 6 8 101

    0.5

    0

    0.5

    1

    Rea

    l Par

    t

    10 8 6 4 2 0 2 4 6 8 101

    0.5

    0

    0.5

    1

    Time (s)

    Imag

    inar

    y Pa

    rt

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 18

  • Example 2: x(t) = ejt

    105

    05

    10

    10.5

    00.5

    1

    1

    0.5

    0

    0.5

    1

    Time (s)

    Complex:Blue Real:Green Imaginary:Red Complex Plane:Yellow

    Real Part

    Imag

    inar

    y Pa

    rt

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 19

  • Example 2: MATLAB Code

    w = j*1;fs = 500; % Sample rate (Hz)t = -10:1/fs:10; % Time index (s)y = exp(w*t);N = length(t);subplot(2,1,1);

    h = plot(t,real(y));box off;grid on;ylim([-1.1 1.1]);ylabel(Real Part);

    subplot(2,1,2);h = plot(t,imag(y));box off;grid on;ylim([-1.1 1.1]);xlabel(Time (s));ylabel(Imaginary Part);

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 20

  • Example 2: MATLAB Code Continued

    figure;h = plot3(t,zeros(1,N),zeros(1,N),k);hold on;

    h = plot3(t,imag(y),real(y),b);h = plot3(t,1.1*ones(size(t)),real(y),r);h = plot3(t,imag(y),-1.1*ones(size(t)),g);hold off;

    grid on;ylabel(Imaginary Part);zlabel(Real Part);title(Complex:Blue Real:Red Imaginary:Green);axis([min(t) max(t) -1.1 1.1 -1.1 1.1]);view(27.5,22);

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 21

  • Discrete-Time Signal Intuition

    x[n] =

    k

    akejkn y[n] =

    k

    akH(ejk

    )ejkn

    It is similarly worthwhile to understand discrete-time complexexponentials as thoroughly as possible

    x[n] = zn||z|=1 = ejn = cos(n) + j sin(n)

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 22

  • Example 3: x[n] = ej0.2n

    10 5 0 5 10 15 20 25 30 35 401

    0.5

    0

    0.5

    1

    Rea

    l Par

    t

    10 5 0 5 10 15 20 25 30 35 401

    0.5

    0

    0.5

    1

    Time (n)

    Imag

    inar

    y Pa

    rt

    J. McNames Portland State University ECE 223 Complex Sinusoids Ver. 1.07 23

  • Example 3: x[n] = ej0.2n