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  • SINUSOIDAL SIGNALS

  • SINUSOIDAL SIGNALS

    x(t) = cos(2ft + )

    = cos(t + ) (continuous time)

    x[n] = cos(2fn + )

    = cos(n + ) (discrete time)

    f : frequency (s1 (Hz)) : angular frequency (radians/s) : phase (radians)

    2

  • Sinusoidal signals

    0 T/2 T 3T/2 2T

    0

    A

    A

    A cos( t)A sin( t)

    xc(t) = A cos(2ft) xs(t) = A sin(2ft)= A cos(2ft /2)

    - amplitude A -amplitude A

    - period T = 1/f -period T = 1/f- phase: 0 -phase: /2

    3

  • WHY SINUSOIDAL SIGNALS?

    Physical reasons:- harmonic oscillators generate sinusoids, e.g., vibrating

    structures- waves consist of sinusoidals, e.g., acoustic waves or

    electromagnetic waves used in wireless transmission

    Psychophysical reason:- speech consists of superposition of sinusoids- human ear detects frequencies- human eye senses light of various frequencies

    Mathematical (and physical) reason:- Linear systems, both physical systems and man-made

    filters, affect a signal frequency by frequency (hence low-pass, high-pass etc filters)

    4

  • EXAMPLE:

    TRANSMISSION OF A LOW-FREQUENCY SIGNAL USING

    HIGH-FREQUENCY ELECTROMAGNETIC (RADIO) SIGNAL

    - A POSSIBLE (CONVENTIONAL) METHOD:

    AMPLITUDE MODULATION (AM)

    Example: low-frequency signal

    v(t) = 5 + 2 cos(2ft), f = 20 Hz

    High-frequency carrier wave

    vc(t) = cos(2fct), fc = 200 Hz

    Amplitude modulation (AM) of carrier (electromagnetic) wave:

    x(t) = v(t) cos(2fct)

    5

  • 8

    6

    4

    2

    0

    2

    4

    6

    8

    v

    0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.18

    6

    4

    2

    0

    2

    4

    6

    8

    x

    t

    Top: v(t) (dashed) and vc(t) = cos(2fct).Bottom: transmitted signal x(t) = v(t) cos(2fct).

    6

  • Frequency contents of transmitted signal

    x(t) = v(t) cos(2fct)

    = (5 + 2 cos(2ft)) cos(2fct)

    = 5 cos(2fct) + 2 cos(2ft) cos(2fct)

    Trigonometric identity:

    cos cos =12

    cos( ) + 12

    cos( + )

    2 cos(2ft) cos(2fct) = cos (2(fc f)t)+cos (2(fc + f)t)

    7

  • x(t) = 5 cos(2fct) + cos (2(fc f)t) + cos (2(fc + f)t)

    Spectrumof v(t)

    Spectrumof vc(t)

    0 20 200

    5

    2

    1

    -

    Frequency

    Spectrum of x(t)

    -

    Frequency0 200

    5

    220

    1

    180

    1

    8

  • We see that a low-frequency signal in frequency range

    0 fs fmax (baseband signal)can be transmitted as a signal in the frequency range

    fc fmax f fc fmax (RF (radio frequency) signal).

    Another analog modulation technique is frequency modulation

    (FM)

    9

  • Representation of sinusoidal signal

    A. Amplitude and phase

    B. Sine and cosine components

    C. Complex exponentials

    10

  • A. Amplitude and phase

    x(t) = A cos(2ft + )

    or

    x(t) = A cos(t + )

    where = 2f is the angular frequency (radians/second)

    The phase describes translation in time compared to cos(t):

    x(t) = A cos(t + ) = A cos((t + /))

    11

  • B. Sine and cosine components

    x(t) = a cos(t) + b sin(t)

    Follows from trigonometric identity:

    cos( + ) = cos() cos() sin() sin()

    x(t) = A cos(t + )

    = A cos() cos(t)A sin() sin(t)

    Hence:

    a = A cos(), b = A sin()and, conversely,

    A =

    a2 + b2, = arctan(b/a)12

  • EXAMPLE

    2 0 2 4 6 8 10 12

    1

    0

    1

    x(t)

    Signal x(t) = cos(t + )Period: T = 10 (s)Frequency: f = 1/T = 0.1 (s1 =Hz (periods/second))Angular frequency: = 2f = 0.628 (radians/s)

    13

  • A. Representation using amplitude and phase

    Amplitude: A = 1From figure we see that x(t) = cos((t 2)).But x(t) = cos(t + ) = cos((t + /)Phase: = 2 = 0.4 radians or 0.4 180/ = 72

    14

  • B. Sine and cosine components

    x(t) = a cos(t) + b sin(t) wherea = cos() = cos(0.4) = 0.3090b = sin() = sin(0.4) = 0.9511

    2 0 2 4 6 8 10 12

    1

    0

    1

    x(t)

    2 0 2 4 6 8 10 12

    1

    0

    1

    Cosine componentSine component

    x(t) = 0.3090 cos(t) + 0.9511 sin(t)

    15

  • C. Representation using complex exponentials

    Background. A common operation is to combine sinusoidal

    components with different phases:

    x(t) = A1 cos(t + 1) + A2 cos(t + 2)

    One way: first write

    A1 cos(t + 1) = a1 cos(t) + b1 sin(t)

    A2 cos(t + 2) = a2 cos(t) + b2 sin(t)

    Then

    x(t) = (a1 + a2) cos(t) + (b1 + b2) sin(t)

    16

  • Streamlining the above procedure

    We have:

    cos(t + ) = cos() cos(t) sin() sin(t)and

    sin(t + ) = sin() cos(t) + cos() sin(t)

    Instead of working with the two signals cos(t) and sin(t) itis convenient to working with the single complex-valued signal

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

    where j =1 (imaginary unit number)

    17

  • -

    6

    Re

    Im

    *

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

    cos(t)t

    1

    Then:

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

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

    (sin() cos(t) + cos() sin(t)

    )

    =(cos() + j sin()

    )(cos(t) + j sin(t)

    )

    = g()g(t)

    18

  • In terms of the complex-valued signal, phase shift with angle

    is equivalent to multiplication by the complex number g():

    g(t + ) = g()g(t) (A)

    cos(t + ) = Re (g()g(t))

    sin(t + ) = Im (g()g(t))

    We will see that using the multiplication (A) to describe

    phase shifts results in significant simplification of formulae

    and calculations

    19

  • Properties of the complex function g() = cos() + j sin():

    - g(0) = 1

    - g(1 + 2) = g(1)g(2)

    - dg()d = sin() + j cos() = jg()The only function which satisfies these relations is

    g() = ej

    e = 2.71828182845904523536 . . . (Eulers number)

    20

  • Hence we have:

    Eulers relation

    ej = cos() + j sin()

    e = 2.71828182845904523536 . . . (Eulers number)

    As ej = cos() j sin() we have

    cos() =12

    (ej + ej

    )

    sin() =12j

    (ej ej)

    21

  • It follows that

    x(t) = a cos(t) + b sin(t)

    =(

    a

    2+

    b

    2j

    )ejt +

    (a

    2 b

    2j

    )ejt

    =12

    (a jb) ejt + 12

    (a + jb) ejt

    or

    x(t) = c ejt + cejt

    where c = 12 (a jb) and c = 12 (a + jb) (complex conjugate)Thus the cosine and sine components are parameterized as the

    real and imaginary components of the complex number c

    22

  • PREVIOUS EXAMPLE:

    The signal

    x(t) = cos(t 0.4)= a cos(t) + b sin(t)

    = 0.3090 cos(t) + 0.9511 sin(t)

    can be represented using complex exponentials as

    x(t) = c ejt + c ejt

    wherec =

    12

    (a jb) = 0.1545 j0.4755

    c =12

    (a + jb) = 0.1545 + j0.4755

    Hence:

    x(t) = (0.1545 j0.4755) ejt + (0.1545 + j0.4755) ejt

    23

  • SUMMARY

    Let us summarize the results so far. We have the following

    alternative representations of a sinusoidal signal:

    Amplitude and phase:

    x(t) = A cos(t + )

    Sine and cosine components:

    x(t) = a cos(t) + b sin(t)

    where

    a = A cos(), b = A sin()or

    A =

    a2 + b2, = arctan(b/a)24

  • Complex exponentials:

    x(t) = c ejt + c ejt

    where

    c =12

    (a jb) , c = 12

    (a + jb)

    or

    a = 2Re(c), b = 2Im(c)

    25

  • EXAMPLE - Signal transmission with reflected component

    Transmitted signal: x(t) = cos(t)

    Received signal: y(t) = x(t) + rx(t )PROBLEM: Determine amplitude and phase of y(t)

    We have

    y(t) = cos(t) + r cos((t ))The problem can be solved in a straightforward way by

    introducing complex exponential:

    x(t) =12

    (ejt + ejt

    )

    26

  • Then:

    y(t) = cos(t) + r cos((t ))=

    12

    (ejt + ejt

    )+ r

    12

    (ej(t) + ej(t)

    )

    =12

    (1 + rej

    )ejt +

    12

    (1 + rej

    )ejt

    Here,

    12

    (1 + rej

    )=

    12

    (1 + r cos() jr sin()

    )

    and

    12

    (1 + rej

    )=

    12

    (1 + r cos() + jr sin()

    )

    27

  • Then, defining

    a = 1 + r cos(), b = r sin()

    we have (cf. above)

    y(t) =12

    (a jb) ejt + 12

    (a + jb) ejt

    = a cos(t) + b sin(t)

    = Ay cos(t + y)

    where

    Ay =

    a2 + b2, y = arctan(b/a)

    28

  • SOME SIMPLIFICATIONS

    As

    a cos(t) + b sin(t) =12

    (a jb) ejt + 12

    (a + jb) ejt

    it follows that it is sufficient to consider the complex-valued

    signal12

    (a jb) ejt

    Here, the complex number

    c =12

    (a jb)

    determines the signals amplitude and phase.

    29

  • To see how this works, consider a linear dynamical system with

    input x(t) and output y(t)

    -- G y(t)x(t)

    For the sinusoidal input

    x(t) = cos(t)

    the output takes the form

    y(t) = A cos(t + ) = a cos(t) + b sin(t)

    and we want to determine the amplitude A and phase .

    30

  • As the phase shift introduces a combination of cosines and

    sines, the problem can be simplified by embedding the input

    signal into a larger class of signals involving both a cosine and

    a sine component.

    It turns out that a convenient way to do this is to consider the

    complex input signal

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

    = ejt

    The output then also has real and imaginary components:

    yc(t) = A cos(t + ) + jA sin(t + )

    = Aej(t+)

    = Aejejt

    31

  • or

    yc(

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