Chapter 2 - Discrete Time Signals and 2 - Discrete...Discrete Time Signals and Systems ... • The complex signal ejnw is an important signal in discrete time signal processing

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  • 2-1

    2. Discrete Time Signals and Systems

    We will review in this chapter the basic theories of discrete time signalsand systems. The relevant sections from our text are 2.0-2.5 and 2.7-2.10.

    The only material that may be new to you in this chapter is the section onrandom signals (Section 2.10 of Text)

    2.1 Discrete Time Signals

    A discrete-time (DT) signal is signal that exists at specific time instants.The amplitude of a discrete-time signal can be continuous though.

    When the amplitude of a DT signal is also discrete, then the signal is adigital signal.

    A DT signal can be either real or complex. While a real signal carries onlyamplitude information about a physical phenomenon, a complex signalcarries both amplitude and phase information.

    Throughout this course, we use square brackets [ ] to denote a DT signaland round brackets ( )i to denote a continuous time signal.

    Example : If the n-th sample of the DT signal [ ]x n is the value of the

    analog signal ( )ax t at t nT= , then

    [ ] ( )ax n x nT=

  • 2-2

    Some common DT signals are

    1. Unit sample

    1 0[ ]

    0 otherwise

    nn

    ==

    2. Unit step

    1 0[ ]

    0 0

    nu n

    n

    =

  • 2-3

    [ ] nx n A=

    where both and A are real. If 0A > and 0 1< < , then [ ]x ndecreases as n increases; see figure below.

    4. Sinusoidal

    ( )[ ] cos ox n A n = +

    where A is the amplitude, o is the frequency, and is the phase.

    5. Periodic

    [ ] [ ]x n N x n+ =

  • 2-4

    for any time index n . Here N denotes the period.

    Exercise: Is the sinusoidal signal defined above periodic in general?

    Example : Express the unit step function in terms of the unit-impulse

    Answers:

    (a) [ ] [ 1] [ ]u n u n n =

    (b) 0

    [ ] [ ]k

    u n n k

    =

    =

    Example : Express

    1 2,3,...,10[ ]

    0 otherwise

    nx n

    ==

    in terms of the unit step function.

    Answer:

    [ ] [ 2] [ 11]x n u n u n=

  • 2-5

    Example : Express the sinusoidal signal in terms of complex exponentialsignals

    Answer:

    ( )( ) ( )

    ( ) ( )1 1 2 2

    [ ] cos

    2

    o o

    o

    j n j n

    n n

    x n A n

    e eA

    A A

    + +

    = +

    +=

    = +

    where

    *1 22

    jAeA A

    = =

    *1 2

    oje = =

    It is also possible to express the signal as

    ( ){ }1 1[ ] 2 Re nx n A =

    Example : Express an arbitrary DT signal in terms of the unit impulse

    Answer

    [ ][ ] [ ]k

    x n x k n k

    =

    =

  • 2-6

    2.2 Discrete Time Systems

    Let [ ]T represents the transformation a discrete time system performedon its input [ ]x n . The corresponding output signal of the system is

    [ ][ ] [ ]y n T x n= .

    The system is linear if

    [ ]1 2 1 2[ ] [ ] [ ] [ ]T ax n bx n ay n by n+ = +

    where 1 2[ ] and [ ]y n y n are the responses of the system to inputs of

    1 2[ ] and [ ]x n x n respectively.

    The above equation illustrates the principle of superposition.

    Assume the system is linear and let [ ]hh n be the output of the system whenthe input is

    [ ] [ ]kp n n k=

    (i.e. a unit-impulse at time n k= ). Then according to the linearity property,

    [ ]y n[ ]T [ ]x n

  • 2-7

    when the input is

    [ ]

    [ ]

    [ ] [ ]

    [ ]

    k

    kk

    x n x k n k

    x k p n

    =

    =

    =

    =

    ,

    the output will be

    [ ][ ] [ ] kk

    y n x k h n

    =

    =

    The system is time-invariant if

    [ ] [ ]kh n h n k= ,

    i.e. the output is delayed if the input is delayed. In this case

    [ ][ ] [ ]

    [ ] [ ]k

    y n x k h n k

    x n h n

    =

    =

    =

    The signal [ ]h n is called the impulse response of the time-invariantsystem.

    While the focus of this course is on linear, time-invariant (LTI) system,there are many real-life applications where the system is non-linear andtime-variant. A good example is a digital FM demodulator operating in themobile radio environment.

  • 2-8

    Example : Provide a physical interpretation of a LTI system whose impulseresponse is

    [ ] [ ]h n u n=

    Answer: The output of the system is

    [ ]

    [ ]

    1

    [ ] [ ]

    [ ]

    [ ]

    [ ] [ ]

    [ 1] [ ]

    k

    k

    n

    k

    n

    k

    y n x k h n k

    x k u n k

    x k

    x k x n

    y n x k

    =

    =

    =

    =

    =

    =

    =

    = +

    = +

    Thus the system is an integrator.

    A system is casual if and only if the output at time n depends only on theinput up to time n . According to the equation

    [ ][ ] [ ]k

    y n h k x n k

    =

    = ,

    this means the impulse response [ ]h n is zero when 0n < .

  • 2-9

    Example : A moving averager computes the signal

    2

    11 2

    1[ ] [ ]

    1

    M

    k M

    y n x n kM M =

    = + +

    from its input [ ]x n . Here 1M and 2M are positive integers. What is theimpulse response of the system? Is the system casual?

    Answer:

    The output can be rewritten as

    1

    2

    1 2

    1 2

    1

    1 2

    1[ ] [ ]

    1

    1 [ ] [ ]

    1

    n M

    m n M

    n M n M

    m m

    y n x mM M

    x m x mM M

    +

    =

    +

    = =

    =+ +

    = + +

    Compared to the output of the integrator, we can deduce that the impulseresponse of the system is

    [ ] [ ]( )1 21 2

    1[ ] 1

    1h n u n M u n M

    M M= +

    + +

    Since the impulse response is non-zero when 0n < , so the system is notcasual.

    A real physical system can not be non-casual, i.e. it can not generate anoutput before there is an input. So in practice what a non-casual system

  • 2-10

    means is that there is a processing delay. For example, you can view themoving averager as a device that computes the local mean of the signal

    [ ]x n at time n after it observes the sample 1[ ]x n M+ . So 1M is the delay.

    A system is stable if a bounded input results in a bounded output. Therequirement for having a stable system can be derived from theinput/output relationship of a LTI system, which states that

    [ ][ ] [ ]k

    y n x k h n k

    =

    =

    This means

    [ ]

    [ ]

    [ ] [ ]max max

    [ ] [ ]

    [ ]

    k

    k

    k m

    y n x k h n k

    x k h n k

    x h n k x h m

    =

    =

    = =

    =

    =

    where is the absolute value operator and maxx is the largest magnitude ofthe input signal.

    So if the impulse response of the system is absolute-summable, i.e. when

    [ ]k

    S h k

    =

    = <

    then the system is stable.

  • 2-11

    Example : Is the integrator a stable system?

    Answer: Since [ ] 1h n = for 0n and zero otherwise, the impulse responseis not absolute-summable. Consequently the system is not stable.

    Example : Is the moving averager a stable system?

    Answer: Yes, because the impulse response consists of only a finitenumber of non-zero samples.

    Finite Impulse Response (FIR) and Infinite Impulse Response (IIR):

    FIR => An impulse response of finite duration, hence a finite number ofnon-zero samples. Always stable.

    IIR => The impulse response is infinitely long. Can be unstable (forexample the integrator).

    Example : Comment on the stability of a LTI system with the exponentialimpulse response

    0[ ]

    0 otherwise

    na nh n

    =

    Solution:

    0 0

    [ ]kk

    k k k

    h k a a

    = = =

    = =

  • 2-12

    This is summable if 1a < . In this case,

    1[ ]

    1kS h k

    a

    =

    = =

    Cascading of LTI systems serial connection of two or more systems; seethe example below.

    As far as the input/output relationship is concerned, it really does notmatter what the order of the concatenation is. For the example above, bothpossibilities yields the same combined impulse response of

    1 2[ ] [ ] [ ]h n h n h n=

  • 2-13

    In many applications, we have to concatenate a system to an existing oneso that the combined system yields the desired response. A good example isthe equalizer used in a digital communication system.

    Many communication channels introduces intersymbol interference (ISI)ef. This means the received signal [ ]r n depends not only on the data bit

    [ ]b n , but also on some adjacent bits. For example,

    1 1[ ] [0] [ ] [1] [ 1]r n h b n h b n= +

    where 1[ ]h n represents the impulse response of the channel. The objectiveof equalizer design is to find a digital filter with an impulse response 2 [ ]h nso that the combined response of the channel and the equalizer,

    1 2[ ] [ ] [ ]h n h n h n= , is the unit-impulse function. This means afterequalization, we have [ ] [ ]y n b n= , i.e. the ISI is removed.

    Exercise: Consider an ISI channel with 1[ ] [ ]nh n a u n= , where 0 1a< < ,

    and [ ]u n is the unit step function. Determine the equalizer that completelyremoves the ISI.

    Systems governed by the Linear Constant Coefficient Difference Equation(LCCDE):

    1 0

    [ ] [ ] [ ]N M

    k jk j

    y n a y n k b x n j= =

    = +

    The above equation suggests that current output of the system depends onthe previous output as well as the current and previous input.

  • 2-14

    In analyzing the above system, we assume the input is applied at time0n = (i.e. [ ] 0x n = for negative n ) and the initial state of the system is

    defined as

    ( )[0] [ 1], [ 2],..., [ ]y y y N= Y

    (a) Zero State Response (ZSR) response of the system to an unitimpulse applied at time 0n = , under the condition that [0]Y is theall-zero vector.

    (b) Zero Input Response (ZIR) response of the system due to a non-zero initial state but no input.

    Example : [ ] [ 1] [ ]y n ay n x n= +

    Let the initial state be [0] [ 1]y b= =Y , then

    2

    3 2

    [0] [0]

    [1] [0] [1]

    [2] [0] [1] [2]

    y ab x

    y a b ax x

    y a b a x ax x

    = +

    = + +

    = + + +

    or in general

    1

    0

    [ ] [ ]n

    n n k

    k

    y n a b a x k+

    =

    = +

    The ZIR is1

    1[ ] [ ]nh n a bu n+=

    and the ZSR is

  • 2-15

    2[ ] [ ]nh n a u n=

    It is clear that the ZIR corresponds to the bias term in [ ]y n . Since it isindependent of the input, the system can NOT be classified as a linearsystem. Note that the response of the system to

    3 1 1 2 2[ ] [ ] [ ]x n w x n w x n= +

    is

    { }

    13

    0

    11 1 2 2

    0

    [ ] [ ]

    [ ] [ ]

    nn n k

    k

    nn n k

    k

    y n a b a x k

    a b a w x k w x k

    +

    =

    +

    =

    = +

    = + +

    ,

    which is different from

    3 1 1 2 2[ ] [ ] [ ]y n w y n w y n= + ,

    where

    11 1

    0

    12 2

    0

    [ ] [ ],

    [ ] [ ]

    nn n k

    k

    nn n k

    k

    y n a b a x k

    y n a b a x k

    +

    =

    +

    =

    = +

    = +

  • 2-16

    2.3 Fourier Transform of Discrete Time Signals

    Consider the sinusoidal signal

    ( )[ ] cosx n A n = +

    It can be written in terms of two complex exponential functions as

    ( ) ( )

    1 2[ ] 2

    j n j nj n j ne ex n A A e A e

    + ++= = +

    where

    *1 22

    jAA e A= =

    The complex signal j ne is an important signal in discrete time signalprocessing it is an eigenfunction of a linear system and it leads us to theconcept of Fourier Transform of a discrete-time signal.

    Again let us use [ ]T to represent the operation a discrete time systemperforms on its input. A signal [ ]f n is an eigenfunction of the system if

    [ ][ ] [ ]T f n a f n= ,

    where the constant a is called an eigenvalue. This definition is consistentwith that in matrix theory where the eigenvector v and the eigenvalue b ofa matrix A is defined as

    b=Av v .

  • 2-17

    Here the matrix A is analogous to our linear system.

    As shown in Section 2.2, the transformation performed by a LTI on itsinput [ ]x n is described by the convolution formula:

    [ ] [ ] [ ]k

    y n h k x n k

    =

    = ,

    where [ ]h n is the impulse response of the system and [ ]y n is thetransformed signal or output of the system. If

    [ ] j nx n e = ,

    then the output signal becomes

    ( )

    ( )

    [ ] [ ]

    [ ]

    [ ]

    j n k

    k

    j n j k

    k

    j n j k

    k

    j n j

    y n h k e

    h k e e

    e h k e

    e H e

    =

    =

    =

    =

    =

    =

    =

    ,

    where

    ( ) [ ]j j kk

    H e h k e

    == .

  • 2-18

    It is clear from the above analysis that j ne is indeed an eigenfunction of a

    discrete-time LTI system with ( )jH e being the corresponding eigenvalue.

    In the linear system literature, ( )jH e is called the frequency response of adiscrete-time LTI system.

    In general, the expression

    ( ) [ ]j j kk

    X e x k e

    ==

    is called the Fourier Transform of the discrete-time signal [ ]x n .

    One important property of the Fourier Transform of a discrete time signalis that it is periodic in with a period of 2 . This is quite different fromthe Fourier Transform of a continuous time signal, which in general is notperiodic.

    Example: Express the output of a LTI system in terms of its frequencyresponse when the input is the sinusoid ( )[ ] cosx n A n = + . Assume theimpulse response of the sytem is a real signal.

    Solution:

    - The sinusoidal input can be written as a weighted sum of two complexexponential functions as

    1 2[ ]j n j nx n A e A e = +

  • 2-19

    where *

    1 2/ 2jA Ae A= = are the weighting coefficients.

    - Since the system is linear, the sinusoidal response is

    1 1 2 2[ ] [ ] [ ]y n A y n A y n= +where

    ( )1[ ] j jy n H e e =and

    ( )2[ ] j jy n H e e = ,

    are the outputs of the system when the inputs are 1[ ]j nx n e = and

    2[ ]j nx n e = respectively.

    - Since the impulse response is real,

    ( ) ( )*

    *[ ] [ ]j j k j k jk k

    H e h k e h k e H e

    = =

    = = =

    .

    This means we can write the output of the system as

    ( ) ( )

    ( ) ( )( ){ }

    ( ) ( )

    1 2

    1 2

    [ ] ]

    * *1 1

    1

    [ ]

    2Re

    cos ( )

    j j n j j n

    y n y n

    j j n j j n

    j j n

    j

    y n A H e e A H e e

    A H e e A H e e

    A H e e

    A H e n

    = +

    = +

    =

    = + +

    14243 1442443

  • 2-20

    Note that ( )jH e and ( ) are respectively the magnitude and phase ofthe frequency response, i.e.

    ( ) ( ) ( )j j jH e H e e =In conclusion, when the input is a sinusoid, the output is also a sinusoidat the same frequency but with the amplitude scaled by ( )jH e and withthe phase shifted by an amount ( ) .

    Example : Determine the frequency response of a delay element describedby the impulse response

    [ ] [ ]h n n d=

    Solution

    ( ) [ ] [ ]j j n j n j dn n

    H e h n e n d e e

    = =

    = = =

    This means

    ( ) 1jH e = (constant magnitude response)and

    ( ) d = (linear phase)

    Example : Determine the Fourier Transform of the one-sided exponentialsignal

  • 2-21

    [ ] [ ]nx n a u n=

    where 0 1a< < and [ ]u n is the unit-step function.

    Solution:

    ( ) ( )0 0

    1[ ]

    1

    nj j n n j n j

    jn n n

    X e x n e a e aeae

    = = =

    = = = =

    Since

    ( )

    ( )( ) ( )

    ( ) ( ) ( ){ }( )

    2 2 2

    2 22 2 2 2

    2 2 2

    2

    1 1 cos( ) sin( )

    1 cos( ) sin( ) 1 cos( ) sin ( )

    1 cos( ) sin ( ) 1 cos( ) sin ( )

    1 cos( ) sin ( ) cos ( ) sin ( )

    1 cos( )

    jae a ja

    a aa a j

    a a a a

    a a j

    a a

    = +

    = + + + +

    = + +

    = + ( )2 2sin ( )exp ( )

    j

    wheresin( )

    ( )1 cos( )

    a

    a

    =

    ,

    this means the magnitude of the Fourier transform is

    ( )( )2 2 2

    1

    1 cos( ) sin ( )

    jX ea a

    =

    +

    and the phase is simply

    ( ) ( ) = .

  • 2-22

    Existence of the Fourier Transform:

    - If we set the parameter a in the above example to unity, then the signalbecomes a unit-step. The Fourier Transform in this case, however, doesnot exist in the finite magnitude sense.

    - A sufficient condition for the existence of the Fourier transform (in thefinite-magnitude sense) is that the signal is absolute-summable, i.e.

    [ ]k

    S x k

    =

    = <

    The proof is the same as that we used to proof the stability of a LTIsystem.

    - We can deduce from the above that the Fourier Transform always existsfor signals with finite duration.

    Example : Determine the Fourier Transform of the signal

    1/( 1) 0[ ]

    0 otherwise

    M n Mx n

    + =

    Solution

    ( )( )

    ( ) ( ) ( ){ }{ }

    ( )( )( )

    1

    0

    1 / 2 1 /2 1 / 2

    /2 /2 /2

    12/2

    2

    1 1 1[ ]1 1 1

    1

    1

    sin1

    1 sin

    j MMj j n j n

    jn n

    j M j M j M

    j j j

    Mj M

    eX e x n e e

    M M e

    e e e

    M e e e

    eM

    +

    = =

    + + +

    +

    = = =+ +

    =

    +

    =+

  • 2-23

    The magnitude of the transform is

    ( ) ( )( )( )1

    2

    2

    sin11 sin

    MjX e

    M

    +=

    +

    At a first glance, the phase of the Fourier Transform is ( ) / 2M = .However, the sin( ) / sin( )i function can take on either + or ve value.When there is a sign change in this function, that corresponds to anadditional 180 degree phase shift.

    Plots of the magnitude and phase o...

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