SIGNALS - PowerPoint PPT Presentation

1 / 74
About This Presentation
Title:

SIGNALS

Description:

Signal is a physical quantity that varies with respect to time , ... For these common signals, the z-transform pairs have been tabulated (see Lathi, Table 5.1) ... – PowerPoint PPT presentation

Number of Views:476
Avg rating:3.0/5.0
Slides: 75
Provided by: admi1240
Category:
Tags: signals | lathi

less

Transcript and Presenter's Notes

Title: SIGNALS


1
SIGNALS SYSTEMS
  • Mrs.S.Suganthi., Senior Lecturer/ECE.
  • Mrs. A. Ahila Senior Lecturer/ECE

2
UNIT I
3
SIGNAL
  • Signal is a physical quantity that varies with
    respect to time , space or any other
    independent variable
  • Eg x(t) sin t.
  • the major classifications of the signal are
  • (i) Discrete time signal
  • (ii) Continuous time signal

4
Unit Step Unit Impulse
  • Discrete time Unit impulse is defined as
  • d n 0, n? 0
  • 1, n0
  • Unit impulse is also known as unit sample.
  • Discrete time unit step signal is defined by
  • Un0,n0
  • 1,ngt 0
  • Continuous time unit impulse is defined as
  • d (t)1, t0
  • 0, t ? 0
  • Continuous time Unit step signal is defined as
  • U(t)0, tlt0
  • 1, t0

5

SIGNAL
  • Periodic Signal Aperiodic Signal
  • A signal is said to be periodic ,if it exhibits
    periodicity.i.e., X(t T)x(t), for all values of
    t. Periodic signal has the property that it is
    unchanged by a time shift of T. A signal that
    does not satisfy the above periodicity property
    is called an aperiodic signal
  • even and odd signal ?
  • A discrete time signal is said to be even when,
    x-nxn. The continuous time signal is said
    to be even when, x(-t) x(t) For example,Cos?n
    is an even signal.

6
Energy and power signal
  • A signal is said to be energy signal if it
    have finite energy and zero power.
  • A signal is said to be power signal if it have
    infinite energy and finite power.
  • If the above two conditions are not satisfied
    then the signal is said to be neigther energy
    nor power signal

7
Fourier Series
  • The Fourier series represents a periodic signal
    in terms of frequency components
  • We get the Fourier series coefficients as
    follows
  • The complex exponential Fourier coefficients are
    a sequence of complex numbers representing the
    frequency component ?0k.

8
Fourier series
  • Fourier series a complicated waveform analyzed
    into a number of harmonically related sine and
    cosine functions
  • A continuous periodic signal x(t) with a period
    T0 may be represented by
  • X(t)S8k1 (Ak cos k? t Bk sin k? t) A0
  • Dirichlet conditions must be placed on x(t) for
    the series to be valid the integral of the
    magnitude of x(t) over a complete period must be
    finite, and the signal can only have a finite
    number of discontinuities in any finite interval

9
Trigonometric form for Fourier series
  • If the two fundamental components of a periodic
    signal areB1cos?0t and C1sin?0t, then their sum
    is expressed by trigonometric identities
  • X(t) A0 S8k1 ( Bk 2 Ak 2)1/2 (Ck cos k? t-
    fk) or
  • X(t) A0 S8k1 ( Bk 2 Ak 2)1/2 (Ck sin k? t
    fk)

10
UNIT II
11
Fourier Transform
  • Viewed periodic functions in terms of frequency
    components (Fourier series) as well as ordinary
    functions of time
  • Viewed LTI systems in terms of what they do to
    frequency components (frequency response)
  • Viewed LTI systems in terms of what they do to
    time-domain signals (convolution with impulse
    response)
  • View aperiodic functions in terms of frequency
    components via Fourier transform
  • Define (continuous-time) Fourier transform and
    DTFT
  • Gain insight into the meaning of Fourier
    transform through comparison with Fourier series

12
The Fourier Transform
  • A transform takes one function (or signal) and
    turns it into another function (or signal)
  • Continuous Fourier Transform

13
Continuous Time Fourier Transform
  • We can extend the formula for continuous-time
    Fourier series coefficients for a periodic signal
  • to aperiodic signals as well. The
    continuous-time Fourier series is not defined for
    aperiodic signals, but we call the formula
  • the (continuous time)
  • Fourier transform.

14
Inverse Transforms
  • If we have the full sequence of Fourier
    coefficients for a periodic signal, we can
    reconstruct it by multiplying the complex
    sinusoids of frequency ?0k by the weights Xk and
    summing
  • We can perform a similar reconstruction for
    aperiodic signals
  • These are called the inverse transforms.

15
Fourier Transform of Impulse Functions
  • Find the Fourier transform of the Dirac delta
    function
  • Find the DTFT of the Kronecker delta function
  • The delta functions contain all frequencies at
    equal amplitudes.
  • Roughly speaking, thats why the system response
    to an impulse input is important it tests the
    system at all frequencies.

16
Laplace Transform
  • Lapalce transform is a generalization of the
    Fourier transform in the sense that it allows
    complex frequency whereas Fourier analysis can
    only handle real frequency. Like Fourier
    transform, Lapalce transform allows us to analyze
    a linear circuit problem, no matter how
    complicated the circuit is, in the frequency
    domain in stead of in he time domain.
  • Mathematically, it produces the benefit of
    converting a set of differential equations into a
    corresponding set of algebraic equations, which
    are much easier to solve. Physically, it produces
    more insight of the circuit and allows us to know
    the bandwidth, phase, and transfer
    characteristics important for circuit analysis
    and design.
  • Most importantly, Laplace transform lifts the
    limit of Fourier analysis to allow us to find
    both the steady-state and transient responses
    of a linear circuit. Using Fourier transform, one
    can only deal with he steady state behavior (i.e.
    circuit response under indefinite sinusoidal
    excitation).
  • Using Laplace transform, one can find the
    response under any types of excitation (e.g.
    switching on and off at any given time(s),
    sinusoidal, impulse, square wave excitations,
    etc.

17
Laplace Transform
18
Application of Laplace Transform to Circuit
Analysis
19
system
  • A system is an operation that transforms input
    signal x into output signal y.

20
LTI Digital Systems
  • Linear Time Invariant
  • Linearity/Superposition
  • If a system has an input that can be expressed as
    a sum of signals, then the response of the system
    can be expressed as a sum of the individual
    responses to the respective systems.
  • LTI

21
Time-Invariance Causality
  • If you delay the input, response is just a
    delayed version of original response.
  • X(n-k) y(n-k)
  • Causality could also be loosely defined by there
    is no output signal as long as there is no input
    signal or output at current time does not
    depend on future values of the input.

22
Convolution
  • The input and output signals for LTI systems have
    special relationship in terms of convolution sum
    and integrals.
  • Y(t)x(t)h(t) Ynxnhn

23
UNIT III
24
Sampling theory
  • The theory of taking discrete sample values (grid
    of color pixels) from functions defined over
    continuous domains (incident radiance defined
    over the film plane) and then using those samples
    to reconstruct new functions that are similar to
    the original (reconstruction).
  • Sampler selects sample points on the image plane
  • Filter blends multiple samples together

25
Sampling theory
  • For band limited function, we can just increase
    the sampling rate
  • However, few of interesting functions in
    computer graphics are band limited, in
    particular, functions with discontinuities.
  • It is because the discontinuity always falls
    between two samples and the samples provides no
    information of the discontinuity.

26
Sampling theory
27
Aliasing
28
Z-transforms
  • For discrete-time systems, z-transforms play the
    same role of Laplace transforms do in
    continuous-time systems
  • As with the Laplace transform, we compute forward
    and inverse z-transforms by use of transforms
    pairs and properties

Bilateral Forward z-transform
Bilateral Inverse z-transform
29
Region of Convergence
  • Region of the complex z-plane for which forward
    z-transform converges
  • Four possibilities (z0 is a special case and may
    or may not be included)

30
Z-transform Pairs
  • hn dn
  • Region of convergence entire z-plane
  • hn dn-1
  • Region of convergence entire z-plane
  • hn-1 ? z-1 Hz
  • hn an un
  • Region of convergence z gt a which is the
    complement of a disk

31
Stability
  • Rule 1 For a causal sequence, poles are inside
    the unit circle (applies to z-transform functions
    that are ratios of two polynomials)
  • Rule 2 More generally, unit circle is included
    in region of convergence. (In continuous-time,
    the imaginary axis would be in the region of
    convergence of the Laplace transform.)
  • This is stable if a lt 1 by rule 1.
  • It is stable if z gt a and a lt 1 by rule 2.

32
Inverse z-transform
  • Yuk! Using the definition requires a contour
    integration in the complex z-plane.
  • Fortunately, we tend to be interested in only a
    few basic signals (pulse, step, etc.)
  • Virtually all of the signals well see can be
    built up from these basic signals.
  • For these common signals, the z-transform pairs
    have been tabulated (see Lathi, Table 5.1)

33
Example
  • Ratio of polynomial z-domain functions
  • Divide through by the highest power of z
  • Factor denominator into first-order factors
  • Use partial fraction decomposition to get
    first-order terms

34
Example (cont)
  • Find B0 by polynomial division
  • Express in terms of B0
  • Solve for A1 and A2

35
Example (cont)
  • Express Xz in terms of B0, A1, and A2
  • Use table to obtain inverse z-transform
  • With the unilateral z-transform, or the bilateral
    z-transform with region of convergence, the
    inverse z-transform is unique

36
Z-transform Properties
  • Linearity
  • Right shift (delay)

37
Z-transform Properties
  • Convolution definition
  • Take z-transform
  • Z-transform definition
  • Interchange summation
  • Substitute r n - m
  • Z-transform definition

38
UNIT IV
39
Introduction
  • Impulse response hn can fully characterize a
    LTI system, and we can have the output of LTI
    system as
  • The z-transform of impulse response is called
    transfer or system function H(z).
  • Frequency response at
    is valid if ROC includes and

40
5.1 Frequency Response of LIT System
  • Consider and
    , then
  • magnitude
  • phase
  • We will model and analyze LTI systems based on
    the magnitude and phase responses.

41
System Function
  • General form of LCCDE
  • Compute the z-transform

42
System Function Pole/zero Factorization
  • Stability requirement can be verified.
  • Choice of ROC determines causality.
  • Location of zeros and poles determines the
    frequency response and phase

43
Second-order System
  • Suppose the system function of a LTI system is
  • To find the difference equation that is satisfied
    by the input and out of this system
  • Can we know the impulse response?

44
System Function Stability
  • Stability of LTI system
  • This condition is identical to the condition that
  • The stability condition is equivalent to the
    condition that the ROC of H(z) includes the unit
    circle.

45
System Function Causality
  • If the system is causal, it follows that hn
    must be a right-sided sequence. The ROC of H(z)
    must be outside the outermost pole.
  • If the system is anti-causal, it follows that
    hn must be a left-sided sequence. The ROC of
    H(z) must be inside the innermost pole.

46
Determining the ROC
  • Consider the LTI system
  • The system function is obtained as

47
System Function Inverse Systems
  • is an inverse system for , if
  • The ROCs of must
    overlap.
  • Useful for canceling the effects of another
    system
  • See the discussion in Sec.5.2.2 regarding ROC

48
All-pass System
  • A system of the form (or cascade of these)

49
All-pass System General Form
  • In general, all pass systems have form

real poles
complex poles
Causal/stable
50
All-Pass System Example
Unit circle
z-plane
0.8
0.5
2
51
Minimum-Phase System
  • Minimum-phase system all zeros and all poles are
    inside the unit circle.
  • The name minimum-phase comes from a property of
    the phase response (minimum phase-lag/group-delay)
    .
  • Minimum-phase systems have some special
    properties.
  • When we design a filter, we may have multiple
    choices to satisfy the certain requirements.
    Usually, we prefer the minimum phase which is
    unique.
  • All systems can be represented as a minimum-phase
    system and an all-pass system.

52
UNIT V
53
Example
  • Block diagram representation of

54
Block Diagram Representation
  • LTI systems with rational system function can be
    represented as constant-coefficient difference
    equation
  • The implementation of difference equations
    requires delayed values of the
  • input
  • output
  • intermediate results
  • The requirement of delayed elements implies need
    for storage
  • We also need means of
  • addition
  • multiplication

55
Direct Form I
  • General form of difference equation
  • Alternative equivalent form

56
Direct Form I
  • Transfer function can be written as
  • Direct Form I Represents

57
Alternative Representation
  • Replace order of cascade LTI systems

58
Alternative Block Diagram
  • We can change the order of the cascade systems

59
Direct Form II
  • No need to store the same data twice in previous
    system
  • So we can collapse the delay elements into one
    chain
  • This is called Direct Form II or the Canonical
    Form
  • Theoretically no difference between Direct Form I
    and II
  • Implementation wise
  • Less memory in Direct II
  • Difference when using finite-precision arithmetic

60
Signal Flow Graph Representation
  • Similar to block diagram representation
  • Notational differences
  • A network of directed branches connected at nodes
  • Example representation of a difference equation

61
Example
  • Representation of Direct Form II with signal flow
    graphs

62
Determination of System Function from Flow Graph
63
Basic Structures for IIR Systems Direct Form I
64
Basic Structures for IIR Systems Direct Form II
65
Basic Structures for IIR Systems Cascade Form
  • General form for cascade implementation
  • More practical form in 2nd order systems

66
Example
  • Cascade of Direct Form I subsections
  • Cascade of Direct Form II subsections

67
Basic Structures for IIR Systems Parallel Form
  • Represent system function using partial fraction
    expansion
  • Or by pairingthe real poles

68
Example
  • Partial Fraction Expansion
  • Combine poles to get

69
Transposed Forms
  • Linear signal flow graph property
  • Transposing doesnt change the input-output
    relation
  • Transposing
  • Reverse directions of all branches
  • Interchange input and output nodes
  • Example
  • Reverse directions of branches and interchange
    input and output

70
Example
Transpose
  • Both have the same system function or difference
    equation

71
Basic Structures for FIR Systems Direct Form
  • Special cases of IIR direct form structures
  • Transpose of direct form I gives direct form II
  • Both forms are equal for FIR systems
  • Tapped delay line

72
Basic Structures for FIR Systems Cascade Form
  • Obtained by factoring the polynomial system
    function

73
Structures for Linear-Phase FIR Systems
  • Causal FIR system with generalized linear phase
    are symmetric
  • Symmetry means we can half the number of
    multiplications
  • Example For even M and type I or type III
    systems

74
Structures for Linear-Phase FIR Systems
  • Structure for even M
  • Structure for odd M
Write a Comment
User Comments (0)
About PowerShow.com