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