Title: SIGNALS
1SIGNALS SYSTEMS
- Mrs.S.Suganthi., Senior Lecturer/ECE.
- Mrs. A. Ahila Senior Lecturer/ECE
2UNIT I
3SIGNAL
- 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
4Unit 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
5SIGNAL
- 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.
6Energy 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
7Fourier 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.
8Fourier 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
9Trigonometric 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)
10UNIT II
11Fourier 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
12The Fourier Transform
- A transform takes one function (or signal) and
turns it into another function (or signal) - Continuous Fourier Transform
13Continuous 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.
14Inverse 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.
15Fourier 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.
16Laplace 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.
17Laplace Transform
18Application of Laplace Transform to Circuit
Analysis
19system
- A system is an operation that transforms input
signal x into output signal y.
20LTI 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
21Time-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.
22Convolution
- 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
23UNIT III
24Sampling 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
25Sampling 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.
26Sampling theory
27Aliasing
28Z-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
29Region 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)
30Z-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
31Stability
- 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.
32Inverse 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)
33Example
- 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
34Example (cont)
- Find B0 by polynomial division
- Express in terms of B0
- Solve for A1 and A2
35Example (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
36Z-transform Properties
- Linearity
- Right shift (delay)
37Z-transform Properties
- Convolution definition
- Take z-transform
- Z-transform definition
- Interchange summation
- Substitute r n - m
- Z-transform definition
38UNIT 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
405.1 Frequency Response of LIT System
- Consider and
, then - magnitude
- phase
-
- We will model and analyze LTI systems based on
the magnitude and phase responses.
41System Function
- General form of LCCDE
- Compute the z-transform
42System 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
43Second-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?
44System 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.
45System 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.
46Determining the ROC
- Consider the LTI system
- The system function is obtained as
47System 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
48All-pass System
- A system of the form (or cascade of these)
49All-pass System General Form
- In general, all pass systems have form
real poles
complex poles
Causal/stable
50All-Pass System Example
Unit circle
z-plane
0.8
0.5
2
51Minimum-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.
52UNIT V
53Example
- Block diagram representation of
54Block 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
55Direct Form I
- General form of difference equation
- Alternative equivalent form
56Direct Form I
- Transfer function can be written as
- Direct Form I Represents
57Alternative Representation
- Replace order of cascade LTI systems
58Alternative Block Diagram
- We can change the order of the cascade systems
59Direct 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
60Signal Flow Graph Representation
- Similar to block diagram representation
- Notational differences
- A network of directed branches connected at nodes
- Example representation of a difference equation
61Example
- Representation of Direct Form II with signal flow
graphs
62Determination of System Function from Flow Graph
63Basic Structures for IIR Systems Direct Form I
64Basic Structures for IIR Systems Direct Form II
65Basic Structures for IIR Systems Cascade Form
- General form for cascade implementation
- More practical form in 2nd order systems
66Example
- Cascade of Direct Form I subsections
- Cascade of Direct Form II subsections
67Basic Structures for IIR Systems Parallel Form
- Represent system function using partial fraction
expansion - Or by pairingthe real poles
68Example
- Partial Fraction Expansion
- Combine poles to get
69Transposed 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
70Example
Transpose
- Both have the same system function or difference
equation
71Basic 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
72Basic Structures for FIR Systems Cascade Form
- Obtained by factoring the polynomial system
function
73Structures 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
74Structures for Linear-Phase FIR Systems
- Structure for even M
- Structure for odd M