EC 35 SIGNALS AND SYSTEMS 3 1 0 4 AIM
To study and analyse characteristics of continuous, discrete signals and systems. OBJECTIVES
To study the properties and representation of discrete and continuous signals. To study the sampling process and analysis of discrete systems using ztransforms. To study the analysis and synthesis of discrete time systems.
1. CLASSIFICATION OF SIGNALS AND SYSTEMS 9
Continuous time signals (CT signals), discrete time signals (DT signals) - Step, Ramp, Pulse, Impulse, Exponential, Classification of CT and DT signals - periodic and periodic, random singals, CT systems and DT systems, Basic properties of systems - Linear Time invariant Systems and properties.
2. ANALYSIS OF CONTINUOUS TIME SIGNALS 9
Fourier series analysis, Spectrum of C.T. singals, Fourier Transform and Laplace Transform in Signal Analysis.
3. LINEAR TIME INVARIANT –CONTINUOUS TIME SYSTEMS 9
Differential equation, Block diagram representation, Impulse response, Convolution integral, frequency response , Fourier and Laplace transforms in analysis, State variable equations and matrix representation of systems
4. ANALYSIS OF DISCRETE TIME SIGNALS 9
Sampling of CT signals and aliasing, DTFT and properties, Z-transform and properties of Z-transform.
5. LINEAR TIME INVARIANT - DISCRETE TIME SYSTEMS 9
Difference equations, Block diagram representation, Impulse response, Convolution sum,
LTI systems analysis using DTFT and Z-transforms , State variable equations and matrix representation of systems.
Tutorial = 15
Total No of periods: 45 + 15 = 60 TEXT BOOK:
1. Allan V.Oppenheim, S.Wilsky and S.H.Nawab, Signals and Systems, Pearson Education, 2007.
2. Edward W Kamen & Bonnie’s Heck, “Fundamentals of Signals and Systems”, Pearson Education, 2007.
REFERENCES:
1. H P Hsu, Rakesh Ranjan“ Signals and Systems”, Schaum’s Outlines, Tata McGraw Hill, Indian Reprint, 2007
2. S.Salivahanan, A. Vallavaraj, C. Gnanapriya, Digital Signal Processing, McGraw Hill International/TMH, 2007.
3. Simon Haykins and Barry Van Veen, Signals and Systems John Wiley & sons , Inc, 2004.
4. Robert A. Gabel and Richard A.Roberts, Signals & Linear Systems, John Wiley, III edition, 1987.
UNIT I
CLASSIFICATION OF SIGNALS AND SYSTEMSContinuous time signals (CT signals), discrete time signals (DT signals)
A signal is a function representing a physical quantity or variable, and typically it contains information about the behavior or nature of the phenomenon. For instance, in a RC circuit the signal may represent the voltage across the capacitor or the current flowing in the resistor. Mathematically, a signal is represented as a function of an independent variable t. Usually t represents time. Thus, a signal is denoted by x(t).
A. Continuous-Time and Discrete-Time Signals:
A signal x(t) is a continuous-time signal if t is a continuous variable. If t is a discrete variable, that is, x(t) is defined at discrete times, then x(t) is a discrete-time signal. Since a discrete-time signal is defined at discrete times, a discrete-time signal is often identified as a sequence of numbers, denoted by {x,) or x[n], where n = integer. Illustrations of a continuous-time signal x(t) and of a discrete-time signal x[n] are shown in Fig. 1-1.
A discrete-time signal x[n] may represent a phenomenon for which the independent
variable is inherently discrete. For instance, the daily closing stock market average is by its nature a signal that evolves at discrete points in time (that is, at the close of each day). On the other hand a discrete-time signal x[n] may be obtained by sampling a continuous-time signal x(t) such as
or in a shorter form as
where we understand that
and xn s are called samples and the time interval between them is called the sampling interval. When the sampling intervals are equal (uniform sampling), then
B. Analog and Digital Signals:
If a continuous-time signal x(l) can take on any value in the continuous interval (a, b),
where a may be - 03 and b may be
+
m, then the continuous-time signal x(t) is called an analog signal. If a discrete-time signal x[n] can take on only a finite number of distinct values, then we call this signal a digital signal.C. Real and Complex Signals:
A signal x(t) is a real signal if its value is a real number, and a signal x(t) is a complex signal if its value is a complex number. A general complex signal x(t) is a function of the form
where x,( t ) and x2( t ) are real signals and
Note that in Eq. (I.l) t represents either a continuous or a discrete variable. D. Deterministic and Random Signals:
Deterministic signals are those signals whose values are completely specified for any given time. Thus, a deterministic signal can be modeled by a known function of time I . Random signals are those signals that take random values at any given time and must be characterized statistically. Random signals will not be discussed in this text.
E. Even and Odd Signals:
A signal x ( t ) or x[n] is referred to as an even signal if x ( - t ) = x ( r )
x [ - n ] = x [ n ]
A signal x ( t ) or x[n] is referred to as an odd signal if x ( - t ) = - x ( t )
x [ - n ] = - x [ n ]
. Any signal x(t) or x[n] can be expressed as a sum of two signals, one of which is even and one of which is odd. That is,
where xe(t) = $ { x ( t ) + x ( - t ) ] even part of x ( t )
xe[n] = i { x [ n ] + x [ - n ] ) even part of x [ n ] (1.5) x0(t) = $ { x ( t ) - x ( - t ) ) odd part of x(t )
x0[n] = $ { x [ n ]- x [ - n ] ) odd part of x [ n ] ( 1.6 )
Note that the product of two even signals or of two odd signals is an even signal and that the product of an even signal and an odd signal is an odd signal.
F. Periodic and Nonperiodic Signals:
A continuous-time signal x ( t ) is said to be periodic with period T if there is a positive nonzero value of T for which
x(t
+
T ) = x ( t ) all t (1.7)An example of such a signal is given in Fig. 1-3(a). From Eq. (1.7) or Fig. 1-3(a) it follows
That
signal x(t) (known as a dc signal). For a constant signal x(t) the fundamental period is undefined since x(t) is periodic for any choice of T (and so there is no smallest positive value). Any continuous-time signal which is not periodic is called a nonperiodic (or aperiodic ) signal.
Periodic discrete-time signals are defined analogously. A sequence (discrete-time signal) x[n] is periodic with period N if there is a positive integer N for which x[n + N] =x[n] all n (1.9)
An example of such a sequence is given in Fig. 1-3(b). From Eq. (1.9) and Fig. 1-3(b) it follows that
for all n and any integer m. The fundamental period No of x[n] is the smallest positive integer N for which Eq. (1.9) holds. Any sequence which is not periodic is called a nonperiodic (or aperiodic sequence.
Note that a sequence obtained by uniform sampling of a periodic continuous-time signal may not be periodic (Probs. 1.12 and 1.13). Note also that the sum of two continuous-time periodic signals may not be periodic but that the sum of two periodic sequences is always periodic .
G. Energy and Power Signals:
Consider v(t) to be the voltage across a resistor R producing a current dt). The instantaneous power p( t) per ohm is defined as
For an arbitrary continuous-time signal x(t), the normalized energy content E of x(t) is defined as
The normalized average power P of x(t) is defined as
Similarly, for a discrete-time signal x[n], the normalized energy content E of x[n] is defined as
The normalized average power P of x[n] is defined as
Based on definitions (1.14) to (1.17), the following classes of signals are defined:
1. x(t) (or x[n]) is said to be an energy signal (or sequence) if and only if 0 < E < m, and so P = 0.
2. x(t) (or x[n]) is said to be a power signal (or sequence) if and only if 0
<
P < m, thus implying that E = m.3. Signals that satisfy neither property are referred to as neither energy signals nor power signals.
Note that a periodic signal is a power signal if its energy content per period is finite, and then the average power of this signal need only be calculated over a period.
A. The Unit Step Sequence:
The unit step sequence u[n] is defined as
which is shown in Fig. 1-10(a). Note that the value of u[n] at n = 0 is defined [unlike the continuous-time step function u(f) at t = 01 and equals unity. Similarly, the shifted unit step sequence ii[n - k] is defined as
B. The Unit Impulse Sequence:
The unit impulse (or unit sample) sequence 6[n] is defined as
which is shown in Fig. 1-ll(a). Similarly, the shifted unit impulse (or sample) sequence 6[n - k] is defined as
which is shown in Fig. 1-1 l(b).
C. Complex Exponential Sequences:
The complex exponential sequence is of the form
Again, using Euler's formula, x[n] can be expressed as
Thus x[n] is a complex sequence whose real part is cos Ron and imaginary part is sin
Ron
. General Complex Exponential Sequences:
The most general complex exponential sequence is often defined as
where C and α are in general complex numbers. Note that Eq. (1.52) is the special case of
Real Exponential Sequences:
If C and a in Eq. (1.57) are both real, then x[n] is a real exponential sequence. Four distinct cases can be identified: a > 1,0 < a
< 1,
- 1 < a < 0, and a < - 1. These four real exponential sequences are shown in Fig. 1-12. Note that if a = 1, x[n] is a constant sequence, whereas if a = - 1, x[n] alternates in value between +C and -C.D. Sinusoidal Sequences:
A sinusoidal sequence can be expressed as
If n is dimensionless, then both R, and 0 have units of radians. Two examples of sinusoidal sequences are shown in Fig. 1-13. As before, the sinusoidal sequence in Eq. (1.58) can be expressed as
SYSTEMS AND CLASSIFICATION OF SYSTEMS
A. System Representation:
A system is a mathematical model of a physical process that relates the input (or excitation) signal to the output (or response) signal.
Let x and y be the input and output signals, respectively, of a system. Then the system is viewed as a transformation (or mapping) of x into y. This transformation is represented by the mathematical notation
where T is the operator representing some well-defined rule by which x is transformed into y. Relationship (1.60) is depicted as shown in Fig. 1-14(a). Multiple input and/or output signals are possible as shown in Fig. 1-14(b). We will restrict our attention for the most part in this text to the single-input, single-output case
B. Continuous;Time and Discrete-Time Systems:
If the input and output signals x and p are continuous-time signals, then the system is
called a continuous-time system [Fig. I - 15(a)]. If the input and output signals are discrete-time signals or sequences, then the system is called a discrete-time system [Fig. I - 15(b)].
Causal and Noncausal Systems:
A system is called causal if its output y ( t ) at an arbitrary time t = t,, depends on only the input x ( t ) for t It o. That is, the output of a causal system at the present time depends on only the present and/or past values of the input, not on its future values. Thus, in a causal system, it is not possible to obtain an output before an input is applied to the
system. A system is called noncausal if it is not causal. Examples of noncausal systems are
Note that all memoryless systems are causal, but not vice versa . Linear Systems and Nonlinear Systems:
If the operator T in Eq. (1.60) satisfies the following two conditions, then T is called a linear operator and the system represented by a linear operator T is called a linear system:
1. Additivity:
for any signals x, and x2. 2. Homogeneity (or Scaling):
for any signals x and any scalar α.
Any system that does not satisfy Eq. (1.66) and/or Eq. (1.67) is classified as a
nonlinear system. Equations (1.66) and ( 1.67) can be combined into a single condition as
where α1 and α2 are arbitrary scalars. Equation (1.68) is known as the superposition property. Examples of linear systems are the resistor [Eq. (1.6111 and the capacitor [Eq.
( 1.62)]. Examples of nonlinear systems are
Note that a consequence of the homogeneity (or scaling) property [Eq. (1.6711 of linear systems is that a zero input yields a zero output. This follows readily by setting α= 0 in Eq.(1.67). This is another important property of linear systems.
F. Time-Invariant and Time-Varying Systems:
A system is called rime-inuariant if a time shift (delay or advance) in the input signal causes the same time shift in the output signal. Thus, for a continuous-time system, the system is time-invariant if
for any real value of T. For a discrete-time system, the system is time-invariant (or shift-incariant ) if
for any integer k. A system which does not satisfy Eq. (1.71) (continuous-time system) or Eq. (1.72) (discrete-time system) is called a time-varying system. To check a system for time-invariance, we can compare the shifted output with the output produced by the shifted input (Probs. 1.33 to 1.39).
G. Linear Time-Invariant Systems
If the system is linear and also time-invariant, then it is called a linear rime-invariant (LTI) system.
H. Stable Systems:
A system is bounded-input/bounded-output (BIBO) stable if for any bounded input x defined by
the corresponding output y is also bounded defined by
PROPERTIES OF CONTINUOUS-TIME LTI SYSTEMS A. Systems with or without Memory:
Since the output y(t) of a memoryless system depends on only the present input x(t), then, if the system is also linear and time-invariant, this relationship can only be of the form
where K is a (gain) constant. Thus, the corresponding impulse response h(f) is simply
Therefore, if h(tJ # 0 for I,, # 0, the continuous-time LTI system has memory. B. Causality:
As discussed in Sec. 1.5D, a causal system does not respond to an input event until that event actually occurs. Therefore, for a causal continuous-time LTI system, we have
Applying the causality condition (2.16) to Eq. (2.101, the output of a causal continuous-time LTI system is expressed as
Alternatively, applying the causality condition ( 2.16) to Eq. (2.61, we have
Equation (2.18) shows that the only values of the input x(t) used to evaluate the output y( t ) are those for r 5 t.
Based on the causality condition (2.161, any signal x(t) is called causal if
and is called anticausal if
Then, from Eqs. (2.17), (2. I8), and (2. Iga), when the input x(t) is causal, the output y(t )
of a causal continuous-time LTI system is given by
C. Stability:
The BIBO (bounded-input/bounded-output) stability of an LTI system (Sec. 1.5H) is readily ascertained from its impulse response. It can be shown (Prob. 2.13) that a continuous-time LTI system is BIBO stable if its impulse response is absolutel integrable, that is,
PROPERTIES OF DISCRETE-TIME LTI SYSTEMS A. Systems with or without Memory:
Since the output y[n] of a memoryless system depends on only the present input x[n], then, if the system is also linear and time-invariant, this relationship can only be of the form
Therefore, if h[n,] # 0 for n, # 0, the discrete-time LTI system has memory B. Causality:
Similar to the continuous-time case, the causality condition for a discrete-time LTI system is
Applying the causality condition (2.44) to Eq. (2.391, the output of a causal discrete-time LTI system is expressed as
Alternatively, applying the causality condition (2.44) to Eq. (Z..V), we have
Equation (2.46) shows that the only values of the input x[n] used to evaluate the output
y[n] are those for k I n.
As in the continuous-time case, we say that any sequence x[n] is called causal if and is called anticausal if
Then, when the input x[n] is causal, the output y[n] of a causal discrete-time LTI system is given by
C. Stability:
It can be shown (Prob. 2.37) that a discrete-time LTI system is BIB0 stable if its impulse response is absolutely summable, that is,
UNIT II
Analysis of continuous time signals
Fourier series analysis:
Fourier series: a complicated waveform analyzed into a number of harmonically related sine and cosine functions
A two parts tutorial on Fourier series. In the first part an example is used to show how Fourier coefficients are calculated and in a second part you may use an applet to further explore Fouries series of the same function.
Fourier series may be used to represent periodic functions as a linear
combination of sine and cosine functions. If f(t) is a periodic function of period T, then under certain conditions, its Fourier series is given by:
where n = 1 , 2 , 3 , ... and T is the period of function f(t). an and bn are called
Fourier coefficients and are given by
Solution to the above example
Coefficient a0 is given by
Coefficients an is given by
And coefficients bn is given by
A computation of the above coefficients gives
a0 = 0 , an = 0 and bn = [ 2 / (n*pi) ] [ 1 - cos (n pi) ]
Note that cos (n pi) may be written as
cos (n pi) = (-1)n
and that bn = 0 whenever n is even.
The given function f(t) has the following Fourier series
The Fourier expansion coefficient ( in OWN) of a periodic signal
is
and the Fourier expansion of the signal is:
which can also be written as:
where is defined as
When the period of approaches infinity , the periodic signal
becomes a non-periodic signal and the following will result:
Interval between two neighboring frequency components becomes zero:
Summation of the Fourier expansion in equation (a) becomes an integral:
the second equal sign is due to the general fact:
Time integral over in equation (b) becomes over the entire time axis:
In summary, when the signal is non-periodic , the Fourier expansion becomes Fourier transform. The forward transform (analysis) is:
and the inverse transform (synthesis) is:
Comparing Fourier coefficient of a periodic signal with with Fourier spectrum of
a non-periodic signal :
we see that the dimension of is different from that of :
If represents the energy contained in the kth frequency component of a periodic
signal , then represents the energy density of a non-periodic signal
distributed along the frequency axis. We can only speak of the energy contained in
a particular frequency band :
Note on notations:
The spectrum of a time signal can be denoted by or to emphasize the fact that the spectrum represents how the energy contained in the signal is distributed as a
function of frequency or . Moreover, if is used, the factor in front of the inverse transform is dropped so that the transform pair takes a more symmetric form. On the other hand, as Fourier transform can be considered as a special case of Laplace
it is also natural to denote the spectrum of by (in OWN).
Example 0:
Consider the unit impulse function:
Example 1:
If the spectrum of a signal is a delta function in frequency domain
, the signal can be found to be:
i.e.,
The spectrum is
This is the sinc function with a parameter , as shown in the figure.
Note that the height of the main peak is and it gets taller and narrower as gets larger. Also note
When approaches infinity, for all , and the spectrum becomes
which represents the energy contained in the signal at (DC component at zero
frequency), and the spectrum is the energy density or distribution which is infinity at zero frequency.
The integral in the above transform is an important formula to be used frequently later:
which can also be written as
Switching and in the equation above, we also have
representing a superposition of an infinite number of cosine functions of all frequencies, which cancel each other any where along the time axis except at where they add up to infinity, an impulse.
Example 3:
The spectrum of the cosine function is
can be similarly obtained to be
Again, these spectra represent the energy density distribution of the sinusoids, while the corresponding Fourier coefficients
and
represent the energy contained at frequency .
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 ω0k by the weights Xk and summing:
We can perform a similar reconstruction for aperiodic signals
These are called the inverse transforms.
Fourier Transform of Impulse Functions
Find the Fourier transform of the Dirac delta function:
Find the DTFT of the Kronecker delta function:
1 0 0 ) ( p k n ik ke X nx
k t ik ke X t
x( ) 0
X e d nx ( ) i n
2 1 ) (
X e d
t
x ( ) i t
2 1 ) ( 1 ) ( ) ( )
( 0
i t i t i
e dt e t dt e t x X 1 ) ( ) ( )
( 0
i n n i n n i e e n e n x X 1 ) ( ) ( )( 0
The delta functions contain all frequencies at equal amplitudes.
Roughly speaking, that’s why the system response to an impulse input is important: it tests the system at all frequencies.
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.
UNIT III
LINEAR TIME INVARIANT –CONTINUOUS TIME SYSTEMS
Differential equation, Block diagram representation, Impulse response, Convolution integral,frequency response, State variableequations and matrix representation of systems.
System:
A system is an operation that transforms input signal x into output signal y.
LTI Systems
• Time Invariant
– X(t) y(t) & x(t-to) y(t-to)
• Linearity
– a1x1(t)+ a2x2(t) a1y1(t)+ a2y2(t)
– a1y1(t)+ a2y2(t)= T[a1x1(t)+a2x2(t)]
• Meet the description of many physical systems
• They can be modeled systematically
Differential equation:
• This is a linear first order differential equation with constant coefficients (assuming a and b are constants)
The general nth order linear DE with constant equations is
Linear constant-coefficient differential equations
In RC circuit
Block diagram representations
Block diagram representations of first-order systems described by differential and difference equations
Impulse Response
δ (t)
h(n)=H[ δ (t)]
A very important way to analyse a system is to study the output signal when a unit impulse signal is used as an input.
This impulse response signal can be used to infer properties about the system’s structure (LHS of difference equation or unforced solution).
The system impulse response, h(t) completely characterises a linear, time invariant system
Properties of System Impulse Response
Stable
A system is stable if the impulse response is absolutely summable
Causal
A system is causal if
h(t)=0 when t<0 Finite/infinite impulse response
The system has a finite impulse response and hence no dynamics in y(t) if there exists T>0, such that:
h(t)=0 when t>T Linear
ad(t) ah(t) Time invariant
d(t-T) h(t-T)
Convolution Integral
• An approach (available tool or operation) to describe the input-output relationship for LTI Systems
• In a LTI system
– d(t) h(t)
– Remember h(t) is T[d(t)]
– Unit impulse function the impulse response
• It is possible to use h(t) to solve for any input-output relationship
• Any input can be expressed using the unit impulse function
Convolution Integral - Properties
• Commutative
x
(
t
)
*
h
(
t
)
h
(
t
)
*
x
(
t
)
• Associative [x(t)*h1(t)]*h2(t)x(t)*[h1(t)*h2(t)]
t d x
t
x( )
( ) ( )
t
• Distributive x(t)*[h1(t)h2(t)][x(t)*h1(t)][x(t)*h2(t)]
• Thus, using commutative property:
State variables and Matrix representation
• State variables represent a way to describe ALL linear systems in terms of a common set of equations involving matrix algebra.
• Many familiar properties, such as stability, can be derived from this common representation. It forms the basis for the theoretical analysis of linear systems.
• State variables are used extensively in a wide range of engineering problems, particularly mechanical engineering, and are the foundation of control theory.
• The state variables often represent internal elements of the system such as voltages across capacitors and currents across inductors.
• They account for observable elements of the circuit, such as voltages, and also account for the initial conditions of the circuit, such as energy stored in
capacitors. This is critical to computing the overall response of the system.
• Matrix transformations can be used to convert from one state variable representation to the other, so the initial choice of variables is not critical.
• Software tools such as MATLAB can be used to perform the matrix manipulations required.
• Let us define the state of the system by an N-element column vector, x(t):
Note that in this development, v(t) will be the input, y(t) will be the output, and x(t) is used for the state variables.
• Any system can be modeled by the following state equations:
• This system model can handle single input/single output systems, or multiple inputs and outputs.
• The equations above can be implemented using the signal flow graph shown to the below
h t d h x t d x
t
x( )
( ) ( )
( ) ( )
tN N t x t x t x t x t x t x
t ( ) ( ) ( )
) ( ) ( ) ( )
( 2 1 2
• Consider the CT differential equations:
• A second-order differential equation requires two state variables:
• We can reformulate the differential equation as a set of three equations:
• We can write these in matrix form as:
• This can be extended to an Nth-order differential equation of this type:
The state variables are defined as
The resulting state equation is
) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 0 1 0 1 1 3 2 2 1 t x t y t v b t x a t x t x t x t x t x t x t x N i i i N N N
Matrix representation ) ( ) ( ) ( )(t a1y t a0y t b0v t
y
) ( ) ( ) ( ) ( 2
1 t y t x t y t
x
) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 0 2 1 1 0 2 2 1 t x t y t v b t x a t x a t x t x t x
) ( ) ( 0 1 ) ( ) ( 0 ) ( ) ( 1 0 ) ( ) ( 2 1 0 2 1 1 0 2 1 t x t x t y t v b t x t x a a t x t x ) ( ) ( ) ( 0 1 0 t v b t y a t y N i i iN
t i N yt
1 0 0 0
0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1 2 1 0 D C B A b a a a a NUNIT IV
ANALYSIS OF DISCRETE TIME SIGNALS
SAMPLING OF CT SIGNALS AND ALIASING,DTFT AND
PROPERTIES,Z-TRANSFORM AND PROPERTIES OF Z-PROPERTIES,Z-TRANSFORM SAMPLING
Sampling theory
Let x(t) be a continuous signal which is to be sampled, and that sampling is performed by measuring the value of the continuous signal every T seconds, which is called the
sampling interval. Thus, the sampled signal x[n] given by:
x[n] = x(nT), with n = 0, 1, 2, 3, ...
The sampling frequency or sampling rate fs is defined as the number of samples obtained in one second, or fs = 1/T. The sampling rate is measured in hertz or in samples per second.
The frequency equal to one-half of the sampling rate is therefore a bound on the highest frequency that can be unambiguously represented by the sampled signal. This frequency (half the sampling rate) is called the Nyquist frequency of the sampling system.
distinguished from other components with frequencies NfN + f and NfN – f for nonzero integers N. This ambiguity is called aliasing. To handle this problem as gracefully as possible, most analog signals are filtered with an anti-aliasing filter (usually a low-pass filter with cutoff near the Nyquist frequency) before conversion to the sampled discrete representation.
► 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
► Sampling theory
Sampling Theorem:
bandlimited signal can be reconstructed exactly if it is sampled at a rate atleast twice the maximum frequencycomponent in it."
► Consider a signal g(t) that is bandlimited.
► Sampling theory
The maximum frequency component of g(t) is fm. To recover the signal g(t) exactly from its samples it has to be sampled ata rate fs _ 2fm.
The minimum required sampling rate fs = 2fm is called nyquist rate
Aliasing
Aliasing is a phenomenon where the high frequency components of the sampled signal interfere with each other because of inadequate sampling ωs < 2ωm.
Aliasing leads to distortion in recovered signal. This is the reason why sampling
frequency should be atleast twice the bandwidth of the signal
.
DISCRETE TIME FOURIER TRANSFORM
In mathematics, the discrete-time Fourier transform (DTFT) is one of the specific forms of Fourier analysis. As such, it transforms one function into another, which is called the frequency domain representation, or simply the "DTFT", of the original function (which is often a function in the time-domain). But the DTFT requires an input function that is discrete. Such inputs are often created by sampling a continuous function, like a person's voice.
Often the sequence represents the values (aka samples) of a continuous-time
function, , at discrete moments in time: , where is the sampling interval
(in seconds), and is the sampling rate (samples per second). Then the DTFT provides an approximation of the continuous-time Fourier transform:
To understand this, consider the Poisson summation formula, which indicates that a
periodic summation of function can be constructed from the samples of function
The result is:
(Eq.2)
The right-hand sides of Eq.2 and Eq.1 are identical with these associations:
comprises exact copies of that are shifted by multiples of ƒs and combined by addition. For sufficiently large ƒs, the k=0 term can be observed in the region [−ƒs/2, ƒs/2] with little or no distortion (aliasing) from the other terms.
Inverse transform
The following inverse transforms recover the discrete-time sequence:
integration change the transform into a continuous-time Fourier transform [inverse], which produces a sequence of Dirac impulses. That is:
Properties
This table shows the relationships between generic discrete-time Fourier transforms. We use the following notation:
is the convolution between two signals
is the complex conjugate of the function x[n] represents the correlation between x[n] and y[n].
Property Time domain Frequency domain Remarks
Linearity
Shift in time integer k
Shift in frequency (modulation )
real number a
Time reversal Time conjugation Time reversal & conjugation
Integral in frequency
Convolve in time
Multiply in time
Correlation
Parseval's theorem
SYMMETRY PROPERTIES
The Fourier Transform can be decomposed into a real and imaginary part or into an even and odd part.
or
Time Domain Frequency Domain
Z
-transforms
► Definition: The Z – transform of a discrete-time signal x(n) is defined as the power series:
( )
( )
kk
X z
x n z
X z
( )
Z x n
[ ( )]
where z is a complex variable. The above given relations are sometimes called the direct Z - transform because they transform the time-domain signal x(n) into its complex-plane representation X(z). Since Z – transform is an infinite power series, it exists only for those values of z for which this series converges. The region of convergence of X(z)is the set of all values of z for which X(z) attains a finite value.
► For discrete-time systems, z-transforms play the same role of Laplace transforms do in continuous-time systems
Bilateral forward Z transform
Bilateral inverse Z transform
n
n
z n h z
H[ ]
R
n
dz z z H j n
h [ ] 1
2 1 ] [
Z
-transform Pairs
► h[n] = d[n]Region of convergence: entire z-plane
1] [ 0 0
n n n n z n z n zH
► h[n] = d[n-1]
Region of convergence: entire z-plane
h[n-1] z-1 H[z]
11 1 1 1 ] [
n z
n z zz H n n n n
►
Inverse
z
-transform
F
z z dzj n f n j c j c 1 2 1
► 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 we’ll 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)
Z-transform Properties Properties of z - transform
1. Linearity
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UNIT V
LINEAR TIME INVARIANT DISCRETE TIME SYSTEMS DIFFERENCE EQUATIONS,BLOCK DIAGRAM
REPRESENTATION,IMPULSE RESPONSE,CONVOLUTION SUM,LTI SYSTEMS ANALYSIS USING DTFT AND Z-TRANSFORMS,STATE VARIALE AND MATRIX REPRESENTATION OF SYSTEMS
DIFFERENCE EQUATIONS
A discrete-time system is anything that takes a discrete-time signal as input and generates a discrete-time signal as output.1 The concept of a system is very general. It may be used to
model the response of an audio equalizer .
In electrical engineering, continuous-time signals are usually processed by electrical circuits described by differential equations.
For example, any circuit of resistors, capacitors and inductors can be analyzed using mesh analysis to yield a system of differential equations.
The voltages and currents in the circuit may then be computed by solving the equations. The processing of discrete-time signals is performed by discrete-time systems.
Similar to the continuous-time case, we may represent a discrete-time system either by a set of
difference equations or by a block diagram of its implementation. For example, consider the following difference equation.
y(n) = y(n-1)+x(n)+x(n-1)+x(n-2)
This equation represents a discrete-time system. It operates on the input signal x(n)x(n) to produce the output signal y(n).
BLOCK DIAGRAM REPRESENTATION • Block diagram representation of
y
n a y
n a y
n
b x
n0 2
1 1 2
• 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
Direct Form I
General form of difference equation
M
k k N
k
kyn k b xn k
a
0 0
Alternative equivalent form
Direct Form II
► Cascade form
General form for cascade implementation
M k k N kkyn k b xn k
Parallel form
► Represent system function using partial fraction expansion
IMPULSE RESPONSE
► Impulse response h[n] 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).
CONVOLUTIO N SUM
The convolution sum provides a concise, mathematical way to express
the output of an LTI system based on an arbitrary discrete-time input signal and the system's response. The convolution sum is expressed as
Linear time-invariant systems can be described by the convolution sum • Convolution is conmutative
x[n] h[n] = h[n] x[n] • Convolution is distributive
x[n] (h1[n] + h2[n]) = x[n] h1[n] + x[n] h2[n] • Cascade connection:
y[n] = h1[n] [ h2[n] x[n] ] = [ h1[n] h2[n] ] x[n] • Parallel connection
y[n] = h1[n] x[n] + h2[n] x[n] ] = [ h1[n] + h2[n] ] x[n] • LTI systems are stable iff
P NP P
k
N
k k k
k k k k N k k k z d z d z e B z c A z C z H 1 1 1 1 1 1
0 1 1
1 1
n xn hny
z X
z H z.Y
[ ] [ ] [ ]
[ ] [ ] [ ]
k
y n x k h n k
• LTI systems are causal if
h[n] = 0 n < 0 LTI SYSTEMS ANALYSIS USING DTFT
• Consider and , then
– magnitude
– phase
Frequency response at is valid if ROC includes
LTI SYSTEMS ANALYSIS USING Z-TRANSFORM
• The z-transform of impulse response is called transfer or system function H(z).
• General form of LCCDE
• Compute the z-transform
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
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