63864722 Signals and Systems Lecture Notes

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EC 35 SIGNALS AND SYSTEMS 3 1 0 4AIMTo study and analyse characteristics of continuous, discrete signals and systems.OBJECTIVESTo 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 9Continuous 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 Timeinvariant Systems and properties.2. ANALYSIS OF CONTINUOUS TIME SIGNALS 9Fourier series analysis, Spectrum of C.T. singals, Fourier Transform and LaplaceTransform in Signal Analysis.3. LINEAR TIME INVARIANT CONTINUOUS TIME SYSTEMS 9Differential equation, Block diagram representation, Impulse response, Convolutionintegral, frequency response , Fourier and Laplace transforms in analysis, State variable

equations and matrix representation of systems4. ANALYSIS OF DISCRETE TIME SIGNALS 9Sampling of CT signals and aliasing, DTFT and properties, Z-transform and properties ofZ-transform.5. LINEAR TIME INVARIANT - DISCRETE TIME SYSTEMS 9Difference equations, Block diagram representation, Impulse response, Convolutionsum,LTI systems analysis using DTFT and Z-transforms , State variable equations and matrixrepresentation of systems.Tutorial = 15Total No of periods: 45 + 15 = 60TEXT BOOK:

1. Allan V.Oppenheim, S.Wilsky and S.H.Nawab, Signals and Systems, PearsonEducation, 2007.2. Edward W Kamen & Bonnies Heck, Fundamentals of Signals and Systems,Pearson Education, 2007.REFERENCES:1. H P Hsu, Rakesh Ranjan Signals and Systems, Schaums Outlines, Tata McGraw Hill, Indian Reprint, 20072. S.Salivahanan, A. Vallavaraj, C. Gnanapriya, Digital Signal Processing, McGraw HillInternational/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, 1987.5. Rodger E. Ziemer, William H. Tranter, D. Ronald Fannin. Signals & systems, FourthEdition, Pearson Education, 2002.

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UNIT ICLASSIFICATION 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 flowingin the resistor. Mathematically, a signal is represented as a function of an independentvariable 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 discretevariable, that is, x(t) is defined at discrete times, then x(t) is a discrete-time signal. Since adiscrete-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 acontinuous-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 independentvariable is inherently discrete. For instance, the daily closing stock market average is by itsnature a signal that evolves at discrete points in time (that is, at the close of each day). Onthe other hand a discrete-time signal x[n] may be obtained by sampling a continuous-timesignal 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 samplinginterval. When the sampling intervals are equal (uniform sampling), then

where the constant T, is the sampling interval.

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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 ananalog signal. If a discrete-time signal x[n] can take on only a finite number of distinctvalues, then we call this signal a digital signal.

C. Real and Complex Signals:A signal x(t) is a real signal ifits 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 theform

where x,( t ) and x2( t ) are real signals andNote 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 anygiven time. Thus, a deterministic signal can be modeled by a known function of time I .

Randomsignals are those signals that take random values at any given time and must becharacterized statistically. Random signals will not be discussed in this text.

E. Even and Odd Signals:

A signal x ( t ) orx[n] is referred to as an even signal ifx ( - t ) =x ( r )x [ - n ] =x [ n ]A signal x ( t ) orx[n] is referred to as an odd signal ifx ( - t ) = - x ( t )x [ - n ] = - x [ n ]Examples of even and odd signals are shown in Fig. 1-2

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. Any signal x(t) orx[n] can be expressed as a sum of two signals, one of which is evenand one of which is odd. That is,

where xe(t) = $ { x ( t ) +x ( - t ) ] even part ofx ( t )xe[n] = i { x [ n ] + x [ - n ] ) even part ofx [ n ] (1.5)

x0(t) = $ { x ( t ) - x ( - t ) ) odd part ofx(t )x0[n] = $ { x [ n ]- x [ - n ] ) odd part ofx [ n ] ( 1.6 )Note that the product of two even signals or of two odd signals is an even signal andthat 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 periodicwith periodT if there is a positivenonzero value ofT 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

for all t and any integerm. The fundamental periodT, ofx ( t ) is the smallest positivevalue ofT for which Eq. (1.7) holds. Note that this definition does not work for a constant

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signal x(t) (known as a dc signal). For a constant signal x(t) the fundamental period isundefined since x(t) is periodic for any choice of T (and so there is no smallest positivevalue). 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-timesignal) x[n] is periodic with period N if there is a positive integerN 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 thatfor all n and any integer m. The fundamental period Noof x[n] is the smallest positive

integerN for which Eq. (1.9) holds. Any sequence which is not periodic is called anonperiodic (or aperiodic sequence.Note that a sequence obtained by uniform sampling of a periodic continuous-timesignal may not be periodic (Probs. 1.12 and 1.13). Note also that the sum of twocontinuous-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 resistorRproducing a current dt). Theinstantaneous power p( t) per ohm is defined as

Total energy E and average powerP on a per-ohm basis are

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For an arbitrary continuous-time signal x(t), the normalized energy content E of x(t) is

defined as

The normalized average powerP of x(t) is defined as

Similarly, for a discrete-time signal x[n], the normalized energy content E of x[n] isdefined as

The normalized average powerP 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) ifand only if0 < E < m, andso P = 0.

2. x(t) (or x[n]) is said to be a power signal (or sequence) if and only if0 < P < m, thus

implying that E = m.3. Signals that satisfy neither property are referred to as neither energy signals nor powersignals.

Note that a periodic signal is a power signal ifits energy content per period is finite, andthen 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 stepsequence ii[n - k] is defined as

which is shown in Fig. 1-lO(b).

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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) sequence6[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 sinRon. General Complex Exponential Sequences:The most general complex exponential sequence is often defined as

where C and are in general complex numbers. Note that Eq. (1.52) is the special case

of

Eq. (1.57) with C = 1 and

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Real Exponential Sequences:IfC 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 realexponential sequences are shown in Fig. 1-12. Note that ifa = 1, x[n] is a constant

sequence, whereas ifa = - 1, x[n] alternates in value between +Cand -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

As we observed in the case of the complex exponential sequence in Eq. (1.52), the same G.

observations [Eqs. (1.54) and (1.5611 also hold for sinusoidal sequences. For instance, the

sequence in Fig. 1-13(a) is periodic with fundamental period 12, but the sequence in Fig.l-13( b) is not periodic.

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SYSTEMS AND CLASSIFICATION OF SYSTEMS

A. System Representation:

A system is a mathematical model of a physical process that relates the input (orexcitation) signal to the output (or response) signal.

Let x and ybe the input and output signals, respectively, of a system. Then the systemis viewed as a transformation (or mapping) ofx 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/oroutput signals are possible as shown in Fig. 1-14(b). We will restrict our attention for themost part in this text to the single-input, single-output case

B. Continuous;Time and Discrete-Time Systems:Ifthe input and output signals x andp 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 onlythe input x ( t ) fort It o. That is, the output of a causal system at the present time dependson only the present and/or past values of the input, not on its future values. Thus, in acausal 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:Given that Tx, = y, and Tx, = y,, then

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for any signals x, and x2.2. Homogeneity(orScaling):

for any signals x and any scalar.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 1 and 2 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 = 0 inEq.(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, thesystem is time-invariant if

for any real value ofT. For a discrete-time system, the system is time-invariant (orshift-incariant ) if

for any integerk. A system which does not satisfy Eq. (1.71) (continuous-time system) orEq. (1.72) (discrete-time system) is called a time-varying system. To check a system fortime-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 iffor any bounded

input x defined by

the corresponding output y is also bounded defined by

where k , andk, are finite real constants. Note that there are many other definitions ofstability. (See Chap. 7.)

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PROPERTIES OF CONTINUOUS-TIME LTI SYSTEMS

A. Systems with or without Memory:

Since the output y(t) ofa 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 forI,, # 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 thatevent 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 forr5 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 outputy[n] of a memoryless system depends on only the present inputx[n],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 is simply

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Therefore, ifh[n,] # 0 forn, # 0, the discrete-time LTI system has memory

B. Causality:

Similar to the continuous-time case, the causality condition for a discrete-time LTIsystem is

Applying the causality condition (2.44) to Eq. (2.391, the output of a causal discrete-timeLTI 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 forkI 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 outputy[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 itsimpulse response is absolutely summable, that is,

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UNIT II

Analysis of continuous time signals

Fourier series analysis:

Fourier series: a complicated waveform analyzed into a number ofharmonically related sine and cosine functions

A two parts tutorial on Fourier series. In the first part an example is usedto show how Fourier coefficients are calculated and in a second part you mayuse an applet to further explore Fouries series of the same function.

Fourier series may be used to represent periodic functions as a linearcombination 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). a n and bn are calledFourier coefficients and are given by

Example : Find the Fourier series of the periodic function f(t) defined by

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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

Continuous Time Fourier Transform

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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:

Discrete frequency becomes continuous frequency:

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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 Fourierexpansion becomes Fourier transform. The forward transform (analysis) is:

and the inverse transform (synthesis) is:

Note that is denoted by in OWN.

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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 factthat 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

transform when the real part of the complex argument is zero:

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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.,

Example 2:

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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

Recall that the Fourier coefficient of is

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which represents the energy contained in the signal at (DC component at zero

frequency), and the spectrum is the energy density or distributionwhich 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 addup to infinity, an impulse.

Example 3:

The spectrum of the cosine function is

The spectrum of the sine function

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can be similarly obtained to be

Again, these spectra represent the energy density distribution of the sinusoids, while thecorresponding 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 canreconstruct 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.

Fourier Transform of Impulse FunctionsFind the Fourier transform of the Dirac delta function:

Find the DTFT of the Kronecker delta function:

1

0

0)(p

k

nik

keXnx

k

tik

keXtx0)(

deXnx ni)(

2

1)(

deXtx ti)(

2

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ititi edtetdtetxX

1)()()( 0

in

ni

n

ni eenenxX

1)()()( 0

in

ni

n

ni eenenxX

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The delta functions contain all frequencies at equal amplitudes.

Roughly speaking, thats why the system response to an impulse input is important: ittests 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. LikeFourier 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 timedomain.

Mathematically, it produces the benefit of converting a set of differential equations intoa 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, andtransfer characteristics important for circuit analysis and design.

Most importantly, Laplace transform lifts the limit of Fourier analysis to allow us tofind 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 underindefinite 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.

Laplace Transform

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Application of Laplace Transform to Circuit Analysis

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UNIT IIILINEAR TIME INVARIANTCONTINUOUS TIME SYSTEMSDifferential equation,Block diagram representation, Impulse response,Convolution

integral,frequency response,State variableequations and matrix representation ofsystems.

System:A system is an operation that transforms input signalx into output signaly.

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

Non-LTI systems typically have no general mathematical procedure to

obtain solution

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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 equationsIn RC circuit

To introduce some of the important ideas concerning systems specified by linear

constant-coefficient differential equations ,let us consider a first-order differential

equations:

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Block diagram representations

Block diagram representations of first-order systems described by differential anddifference equations

Impulse Response

(t) h(n)=H[ (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 systems structure

(LHS of difference equation or unforced solution).The system impulse response, h(t) completely characterises a linear, time invariant

system

LTI System

H[.]

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Properties of System Impulse ResponseStable

A system is stable if the impulse response is absolutely summable

Causal

A system is causal ifh(t)=0 when t0, such that:

h(t)=0 when t>T

Linearad(t)ah(t)

Time invariantd(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 )(*)()(*)( txththtx

Associative )](*)([*)()(*)](*)([ 2121 ththtxththtx

dtxtx )()()(

t

th )(

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Distributive )](*)([)](*)([)]()([*)( 2121 thtxthtxththtx

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 anN-element column vector,x(t):

Note that in this development, v(t) will be the input,y(t) will be the

output, andx(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

dtxhdthxtx )()()()()(

tN

N

txtxtx

tx

tx

tx

t )()()(

)(

)(

)(

)( 212

1

x

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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 anNth-order differential equation of this type:

The state variables are defined as

The resulting state equation is

)()(

)()()(

)()(

)()(

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1

0

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1

32

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txty

tvbtxatx

txtx

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iiN

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Matrix representation

)()()()( 001 tvbtyatyaty

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)()()()(

)()(

1

021102

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i

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Nitytx ii ...,,2,1,)()1(

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00001

0

0

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1000

100

0010

01210

DC

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baaaa N

UNIT IV

ANALYSIS OF DISCRETE TIME SIGNALS

SAMPLING OF CT SIGNALS AND ALIASING,DTFT AND PROPERTIES,Z-

TRANSFORM AND PROPERTIES OF Z-TRANSFORMSAMPLING

Sampling theory

Letx(t) be a continuous signal which is to be sampled, and that sampling is performed by

measuring the value of the continuous signal every Tseconds, which is called the

sampling interval. Thus, the sampled signalx[n] given by:

x[n] =x(nT), with n = 0, 1, 2, 3, ...

Thesampling frequencyor sampling ratefs is defined as the number of samples obtained

in one second, orfs = 1/T. The sampling rate is measured inhertzor in samples persecond.

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 theNyquist frequencyof the sampling system.

Frequencies above the Nyquist frequencyfNcan be observed in the sampled signal, buttheir frequency is ambiguous. That is, a frequency component with frequencyfcannot be

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distinguished from other components with frequenciesNfN+fandNfNffor nonzerointegersN. This ambiguity is calledaliasing. To handle this problem as gracefully as

possible, most analog signals are filtered with ananti-aliasingfilter (usually alow-passfilterwith 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

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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

AliasingAliasing is a phenomenon where the high frequency components of the sampled signal

interfere with each other because of inadequate sampling s < 2m.

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

Inmathematics, the discrete-time Fourier transform (DTFT) is one of the specificforms ofFourier analysis. As such, it transforms one function into another, which is

called thefrequency domainrepresentation, or simply the "DTFT", of the originalfunction (which is often a function in thetime-domain). But the DTFT requires an input

function that isdiscrete. Such inputs are often created bysamplinga continuous function,like a person's voice.

Given a discrete set of real or complex numbers: (integers), the discrete-

time Fourier transform (orDTFT) of is usually written:

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Often the sequence represents the values (akasamples) 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 DTFTprovides an approximation of thecontinuous-time Fourier transform:

To understand this, consider thePoisson summation formula, which indicates that a

periodic summationof function can be constructed from the samples of function

The result is:

(Eq.2)

The right-hand sides ofEq.2and Eq.1are identical with these associations:

comprises exact copies of that are shifted by multiples ofs and combined

by addition. For sufficiently large s, the k=0term can be observed in the region [s/2,s/2] with little or no distortion (aliasing) from the other terms.

Inverse transform

The following inverse transforms recover the discrete-time sequence:

The integrals span one full period of the DTFT, which means that the x[n] samples are

also the coefficients of aFourier series expansionof the DTFT. Infinite limits of

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integration change the transform into acontinuous-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. Weuse the following notation:

is theconvolutionbetween two signals

is thecomplex conjugateof the functionx[n]

represents thecorrelationbetweenx[n] andy[n].

Property Time domain Frequency domainRemark

s

Linearity

Shift in time integerk

Shift in

frequency(modulation

)

realnumbera

Timereversal

Time

conjugation

Timereversal &

conjugation

Derivative

in frequency

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Integral infrequency

Convolve intime

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: TheZtransform of a discrete-time signalx(n) is defined as the power

series:

( ) ( ) k

k

X z x n z

( ) [ ( )]X z Z x n

wherezis a complex variable. The above given relations are sometimes called thedirect Z - transformbecause they transform the time-domain signalx(n) into itscomplex-plane representationX(z). SinceZtransform is an infinite power series, itexists only for those values ofzfor which this series converges. The region of

convergence ofX(z)is the set of all values ofzfor whichX(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

nznhzH ][

R

n dzzzHj

nh 1][2

1][

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Z-transform Pairs h[n] = d[n]

Region of convergence: entirez-plane

1][0

0

n

n

n

n znznzH

h[n] = d[n-1]

Region of convergence: entirez-plane

h[n-1] z-1H[z] 1

1

1

11][

zznznzHn

n

n

n

Inverse z-transform

dzzzFj

nf njc

jc

1

21

Using the definition requires a contour integration in the complexz-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, thez-transform pairs have been tabulated (see

Lathi, Table 5.1)

Z-transform PropertiesProperties of z - transform

1. Linearity

))(())(())()(( 2121 nTxZnTxZnTxnTxZ

2. Initial Value )(lim)0( zXxz

1)1()0()( zxxzX

3. Final value )()1(lim)(

1

1 zXzx z

)(lim)(

0ssXx

s

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)()1(lim)(lim

)()1()(

11

1110

1

10

1

1

1

zXzssX

zXzssX

zszs

zs

zs

UNIT V

LINEAR TIME INVARIANT DISCRETE TIME SYSTEMS

DIFFERENCE EQUATIONS,BLOCK DIAGRAMREPRESENTATION,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. Itmay 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 ofdifference 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

nxbnyanyany 021 21

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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 IGeneral form of difference equation

M

k

k

N

k

k knxbknya00

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Alternative equivalent form

Direct Form II

Cascade form

General form for cascade implementation

M

k

k

N

k

k knxbknyany01

21

21

1

11

1

1

1

11

1

1

111

111

N

k

kk

N

k

k

M

k

kk

M

k

k

zdzdzc

zgzgzf

AzH

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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 functionH(z).

CONVOLUTIO N SUMThe convolution sum provides a concise, mathematical way to expressthe output of an LTI system based on an arbitrary discrete-time input signal and the system'sresponse. 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 PP N

k

N

k kk

kk

k

k

N

k

k

kzdzd

zeB

zc

AzCzH

1 111

1

10 11

1

1

nhnxny

.zHzXzY

[ ] [ ] [ ]

[ ] [ ] [ ]

ky n x k h n k

y n x n h n

[ ]k

h k

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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 functionH(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

)()()( jjj eHeXeY

)()()( jjj eHeXeY

)()()(

jeXjjj

eeXeX

)()()(

jeHjjj eeHeH

,1z 1

z

j zHeH

jjj eHeXeY

.zHzXzY

knxbknyaM

k

k

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k

k 00

zXzbzYza kM

k

k

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k

kk

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k

k

k

kM

k

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zb

zX

zYzH

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63864722 Signals and Systems Lecture Notes

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