A signal is defined as any physical a quantity that vaies witin time, space or any

other independent variable or variables. Mathematically , describe a signal as a

function of one or more independent variable. For example the function:

S1 (t) = 5t

S2 (t) = 20t2 (1.1.1)

Describe two signal, one that varies linerly with the independent variable t (time)

and a second that varies quadratically with t. As another example consider the

function :

S(x,y) = 3x + 2xy + 10y2 (1.1.2)

This fnction desribes a signal of two independent variables x and y that could

represent the two coordinate in a plane.

The signals describes by (1.1.1) and (1.1.2) belong to a class of signals that are

precisely defined by specifying the functional dependence on the independent

variable. However, there are cases where such a functional relationship is unknown

or too highly complicated to be of any practiccal use. For example a speech signal

(1.1)(belum jelas) cannot be described functionally by expressions such as (1.1.1).

in general , a segment of speech may represented to a high degree of accuracy as

asum of several sinusoids of different amplitudes and frequencies that is as :

∑i=0

n

Ai (t ) sin [2π fi ( t ) t+θ ( t )] (1.1.3)

Where { Ai (t) }, { fi (t) } and { θi (t) } are the sets of (possibility time-varying)

amplitudes frequncies and phases, respectically of the sinusoids. In fact, one way to

interpret the information content or message conveyed by any short time segment

of the speech signal is a meassure the amplitudes frequencies and phases contained

in the short time segment of the signal.

A system may also be defined as a physicaly devices that perform an operation an a

signal. For example a filter used to reduce the noise interference corrupting a

desired informarion bearing signal is called a system. In this case the filter

performs some operation (s) (filtering ) the noise and interference from the desired

information bearing signal.

When we pass a signal through a system as in filtering , we say that we have

processed the signal. In this case the processing of the signla involves filtering the

noise and interference from desired signal. In general the system in characterized

by type of operation that it performs in the signal. For example if the operation is

linier, the system is called linear, if the operation on the signal is non linier the

system is said to be nponlinier and so forth such operations are usually referred to

as signal processing.

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

Y = T x

Where T is the operator representing some welldefined rule by which x

transformed into y. Relationship is depiced as shown in left figure. Multiple input

and or output signals are possible as shown in right figure. We will rstrict our

attention for the most part in this text to the single-input, single-output case.

System with single or multiple input output signals

Along with the classification of system below, it is also important to understand the

clssification of signals:

1. Continuous-time and discrete-time systems

This may be the simplest classification to understand as the idea of discrete-

time and continuous-time i one of the most fundamental properties to all of

signal and system. A system where the input and output signals are

continuous is a continuous system and one where the input and output

signals are discrete is a discrete system. If the input and output signals x and

y are continuous-time signals, then the system is called a continuous-time

system. If the input and output signals are discrete-time signals or

sequences, then the system is called a discrete-time system.

Continuous-time system(left) and dicrete-time system(right)

2. Liniear vs nonliniear

A liniear system is any system that obey the properties of scaling

(homogenneity) and superposition (additivity), while a nonliniear system is

any system that does not obey one of these.

To show that a sysem H obeys the scaling property is show that

H (kf(t)) = kH (f(t))

A block diagram demonstrating the scaling property of linearty

A block diagram demonstrating the superposition property of linearty

To demonstrate that a system H obeys the superposition property of

linearty is to show that:

H(f1(t) + f2(t)) = H (f1(t)) + H (f2(t))

It is possible to check asystme for liniearty in a single (though larger) step.

To do this,simply combine the first two steps to get

H(k1f1(t) + k2f2(t)) = k2H (f1(t)) + k2H (f2(t))

The general class of the systems can also be subdivided into linear

systems. A linear system is one that satisfies the superposition principle.

Simply stated, the principle of superposition requires that the response of

the system to weighted sum of signals be equal to the correspnding

weighted sum of the responses (outputs) of the system to each of the

individual input signals. Hence we have the following definition of

linearity.

Definition. A relaxed T system is linear if and only if

T[a1x1(n)+a2x2(n)] = a1T[x1(n)]+ a2T[x2(n)]

For any arbitrary input sequnces x1(n) and x2(n), and any arbitrary

constants a1 and a2.

gambar

The superposition principle in the relation up(diatas) can be separated into

two pars. First,suppose that a2 = 0 then reduces to

T[a1x1(n)] = a1T[x1(n)] = a1y1(n)

where

y1(n) = T [x1(n)]

the relation up(diatas) demonstrates the multiplicative or scaling property

of a linear system. That is, if the response of the system to the input x1(n)

is y1(n), the response to a1x1(n) is simply a1y1(n). Thus any scaling of

the input results in an identical scaling of the corresponding output.

Second, suppose that a1 = a2 = 1 in T[a1x1(n)+a2x2(n)] = a1T[x1(n)]+

a2T[x2(n)] then

T[x1(n)+ x2(n)] = T[x1(n)]+ T[x2(n)]

= y1(n) + y2(n)

This relation demonstrates the additivity of a linear system. The additivity

and multiplicative properties contitute the superposition principle as it

applies to linear systems.

The linearity condition embodied in T[a1x1(n)+a2x2(n)] = a1T[x1(n)]+

a2T[x2(n)] Can be extended arbirarily to any weighted linear

combination of signals by induction. In general we have

x(n) = ∑k=1

M−1

akxk (n )→y (n)=∑k=1

M−1

akyk (n)

where

yk(n) = T [xk(n)] k= 1,2, . . . . .,M – 1

we observe from T[a1x1(n)] = a1T[x1(n)] = a1y1(n) that if a1 = 0 , then

y(n) = 0, in other words, a relaxed, linear system with zero input produces

a zero output. If a system produces a nonzero output with a zero input, the

system may be either nonrelaxed or non linear. If a relaxed system does

not satisfy the superposition above, it is called non linear.

3. Time invariant and time variant system

A time invariant system is one that does nnot depend on when it occurs: the

shape the output does not change with a delay of the input. That is to say

that for a system H where H(f(t)) = y(t), H is time invariant it for all T

H(f(t - T)) = y(t - T)

This block diagram shows what the condition for time invariance. The output is the same weather the delay is put on the input or the output.

When this property does not holdfor a system, then it is to be time variant , or time varying.

We can subdivide the general class of systems into the two broad categories, time invariant systems and time variant systems. A system is called time –invariant if its input-output characteristics do not change with time. To elaborate, suppose that we have a system T ia a relaxed state which signal y(n). Thus we write :

y(n) = T [x(n)]

now suppose that the same input signal is delayed by k units of time to yield x(n-k), and again applied to the same system. If the characteristics of the system do not change with time, the output of the relaxed system will be y(n-k). That is, the output will be the same as the response to x(n), except that it will be delayed by the same k units in time that the input was delayed. This leads us to define a time-invariant or shift-invariant system as follows.

Definition. A relaxed system T is time invariant or shift invariant if and only if

x(n)→y(n)(panah menunjukkan perubahan T)

implies that

x(n-k)→y(n-k)

for every input signal x(n) and every time shift k.

To determinate if any given system is time invariant ,we need to perform the test specified by the preceding definition. Basically, we excite the system with an arbitrary input sequence x(n), which produces an output denoted as y(n). Next we delay the input sequence by the same amount k and recompute the output. In general, we can write the output as

y(n,k) = T [x(n-k)]

now if this output y(n,k) = y(n-k), for all possible values of k, the system is time invariant. On the other hand, if the output y y(n,k)≠ y(n-k),even for one value of k, the system is time variant

differentiator,time multiplier,folder,modulator

4. Causal vs Noncausal systems

A causal system is one the that is nonanticipative : that is, the output may

depend on the current and past inputs, but not future inputs. All “realtime”

systems must be causal, since they can not have future inputs available to

them.

One may think the idea of future inputs does not seem to make much

physical sense;however, we have only been dealing with time as our

dependent variable so far, which is not alwasy the case. Imagine rather that

we wanted to do image processing. Then the dependent variable might

represent pixels to the left and right (the “future”) of the current position on

the image, and we would have anoncausal system.

We begin with the definition of causal discrete-time systems.

Definition a system is said to be causal if the output of the system at any

time n[i.e., y(n)] depends only on present adn past inputs [i.e, x(n),x(n-1), . .

. ], but does not depend on future inputs [i.e, x(n+1),x(n+2), . . . ]. In

mathematical terms, the output of a causal system satisfies an equation of

the form

y(n) = F[x(n),x(n-1),x(n-2), . . . .]

where F[.] is some arbitrary function.

If a system does not satisfy this definition, it is called noncausal. Such a

system has an output that depends not only on present and past inputs but

also on future inputs,

It is apparent that in real-time signal processing applications we cannot

observe future values if the signal, and hence a noncausal system is

physically unrrealizable(it cannot be implemented). In the other hand , if the

signal is recorded so that the processing is done off-line(nonreal time), it is

possible to implement a noncausal system , since all values of the signal are

valiable at the time of processing . this is often the case in the processing of

geophysical signal and images.

5. Stable vs Unstable

A stable system is one where the output does not diverge as long as the

input does not diverge. A bounded input prodices a bounded output. It is

from this property that this type of system is referred to as bounded input

bounded output (BIBO) stable.

Representing this in a mathematical way, a stablesystem must have the

following property, where x (t) is the input and y (t)is the output. The

output must satisfy the condition :

|y (t)| ≤My<∞

When we have an input to the system that can be described as

|x (t)| ≤Mx<∞

Mx ND My both represent a set of finite positive numbers and these

relationship hold for all of t.

If these condition are not met,i.e. a system’s output grows without limit

(diverges) from a bounded input, then the system is unstable

6. Sytems with memory and without memory

A system is said to be momoryless if the output at any time depends on only

the input at the same time. Othewise, the system is said to have memory.

An example of a memoryless system is a resistor R with the input x(t) taken

as the current and the voltage taken as the output y(t). The input output

relationship (ohm’s law) of a resistor is

y(t) = Rx(t)

an example of a system with memory is a capasitor C with the current as the

input x(t) and the voltage as the output y(t); then

y(t) = 1C∫−∞

t

x (τ )dτ

a second example of a system with memory is a discrete-time system

whose input andoutput sequences are related by

y[n] = ∑k=−∞

n

x [k ]

Discrete-time signals

So far, we have treated what are what are known as analog signals and system.

Mathematically, analog signals are functions having countinuous quantities as their

independent variables, such as space and time. Discrete-time signals are functions

defined on the integers; they are sequences. One of the fundamentals results of

signal theory will detail conditions under which an analog signal can be converted

into a discrete-time signals can be manipulated by systems instantited as computer

programs. Subsquent modules describe how virtually all analog signal processing

can be performed with software.

Discrete-time systems in the time-domain

A discrete-time signal s(n) is delayed by n0 samples when we write s(n-n0), with

n0>0. Choosing n0 to be negative advances the signal analog the integers.

State space analaysis

So far we have studied liniear time-variant systems based on their input-output

relationships, which are known as external descriptionof the systems. In this

chapter we discuss the method of state space representation of systems, which are

known as the internal descriptions of the systems. The representation of systems in

this form has many advantages:

1. It provides an insight into the behavior of the system.

2. It allows us to hanle systems with multiple inputs and outputs in a unified

way.

3. It can be extended to nonliniear and time-varying systems.

Since the state space representation is given in terms of matrix equations,

the reader should hae some familiarity with matrix or linear algebra. A brief

review is given in App. A.

Definition : the state of a system at time t0 (or n0) is defined as the minimal

information that is suffcient to determine the state and the output of the system for

all times t ≥ t0 (n≥ n0) when the input to the system is also known for all times t ≥

t0 (n≥ n0). The variables that contain this information are called the state variables.

Note that this definition of the state of the system appies only to causal systems.

Cosider a single-input single-output LTI elecric network whose structure is known.

Then the complete knowledge of the input x (t) over the time interval -∞to t is

sufficient to determine the output y(t) over the same time interval. However, if the

input x(t) is known over only the time interval t0 to t, then the current through the

inductors and the voltage across the capacitors at some time t0 must be known in

order to determine the output y(t) over the time interval t0 to t. These current and

voltages constitute the “state” of the network at time t0. In this sense, the of the

state of the network is related to the memory of the network.

Selection of state variables :

Since the state variables of a systems can be interpreted as the “memory elemens”

of the system, for discrete-time systems which are formed by unit-delay elements,

amplifiers, and adders, we choose the outputs of the unit-delay elements as the state

variables of the system. For continuous-time systems containing physical energy-

storing elements, the outputs of these memory elements can be chosen to be the

state variaables of the system. If the system is described by difference or

differential equation, the state variables can be chosen as shown in the following

sections. Note that the choice of state variables of a system is not unique. There are

infinitely many choices for any given system.

State space representation of discrete-time LTI systems

a. Systems described by difference equation

b. Similarity transformation

c. Multiple-Input multiple-Output systems

State space representation of continuous-time LTI systems

a. Systems described by differential equations

b. Multiple-Input multiple-Output systems

Solutions of state equations for dicrete-time LTI systems

a. Solution in the time domain

b. Determination of A”

c. The z-traansform solution

d. System funtion H(z)

e. Stability

Solutions of state equations for continuous-time LTI systems

a. Laplace transform method

b. The function H(s)

c. Solution in time domain

d. Evaluation of e At

e. Stability

Liniear time-invariant systems

Two most important attributes of systems are linearity and time-invariance.

Response of a continuous-time LTI system and the convolution integral

a. Impulse response : the inpulse response h(t) of a continuous-time LTI

system (represented by T) is defined to be the response of the system when

the input is δ(t), that is,

h() = T {δ (t)}

b. Response to an arbitrary input :

x() = ∫−∞

∞

x (τ ) δ (t−τ )dτ

c. Convolution integral : defines the convolution of two continuous-time

signals x(t) and h(t) denoted by

y(t) = x(t) * h (t) = ∫−∞

∞

x (τ ) δ (t−τ )dτ

equation is commonly called the convolution integral. Thus, we have the

fundamental result that the output of any continuous-time LTIsystem is the

convolution of the input x(t) with the impulse response h(t) of the system.

Continuous-time LTI system

d. Properties of the convolution integral : the convolution integral has the

following :

1. Commutative :

x(t) * h(t) = h(t) * x(t)

2. Associative :

{ x(t) * h(t) } * h2(t) = x(t) * {h(t) * h2(t)}

3. Distributive :

x(t) * {h(t) + h2(t)} = x(t) * h(t) * h2(t)

e. Properties of continuous LTI systems

1. Systems with or without memory

y(t) = kx(t)

where k is a (gain) constant. Thus, the corresponding impulse

response h(t is simply)

h(t) = kδ (t)

therefore, if h(t0) ≠ 0 for t0 ≠ , the continuous-time LTI system has

memory.

2. Causality

h(t) = 0 t¿0

LTI system is expresses as

y(t) = ∫0

∞

x (τ ) δ (t−τ )dτ

y(t) = ∫−∞

t

x (τ ) δ (t−τ )dτ

anti causal if

h(t) = 0 t¿0

when the input x(t)

y(t) = ∫0

t

x (τ ) δ (t−τ )dτ

3. Stability

∫−∞

∞

¿h ( τ )∨dτ<∞

Response of LTI systems to arbitrary inputs: the convolution sum

Having resolved an arbitrary input signal x(n) into a weighted sum of impulses we

are now ready to determine the response of any relaxed linear system to any input

signal. First, we denote the response y(n,k), −∞<k<∞ . That is ,

y(n,k) ≡ h(n,k) = T[δ (n−k )]

in y(n,k) ≡ h(n,k) = T[δ (n−k )] we note that n is the time index and k is a

parameter showing the location of the input impulse. If the impulse at the input

scaled by an amount ckh(n,k) = x(k)h(n,k)

finally, if the input is the arbitrary signal x(n) that is expressed as a sumof weighted

impulses, that is,

x(n) = ∑k=−∞

∞

x (k ) δ(n−k )

then the response of the system to x(n) is the corresponding sum of weighted

outputs, that is,

y(n) = T[x(n)] = T[ ∑k=−∞

∞

x (k ) δ (n−k )]= [ ∑

k=−∞

∞

x (k ) δ (n−k )]= ∑k=−∞

∞

x (k )h (n , k )

Clearly ∑k=−∞

∞

x (k )h (n , k ) follows from the superposition property of linear systems,

and is known as the superposition summation.

We note that ∑k=−∞

∞

x (k )h (n , k )is an expresssion for the response of linear system to

any arbitrary input sequnce x(n).this expressin is a function of both x(n) and the

responses h(n,k) of the system to the unit impulses δ (n-k) for −∞<k<∞.

In deriving ∑k=−∞

∞

x (k )h (n , k )we used the linearity property of the system but not its

time-invariance property. Thus the expression in ∑k=−∞

∞

x (k )h (n , k )applies to any

relaxed linear (time-variant) system.

If, in addition, the system is time invariant, the formula in ∑k=−∞

∞

x (k )h (n , k )

simplifies considerably. In fact, if the response of the LTI system to the unit sample

sequence δ(n-k )is

h(n-k) = T [δ(n-k)]

consequently, the formula in ∑k=−∞

∞

x (k )h (n , k )reduces to

y (n) = ∑k=−∞

∞

x (k )h (n−k )

now we observe that the relaxed LTI system is compltely characterized by a single

function h(n), namely ,its response to the unit sample sequence δ(n). In contrast,

the general characterization of the output of a time-variant, linear system requires

an infinite number of unit sample response function, h(n,k), one for each possible

delay.

The formula y (n) = ∑k=−∞

∞

x (k )h (n−k )

That gives the response y(n) of the LTI system as a function of the input signal

x(n) and the unit sample (impulse) response h(n) is called a convolution sum. We

say that the input x(n) is convolved with the impulse response h(n) to yield. We

shall now explain the procedure for computing the response y(n), both

mathematically and gra[hically, given the input x(n) and the impulse response h(n)

of the system.

Suppose that we wish to caompute the output of the system at some time instant,

say n =n0. According to y (n) = ∑k=−∞

∞

x (k )h (n−k ) the response at n = n0 is given as

y(n0) = y (n) = ∑k=−∞

∞

x (k )h (n0−k )

our first observation is that the index in the summation is k, and hence both the

input signal x(k) and the impulse response h(n0 - k) are function of k. Second , we

observe that the sequnces x(k) and h(n0-k) are multiplied together to form a

product sequences. The output y(n0) is simply the sum over all values of the

product sequence. The sequences h(n0 - k) is obtained from h(k) by, first,folding

h(k) about k = 0 (the same origin ), which result in the the sequence h(-k). The

folded sequences is then shifted by n0 to yield h(n0-k). To summarize, the process

of computing the covolution between x(k) and h(k) involves the following four

steps.

1. Folding. Fold h(k) about k – 0 to obtain h(-k)

2. Shifting. Shift h(-k) by n0 to the rigt (left) if n0 is positive

(negative), to obtain h(n0-k)

3. Multiplication. Multiply x(k) by h(n0 - k) to obtain the product

sequnces vn0(k) ≡ x(k)h(n0 - k)

4. Summation. Sum all the values of the product sequence vn0(k) to

obtain the value of the output at time n = n0