byst cpe200-w2003: introduction 1 cpe200 signals and systems chapter 1: introduction

BYSTBYSTCPE200-W2003: IntroductionCPE200-W2003: Introduction 1

CPE200 Signals CPE200 Signals and Systemsand Systems

Chapter 1: Introduction


Signals and Systems

To study and analyze some physical phenomena, we need to determine the mean allowing us to understand and describe such phenomena in a systematic way. In practice, we can accomplish this goal by describing a physical phenomenon as a mathematical function called “Signal”.

1.1

We encounter so many signals in our daily life which are generated by natural means. For example, the speech or the sound coming into our ears is a mechanical signal representing the air pressure. The picture that we see is a light signal representing the intensity of light. All electrical devices are associated with voltage and current signals.


Definition 1: Signal is a pattern of variations of a measurable quantity that is a function of one or more independent variables such as time (t) and space (x and y).

Signal can be represented mathematically as:

S(t) = atS(t) = at2 + bt + c

S(x,y) = ax + by + cxyS(t) = Asin(t+)

(1.1)

Where a, b, c, A, , and are constant values.

The function which is used to describe a signal is called the representation of the signal.


There are 3 components to signal theory:

Typically, a signal carries information about the behavior or nature of the phenomenon.

Modeling: A process to determine a representation of the signal. Analysis: A process to extract information carried by the signal (Signal Processing). Design: A process to synthesize a physical process that is described by the signal.

Examples of signals can be shown as following:


0 200400600800100012001400160018002000

-1

-05

0

05.

1

15. 1x0

4

A Speech A Speech SignalSignal

A speech signal is a mechanical signal representing the air pressure and carries voice information.

Am

plitu

de

Time

A Speech Waveform

Figure 1.1 Speech waveform.


A Digital Image (2-D A Digital Image (2-D Signal)Signal)

A digital image is a light signal representing the light intensity and carries visual information .

The intensity I(x,y) at any location of a digital image is a function of two spatial independent variables (x and y).

Figure 1.2 Digital image (2-D signal).


-0.01 -0.005 0 0.005 0.01-10

-8

-6

-4

-2

0

2

4

6

8

10

A continuous-time (c-t) signal is a signal that is present for all instants in time or space.

1.1.1 Types of Signals

Continuous-Time ( c-t or analog ) Signal

Time

Am

pli

tud

e1.

Figure 1.3 Continuous-time (c-t) signal.


A discrete-time (d-t) signal is a signal that is present only at certain specific values of time or space.

2. Discrete-Time (d-t) Signal

-20 -15 -10 -5 0 5 10 15 20-10

-8

-6

-4

-2

0

2

4

6

8

10

Am

pli

tud

e

Time

Figure 1.4 Discrete-time (d-t) signal.


-20 -15 -10 -5 0 5 10 15 20-3

-2

-1

0

1

2

3

A digital signal is a discrete-time signal having a set of discrete values.

3. Digital SignalA

mp

litu

de

Time

Figure 1.5 A digital signal


Typically, we will denote continuous-time signals with the continuous-time independent variable, t. We will also enclose the independent variable in parentheses ( • ). For example, x(t), y(t), o(t), etc.

In the case of discrete-time signals, we typically denote them with the discrete-time independent variable, n and enclose the independent variable in brackets [ • ]. For example, x[n], y[n], o[n], etc.


System is a physical device that performs an operation on a signal. We can consider a system as anything that takes an input signal, operates on it, and produces an output signal.

System

x(t) y(t)

Input (or Excitation) signal

Output (or Response) signal

Let x(t) and y(t) be the input and output signals, respectively, of a system. The system can be represented mathematically as:

y(t) = [x(t)] (1.2)

1.1.2 Systems

Figure 1.6 A block diagram representation of a continuous-time system.


Examples of Systems

RC Circuit

x(t) = i(t)R C +

-Vc(t) = y(t)

Mass-Spring-Damper System

M

K

D

f(t)

y(t)

cdy(t)

dt1R

+ y(t) = x(t)

Figure 1.7 RC Network.

Figure 1.8 Mass-Spring-Damper System.


D = damping coefficient

M = massK = spring constant

f(t) = force

y(t) = displacement of the mass

Md2y(t)

dt2

dy(t)dt

+ D + K y(t) = f(t)

Communication System

Transmitter

Message

Signal Channel

TransmittedSignal

ReceiverReceived

Signal

ReceivedMessage Sig

nal

Figure 1.9 Element of the communication system.


Signal processing is a method to extract useful information carried by the signal.

Analog SignaAnalog Signal Processorl Processor

C-T input signal

C-T output signal

There are two types of signal processing systems: analog signal processing and digital signal processing.

1.1.3 Signal Processing

1. Analog signal processing system. This is a system that processes the input signal directly on its analog form.

Figure 1.10 A block diagram representation of an analog signal processing system.


2. Digital signal processing system. The digital signal processing system, on the other hand, is a system that processes the input signal on its digital form.

Fig. 1.11 shows a block diagram of the digital signal processing system. To perform the processing digitally, the digital signal processing system requires two additional steps. The first step is the step to convert a continuous-time signal into a discrete-time discrete valued (digital) signal which is called the analog-to-digital (A/D) conversion. A digital signal produced from the digital signal processor is converted back to continuous-time form by the other additional step called the digital-to-analog (D/A) conversion.


Digital SignaDigital Signal Processorl Processor

C-T input signal

C-T output signal

Analog-to- DiAnalog-to- Digital Convertgital Convert

erer

Digital-to- AnDigital-to- Analog Convertealog Converte

rr

Digital signal

Digital signal

Figure 1.11 A block diagram representation of a digital signal processing system.


Analog-to-Digital Conversion1.2The digital signal is generated from the analog (continuous-time) signal using these following two-step process:

Analog SignalAnalog Signal

Discrete-Time Discrete-Time SignalSignal

Digital SignalDigital Signal

1. Sampling Process1. Sampling Process

2. Quantization Process2. Quantization Process

A sampling process is the process to sample a continuous-time (c-t) signal at a certain period of time called the sampling interval.

Figure 1.12 A two-step process to generate a digital signal from an analog signal.


Thus, the sampling process will convert a c-t signal into a discrete-time (d-t) signal.

Where T = the sampling period

i.e. (1.3)sF

nnTta )t(x)n(x

Fs = the sampling frequency

= 1/Tn = 0, ±1, ±2,…

xa(t)

Fs = 1/T

Sampler

x(n) = xa(nT)

Analog Signal

Discrete-time Signal

Figure 1.13 Periodic sampling of an analog signal.


A quantization process is the process to round up the values of the d-t signal to a finite set of possible values. Thus, the quantization process will convert a d-t continuous-valued signal into a d-t discrete-valued (digital) signal.

A digitization process is the process to convert a analog signal into an encoded digital signal. This process is usually called analog-to-digital (A/D) conversion and is illustrated in this Fig. 1.14.

1.2.1 Digitization


SamplerSampler

QuantizerQuantizer

EncoderEncoder

Figure 1.14 A block diagram of a digitization process.

Analog signal

Discrete-time continuous-valued signal

Discrete-time discrete-valued (digital) signal

Encoded-digital signal


Digital Sound RecordingDigital Sound Recording

Sampling

Quantizing

Encoding

An Example of Digital Signal ProAn Example of Digital Signal Processingcessing


The Concept of Frequency in Continuous-Time and Discrete-Time Signals

1.3

Acos()

xa(t)

Tp

xa(t) = Acos(t + )

An analog sinusoidal signal xa(t) can be represented as:

t

(1.4)

Figure 1.15 An analog sinusoidal signal.

Where

t


=

Tp = the fundamental period (sec)1F

The analog sinusoidal signal described by Eq. 1.3 has these following properties:

1. Periodical: i.e.

2. Distinction: C-t sinusoidal signals with different frequencies are themselves distinct.

xa(t + Tp) = xa(t)

3. The rate of oscillation of the signal will be increased if the frequency F is increased.

= the angular frequency (rad/sec)

= 2FF = the frequency (cycles/sec)

= the phase

(1.5)


x(n) = Acos(n + )

n

Figure 1.16 A discrete-time sinusoidal signal.

x[n] = Acos(n + )

A discrete-time sinusoidal signal x(n) can be represented as:

(1.6)

= the angular frequency (rad/sample)

Where

= 2f

n


f = the frequency (samples/sec)

= the phase

The discrete-time sinusoidal signal described by Eq. 1.6 has these following properties:

1.

x[n + N] = x[n] for all n

A discrete-time sinusoid is periodic only if the frequency of the d-t signal is a rational number.By definition, a d-t signal x(n) is periodic with period N (N>0) if an only if

(1.7)

The smallest possible N satisfying the above condition is called “fundamental period” of the d-t signal.


= 2f0

Eq. 1.8 will be equal to Eq. 1.6 only if sinN = 0 and cosN = 1 which can be satisfied if and only if:

If x[n] = Acos(n + ) , x[n+N] will be defined as:

x[n+N] = Acos(n+N) + )

= A[cos(n+ )cos - sin(n+ )sin)]

(1.8)

Where = the fundamentalangular frequency

f0 = the fundamentalfrequency


For a d-t sinusoidal x[n] with frequency f0 to be periodic, this following condition must exist for any integer k:

f0 =kN (1.9)

i.e.

0 =2kN

or

N = 2k

Where k = any integer number


2. Non-distinction: Discrete-time sinusoids that have frequencies separated by an integer multiple of 2 are identical. Hence the frequency range for d-t sinusoids is finite with duration 2. Usually, the range:

3. At = or - (f=0.5 or -0.5), the highest frequency in a discrete-time signal is obtained.

or20

is used and it is called the fundamental range.

)2

1f

2

1or1f0(


To establish the relationship between the frequency F (or ) of analog signals and the frequency f (or ) of d-t signals, we start from considering an analog sinusoidal signal expressed as shown in Eq. (1.4):

xa(t) = Acos(t + )

When we sample this analog signal at a rate of Fs samples per second, the d-t signal x(n) can be expressed as follows:

xa(nT) x(n) = Acos(nT + ) (1.10)

By comparing Eq. (1.10) with Eq. (1.4), the relationship between the angular frequency of a d-t signal and the angular frequency of an analog signal can be expressed as shown in Eq. (1.11):


(1.11) = T

(1.12)f = F/Fs

or

The conversion from the analog frequency to the digital frequency or the conversion from the digital frequency to the analog frequency can be summarized as illustrated in Table 1.1.

From Table 1.1, we can notice that the basic difference between c-t and d-t signals is in their range of frequency values. C-T signals have the infinite frequency range. D-T signals, on the other hand, have the finite frequency range. Thus there is a possibility that more than one analog frequencies are mapped into the same digital frequency.


Table 1.1 Relations among frequency variables

C-T Signals D-T Signals

= 2F = 2f

= T and f = F/Fs

= T and F = fFs

- < < - < F <

<_- <_

-1/2 <_

<_

<_f 1/2

<_ --Fs/2 <_ <_F Fs/2


The highest frequency in c-t signal depends on the sampling frequency. Since the highest frequency in a d-t signal is:

TFsmax

(1.14)

(1.13)max = or fmax = 1/2,

the corresponding highest values of analog frequency must be:

T2

1

2

FF s

max (1.15)or


The Sampling Theorem

An analog signal xa(t) can be reconstructed from its sample values xa(nT) if the sampling rate 1/T is greater than twice the highest frequency Fmax presenting in xa(t).

Definition: The sampling rate 2Fmax for an analog band-limited signal is referred to as the Nyquist rate.

Therefore, any frequency above Fs/2 or below -Fs/2 results in samples that are identical to corresponding frequency in the range:

S SF FF

2 2


Thus, we must have some knowledge about the frequency content of any given analog signal in order to sample it with an appropriate sampling rate. However, this detailed knowledge of the characteristics of such signals is normally not available prior to obtaining the signal. In fact, it is the information that we would like to extract in digital signal processing.

From the sampling theorem, we must sample a c-t signal with Fs > 2Fmax to ensure that all the sinusoidal components in the c-t signal are mapped into corresponding d-t frequency components with frequency in the fundamental interval.


Classification of Signals1.3

T

T

22

Tdt|)t(x|dt|)t(x|limE (1.16)

dt|)t(x|T2

1limP

T

T

2

T

(1.17)

For a c-t signal,

and

Let x(t) be a c-t signal and x[n] be a d-t signal.

The total energy, E, and power, P, of x(t)over an infinite time interval time interval can be determined as follows:

1.3.1 Energy and Power Signals


(1.18)

N

Nn

22

N|]n[x||]n[x|limE

(1.19)

N

Nn

2

N|]n[x|

1N2

1limP

and

Similarly, E and P for a d-t signal can be defined as follows:

A signal x(t)/x[n] is called an “energy signal” if and only if the total energy is finite which means the power is equal to zero.

A signal x(t)/x[n] is called a “power signal” if and only if the total energy is infinite and the power is finite.


Energy signal

Power signal

0

P0

i.e.

E P

1.3.2 Periodic and Aperiodic Signals

A signal x(t) is periodic with period T (T>0) if and only if:

x(t+kT) = x(t) (1.20)

Where k is any integer.

The smallest value of T for which Eq. 1.20 holds is called the “fundamental period”.


A signal x(t) that is not periodic will be called an “aperiodic” signal.

On the other hand, if there is no value of N satisfying Eq. 1.21, the signal is called an “aperiodic” (non-periodic) signal.

In case of a d-t signal, a signal x[n] is periodic with period N (N>0) if and only if:

x[n+kN] = x[n] for all n (1.21)

Where k is any integer.

The smallest value of N for which Eq. 1.21 holds is called the “fundamental period”.


A signal x(t) or x[n] is called a conjugate- symmetric signal if

1.3.3 Classification Based on Symmetry

x(t) = x*(-t)

x[n] = x*[-n]

or (1.22)

On the other hand, signal x(t) or x[n] is called a conjugate- antisymmetric signal if

x(t) = -x*(-t)

x[n] = -x*[-n]

or (1.23)


In the case of real-value signal, a conjugate- symmetric signal is called an “even signal” and a conjugate-antisymmetric signal is called an “odd signal”. Thus

x(t) = x (-t)

x[n] = x [-n]or (1.24)

x(t) = -x(-t)

x[n] = -x[-n]or (1.25)

Even signal:

Odd signal:

From Eq. 1.25, x(0) = -x(0); therefore, an odd signal must be 0 at t=0 or n=0.


Any complex signal x(t) or x[n] can be expressed as a sum of its conjugate- symmetric part xcs(t) or xcs[n] and its conjugate-antisymmetric part xca(t) or xca[n] as show in Eq. 1.26.

x(t) = xcs(t) + xca(t)

x[n] = xcs[n] + xca[n]or (1.26)

or (1.27)

Where

xcs(t) = (x(t) + x*(-t))12

xcs[n] = (x[n] + x*[-n])12

and


(1.28)or

xca(t) = (x(t) - x*(-t))12

xca[n] = (x[n] - x*[-n])12

Similarly, any real signal x(t) or x[n] can be expressed as a sum of its even part xev(t) or xev[n] and its odd part xod(t) or xod[n] as show in Eq. 1.29.

x(t) = xev(t) + xod(t)

x[n] = xev[n] + xod[n]or (1.29)

or (1.30)

Where

xev(t) = (x(t) + x(-t))12

xev[n] = (x[n] + x[-n])12


(1.31)or

xod(t) = (x(t) - x(-t))12

xod[n] = (x[n] - x[-n])12

and

An example of the even signal is a sinusoidal signal expressed as the cosine function. A sinusoidal signal expressed as the sine function is an example of the odd signal.

Basic Operation on Signals1.4

Typically, in the area of signal processing, it is required some basic operations to perform a simply process such as addition, multiplication, amplify, etc. on the signals.Note: Since all operations discussed later will perform the same results to c-t and d-t signals, only the d-t signals will be discussed or illustrated for convenience.


1.4.1 A Signal Multiplier

A signal multiplier will form the product of values of x1(t)/x1[n] and x2(t)/x2[n] signals at each instant as illustrated in Fig. 1.17.

Figure 1.17 A signal multiplier.

For a signal multiplier having a sinusoidal signal as one of its input, this operation will be called “modulation”. The device performs the modulation operation is known as a “modulator”.

Xx1[n]

x2[n]

y[n] = x1[n]x2[n]


1.4.2 An AdderAn adder will form the addition of values of x1(t)/x1[n] and x2(t)/x2[n] signals at each instant as illustrated in Fig. 1.18.

Figure 1.18 An adder.

An output y(t) of a scalar multiplier will be the result of multiplication a input signal x(t) with a scalar A as illustrated in Fig. 1.19.

1.4.3 A Scalar Multiplier

x1[n]

x2[n]

y[n] = x1[n] + x2[n]+


Figure 1.19 A scalar Multiplier

An operation that performs a time delaying and advancing on signal is known as “time-shifting” operation.

1.4.4 A Time Shifting

Let x(t) or x[n] be an input signal and y(t) or y[n] be an output signal resulting from a time-shifting operation. y(t) and y[n] will be defined as follows:

y(t) = x(t - t0)

where t0 is the time shift and N is an integer

(1.32)

x[n] y[n] = Ax[n]A

y[n] = x[n - N]


t0 > 0Delaying (Shift to the right)

t0 < 0 Advancing (Shift to the left)

N > 0

N < 0

1.4.5 A Time Reversal

A time reversal, which sometimes is called the folding or the reflection operation, is the operation to replace the independent variable “t” or “n” by “-t” or “-n” as shown in Eq. 1.33.

y(t) = x(-t)(1.33)

y[n] = x[-n]


Let TD and TR represent the time-shifting and time-reversal, respectively. Thus, we can represent TD and TR as shown in Eq. 1.34.

TDk[x[n]] = x[n-k] k > 0(1.34)

TR[x[n]] = x[-n]

The combination operation of TD and TR can be expressed as either Eq. 1.35 or Eq. 1.36. We can notice that the result of performing TR before TD (Eq. 1.35) is different from the result of performing TD before TR (Eq. 1.36).

TDk{TR[x[n]]} = TDk{x[-n]}

(1.35)= x[-n+k]


TR{TDk[x[n]]} = TR{x[n-k]}

(1.36)= x[-n-k]

3

2

0

2

1 2

1

3 4-1-2-3-4

4

3

22

1

4

0 1 2 3 4-1-2-3-4

x[n] x[-n]

3

2

0

2

1 2

1

3 4-1-2-3-4

4x[n-2]

5 6

... . .. ... ...

. .. ...

1

2

3

4

0 1 2 3 4-1-2-3-4

y1[n] = FD{TD2[x[n]]} = x[-n-2]

... ...

3

22

1

4

0 1 2 3 4-1-2-3-4

... ...

5 6

2

-5-6 5 6

-5-6

1.Folding

2.Delaying1

.Del

ayin

g

2.Folding

y2[n] = TD2{FD[x[n]]} = x[-n+2]

(b)

(a)

Figure 1.20 Graphical illustration of the different output signal generated by:

(a) Time-delaying the input signal and then folding.

(b) Folding the input signal and then time delaying.


1.4.6 Time Scaling

Let x(t) be a c-t signal and a d-t signal, respectively. The signal y(t) obtained by scaling the independent variable, time t, by a factor of a is defined by

y(t) = x(at) (1.37)

Compressinga > 1

Stretchinga < 1

In the discrete time case, this operation is known as “sampling rate alteration”.Let x[n] be a d-t signal with a sampling rate of Fx Hz. And y[n] be the d-t signal from altering the sampling rate Fx Hz. To a new sampling rate Fy Hz. The sampling rate alteration ration R can be expressed by


(1.37)Fy

Fx

= R

If R is greater than 1, the process is called “interpolation” process and the operation is call “up-sampling”. Thus, the up-sampling operation will increase the number of samples in the input signal.

If R is less than 1, on the other hand, the process is called “decimation” process and the operation is call “down-sampling”. The number of samples in the input signal will be decreased after the down-sampling process.

To perform an up-sampling operation by an integer factor of L (L>1), L-1 new samples between successive values of the input signal will be interpolated.


This interpolation process can be accomplished in a various means. The easiest and very common mean to up-sampling the sequence is performed by adding L-1 zeros between successive values of the input signal x[n]. This up-sampling operation can be expressed by Eq. 1.38.

otherwise0

,L2,L,0n]L/n[x]n[xu

(1.38)

The decimation process by a factor of an integer M (M>1) can be performed by taking only every Mth. Sample of the input signal. This results in a signal with a lower sampling rate. The down-sampling operation by a factor of M is expressed by Eq. 1.39

(1.39)xd[n] = x[Mn]


0 10 20 30 40 50-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Input Sequence

Time index n

Am

plit

ud

e

0 10 20 30 40 50-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Output Sequence

Time index n

Am

plit

ude

(a)

(b)Figure 1.21 Graphical illustration of the up-sampling operation. (a) the input signal. (b) Illustration of the output resulting from up-sampling the input in (a) by 4.


0 10 20 30 40 50

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Output Sequence

Am

plit

ud

e

Time index n

0 10 20 30 40 50

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Input Sequence

Time index n

Am

plit

ud

e

(a)

(b)Figure 1.22 Graphical illustration of the down-sampling operation. (a) the input signal. (b) Illustration of the output resulting from down-sampling the input in (a) by 4.


Elementary Signals1.5There are some elementary signals that are used in studying signals and systems. Such signals are exponential and sinusoidal signals (See section 1.3), the impulse function, the step function, and ramp function.

A exponential signal is of the form:

1.5.1 Exponential Signals

x(t) = Ceat (1.40)

If C and a are complex numbers, a exponential signal x(t) defined by Eq. 1.40 will be called “complex exponential” signal.

In the case that C and a are real, x(t) defined by Eq. 1.40 will be called “real exponential” signal.


There are basically two types of exponential signals which determines by the value of “a”.

Growing exponentiala > 0

Decaying exponentiala < 0

1.5.2 Relationship Between Sinusoidal and Complex Exponential Signals

Consider the complex exponential containing “a” to be purely imaginary defined as follows:

(1.41)tj 0Ce)t(x jAeCwhere and A is real.

Using Euler’s identity, we can write the complex exponential in Eq. 1.41 in term of


sinusoidal signals as:

tsinjtcose 00tj 0

(1.42)

)tsin(jA)tcos(A)t(x 00

}CeRe{)tcos(A tj0

0(1.43)

i.e.

and

}CeIm{)tsin(A tj0

0

where Re{ } and Im{ } denote the real and imaginary part of the complex number, respectively.

The term Acos(t + ) can also represent as:

tjjtjj0

00 ee2

Aee

2

A)tcos(A

(1.44)


Similarly, in the d-t system, we can represent:

}CeRe{)ncos(A nj0

0

(1.45)

and

}CeIm{)nsin(A nj0

0

njjnjj0

00 ee2

Aee

2

A)ncos(A

1.5.3 The D-T Unit Impulse and Unit Step Functions

The d-t unit impulse (or unit sample) signal, [n], plays the most important role in the representation of any d-t signals and in the analysis of a d-t system. This signal is defined as:


0nfor0

0nfor1]n[

The unit impulse signal is a signal that is zero everywhere, except at n=0 where its value is unity.

(1.46)

Any D-T signal is the sum of scaled and shifted unit impulses.

k

x[n] x[k] [n k]

=

x[-1][n+1]

2

-1 + +

+

2

x[1][n-1]

1

x[2][n-2]

2

-2

x[0][n-0]

3

0

(1.47)

2

2

3

-1 0 1

-2

2


The d-t unit step signal is denoted as u[n] and is defined as:

0nfor0

0nfor1]n[u (1.48)

From Eq. 1.46 and Eq. 1.48, the relationship between the unit impulse signal and the unit step signal can be stated as follows:

n

m

]m[]n[u (1.49)

(1.50)[n] = u[n] - u[n-1]

When x[n] in the Eq. 1.47 is equal to u[n], the Eq. 1.47 will be reduced to:

0k

]kn[]n[u (1.51)

(running sum)


1.5.4 The C-T Unit Impulse and Unit Step Functions

The c-t unit step signal is denoted as u(t) and is defined as:

0tfor0

0tfor1)t(u (1.52)

Eq. 1.52 indicates that the c-t unit step u(t) is discontinuous at t=0.

The c-t unit impulse, commonly denoted by (t), is defined by the following pair of relations:

0tfor0)t(

1dt)t(

(1.53)

(1.54)


Eq. 1.53 states that the c-t unit impulse (t) is zero everywhere except at the origin and Eq. 1.54 states that the total area under (t) is unity. The c-t unit impulse (t) is also referred to as the Dirac delta function.

Similar to the d-t case, u(t) the running integral of (t) with respect to time t as illustrated by Eq. 1.55. Conversely, (t) is the derivative of u(t) as illustrated by Eq. 1.56.

t

d)()t(u (1.55)(running integral)

dt

)t(du)t( (1.56)


As we can see that the relationship between (t) and u(t) is harder to visualize than the relationship between [n] and u[n] because u(t) is discontinuous at t=0. One way to visualize this relationship is to consider an approximation to unit step u(t) which slowly increases its values from 0 to 1 in a short time interval of length as illustrate in Fig. 1.23.

t0

u(t)

1

Figure 1.23 Continuous approximation to the unit step, u(t).

Thus

)}t(u{lim)t(u0

(1.57)


Unlike u(t), u(t) is differentiable and the result is illustrate in Fig. 1.24.

t0

(t)

1

Figure 1.24 Derivative of u(t).

dt

)t(du)t(

(1.58)

Thus

Note that we can consider (t) is a short pulse having a unit area for any value of . As is decreased, its amplitude is increased such that the area under the pulse is maintained constant at unity. Finally, if is infinitesimal ( → 0), its amplitude will be


i.e.

(t) has infinite amplitude but finite area.

infinite. However, its area still remain at unity. Thus, we can consider (t) as a pulse having infinitesimal width.

)}t({lim)t(0

(1.59)

It is convention to graphically illustrate (t) as an arrow having a unit area (Fig. 1.25).

t0

(t)

11

Figure 1.24 Derivative of u(t).

Area = 1(Not an Amplitude)

Area is concentrated at t=0.


1.5.5 Ramp Functions

The ramp signal is denoted as ur(t) and is defined as:

0tfor0

0tfort)t(u r (1.60)

The ramp signal is similar to the unit step signal, except its value is equal to t when t is greater than or equal to 0. The ramp signal is considered as the integral of the unit step function u(t).

Equivalently, we may defined the ramp signal as:

ur(t) = tu(t) (1.61)


(1.62)

For a d-t ramp signal,

0nfor0

0nforn]n[u r

or, equivalently,

ur[n] = nu[n] (1.63)

Basic System Properties1.6

The properties of a system describe the characteristic of the operator (See 1.1.2) representing the system. In this section we will consider the most basic properties of systems such as static/dynamic, linearity, causality, time-invariance, and stability.


1.6.1 Static and Dynamic System

Static System: A system that the output depends only on the currecurrentnt input.

=

Thus, a static system will be considered as a system without memory (memorylessmemoryless).

Dynamic System: A system that the output depends not only on the currentcurrent input but also on the pastpast or the ffutureuture inputs.

=

Thus, a dynamic system will be considered as a system with memorymemory.

Note that “memorymemory”, in many physical systems, is referred to the storage of energystorage of energy.


1.6.2 Invertibility and Inverse System

InputInput OutputOutput

1 to 1 Mapping1 to 1 Mapping

InvertibleInvertible

For each input, the system will produce a unique output.

A system is said to be invertibleinvertible if the inputinput of the system can be recoveredrecovered from the system output. In other words, If a system is invertible, there existsexists an inverse systeminverse system such that when cascaded with the original system, yields an output equal to the original input (Fig. 1.25).


System

InverseSystem-1

x(t) y(t)

Figure 1.25 A general invertible system.

w(t) = x(t)

From Fig. 1.25, the second c-t system is called an “inverse systeminverse system” and represented by the operator -1-1. This operator is called the “inverse operatorinverse operator”. Note that, here, -1-1 is not the reciprocal of the operator .

i.e. In this case,

11


1.6.3 Linearity

Theorem A system is linearlinear if and only if

{a{a11xx11[n] + a[n] + a22xx22[n]} = a[n]} = a11{x{x11[n]} + a[n]} + a22{x{x22[n]}[n]} (1.64)

where

a1 and a2 = any constants.

x1[n] and x2[n] = any input signals,

it satisfies the superposition principlesuperposition principle defined by Eq 1.64:

Let y[n] be the output of a d-t system when the input signal is x[n]. Thus

y[n] = {x[n]}.


2. AdditivityAdditivity property.

Then, Eq. 1.64 simply states that a linear system requires two properties:

1. ScalingScaling property (MultiplicativeMultiplicative)

{ax[n]} = a{ax[n]} = a{x[n]} = ay[n]{x[n]} = ay[n]

If the input signal x[n] is scaled by “a”, the output of a linear system will be scaled by the same factor.

Thus,

(1.65)

This property can be simply demonstrated by Fig. 1.26 or Eq. 1.66 as follows:

{x{x11[n] + x[n] + x22[n]} = [n]} = {x{x11[n]} + [n]} + {x{x22[n]}[n]}

(1.66)


The scaling and additivity properties of a linear system can be depicted by Fig. 1.26 as follows:

x1[n]

x2[n]

a1

a2

+y[n]

x1[n]

x2[n]

a1

a2

+y’[n]

Figure 1.26 Graphical representation of the superposition principle.


1.6.4 Time Invariance

Theorem A system is time invarianttime invariant or

or shift invariantshift invariant if and only if

y[n] = {x[n]}

implies that

y[n-k] = {x[n-k]}.

for every input signal x[n] and every time shtime shift kift k.

where

y[n] = the response of a system when the input is x[n],

y[n,k] = the response of a system when the input is x[n-k],

and

(1.67)


Eq. 1.67 implies that the output (response) signal of a time-invariant system does not depend on the time when the input is applied, k. In the other words, the characteristicscharacteristics of a time-invariant system is not a function of not a function of timetime.

In contrast, the system is said to be time varitime variantant if its characteristics is a function of time.

y[n-k] = y[n] which is delayed by k.

= TDk{y[n]}.

Therefore, if

y[n,k] = y[n-k]for all k,

Time-Invariant System


y[n,k] ≠ y[n-k]for all k,

If

Time-Variant System

1.6.5 Causality

A system is causalcausal if the response at any time does not depend on values of the future inputs. Thus, in a causal system, the n0th (t0th) output signal depends only on input signals x[n] (x(t)) for n ≤ n0 (t ≤ t0).

i.e. For a casual system,

y[n] = F{x[n], x[n-1], x[n-2], …}

ory(t) = F{x(t), x(t-1), x(t-2), …}


1.6.6 Stability

A system is said to be bounded input- bound bounded input- bounded output (BIBO) stableed output (BIBO) stable if and only if every bounded input results in a bounded output.

x[n] is the bounded input if

|x[n]| ≤ Mx < ∞ for all n

or|x(t)| ≤ Mx < ∞ for all t

(1.68)

where

Mx = some finite positive numbers

Therefore, whenever, the input signals satisfy Eq. 1.68, the output of BIBO stable system must satisfies the following condition:


|y[n]| ≤ My < ∞ for all n

or

|y(t)| ≤ My < ∞ for all t

(1.69)

where

My = some finite positive numbers

byst cpe200-w2003: introduction 1 cpe200 signals and systems chapter 1: introduction

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