Classification of SignalsIand Systems1Syllabus:Introduction,
Continuous Time and discrete time signals, classification of
signals, simple manipulations of discrete time signals, amplitude
and phase spectra, classification of systems, analog to digital
conversion of signals.I;ContentsPage No.1.1 Introduction1-21.2
Concept of Signal and Signal Processing1 -21.3 Block Diagram
Representation of DSP System1-31.4 Analog to Digital Conversion of
Signals1 -41.5 Classification of Signals1 -51.6 Representation of
DT Signals1-51.7 Basic Sequences1-61.8 Simple Manipulation of
Discrete Time Signals1 -61.9 Classification of Signals1 -111.10
Representation of DT Signals1 -281.11 Basic Sequences1-301.12
Simple Manipulation of Discrete Time Signals1-361.13 Relationship
between Unit Step and Unit Impulse1-561.14 1.1introduction :The
world of science and engineering is filled with signals such as
images from remote space probes, voltages generated by the heart
and brain and countless other applications.1.1.1 What is
DSP?Digital signal processing is used in a wide variety of
applications. It is hard to get exact definition of DSP.Let us
first look at the dictionary meanings of these words:Digital:
Operating by the use of discrete signals to represent data in the
form of numbers.Signal: A variable parameter by which information
is conveyed through an electronic circuit.Processing : To perform
operations on data according to programmed instructions. This leads
to a simple definition of DSP.Definition of DSP : DSP is defined as
changing or analysing information which is measured as discrete
sequences of numbers.1.2Concept of Signal and Signal Processing :
In a communication system, the word 'signal' is very commonly used.
Therefore we must know its exact meaning. Mathematically, signal is
described as a function of one or more independent variables.
Basically it is a physical quantity. It varies with some dependent
or independent variables. So the term signal is defined as "A
physical quantity which contains some information and which is
function of one or more independent variables." The signals can be
one-dimensional or multidimensional.One dimensional signals: When
the function depends on a single variable, the signal is said to be
one dimensional. Example of one dimensional signal is speech signal
whose amplitude varies with time.Multidimensional signals: When the
function depends on two or more variables, the signal is said to be
multidimensional. The example of a multidimensional signal is an
image because it is a two dimensional signal with horizontal and
vertical co-ordinates.
J1.2.1 System :A system is defined as the entity that operates
onone or more signals to accomplish a function, toproduce new
signals.Fig. 1.2.1 demonstrates the interaction betweensignals and
system.The types of input and output signals depends on the type of
system being used.
1.2.2 Types of Systems :Signals and systems have several
applications. Some of the important types of systems are asfollows
:1. Communication system.4. Biomedical signal processing.2. Control
system.5. Auditory system.3. Remote sensing system.1.3 Block
Diagram Representation of DSP System :Fig. 1.3.1 shows that the
basic elements of digital signal processing system
Fig. 1.3.1The different blocks of this system are as follows
:1.Input signal:It is the signal generated from some transducer or
from some communication system. It may be biomedical signal like
ECG or EEG. Generally input signal is analog in nature. It is
denoted by x(t).2.Anti-aliasing filter:Anti aliasing filter is
basically a low pass filter. It is used for the following purposes
:(a) It removes the high frequency-noise contain in input
signal.(b) As the name indicates; it avoids aliasing effect. That
means it is used to band limit the signal.3.Sample and hold
circuit:As the name indicates; this block takes the samples of
input signal. It keeps the voltage level of input signal relatively
constant which is the requirement of ADC.Sometimes amplifiers are
used to bring the voltage level of input signal upto the required
voltage level of ADC.4.Analog to digital converter (ADC):As the
name indicates; this block is used to convert analog signal into
digital form. This is required because digital signal processor
accepts the signal which is digital in nature.5.Digital signal
processor :It processes input signal digitally. In a simple
languages processing of input signal making modifying the signal as
per requirement. For this purpose DSP processors like ADSP 2100 or
TMS 320 can be used.
6.Digital to analog converter (DAC):The output of digital signal
processor is digital in nature. But the required final output is
analog in nature. So to convert digital signal into analog signal
DAC is used.7.Reconstruction filter:Output signal of DAC is analog,
that means it is a continuous signal. But it may contain high
frequency components. Such high frequency components are unwanted.
To remove these components; reconstruction filter is used.1.4
Advantages of Digital over Analog Signal Processing :
1. Versatility : Digital systems can be reprogrammed for other
applications (where programmable DSP chips are used). Moreover,
digital systems can be ported to different hardware.2.
Repeatability : Digital systems can be easily duplicated. These
systems do not depend upon component tolerances and temperature.3.
Simplicity : It is easy to built any digital system as compared to
an analog one.4. Accuracy : To design analog system; analog
components like resistors, capacitors and inductors are used. The
tolerance of these components reduce accuracy of analog system.
While in case of DSP ; much better accuracy is obtained.5. Remote
processing : Analog signals are difficult to store because of
problems like noise and distortion. While digital signal can be
easily stored on storage media like magnetic tapes, disks etc. Thus
compared to analog signals; digital signals can be easily
transposed. So remote processing of digital signal can be done
easily.6. Implementation of algorithms : The mathematical
processing algorithms can be easily implemented in case of digital
signal processing. But such algorithms are difficult to implement
in case of analog signals.7. Easy upgradations : Because of the use
of software; digital signal processing systems can be easily
upgraded compared to analog system.8. Compatibility : In case of
digital systems; generally all applications needs standard
hardware. Thus operation of dsp system is mainly dependent on
software. Hence universal compatibility is possible compared to
analog systems.9. Cheaper : In many applications; the digital
systems are comparatively cheaper than analog systems.10. 1.5
Disadvantages of Digital Over Analog Signal Processing :The digital
signal processing systems have many advantages. Even though there
are certaindisadvantages as follows :
1. System complexity : The digital signal processing system,
makes use of converters like ADC and DAC. This increases the system
complexity compared to analog systems. Similarly in many
applications; the time required for this conversion is more.2.
Bandwidth limitation : In case of DSP system; if input signal is
having wide bandwidth then it demands for high speed ADC. This is
because, to avoid aliasing effect, the sampling rate should be
atleast twice the bandwidth. Thus such signals require fast digital
signal processors. But always there is a practical limitation in
the speed of processors and ADC.3. Power consumption : A typical
digital signal processing chip contains more than 4 lakh
transistors. Thus power dissipation is more in dsp systems compared
to analog systems.4. Cost: For small applications digital signal
processing systems are expensive compared to analog systems.1.6
Comparison between Digital and Analog Signal Processing :Table
1.6.1 shows comparison between digital and analog signal
processingTable 1.6.1: Comparison between digital and analog signal
processing
1.7 Applications of DSP :1.8 Analog to Digital Conversion of
Signals : An incoming signal may come from a digital or analog
source. If it is coming from a digital source then it is in the
right form for processing digitally. But input signal can be analog
in nature, (e.g. speech signal or video signal). Then it has to be
converted into digital form before it can be processed by a digital
system. This type of conversion is performed using analog to
digital converters. (A/ D)The simple block diagram of ADC is shown
in Fig. 1.8.1.
Fig. 1.8.1: Analog to digital conversion1.8.1 Sampling: In order
to represent the original message signal "faithfully" (without loss
of information), it is necessary to take as many samples of the
original signal as possible. Higher the number of samples, closer
is the representation. The number of samples depends on the
"sampling rate" and the maximum frequency of the signal to be
sampled. Sampling theorem was introduced to the communication
theory in 1949 by Shannon. Therefore this theorem is also called as
"Shannon's sampling theorem". The statement of sampling theorem in
time domain, for the bandlimited signals of finite energy is as
follows :Statement:(i) If a finite energy signal x(t) contains no
frequencies higher than "W" Hz (i.e. it is a bandlimited signal)
then it is completely determined by specifying its values at the
instants of time which are spaced (1/2W) seconds apart. ii) If a
finite energy signal x (t) contains no frequency components higher
than "W" Hz then it may be completely recovered from its samples
which are spaced (1/2W) seconds apart.
1.8.2 Quantization : Quantization is a process of approximation
or rounding off. The sampled signal is applied to the quantizer
block. Quantizer converts the sampled signal into an approximate
quantized signal which consists of only a finite number of
predecided voltage levels. Each sampled value at the input of the
quantizer is approximated or rounded off to the nearest standard
predecide voltage level.These standard levels are known as the
"quantization levels."The quantization process takes place as
follows :The input signal x (t) is assumed to have a peak to peak
swing of VL to VH volts. This entirevoltage range has been divided
into "Q" equal intervals each of size "S".
At the center of these steps, the quantization levels q0, qv ...
q7 are located. xq (t) represents the quantized version of x (t).
We obtain xq (t) at the output of the quantizer. When x (t) is in
the range A0, then corresponding to each value of x (t), the
quantizer output will be equal to "q0". Similarly for all the
values of x (t) in the range Al5 the quantizer output is constant
equal to "qj". Thus in each range from A0 to A7 , the signal x (t)
is rounded off to the nearest quantization level and the quantized
signal is produced. The quantized signal x (t) is thus an
approximation of x (t). The difference between them is called as
quantization error or quantization noise. This error should be as
small as possible. To minimize the quantization error we need to
reduce the step size "S" by increasing the number of quantization
levels Q.Why is quantization required ? If we do not use the
quantizer block, then we will have to convert each and every
sampled value into a unique digital word. This will need a large
number of bits per word (N). This will increase the bit rate and
hence the bandwidth requirement of the channel. To avoid this, if
we use a quantizer with only 256 quantization levels then all the
sampled values will be finally approximated into only 256 distinct
voltage levels. So we need only 8 bits per word to represent each
quantized sampled value. Thus the number of bits per word can be
reduced. This will eventually reduce the bit rate and bandwidth
requirement.Quantization error or quantization noise e :The
difference between the instantaneous values of the quantized signal
and input is called asquantization error or quantization noise.e =
xq(t)-x(t)...(1.8.2)The quantization error is shown by shaded
portions of the waveform in Fig. 1.8.2.The maximum value of
quantization error is S/2 where S is step size. Therefore to reduce
thequantization error we have to reduce the step size by increasing
the number of quantization levelsi.e. Q.The mean square value of
the quantization is given by,S2Mean square value ofquantization
error = tx...(1.8.3)The relation between the number of quantization
levels Q and the number of bits per word (N) inthe transmitted
signal can be found as follows :Because each quantized level is to
be converted into a unique N bit digital word, assuming a binary
coded output signal.The number of quantization levels Q = Number of
combinations of bits/word.I/i Q = 2N...(1.8.4)4Thus if N = 4 i.e. 4
bits per word then the number of quantization levels will be 2 i.e.
16. 1.8.3 Encoding: Our final aim is to convert the signal into the
binary form. So after quantizing, the signal is applied to encoder
block. Encoder assigns unique binary number to each quantization
level. That means each quantization level is converted into the
binary digits. The bits in the binary digit are denoted by 'b'. The
number of bits in the binary digit depends on the number of levels
(L). This relation is 2 > LThusb > log2LSolved Problems on
Sampling Theorem :Ex. 1.8.1 : Two signals x^t) = cos 20 nt and x2
(t) = cos 100 ret are sampled with sampling frequency 40 Hz. Obtain
the associated discrete time signals x,(n) and x2(n) and comment on
the result. Soln. :Given signal is,xt(t) = cos 20nt...(1)
Compare Equation (1) with standard equationx^t) =
cos2rcF1t...(2).-. 2tcF, = 20 re ^> Ft = 10 Hz.Now discrete
signal Xj(n) is obtained by replacing 't' in Equation (2) by J;here
fs = Sampling frequency = 40 Hz. Thus Equation (2) becomes,Xj(n) =
cos 2TCFJ tx^n) = cos 2 Tt "TT"Xj(n) = cos 2 Tt (7J n
(ii) The given signal is,x2(t) = coslOOJtt...(3)Compare it
with,x2(t) = cos2rcF2t...(4).-. 2nF2 = 100 7i => F2 = 50 HzNow
discrete time signal x2(n) is obtained by putting t = T. Thus
Equation (4) becomes,x2(n) = cos 2 Tt -~7frx2(n) = cos 2 Tt (7)
nx2(n) = cos2n( 1 + tJ n = cos (2rcn + 2tc 7 nJ(5)1Now we have cos
(2rcn + 0) = cos 9. Thus Equation (5) becomes,x2(n) = cos 2tc 7
n
Comment:Given sampling frequency, fs = 40 Hz. Thus the frequency
contained in signal should be less than Ifs or equal to ~x; that
means < 20 Hz. But this is not the case in this example. So
aliasing takes place. Here Iboth the sequences Xj(n) and x2(n) are
equal; due to aliasing effect.
Ex. 1.8.2 : For an analog signal,xa(t) = 3 cos 50 Ttt + 10 sin
300 7it - cos 100 7it. Calculate Nyquist rate.ISoln.: The given
equation can be written as,xa(t) = 3cos(2nx25t) + 10cos(2rcx
150t)-cos(2n x50t)...(1)Now we can write,xa(t) = 3 cos (2 7T Ft t)
+ 10 cos (2 n F21) - cos (2 n F31)...(2)Comparing Equations (1) and
(2) wee get,F1 = 25Hz,F2=150Hzand F3 = 50Hz.Thus Fmax = 150 Hz. Now
Nyquist rate = 2 Fmax = 2 x 150 Hz
1.9 Classification of Signals :There are various types of
signals. Every signal is having its own characteristic. The
processing of signal mainly depends on the characteristics of that
particular signal. So classification of signal is necessary.
Broadly the signals are classified as follows : Continuous and
discrete time signals Continuous valued and discrete valued signals
Periodic and non-periodic signals Even and odd signals Energy and
power signals Deterministic and random signals Multichannel and
multidimensional signals.1.9.1 Continuous and Discrete Time Signals
: Continuous signal:A signal of continuous amplitude or time is
known as continuous signal or analog signal. This signal is having
some value at every instant of time.Examples:Sinewave, cosinewave,
triangular wave etc. Similarly certain electrical signals derived
in proportion with physical quantity such as temperature, pressure,
sound etc. are also examples of continuous signal. Some of the
continuous signals are as shown in Fig. 1.9.1.
Mathematical expression :Mathematically a continuous signal (eg.
sinewave) can be expressed as,x(t) = Asin(ci)t + 6) Here A =
Amplitude of signal0) = Angular frequency = 2 rcf9 = Phase
shiftCharacteristics: For every fix value of t, x (t) is periodic
in nature. If the frequency ( ~ J is increased then the rate of
oscillation also changes.Discrete time signal:
In this case the value of signal is specified only at specific
time. So the signal represented at "discrete interval of time" is
called as discrete time signal. The discrete time signal is
generated from continuous time signal by using the sampling
operation. This process is shown in Fig. 1.9.2. Consider a
continuous analog signal as shown in Fig. 1.9.2(a). This signal is
continuous in nature from - to + . The sampling pulses are shown in
Fig. 1.9.2(b). These are train of pulses. Here the samples are
taken at Ts, 2 Ts, 3 Ts... and Ts is the sampling time. Fig.
1.9.2(c) shows discrete time signal. Observe that this signal takes
the value, only where the sampling pulse is present. In between the
two sampling pulses the signal is absent. So this is called as
discrete time signal. In Fig. 1.9.2(a), on X-axis time (t) is
plotted. On Y-axis the amplitude is plotted. So continuous time
signal is represented by x (t). Observe Fig. 1.9.2(c). On X-axis
index n is plotted. Here n is the number of corresponding sample.
So discrete time signal is denoted by x ( n). For signal in Fig.
1.9.2(a), the expression is,x(t) = A cos tot and for signal shown
in Fig. 1.9.2(c), the expression is,x (n) = A cos
conCharacteristics: Discrete time sinusoidal signals are identical
when their frequencies are separated by integer multiple of 2 n. If
the frequency of discrete time sinusoidal is a rational number,
then such signal is periodic in nature. For the discrete time
sinusoidal, the highest oscillation is obtained when angular
frequency to = n.1.9.2 Continuous Valued or Discrete Valued Signals
: Continuous valued signal:If the variation in the amplitude of
signal is continuous then, it is called as continuous valuedsignal.
Such signal may be continuous or discrete in nature.Such signals
are as shown in Figs. 1.9.2(a) and (c).Discrete valued signal: If
the variation in the amplitude of signal is not continuous; but the
signal has certain discrete amplitude levels then such signal is
called as discrete valued signal. Such signal may be again
continuous or discrete in nature as shown in Figs. 1.9.3(a) and
1.9.3(b).
(b) Discrete amplitude signal discrete in nature Fig. 1.9.3 As
shown in Fig. 1.9.3(a), the signal is defined at all instants of
time. So it is continuous signal. But it takes only certain
discrete amplitude levels. The amplitude is not continuously
changing with time. So it is discrete amplitude signal continuous
in nature. As shown in Fig. 1.9.3(b), the signal is defined only at
discrete intervals of time. So it is discrete signal. And this
signal takes only certain discrete amplitude levels. So it is
discrete amplitude signal discrete in nature. 1.9.3 Periodic and
Non-periodic Signals :Periodic signal:A signal which repeats itself
after a fixed time period or interval is called as periodic signal.
Theperiodicity of continuous time signal can be defined
mathematically as,x(t) - x(t + T0)...(1.9.1)This is called as
condition of periodicity.. Here T0 is called as fundamental period.
That meansafter this period the signal repeats itself.For the
discrete time signal, the condition of periodicity is,x(n) = x(n +
N)...(1.9.2)Here number 'N' is the period of signal. The smallest
value of N for which the condition ofperiodicity exists is called
as fundamental period.Periodic signals are shown in Figs. 1.9.4(a)
and (b).
1(b) Discrete time periodic signal Fig. 1.9.4Non-periodic
signal:A signal which does not repeat itself after a fixed time
period or does not repeat at all is called asnon-periodic or
aperiodic signal. Thus mathematical expression for non-periodic
signal is,
d...(1.9.3) ...(1.9.4)Sometimes it is said that non-periodic
signal has a period T = as shown in Fig. 1.9.4(c). This
isexponential signal having period, T = .
Fig. 1.9.4(c): A periodic signal having period, T = Condition
for periodicity of a discrete time signal:A discrete time
sinusoidal signal is periodic only if its frequency(f0) is
rational. That means frequency f0 should be in the form of ratio of
two integers.Proof:For the discrete signal, the condition of
periodicity is,x(n + N) = x(n)...(1.9.5)Let x(n) be the cosine
wave. So it can be expressed as,x(n) = A cos(2rcf0 n +
6)...(1.9.6)Here A = Amplitude and 0 = Phase shiftNow the equation
of x(n + N) can be obtained by replacing 'n' by 'n + N' in Equation
(1.9.6)..-. x(n + N) = A cos[2nf0 (n + N) + 0 ]...(1.9.7)According
to condition of periodicity Equation (1.9.5); we can equate
Equations (1.9.5) and(1.9.7).A cos[27tf0 (n + N) + 0 ] = A
cos(27tf0 n + 0)A cos(2nf0 n + 2nf0 N + 0) = A cos(2rcf0 n +
0)...(1.9.STo satisfy this equation,2rcf0N = 27tk...(1.9.9where k
is an integer ....Proved...(1.9.10*Here k and N both are integers.
Thus discrete time (DT) signal is periodic if its frequency f0
isrational.Periodicity condition for x(n) = x.,(n) + x2(n): Here
input sequence x(n) is expressed as summation of two discrete time
sequences. We can calculate the values of fj and f2 corresponding
to Xj(n) and x2(n). Let Xj(n) and x2(n) both be periodic discrete
time signals (sequences). So according to condition of
periodicity,kjk2f, = ^ and f2 = -^N,The resultant signal x(n) is
periodic if "j^~ is ratio of two integers. The period of x(n) will
be leastcommon multiple of Nj and N2.Similarly if continuous time
signals is,x(t) = x^O + x^t)We can calculate the values of T; and
T2 corresponding to Xj(t) and x2(t). Then the resultantTi x(t) is
periodic if ~~zr is ratio of two integers. The fundamental period
of x(t) will be leastcommon multiple of Tl and T2.Solved
examples:Ex. 1.9.1 : Prove that the sinewave shown in Fig. P. 1.9.1
is a periodic signal.
Fig. P. 1.9.15-: n. : The sinewave shown in the Fig. P. 1.9.1
can be mathematically represented as,x (t) = A sin 0)o t...(1)Now,
let us test if it satisfies the condition for periodicity i.e.
if,x(t) = x(t + T0)...(2)So, let us find the expression for x (t +
T0)x(t + T0) = Asinco0(t + T0)= Asin[o)0t + co0T0].-(3)But 0)o =
27t f0 and T0 = f. Therefore Cfl0 T0 = 2n f0 x t = 2n. Substitute
this in Equation (3), to
x (t + T0) = A sin [oo01 + 2n ]= A [ sin(co01) cos 2% +
cos(co01) sin 2n ]:. x(t + T0) = A sin co01 = x (t)...(4)Therefore
the sinewave shown in Fig. P. 1.9.1 is a periodic signal.-Ex. 1.9.2
: Prove that the exponential signal shown in Fig. P. 1.9.2 is
non-periodic.
Fig. P. 1.9.2Soln.: The exponential signal shown in Fig. P.
1.9.2 is expressed mathematically as,x(t) = e-at...(1)Substitute t
= (t + T0) to get,,T ,-a(t + T) -at -oTx(t + T0) = e=e e ButT0 =
ooe = e =0 .-. x(t + T0) = e~at-0 = 0 .-. x(t) * x(t + T0) Hence
the exponential signal shown in Fig. P. 1.9.2 is a non-periodic
signal.Ex. 1.9.3: What is the fundamental frequency of the waveform
shown in Fig. P. 1.9.3, in Hz and rad/sec ?
Soln.:One cycle corresponds to 0.2 sec. Hence T0 = 0.2 sec.1
1.-. Frequency f0 = -j- = ttz = 5 Hz...Ans.Frequency in rad/sec. =
co0 = 2ji f0 = 2x3.14x5 = 31.4 rad/s...Ans.Ex. 1.9.4 : What is the
fundamental frequency of the D.T. square wave shown in Fig. P.
1.9.4.
Fig. P. 1.9.4Soln. :The fundamental angular frequency or simply
fundamental frequency of x (n) is given byIn 22" " NWhen N = a
positive integer indicating number of samples in one cycle. For the
given signal N = 8.2n it.. Q = "o" = T radians...Ans.Ex. 1.9.5 :
State whether the following signals x(t) is periodic or not, giving
reasons. If it is periodic, find the corresponding period, x (t) =
2 cos 100 n t + 5 sin 501 Sofa.: The given signal is,x(t) = 2cosl00
7tt + 5sin50t...(1)ILetx(t) = XjW + x-jCt)...(2)Here x, (t) = 2 cos
100 m t...(3)andx2(t) = 5 sin 501...(4)The standard equation can be
expressed as,xl (t) = A cos ! t...(5)
12ti 6 ' t[ " 2rc~lj T' = | = l-Similarly comparing Equations
(3) and (4) we get,co2 = . 18 7t,.-. 2jtf2 = 18 rc - -z2 7t' T2 =
"t2 = 9-(6)The resultant signal x (t) is periodic if Tf is the
ratio of two integers. From Equations (5) and (6)-eget,li1/6_I
9_9T2 _ 1/9 " 6' 1 ~ 6It is the ratio of two integers. Thus x (t)
is periodic. Now the fundamental period of x (t) is least;: smon
multiple of Tx and T2. Thus fundamental period is t sec.Givenx (t)
=3sin4t...(7)We have the standard equation,x(t) =A sin
cot...(8)Comparing Equations (7) and (8) we get,co =4.-. 2nf
=4!'-'-It is not the ratio of integer values. Thus this signal is
non-periodic in nature.Givenx(t) = 3 + t2...(9)We know that a
continuous time signal is periodic in nature if it satisfies the
equation,x(t) = x(t + T0)...(10)Where T0 is the fundamental period
of repetation. >From Equation (9) we can write,x(t + T0) = 3 +
(t + T0)2i.-. x(t + T0) = 3 + t2 + 2tT0 + To...(11)For any value of
'T0' Equations (9) and (11) cannot be made equal. Thus given signal
is -periodic.
Ex. 1.9.7 : Few discrete time sequence are given below :(i) cos
(0.01 n n )(ii) cos (3 7t n )(iii) sin (3 n )Determine whether they
are periodic or non periodic. If a sequence is periodic, determine
its fundamental period.Soln.:(i) Given sequence isx(n) = cos
(0.017tn)...(1)We have the standard equation,x (n) = cos
con...(2)Comparing Equations (1) and (2) we get,co = 0.017iButco =
27tf27tf = 0.01710.017t 0.01f " 27t - 2f = 2Qn cycles per
sample...(3) ISince frequency 'f is expressed as the ratio of two
integers; this sequence is periodic. Now we I have the condition of
periodicity,Here 'N' indicates, the fundamental period.Comparing
Equations (3) and (4)Fundamental period = N = 200 samples (ii)
Given equation isx(n) = cos(37tn)...(5Comparing with Equation (2)
we get,/. 27tf = 3 7Tf = x cycles/sample
Since 'f is ratio of two integers; the given sequence is
periodic. Comparing Equations (4) and(6) we get,Fundamental period
= N = 2 samples (iii) Given sequence is,x (n) = sin 3
n...(7)Comparing with Equation (2) we get, (0=3 .-. 2n f = 3" f =
JnHere 2 7t is not an integer. That means ' f' cannot be expressed
as the ratio of two integers. Thus the given sequence is
non-periodic.1.9.4 Even and Odd Signals :Even signals:An even
signal is also called as symmetrical signal. A continuous time
(C.T.) signal x (t) is saidto be symmetrical or even if it
satisfies the following condition :Cor*-t)...forC.T. signal.Here x
( - t) indicates that the signal is present for negative time
period. That means x ( -1) isthe signal which is reflected about
vertical (Y) axis. So even signals are symmetric about verticalaxis
or at t = 0.Odd (Antisymmetric) signal:A continuous time (C.T.)
signal x ( t ) is said to be antisymmetric or odd if it satisfies
thefollowing conditionConditio-...for C.T. signalHere x (-1)
indicates that the signal is present for negative time period.
While - x (t) indicates r i.: the amplitude of signal negative.
Thus antisymmetric signal is not symmetric about vertical
axis.Energy and Power Signals :
In Equation (1.9.11), it is expected that N 1. The power signal
is as shown in Fig. 1.9.5(a). Energy signal:The total normalized
energy for a "real" signal x (t) is given by,COE = f
x2(t)dt...(1.9.12) COBut if the signal x (t) is complex then
Equation (1.9.12) is modified as,COE = f |x(t)|2dt...(1.9.13)oo
The energy signal is as shown in Fig. 1.9.5(b).Note:
Ex. 1.9.8 : What is the total energy of the rectangular pulse
shown in Fig. P. 1.9.8 ?Fig. P. 1.9.9Sotn.: Given signal is
periodic. So consider one cycle from 0 to T.
1.9.5 Deterministic and Random Signals :Deterministic Signal:A
signal which can be described by a mathematical expression, loop-up
table or some welldefined rule is called as the deterministic
signal.Examples: Sine wave, cosine wave, square wave etc. Fig.
1.9.6(a) shows C.T. sine wave signal, which is deterministic
signal. Because it can be represented mathematically as,x(t) =
Asin(2rcft) Here A = Amplitude of signal f = Frequency of
signal.Similarly for D.T. wave we have,x (n) = A sin (2jtfn)
Random signal:A signal which cannot be described by any
mathematical expression is called as random signalDue to this it is
not possible to predict about the amplitude of such signals at a
given instant of Itime.Example:A good example of random signal is
"noise" in the communication system. Such a random signal Iis as
shown in Fig. 1.9.6(b).
1.9.6 Multichannel and Multidimensional Signals :Multichannel
signals : As the name indicates, multichannel signals are generated
by multiple sources or multiple I sensors. The resultant signal is
the vector sum of signals from all channels. Example:A common
example of multichannel signal is ECG waveform. To generate ECG
waveform; different leads are connected to the body of a patient.
Each lead is acting as individual channel. Since there are 'n'
number of leads; the final ECG waveform is a result of multichannel
signal.Mathematically final wave is expressed as,
Multidimensional signals:If a signal is a function of single
independent variable, the signal is called as one-dimensional
signal. On the other hand, if the signal is a function of multi
(many) independent variables then it is called as multidimensional
signal.A good example of multidimensional signal is the picture
displayed on the TV screen. To locate a pixel (a point) on the TV
screen two co-ordinates namely X and Y are required. Similarly this
point is a function of time also. So to display a pixel, minimum
three dimensions are required; namely x, y and t. Thus this is
multidimensional signal. Mathematically it can be written as, P
(
(x, y, t). -"parison of Multichannel and Multidimensional
Signal:
1.10 Representation of DT Signals :The discrete time sequence is
denoted by x (n ). Consider such a discrete time signal as shown in
Fig. 1.10.1.
On the X-axis index 'n' is plotted. Here 'n' is corresponding
number of the sample. In the given I diagram value of n varies from
- 3 to + 3. On the Y-axis, amplitude of signal is plotted. The
signal a having some amplitude at each value of n. Now the
different methods used to represent the signal x (n) are as follows
:1. Functional Representation2. Tabular Representation3. Sequence
Representation1. Functional Representation : For the signal shown
in Fig. 1.10.1, the functional representation of signal is as
follows :
Here the amplitude of signal is written below the corresponding
value of n.3. Sequence Representation : The sequence representation
of given signal is as follows :x(n) = {1,2,-1, 1,2,0,1}tHere all
the amplitudes of signal are written sequentially starting from the
leftmost amplitude.ARROW ALWAYS INDICATES THE AMPLITUDE OF SAMPLE
AT n = 0. If arrow is no:shown in the sequence then by default it
is at first position.e.g.:x(n) = {1,2,3,4,5}Here arrow is not
shown; so by default it is at first sample.
That means we can write,x (n ) = {1 , 2, 3,4, 5}t Number of
samples contained in the given sequence is called as the length of
sample, To adjust the length of sequence we can add any number of
zeros at the beginning or at the end sequence. This is called as
ZERO PADDING.e.g.:Ifx(n) = {1,2,0,1,2}tThen we can write,x(n) =
{0,0,1,2,0,1,2} or x (n) = {1, 2, 0 , 1, 2, 0, 0}tTRemember that
the position of pointer (arrow) does not change.Ex. 1.10.1 :
Represent the following signals graphically :(i) x(n) = {1, 2,
0,-1,1}T(ii) x(n) = {0,0,-1,2,3}T(iii) x(n) = {0,1,-1,1,-1)}tloin.
: These signals are as shown in Fig. P. 1.10.1(a), (b) and (c)
respectively.
1.11 Basic Sequences :In the analysis of communication systems,
standard test signals play a vital role. Such signals are used to
check the performance of a system. Applying such signals at the
system; the output is checked. Now depending on the input-output
characteristics of that particular system; study of different
properties of a system can be done. Some standard test signals are
as follows : Delta or unit impulse function. Unit step signal Unit
Ramp signal Exponential signal Sinusoidal signal1.11.1 Delta or
Unit Impulse Function :A discrete time unit impulse function is
denoted by 8 ( n ). Its amplitude is 1 at n = 0 and for allother
values of n; its amplitude is zero.
The graphical representation of delta function is as shown in
Fig. 1.11.1(a).
A continuous time delta function is denoted by 8 (t).
Mathematically it is expressed as,B" N Jlfort = 05(t) = lOfort^OIt
is as shown in Fig. 1.11.1(b).Fig. 1.11.1(b): Unit impulse function
8 (t) 1.11.2 Unit Step Signal:A discrete time unit step signal is
denoted by u(n). Its value is unity (1) for all positive values
ofn. That means its value is one for n > 0. While for other
values of n; its value is zero., , flforn>0 ' u(n) = l0forn0
U(t) = 10fort0...(4)
Ji To obtain Cn for the unit impulse train : We can obtain the
value of "Cn" for the unit impulse train by applying Equation (4)
to Soti- 3n (1) as follows :
tThis is the value of Cn for the unit impulse train. To obtain
the exponential fourier series : Substitute the value of "Cn" from
Equation (5) into the standard expression of exponential
fourier
This is the required fourier series for the unit impulse train
Amplitude spectrum :
-Amplitude spectrum:This means that for every value of "n" the
value of Cn is going to be the same, equal to ( IT The amplitude
spectrum also is train of impulses each having amplitude of ( 1/T0
), as showx 1 Fig. P. 2.2.4(c).Phase spectrum :The phase spectrum
0, that for a > 0. Thus ROC is o>0. It has a pole at s = 0
that means at origin. The sketch of ROC is startFig. 3.2.4.
It is defined as, r(t) = t fort>0 = 0 otherwise
Positive sided growing exponential pulse :i is also called as
right handed growing exponential pulse I a pulse is represented as
F (t) = ea u (t). Here a is arbitraryF"
We have, X(s) = /x(t)e~stdt...(3.2.19) ooHerex(t) =
eat-u(t)...(3.2.20)v'..:tiplication by unit step u (t) indicates
that the exponential pulse ea' is present in the range"": -5 limits
of integration in Equation (3.2.19) will be from t = 0 to t = .
ROC:The laplace transform is _ . It has pole as + 'a'.Thus ROC
is Re {s - a} > 0, that means Re {s} > a. But Re {s} means
a..-. ROC is a > aThis ROC is shown in Fig. 3.2.8.
5. Positive sided decaying exponential signal:This is also
called as right handed decaying exponential signal. It is given by
x (t) = e~a u (t). Here 'a' is some positive arbitrary constant.
Such a function is shown in Fig. 3.2.9.
According to definition of laplace transform,
We have x (t) = e at u (t). Multiplication by unit step u (t)
indicates that the signal is onl; I range t = 0 to t = . Thus
Equation (3.2.22) becomes,
The ROC is,Re {s + a}> 0 ; that means Re {s} > - a.-.
ROCisa>-aThis ROC is shown in Fig. 3.2.10.
I Negative sided (left handed) exponential signal: As the name
indicates; such exponential signal is in the range t = - to t = 0.
This signal is given x(t) = -e-atu(-t) Here 'a' is some arbitrary
constant. For a > 0 the signal is shown in Fig. 3.2.11(a) and
for a < 0, lie :-_" is shown in Fiff. 3.2.1 \(b\
Since the eiven signal is in the ranee t = - to t = 0 we
get,
In Equation (3.2.24) if the power of exponent of second term is
negative then we will get tha I means this term becomes zero. Thus
we can write the laplace transform.
The pole is at s = - aROC:To obtain this laplace transform, the
condition is Re {(s + a)} < 0. That means Re {s} < - a..-.
ROC is o X(s),ROC:^LTand h (t) H (s)ROC : R2LTthen x; (t) * h (t) X
(s) H (s) ROC : Intersection of R[ and R-3.2.5Initial Value Theorem
:LTIf x (t) < X (s)
1.2.6 Final Value Theorem :2.7 Laplace Transform of Periodic
Signal:Satement:If x (t) is the periodic function with fundamental
period T0' that means x (t) = x (t + T0) then
We know signal x (t) is periodic
tanf : From the definition for unilateral Laplace transform
3.2.8 Table of Laplace transform :Table 3.2.1 shows the standard
laplace transform pairs.Table 3.2.1
I Summary of laplace transform properties : Table 3.2.2 shows
the summary of laplace transform properties. Table 3.2.2
Solved Problems:
i i i According to differentiation in s domain property
in I Given y (t) = x (t) cos 7tOne of the important properties
of Laplace transform is Time cos property. It states that
El. 3.2.2 : Find Lapface transform of following signafs. Draw
ROC in each case
Here ROC remains unchanged.Thus ROC is 6 > 0. It is shown in
Fig. P. 3.2.2.ROC of X (s) is the combination of two ROCs. Thus
combined ROC is a > 0. It is same as ROC shown in Fig. P.
3.2.2.3.3 Laplace Transform used for Waveform Synthesis :Any time
domain function can be expressed in terms of singular functions.
There are additions a subtractions of a singular function to an
existing function. There are three possibilities.(1) Step Step(2)
Step Ramp(3) Ramp Ramp Case 1: Step Step :The addition or
subtraction of a step to a step results in a step function. The
magnitude of n resultant is the algebraic addition or subtraction
respectively of the two steps. The change in ll magnitude occurs at
the instant of addition.
DEMOConsider x (t) = u (t) + 3u (t - 2) = Xj (t) + x2 (t)
say
Fig. 3.3.1For t < 0, X! (t) and x2 (t) have value zero. For 0
< t < 2, x2 (t) is zero. Therefore resultant will ha t only
value of x, (t) i.e. 1. At t = 2, we have addition of case 1 - step
plus step. The resultant is a 'kn The magnitude is the addition of
two steps, i.e. 1 + 3 = 4. The change in the resultant occurs at
-.stant of addition i.e. at t = 2 as shown in Fig. 3.3.1(c).
tlf instead we add a wave - 3u (t - 2) the resultant would have
magnitude 1 + (- 3) = - 2, at . as shown in Fig. 3.3.1(d). The
student should note that in any term such as M u (t - N), M refers
to the magnitude (either or negative) and N to the instant in time.
I : Step Ramp : en the addition of a step is done to a ramp
function, the result is a ramp function, shifted by an 1IIL-: equal
to the step.
DEMOConsiderx (t)=u(t) + r(t-2)Fort