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