Lecture notes Discrete time systems and Signal processing

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EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING A Course Material on DISCRETE TIME SYSTEMS AND SIGNAL PROCESSING By Mrs.R.Thilagavathi ASSISTANT PROFESSOR DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING SASURIE COLLEGE OF ENGINEERING VIJAYAMANGALAM – 638 056 SCE 1 DEPT OF EEE EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING QUALITY CERTIFICATE This is to certify that the e-course material Subject Code : EE6403 Subject : DISCRETE TIME SYSTEMS AND SIGNAL PROCESSING Class : II Year EEE being prepared by me and it meets the knowledge requirement of the university curriculum. Signature of the Author Name: Designation: This is to certify that the course material being prepared by Mrs.R.Thilagavathi is of adequate quality. He has referred more than five books among them minimum one is from aborad author. Signature of HD Name: SEAL SCE 2 DEPT OF EEE EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING EE6403 DISCRETE TIME SYSTEMS AND SIGNAL PROCESSING L T P C3 0 0 3 UNIT I INTRODUCTION 9 Classification of systems: Continuous, discrete, linear, causal, stable, dynamic, recursive, time variance; classification of signals: continuous and discrete, energy and power; mathematical representation of signals; spectral density; sampling techniques, quantization, quantization error, Nyquist rate, aliasing effect. UNIT II DISCRETE TIME SYSTEM ANALYSIS 9 Z-transform and its properties, inverse z-transforms; difference equation – Solution by z-transform, application to discrete systems - Stability analysis, frequency response – Convolution – Discrete Time Fourier transform , magnitude and phase representation. UNIT III DISCRETE FOURIER TRANSFORM & COMPUTATION 9 Discrete Fourier Transform- properties, magnitude and phase representation - Computation of DFT using FFT algorithm – DIT &DIF using radix 2 FFT – Butterfly structure. UNIT IV DESIGN OF DIGITAL FILTERS 9 FIR & IIR filter realization – Parallel & cascade forms. FIR design: Windowing Techniques – Need and choice of windows – Linear phase characteristics. Analog filter design – Butterworth and Chebyshev approximations; IIR Filters, digital design using impulse invariant and bilinear transformation - Warping, pre warping. UNIT V DIGITAL SIGNAL PROCESSORS 9 Introduction – Architecture – Features – Addressing Formats – Functional modes - Introduction to Commercial DSProcessors. TOTAL : 45 PERIODS OUTCOMES:  Ability to understand and apply basic science, circuit theory, Electro-magnetic field theory control theory and apply them to electrical engineering problems. TEXT BOOKS: 1. J.G. Proakis and D.G. Manolakis, ‗Digital Signal Processing Principles, Algorithms and Applications‘, Pearson Education, New Delhi, PHI. 2003. 2. S.K. Mitra, ‗Digital Signal Processing – A Computer Based Approach‘, McGraw Hill Edu, 2013. 3. Robert Schilling & Sandra L.Harris, Introduction to Digital Signal Processing using Matlab‖, Cengage Learning,2014. REFERENCES: 1. Poorna Chandra S, Sasikala. B ,Digital Signal Processing, Vijay Nicole/TMH,2013. 2. B.P.Lathi, ‗Principles of Signal Processing and Linear Systems‘, Oxford University Press, 2010 3. Taan S. ElAli, ‗Discrete Systems and Digital Signal Processing with Mat Lab‘, CRC Press, 2009. 4. Sen M.kuo, woonseng…s.gan, ―Digital Signal Processors, Architecture, Implementations & Applications, Pearson,2013 5. Dimitris G.Manolakis, Vinay K. Ingle, applied Digital Signal Processing,Cambridge,2012 6. Lonnie C.Ludeman ,‖Fundamentals of Digital Signal Processing‖,Wiley,2013 SCE 3 DEPT OF EEE EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING DISCRETE SYSTEMS & SIGNAL PROCESSING 1. BASICS OF DIGITAL SIGNALS & SYSTEMS 7 2. ANALYSIS OF LSI SYSTEMS- Z TRANSFORM 27 3. ANALYSIS OF FT, DFT AND FFT 32 4. DIGITAL FILTER CONCEPTS & DESIGN 69 5. DSP ARCHITECTURES AND APPLICATIONS OF DSP 101 SCE 4 DEPT OF EEE EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING CONTENT PG.NO CHAPTER -1 SIGNALS AND SYSTEMS 1.0 Introduction 7 1.1 Classification of signals processing 8  Advantages of DSP over ASP  Disadvantages of DSP over ASP 1.2 Classification of signals 11 1.3 Discrete time systems 11 1.4 Standard signal sequences 12 1.5 Properties Discrete time signals 12 1.6 Symbols used in Discrete time systems 12 1.7 Classification of Discrete time systems 13 1.8 Stability for LTI systems 13 1.9 Analysis of discrete LTI systems 16  Types of convolution  Properties of convolution  Causality of LSI systems  Stability for LTI systems 1.10 Correlation 19  Types of correlation  Properties of correlation 22 1.11 A/D Conversion 1.12 Differential Equations 23 CHAPTER – 2 DISCRETE TIME SYSTEMS ANALYSIS 2.0 Introduction to Z Transform 26  Advantages of Z Transform  Advantages of ROC 2.1 Z Transform plot 26  Properties of Z -Transform 2.2 Relationship between FT & Z Transform 29 2.3 Inverse Z – Transform 31 2.4 Pole Zero Plot 34 2.5 One sided Z – Transform 36 2.6 Solution of Differential Equations 37 2.7 Jury stability criterian 38 CHAPTER – 3 SCE 5 DEPT OF EEE EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING DISCRETE FOURIER TRANSFORM & COMPUTATION 3.0 Introduction 39 3.1 Difference between FT & DFT 39 3.2 Calculation of DFT 3.3 Difference between DFT & IDFT 41 3.4 Properties of DFT 3.5 Application of DFT 47 3.6 FFT 50  Radix 2 FFT Algorithm 3.7 GEOERZEL Algorithm 59 CHAPTER – 4 DESIGN OF DIGITAL FILTERS 4.1 Introduction 61 4.2 Types of digital filters 65 4.3 Structure for FIR system 65 4.4 Structure for IIR system 68 4.5 IIR Filter design (Impulse invariance) 73 4.6 IIR Filter design (BZT) 76 4.7 Butterworth approximation 80 4.8 Frequency Transformation 87 4.9 FIR Filter design 89 4.10 Design filter for pole zero placement 93 CHAPTER – 5 DIGITAL SIGNAL PROCESSOR 5.1 Requirements of DSP Processor 97 5.2 Microprocessor Architecture 100 5.3 Core Architecture of ADSP - 21XX 101  ADSP- 21XX Development tools 5.4 Application of DSP 106 5.6 Figure for finite duration 110 5.7 Figure for infinite duration 111 5.8 Figure for pole zero plot 112  References  Glossary Terms  Tutorial Problems  Worked Out Problems  Question Bank  Question paper SCE 6 DEPT OF EEE EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING CHAPTER 1 SIGNALS AND SYSTEM 1.0 PREREQISTING DISCUSSION A Signal is defined as any physical quantity that changes with time, distance, speed, position, pressure, temperature or some other quantity. A Signal is physical quantity that consists of many sinusoidal of different amplitudes and frequencies. Ex x(t) = 10t 2 X(t) = 5x +20xy+30y A System is a physical device that performs an operations or processing on a signal. Ex Filter or Amplifier. 1.1 CLASSIFICATION OF SIGNAL PROCESSING 1) ASP (Analog signal Processing) : If the input signal given to the system is analog then system does analog signal processing. Ex Resistor, capacitor or Inductor, OP-AMP etc. ANALOG Analog Analog SYSTEM Input Output 2) DSP (Digital signal Processing) : If the input signal given to the system is digital then system does digital signal processing. Ex Digital Computer, Digital Logic Circuits etc. The devices called as ADC (analog to digital Converter) converts Analog signal into digital and DAC (Digital to Analog Converter) does vice-versa. ADC DAC Analog DIGIT AL Analog signal signal SYSTEM Most of the signals generated are analog in nature. Hence these signals are converted to digital form by the analog to digital converter. Thus AD Converter generates an array of samples and gives it to the digital signal processor. This array of samples or sequence of samples is the digital equivalent of input analog signal. The DSP performs signal processing operations like filtering, multiplication, transformation or amplification etc operations over this digital signals. The digital output signal from the DSP is given to the DAC. 1.1.1 ADVANTAGES OF DSP OVER ASP 1. Physical size of analog systems are quite large while digital processors are more compact and light in weight. 2. Analog systems are less accurate because of component tolerance ex R, L, C and active components. Digital components are less sensitive to the environmental changes, noise and disturbances. 3. Digital system are most flexible as software programs & control programs can be easily modified. 4. Digital signal can be stores on digital hard disk, floppy disk or magnetic tapes. Hence becomes transportable. Thus easy and lasting storage capacity. 5. Digital processing can be done offline. SCE 7 DEPT OF EEE EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING 6. Mathematical signal processing algorithm can be routinely implemented on digital signal processing systems. Digital controllers are capable of performing complex computation with constant accuracy at high speed. 7. Digital signal processing systems are upgradeable since that are software controlled. 8. Possibility of sharing DSP processor between several tasks. 9. The cost of microprocessors, controllers and DSP processors are continuously going down. For some complex control functions, it is not practically feasible to construct analog controllers. 10. Single chip microprocessors, controllers and DSP processors are more versatile and powerful. 1.1.2 Disadvantages Of DSP over ASP 1. Additional complexity (A/D & D/A Converters) 2. Limit in frequency. High speed AD converters are difficult to achieve in practice. In high frequency applications DSP are not preferred. 1.2 CLASSIFICATION OF SIGNALS 1. Single channel and Multi-channel signals 2. Single dimensional and Multi-dimensional signals 3. Continuous time and Discrete time signals. 4. Continuous valued and discrete valued signals. 5. Analog and digital signals. 6. Deterministic and Random signals 7. Periodic signal and Non-periodic signal 8. Symmetrical(even) and Anti-Symmetrical(odd) signal 9. Energy and Power signal 1) Single channel and Multi-channel signals If signal is generated from single sensor or source it is called as single channel signal. If the signals are generated from multiple sensors or multiple sources or multiple signals are generated from same source called as Multi-channel signal. Example ECG signals. Multi-channel signal will be the vector sum of signals generated from multiple sources. 2) Single Dimensional (1-D) and Multi-Dimensional signals (M-D) If signal is a function of one independent variable it is called as single dimensional signal like speech signal and if signal is function of M independent variables called as Multi-dimensional signals. Gray scale level of image or Intensity at particular pixel on black and white TV are examples of M-D signals. 3) Continuous time and Discrete time signals. S. No Continuous Time (CTS) Discrete time (DTS) 1 This signal can be defined at any time This signal can be defined only at certain instance & they can take all values in specific values of time. These time instance need the continuous interval(a, b) where a not be equidistant but in practice they are usually takes at equally spaced intervals. can be -∞ & can be ∞ 2 These are described by differential These are described by difference equation. equations. 3 This signal is denoted by x(t). These signals are denoted by x(n) or notation x(nT) can also be used. 4 The speed control of a dc motor using a Microprocessors and computer based systems techogenerator feedback or Sine or uses discrete time signals. SCE 8 DEPT OF EEE EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING exponential waveforms. 4) Continuous valued and Discrete Valued signals. S. No Continuous Valued Discrete Valued 1 If a signal takes on all possible values on If signal takes values from a finite set of a finite or infinite range, it is said to be possible values, it is said to be discrete valued continuous valued signal. signal. 2 Continuous Valued and continuous time Discrete time signal with set of discrete signals are basically analog signals. amplitude are called digital signal. 5) Analog and digital signal S. No Analog signal Digital signal 1 These are basically continuous time & These are basically discrete time signals & continuous amplitude signals. discrete amplitude signals. These signals are basically obtained by sampling & quantization process. 2 ECG signals, Speech signal, Television All signal representation in computers and signal etc. All the signals generated from digital signal processors are digital. various sources in nature are analog. Note: Digital signals (DISCRETE TIME & DISCRETE AMPLITUDE) are obtained by sampling the ANALOG signal at discrete instants of time, obtaining DISCRETE TIME signals and then by quantizing its values to a set of discrete values & thus generating DISCRETE AMPLITUDE signals. Sampling process takes place on x axis at regular intervals & quantization process takes place along y axis. Quantization process is also called as rounding or truncating or approximation process. 6) Deterministic and Random signals S. No Deterministic signals Random signals 1 Deterministic signals can be represented or Random signals that cannot be described by a mathematical equation or lookup represented or described by a table. mathematical equation or lookup table. 2 Deterministic signals are preferable because for Not Preferable. The random signals can analysis and processing of signals we can use be described with the help of their mathematical model of the signal. statistical properties. 3 The value of the deterministic signal can be The value of the random signal cannot evaluated at time (past, present or future) without be evaluated at any instant of time. certainty. 4 Example Sine or exponential waveforms. Example Noise signal or Speech signal 7) Periodic signal and Non-Periodic signal The signal x(n) is said to be periodic if x(n+N)= x(n) for all n where N is the fundamental period of the signal. If the signal does not satisfy above property called as Non-Periodic signals. Discrete time signal is periodic if its frequency can be expressed as a ratio of two integers. f= k/N where k is integer constant. SCE 9 DEPT OF EEE EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING Tutorial problems: a) cos (0.01 ∏ n) Periodic N=200 samples per cycle. b) cos (3 ∏ n) Periodic N=2 samples c) sin(3n) Non-Periodic d) cos(n/8) cos( ∏n/8) Non-Periodic 8) Symmetrical(Even) and Anti-Symmetrical(odd) signal A signal is called as symmetrical(even) if x(n) = x(-n) and if x(-n) = -x(n) then signal is odd. X1(n)= cos(ωn) and x2(n)= sin(ωn) are good examples of even & odd signals respectively. Every discrete signal can be represented in terms of even & odd signals. X(n) signal can be written as 𝑋 𝑛 + 𝑋 (𝑛 ) 𝑋 𝑛 = 2 Rearranging the above terms we have 𝑋 𝑛 − 𝑋 (𝑛 ) 𝑋 𝑛 = 2 Thus X(n)= X (n) + X (n) e o Even component of discrete time signal is given by 𝑋 𝑛 + 𝑋 (𝑛 ) 𝑋 𝑒 𝑛 = 2 Odd component of discrete time signal is given by 𝑋 𝑛 − 𝑋 (𝑛 ) 𝑋 𝑜 𝑛 = 2 Test whether the following CT waveforms is periodic or not. If periodic find out the fundamental period. a) 2 sin(2/3)t + 4 cos (1/2)t + 5 cos((1/3)t Ans: Period of x(t)= 12 b) a cos(t √2) + b sin(t/4) Ans: Non-Periodic a) Find out the even and odd parts of the discrete signal x(n)=2,4,3,2,1 b) Find out the even and odd parts of the discrete signal x(n)=2,2,2,2 9) Energy signal and Power signal Discrete time signals are also classified as finite energy or finite average power signals. The energy of a discrete time signal x(n) is given by ∞ 2 E= 𝑥 (𝑛 ) 𝑛 =−∞ The average power for a discrete time signal x(n) is defined as 1 ∞ 2 𝑃 = lim 𝑥 (𝑛 ) 𝑁 →∞ 𝑛 =−∞ 2𝑁 +1 SCE 10 DEPT OF EEE EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING If Energy is finite and power is zero for x(n) then x(n) is an energy signal. If power is finite and energy is infinite then x(n) is power signal. There are some signals which are neither energy nor a power signal. Tutorial problems: a) Find the power and energy of u(n) unit step function. b) Find the power and energy of r(n) unit ramp function. n c) Find the power and energy of a u(n). 1.3 DISCRETE TIME SIGNALS AND SYSTEM There are three ways to represent discrete time signals. 1) Functional Representation 4 for n=1,3 x(n)= -2 for n =2 0 elsewhere 2) Tabular method of representation n -3 -2 -1 0 1 2 3 4 5 x(n) 0 0 0 0 4 -2 4 0 0 3) Sequence Representation X(n) = 0 , 4 , -2 , 4 , 0 ,…… n=0 1.4 STANDARD SIGNAL SEQUENCES 1) Unit sample signal (Unit impulse signal) δ (n) = 1 n=0 0 n=0 i.e δ(n)=1 2) Unit step signal u(n) = 1 n≥0 0 n0 3) Unit ramp signal u (n) = n n≥0 r 0 n0 4) Exponential signal n j Ø n n j Ø n n x(n) = a = (re ) = r e = r (cos Øn + j sin Øn) 4) Sinusoidal waveform x(n) = A Sin wn 1.5 PROPERTIES OF DISCRETE TIME SIGNALS SCE 11 DEPT OF EEE EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING 1) Shifting : signal x(n) can be shifted in time. We can delay the sequence or advance the sequence. This is done by replacing integer n by n-k where k is integer. If k is positive signal is delayed in time by k samples (Arrow get shifted on left hand side) And if k is negative signal is advanced in time k samples (Arrow get shifted on right hand side) X(n) = 1, -1 , 0 , 4 , -2 , 4 , 0 ,…… n=0 Delayed by 2 samples : X(n-2)= 1, -1 , 0 , 4 , -2 , 4 , 0 ,…… n=0 Advanced by 2 samples : X(n+2) = 1, -1 , 0 , 4 , -2 , 4 , 0 ,…… n=0 2) Folding / Reflection : It is folding of signal about time origin n=0. In this case replace n by –n. Original signal: X(n) = 1, -1 , 0 , 4 , -2 , 4 , 0 n=0 Folded signal: X(-n) = 0 , 4 , -2 , 4 , 0 , -1 , 1 n=0 3) Addition : Given signals are x1(n) and x2(n), which produces output y(n) where y(n) = x1(n)+ x2(n). Adder generates the output sequence which is the sum of input sequences. 4) Scaling: Amplitude scaling can be done by multiplying signal with some constant. Suppose original signal is x(n). Then output signal is A x(n) 5) Multiplication : The product of two signals is defined as y(n) = x1(n) x2(n). 1.6 SYMBOLS USED IN DISCRETE TIME SYSTEM 1. Unit delay -1 Z x(n) y(n) = x(n-1) 2. Unit advance +1 Z x(n) y(n) = x(n+1) SCE 12 DEPT OF EEE EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING 3. Addition x1(n) + y(n) =x1(n)+x2(n) x2(n) 4. Multiplication x1(n) × y(n) =x1(n)x2(n) x2(n) 5. Scaling (constant multiplier) A x(n) y(n) = A x(n) 1.7 CLASSIFICATION OF DISCRETE TIME SYSTEMS 1) STATIC v/s DYNAMIC S. No STATIC DYNAMIC (Dynamicity property) 1 Static systems are those systems whose output at any Dynamic systems output instance of time depends at most on input sample at same depends upon past or future time. samples of input. 2 Static systems are memory less systems. They have memories for memorize all samples. It is very easy to find out that given system is static or dynamic. Just check that output of the system solely depends upon present input only, not dependent upon past or future. S. No System y(n) Static / Dynamic 1 x(n) Static 2 A(n-2) Dynamic 2 3 X (n) Static 2 4 X(n ) Dynamic 2 5 n x(n) + x (n) Static 6 X(n)+ x(n-2) +x(n+2) Dynamic 2) TIME INVARIANT v/s TIME VARIANT SYSTEMS S. No TIME INVARIANT (TIV) / TIME VARIANT SYSTEMS / SHIFT INVARIANT SHIFT VARIANT SYSTEMS (Shift Invariance property) 1 A System is time invariant if its input output A System is time variant if its input output characteristic do not change with shift of time. characteristic changes with time. 2 Linear TIV systems can be uniquely No Mathematical analysis can be characterized by Impulse response, frequency performed. response or transfer function. 3 a. Thermal Noise in Electronic components a. Rainfall per month b. Printing documents by a printer b. Noise Effect SCE 13 DEPT OF EEE EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING It is very easy to find out that given system is Shift Invariant or Shift Variant. Suppose if the system produces output y(n) by taking input x(n) x(n)  y(n) If we delay same input by k units x(n-k) and apply it to same systems, the system produces output y(n-k) x(n-k)  y(n-k) 3) LINEAR v/s NON-LINEAR SYSTEMS S.No LINEAR NON-LINEAR (Linearity Property) 1 A System is linear if it satisfies superposition theorem. A System is Non-linear if it does not satisfies superposition theorem. 2 Let x1(n) and x2(n) are two input sequences, then the system is said to be linear if and only if Ta1x1(n) + a2x2(n)=a1Tx1(n)+a2Tx2(n) a1 x1(n) SYSTEM y(n)= Ta1x1n + a2x2(n) x2(n) a2 x1(n) a1 SYSTEM y(n)=Ta1x1(n)+a2x2(n) x2(n) SY STEM a2 hence T a1 x1(n) + a2 x2(n) = T a1 x1(n) + T a2 x2(n) It is very easy to find out that given system is Linear or Non-Linear. Response to the system to the sum of signal = sum of individual responses of the system. S. No System y(n) Linear or Non-Linear x(n) 1 e Non-Linear 2 2 x (n) Non-Linear 3 m x(n) + c Non-Linear 4 cos x(n) Non-Linear 5 X(-n) Linear 6 Log (x(n)) Non-Linear 10 SCE 14 DEPT OF EEE EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING 4) CAUSAL v/s NON CAUSAL SYSTEMS S.No CAUSAL NON-CAUSAL (Causality Property) 1 A System is causal if output of system at any A System is Non causal if output of time depends only past and present inputs. system at any time depends on future inputs. 2 In Causal systems the output is the function of In Non-Causal System the output is the x(n), x(n-1), x(n-2)….. and so on. function of future inputs also. X(n+1) x(n+2) .. and so on 3 Example Real time DSP Systems Offline Systems It is very easy to find out that given system is causal or non-causal. Just check that output of the system depends upon present or past inputs only, not dependent upon future. S.No System y(n) Causal /Non-Causal 1 x(n) + x(n-3) Causal 2 X(n) Causal 3 X(n) + x(n+3) Non-Causal 4 2 x(n) Causal 5 X(2n) Non-Causal 6 X(n)+ x(n-2) +x(n+2) Non-Causal 5) STABLE v/s UNSTABLE SYSTEMS S.No STABLE UNSTABLE (Stability Property) 1 A System is BIBO stable if every bounded A System is unstable if any bounded input input produces a bounded output. produces a unbounded output. 2 The input x(n) is said to bounded if there exists some finite number M such that x(n) ≤ M x x ∞ The output y(n) is said to bounded if there exists some finite number M such that y(n) ≤ y M ∞ y 1.8 STABILITY FOR LTI SYSTEM It is very easy to find out that given system is stable or unstable. Just check that by providing input signal check that output should not rise to ∞. The condition for stability is given by ∞ 𝑕 (𝑘 ) ∞ 𝑘 =−∞ S.No System y(n) Stable / Unstable 1 Cos x(n) Stable 2 x(-n+2) Stable 3 x(n) Stable 4 x(n) u(n) Stable SCE 15 DEPT OF EEE EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING 5 X(n) + n x(n+1) Unstable 1.9 ANALYSIS OF DISCRETE LINEAR TIME INVARIANT (LTI/LSI) SYSTEM 1) CONVOLUTION SUM METHOD 2) DIFFERENCE EQUATION 1.9.1 LINEAR CONVOLUTION SUM METHOD 1. This method is powerful analysis tool for studying LSI Systems. 2. In this method we decompose input signal into sum of elementary signal. Now the elementary input signals are taken into account and individually given to the system. Now using linearity property whatever output response we get for decomposed input signal, we simply add it & this will provide us total response of the system to any given input signal. 3. Convolution involves folding, shifting, multiplication and summation operations. 4. If there are M number of samples in x(n) and N number of samples in h(n) then the maximum number of samples in y(n) is equals to M+n-1. Linear Convolution states that y(n) = x(n) h(n) ∞ ∞ y(n) = ∑ x (k) h(n – k ) = ∑ x (k) h -(k-n) k= -∞ k= -∞ Example 1: h(n) = 1 , 2 , 1, -1 & x(n) = 1, 2, 3, 1 Find y(n) METHOD 1: GRAPHICAL REPRESENTATION Step 1) Find the value of n = n + n = -1 (Starting Index of x(n)+ starting index of h(n)) x h Step 2) y(n)= y(-1) , y(0) , y(1), y(2), …. It goes up to length(xn)+ length(yn) -1. i.e n=-1 y(-1) = x(k) h(-1-k) n=0 y(0) = x(k) h(0-k) n=1 y(1) = x(k) h(1-k) …. ANSWER : y(n) =1, 4, 8, 8, 3, -2, -1 METHOD 2: MATHEMATICAL FORMULA Use Convolution formula ∞ y(n) = ∑ x (k) h(n – k ) k= -∞ k= 0 to 3 (start index to end index of x(n)) y(n) = x(0) h(n) + x(1) h(n-1) + x(2) h(n-2) + x(3) h(n-3) METHOD 3: VECTOR FORM (TABULATION METHOD) X(n)= x1,x2,x3 & h(n) = h1,h2,h3 SCE 16 DEPT OF EEE EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING X1 x2 x3 h1 h1x1 h1x2 h1x3 h2 h2x1 h2x2 h2x3 h3 y(-1) = h1 x1 y(0) = h2 x1 + h1 x2 h3x1 h3x2 h3x3 y(1) = h1 x3 + h2x2 + h3 x1 ………… METHOD 4: SIMPLE MULTIPLICATION FORM X(n)= x1,x2,x3 & h(n) = h1,h2,h3 x1 x2 x3 y(n) = × y1 y2 y3 1.9.2 PROPERTIES OF LINEAR CONVOLUTION x(n) = Excitation Input signal y(n) = Output Response h(n) = Unit sample response 1. Commutative Law: (Commutative Property of Convolution) x(n) h(n) = h(n) x(n) Unit Sample X(n) Response = y(n) = x(n) h(n) Response =h(n) Unit Sample h(n) Response = y(n) = h(n) x(n) Response =x(n) 2. Associate Law: (Associative Property of Convolution) x(n) h1(n) h2(n) = x(n) h1(n) h2(n) Unit Sample X(n) h(n) Response Unit Sample Response=h1(n) Response=h2(n) Unit Sample Response X(n) Response h(n) = h1(n) h2(n) 3 Distribute Law: (Distributive property of convolution) x(n) h1(n) + h2(n) = x(n) h1(n) + x(n) h2(n) SCE 17 DEPT OF EEE EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING 1.9.3 CAUSALITY OF LSI SYSTEM The output of causal system depends upon the present and past inputs. The output of the causal system at n= n depends only upon inputs x(n) for n≤ n . The linear convolution is given as 0 0 ∞ y(n) = ∑ h(k) x(n–k) k=-∞ At n= n ,the output y(n ) will be 0 0 ∞ y(n ) = ∑ h(k) x(n –k) 0 0 k=-∞ Rearranging the above terms... ∞ -∞ y(n ) = ∑ h(k) x(n –k) + ∑ h(k) x(n –k) 0 0 0 k=0 k=-1 The output of causal system at n= n depends upon the inputs for n n Hence 0 0 h(-1)=h(-2)=h(-3)=0 Thus LSI system is causal if and only if h(n) =0 for n0 This is the necessary and sufficient condition for causality of the system. Linear convolution of the causal LSI system is given by n y(n) = ∑ x (k) h(n – k ) k=0 1.9.4 STABILITY FOR LTI SYSTEM A System is said to be stable if every bounded input produces a bounded output. The input x(n) is said to bounded if there exists some finite number M such that x(n) ≤ M ∞. The x x output y(n) is said to bounded if there exists some finite number M such that y(n) ≤ M ∞. y y Linear convolution is given by ∞ y(n) = ∑ x (k) h(n – k ) k=- ∞ Taking the absolute value of both sides ∞ y(n) = ∑ h(k) x(n-k) k=-∞ SCE 18 DEPT OF EEE EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING The absolute values of total sum is always less than or equal to sum of the absolute values of individually terms. Hence ∞ y(n) ≤ ∑ h(k) x(n–k) k=-∞ ∞ y(n) ≤ ∑ h(k) x(n–k) k=-∞ The input x(n) is said to bounded if there exists some finite number M such that x(n) ≤ M ∞. x x Hence bounded input x(n) produces bounded output y(n) in the LSI system only if ∞ ∑ h(k) ∞ k=-∞ With this condition satisfied, the system will be stable. The above equation states that the LSI system is stable if its unit sample response is absolutely summable. This is necessary and sufficient condition for the stability of LSI system. SELF-STUDY: Exercise No. 1 Q1) Show that the discrete time signal is periodic only if its frequency is expressed as the ratio of two integers. Q2) Show that the frequency range for discrete time sinusoidal signal is -∏ to ∏ radians/sample or -½ cycles/sample to ½ cycles/sample. Q3) Prove δ (n)= u(n)= u(n-1). n Q4) Prove u(n)= ∑ δ(k) k=-∞ ∞ Q5) Prove u(n)= ∑ δ(n-k) k=0 Q6) Prove that every discrete sinusoidal signal can be expressed in terms of weighted unit impulse. Q7) Prove the Linear Convolution theorem. 1.10 CORRELATION: It is frequently necessary to establish similarity between one set of data and another. It means we would like to correlate two processes or data. Correlation is closely related to convolution, because the correlation is essentially convolution of two data sequences in which one of the sequences has been reversed. Applications are in 1) Images processing for robotic vision or remote sensing by satellite in which data from different image is compared 2) In radar and sonar systems for range and position finding in which transmitted and reflected waveforms are compared. SCE 19 DEPT OF EEE EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING 3) Correlation is also used in detection and identifying of signals in noise. 4) Computation of average power in waveforms. 5) Identification of binary codeword in pulse code modulation system. DIFFERENCE BETWEEN LINEAR CONVOLUTION AND CORRELATION S.No Linear Convolution Correlation 1 In case of convolution two signal sequences In case of Correlation, two signal sequences are input signal and impulse response given by the just compared. same system is calculated 2 Our main aim is to calculate the response given Our main aim is to measure the degree to which by the system. two signals are similar and thus to extract some information that depends to a large extent on the application 3 Linear Convolution is given by the equation Received signal sequence is given as y(n) = x(n) h(n) & calculated as Y(n) = α x(n-D) + ω(n) ∞ Where α= Attenuation Factor D= Delay y(n) = ∑ x (k) h(n – k ) ω(n) = Noise signal k= -∞ 4 Linear convolution is commutative Not commutative. 1.10.1 TYPES OF CORRELATION Under Correlation there are two classes. 1) CROSS CORRELATION: When the correlation of two different sequences x(n) and y(n) is performed it is called as Cross correlation. Cross-correlation of x(n) and y(n) is r (l) which can xy be mathematically expressed as r (l) = ∑ x (n) y(n – l ) xy n= -∞ OR ∞ r (l) = ∑ x (n + l) y(n) xy n= -∞ 2) AUTO CORRELATION: In Auto-correlation we correlate signal x(n) with itself, which can be mathematically expressed as ∞ r (l) = ∑ x (n) x(n – l ) xx n= -∞ OR SCE 20 DEPT OF EEE

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