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# GATE Electronics & Communication Vol-7- Signals & Systems

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PUBLISHED FOR GATE 2018

 Edition 8th Authors R K Kanodia & Ashish Murolia Publisher NODIA Pages 946 Binding Paper Back Language English

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SALIENT FEATURES

• Brief Theory

• Problem Solving Methodology

• Fundamental Concepts & Formulae Review

• Vast Question book with Full Solutions

• Multiple Choice Questions, Memory Based Questions and Numerical Types Questions

• Full width coverage of GATE Syllabus

• Well explained and error free solutions

CHAPTER 1 CONTINUOUS TIME SIGNALS

1.1 CONTINUOUS - TIME AND DISCRETE - TIME SIGNALS

1.2 SIGNAL-CLASSIFICATION

1.2.1 Analog and Discrete Signals

1.2.2 Deterministic and Random Signal

1.2.3 Periodic and Aperiodic Signal

1.2.4 Even and Odd Signal

1.2.5 Energy and Power Signal

1.3 BASIC OPERATIONS ON SIGNALS

1.3.2 Multiplication of Signals

1.3.3 Amplitude Scaling of Signals

1.3.4 Time-Scaling

1.3.5 Time-Shifting

1.3.6 Time-Reversal/Folding

1.3.7 Amplitude Inverted Signals

1.4 MULTIPLE OPERATIONS ON SIGNALS

1.5 BASIC CONTINUOUS TIME SIGNALS

1.5.1 The Unit-Impulse Function

1.5.2 The Unit-Step Function

1.5.3 The Unit-Ramp Function

1.5.4 Unit Rectangular Pulse Function

1.5.5 Unit Triangular Function

1.5.6 Unit Signum Function

1.5.7 The Sinc Function

1.6 MATHEMATICAL REPRESENTATION OF SIGNALS

EXERCISE 1.1

EXERCISE 1.2

EXERCISE 1.3

EXERCISE 1.4

SOLUTIONS 1.1

SOLUTIONS 1.2

SOLUTIONS 1.3

SOLUTIONS 1.4

CHAPTER 2 CONTINUOUS TIME SYSTEMS

2.1 CONTINUOUS TIME SYSTEM & CLASSIFICATION

2.1.1 Linear and Non-Linear System

2.1.2 Time-Varying and Time-Invariant system

2.1.3 Systems With and Without Memory (Dynamic and Static Systems)

2.1.4 Causal and Non-causal Systems

2.1.5 Invertible and Non-Invertible Systems

2.1.6 Stable and Unstable systems

2.2 LINEAR TIME INVARIANT SYSTEM

2.2.1 Impulse Response and The Convolution Integral

2.2.2 Properties of Convolution Integral

2.3 STEP RESPONSE OF AN LTI SYSTEM

2.4 PROPERTIES OF LTI SYSTEMS IN TERMS OF IMPULSE RESPONSE

2.4.1 Memoryless LTI System

2.4.2 Causal LTI System

2.4.3 Invertible LTI System

2.4.4 Stable LTI System

2.5 IMPULSE RESPONSE OF INTER-CONNECTED SYSTEMS

2.5.1 Systems in Parallel Configuration

2.6 CORRELATION

2.6.1 Cross-Correlation

2.6.2 Auto-Correlation

2.6.3 Correlation and Convolution

2.7 TIME DOMAIN ANALYSIS OF CONTINUOUS TIME SYSTEMS

2.7.1 Natural Response or Zero-input Response

2.7.2 Forced Response or Zero-state Response

2.7.3 The Total Response

2.8 BLOCK DIAGRAM REPRESENTATION

EXERCISE 2.1

EXERCISE 2.2

EXERCISE 2.3

EXERCISE 2.4

SOLUTIONS 2.1

SOLUTIONS 2.2

SOLUTIONS 2.3

SOLUTIONS 2.4

CHAPTER 3 DISCRETE TIME SIGNALS

3.1 INTRODUCTION TO DISCRETE TIME SIGNALS

3.1.1 Representation of Discrete Time signals

3.2 SIGNAL CLASSIFICATION

3.2.1 Periodic and Aperiodic DT Signals

3.2.2 Even and Odd DT Signals

3.2.3 Energy and Power Signals

3.3 BASIC OPERATIONS ON DT SIGNALS

3.3.2 Multiplication of DT Signal

3.3.3 Amplitude scaling of DT Signals

3.3.4 Time-Scaling of DT Signals

3.3.5 Time-Shifting of DT Signals

3.3.6 Time-Reversal (folding) of DT signals

3.3.7 Inverted DT Signals

3.4 MULTIPLE OPERATIONS ON DT SIGNALS

3.5 BASIC DISCRETE TIME SIGNALS

3.5.1 Discrete Impulse Function

3.5.2 Discrete Unit Step Function

3.5.3 Discrete Unit-ramp Function

3.5.4 Unit-Rectangular Function

3.5.5 Unit-Triangular Function

3.5.6 Unit-Signum Function

3.6 MATHEMATICAL REPRESENTATION OF DT SIGNALS USING IMPULSE OR STEP FUNCTION

EXERCISE 3.1

EXERCISE 3.2

EXERCISE 3.3

EXERCISE 3.4

SOLUTIONS 3.1

SOLUTIONS 3.2

SOLUTIONS 3.3

SOLUTIONS 3.4

CHAPTER 4 DISCRETE TIME SYSTEMS

4.1 DISCRETE TIME SYSTEM & CLASSIFICATION

4.1.1 Linear and Non-linear Systems

4.1.2 Time-Varying and Time-Invariant Systems

4.1.3 System With and Without Memory (Static and Dynamic Systems)

4.1.4 Causal and Non-Causal System

4.1.5 Invertible and Non-Invertible Systems

4.1.6 Stable and Unstable System

4.2 LINEAR-TIME INVARIANT DISCRETE SYSTEM

4.2.1 Impulse Response and Convolution Sum

4.2.2 Properties of Convolution Sum

4.3 STEP RESPONSE OF AN LTI SYSTEM

4.4 PROPERTIES OF DISCRETE LTI SYSTEM IN TERMS OF IMPULSE RESPONSE

4.4.1 Memoryless LTID System

4.4.2 Causal LTID System

4.4.3 Invertible LTID System

4.4.4 Stable LTID System

4.4.5 FIR and IIR Systems

4.5 IMPULSE RESPONSE OF INTERCONNECTED SYSTEMS

4.5.1 Systems in Parallel

4.6 CORRELATION

4.6.1 Cross-Correlation

4.6.2 Auto-Correlation

4.6.3 Properties of Correlation

4.6.4 Relationship Between Correlation and Convolution

4.6.5 Methods to Solve Correlation

4.7 DECONVOLUTION

4.8 RESPONSE OF LTID SYSTEMS IN TIME DOMAIN

4.8.1 Natural Response or Zero Input Response

4.8.2 Forced Response or Zero State Response

4.8.3 Total Response

4.9 BLOCK DIAGRAM REPRESENTATION

EXERCISE 4.1

EXERCISE 4.2

EXERCISE 4.3

EXERCISE 4.4

SOLUTIONS 4.1

SOLUTIONS 4.2

SOLUTIONS 4.3

SOLUTIONS 4.4

CHAPTER 5 THE LAPLACE TRANSFORM

5.1 INTRODUCTION

5.1.1 The Bilateral or Two-Sided Laplace Transform

5.1.2 The Unilateral Laplace Transform

5.2 THE EXISTENCE OF LAPLACE TRANSFORM

5.3 REGION OF CONVERGENCE

5.3.1 Poles and Zeros of Rational Laplace Transforms

5.3.2 Properties of ROC

5.4 THE INVERSE LAPLACE TRANSFORM

5.4.1 Inverse Laplace Transform Using Partial Fraction Method

5.4.2 Inverse Laplace Transform Using Convolution Method

5.5 PROPERTIES OF THE LAPLACE TRANSFORM

5.5.1 Linearity

5.5.2 Time Scaling

5.5.3 Time Shifting

5.5.4 Shifting in the s-domain(Frequency Shifting)

5.5.5 Time Differentiation

5.5.6 Time Integration

5.5.7 Differentiation in the s-domain

5.5.8 Conjugation Property

5.5.9 Time Convolution

5.5.10 s-Domain Convolution

5.5.11 Initial value Theorem

5.5.12 Final Value Theorem

5.5.13 Time Reversal Property

5.6 ANALYSIS OF CONTINUOUS LTI SYSTEMS USING LAPLACE TRANSFORM

5.6.1 Response of LTI Continuous Time System

5.6.2 Impulse Response and Transfer Function

5.7 STABILITY AND CAUSALITY OF CONTINUOUS LTI SYSTEM USING LAPLACE TRANSFORM

5.7.1 Causality

5.7.2 Stability

5.7.3 Stability and Causality

5.8 SYSTEM FUNCTION FOR INTERCONNECTED LTI SYSTEMS

5.8.1 Parallel Connection

5.8.3 Feedback Connection

5.9 BLOCK DIAGRAM REPRESENTATION OF CONTINUOUS LTI SYSTEM

5.9.1 Direct Form I structure

5.9.2 Direct Form II structure

5.9.4 Parallel Structure

EXERCISE 5.1

EXERCISE 5.2

EXERCISE 5.3

EXERCISE 5.4

SOLUTIONS 5.1

SOLUTIONS 5.2

SOLUTIONS 5.3

SOLUTIONS 5.4

CHAPTER 6 THE Z-TRANSFORM

6.1 INTRODUCTION

6.1.1 The Bilateral or Two-Sided z-transform

6.1.2 The Unilateral or One-sided z-transform

6.2 EXISTENCE OF z-TRANSFORM

6.3 REGION OF CONVERGENCE

6.3.1 Poles and Zeros of Rational z-transforms

6.3.2 Properties of ROC

6.4 THE INVERSE z-TRANSFORM

6.4.1 Partial Fraction Method

6.4.2 Power Series Expansion Method

6.5 PROPERTIES OF z-TRANSFORM

6.5.1 Linearity

6.5.2 Time Shifting

6.5.3 Time Reversal

6.5.4 Differentiation in the z -domain

6.5.5 Scaling in z-Domain

6.5.6 Time Scaling

6.5.7 Time Differencing

6.5.8 Time Convolution

6.5.9 Conjugation Property

6.5.10 Initial Value Theorem

6.5.11 Final Value Theorem

6.6 ANALYSIS OF DISCRETE LTI SYSTEMS USING z-TRANSFORM

6.6.1 Response of LTI Continuous Time System

6.6.2 Impulse Response and Transfer Function

6.7 STABILITY AND CAUSALITY OF LTI DISCRETE SYSTEMS USING z-TRANSFORM

6.7.1 Causality

6.7.2 Stability

6.7.3 Stability and Causality

6.8 BLOCK DIAGRAM REPRESENTATION

6.8.1 Direct Form I Realization

6.8.2 Direct Form II Realization

6.8.4 Parallel Form

6.9 RELATIONSHIP BETWEEN s-PLANE & z-PLANE

EXERCISE 6.1

EXERCISE 6.2

EXERCISE 6.3

EXERCISE 6.4

SOLUTIONS 6.1

SOLUTIONS 6.2

SOLUTIONS 6.3

SOLUTIONS 6.4

CHAPTER 7 THE CONTINUOUS TIME FOURIER TRANSFORM

7.1 DEFINITION

7.1.1 Magnitude and Phase Spectra

7.1.2 Existence of Fourier transform

7.1.3 Inverse Fourier Transform

7.2 SPECIAL FORMS OF FOURIER TRANSFORM

7.2.1 Real-valued Even Symmetric Signal

7.2.2 Real-valued Odd Symmetric Signal

7.2.3 Imaginary-valued Even Symmetric Signal

7.2.4 Imaginary-valued Odd Symmetric Signal

7.3 PROPERTIES OF FOURIER TRANSFORM

7.3.1 Linearity

7.3.2 Time Shifting

7.3.3 Conjugation and Conjugate Symmetry

7.3.4 Time Scaling

7.3.5 Differentiation in Time-Domain

7.3.6 Integration in Time-Domain

7.3.7 Differentiation in Frequency Domain

7.3.8 Frequency Shifting

7.3.9 Duality Property

7.3.10 Time Convolution

7.3.11 Frequency Convolution

7.3.12 Area Under x(t)

7.3.13 Area Under X(jÏ‰)

7.3.14 Parsevalâ€™s Energy Theorem

7.3.15 Time Reversal

7.3.16 Other Symmetry Properties

7.4 ANALYSIS OF LTI CONTINUOUS TIME SYSTEM USING FOURIER TRANSFORM

7.4.1 Transfer Function & Impulse Response of LTI Continuous System

7.4.2 Response of LTI Continuous system using Fourier Transform

7.5 RELATION BETWEEN FOURIER AND LAPLACE TRANSFORM

EXERCISE 7.1

EXERCISE 7.2

EXERCISE 7.3

EXERCISE 7.4

SOLUTIONS 7.1

SOLUTIONS 7.2

SOLUTIONS 7.3

SOLUTIONS 7.4

CHAPTER 8 THE DISCRETE TIME FOURIER TRANSFORM

8.1 DEFINITION

8.1.1 Magnitude and Phase Spectra

8.1.2 Existence of DTFT

8.1.3 Inverse DTFT

8.2 SPECIAL FORMS OF DTFT

8.3 PROPERTIES OF DISCRETE-TIME FOURIER TRANSFORM

8.3.1 Linearity

8.3.2 Periodicity

8.3.3 Time Shifting

8.3.4 Frequency Shifting

8.3.5 Time Reversal

8.3.6 Time Scaling

8.3.7 Differentiation in Frequency Domain

8.3.8 Conjugation and Conjugate Symmetry

8.3.9 Convolution in Time Domain

8.3.10 Convolution in Frequency Domain

8.3.11 Time Differencing

8.3.12 Time Accumulation

8.3.13 Parsevalâ€™s Theorem

8.4 ANALYSIS OF LTI DISCRETE TIME SYSTEM USING DTFT

8.4.1 Transfer Function & Impulse Response

8.4.2 Response of LTI DT system using DTFT

8.5 RELATION BETWEEN THE DTFT & THE Z-TRANSFORM

8.6 DISCRETE FOURIER TRANSFORM (DFT)

8.6.1 Inverse Discrete Fourier Transform (IDFT)

8.7 PROPERTIES OF DFT

8.7.1 Linearity

8.7.2 Periodicity

8.7.3 Conjugation and Conjugate Symmetry

8.7.4 Circular Time Shifting

8.7.5 Circular Frequency Shift

8.7.6 Circular Convolution

8.7.7 Multiplication

8.7.8 Parsevalâ€™s Theorem

8.7.9 Other Symmetry Properties

8.8 FAST FOURIER TRANSFORM (FFT)

EXERCISE 8.1

EXERCISE 8.2

EXERCISE 8.3

EXERCISE 8.4

SOLUTIONS 8.1

SOLUTIONS 8.2

SOLUTIONS 8.3

SOLUTIONS 8.4

CHAPTER 9 THE CONTINUOUS TIME FOURIER SERIES

9.1 INTRODUCTION TO CTFS

9.1.1 Trigonometric Fourier Series

9.1.2 Exponential Fourier Series

9.1.3 Polar Fourier Series

9.2 EXISTENCE OF FOURIER SERIES

9.3 PROPERTIES OF EXPONENTIAL CTFS

9.3.1 Linearity

9.3.2 Time Shifting

9.3.3 Time Reversal Property

9.3.4 Time Scaling

9.3.5 Multiplication

9.3.6 Conjugation and Conjugate Symmetry

9.3.7 Differentiation Property

9.3.8 Integration in Time-Domain

9.3.9 Convolution Property

9.3.10 Parsevalâ€™s Theorem

9.3.11 Frequency Shifting

9.4 AMPLITUDE & PHASE SPECTRA OF PERIODIC SIGNAL

9.5 RELATION BETWEEN CTFT & CTFS

9.5.1 CTFT using CTFS Coefficients

9.5.2 CTFS Coefficients as Samples of CTFT

9.6 RESPONSE OF AN LTI CT SYSTEM TO PERIODIC SIGNALS USING FOURIER SERIES

EXERCISE 9.1

EXERCISE 9.2

EXERCISE 9.3

EXERCISE 9.4

SOLUTIONS 9.1

SOLUTIONS 9.2

SOLUTIONS 9.3

SOLUTIONS 9.4

CHAPTER 10 THE DISCRETE TIME FOURIER SERIES

10.1 DEFINITION

10.2 AMPLITUDE AND PHASE SPECTRA OF PERIODIC DT SIGNALS

10.3 PROPERTIES OF DTFS

10.3.1 Linearity

10.3.2 Periodicity

10.3.3 Time-Shifting

10.3.4 Frequency Shift

10.3.5 Time-Reversal

10.3.6 Multiplication

10.3.7 Conjugation and Conjugate Symmetry

10.3.8 Difference Property

10.3.9 Parsevalâ€™s Theorem

10.3.10 Convolution

10.3.11 Duality

10.3.12 Symmetry

10.3.13 Time Scaling

EXERCISE 10.1

EXERCISE 10.2

EXERCISE 10.3

EXERCISE 10.4

SOLUTIONS 10.1

SOLUTIONS 10.2

SOLUTIONS 10.3

SOLUTIONS 10.4

CHAPTER 11 SAMPLING AND SIGNAL RECONSTRUCTION

11.1 THE SAMPLING PROCESS

11.2 THE SAMPLING THEOREM

11.3 IDEAL OR IMPULSE SAMPLING

11.4 NYQUIST RATE OR NYQUIST INTERVAL

11.5 ALIASING

11.6 SIGNAL RECONSTRUCTION

11.7 SAMPLING OF BAND-PASS SIGNALS

EXERCISE 11.1

EXERCISE 11.2

EXERCISE 11.3

EXERCISE 11.4

SOLUTIONS 11.1

SOLUTIONS 11.2

SOLUTIONS 11.3

SOLUTIONS 11.4

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