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Your Question. Enter the code in the box below. Write a review Your Name. Your Review Note: HTML is not translated! The book is featured as The book is categorized into Units Subjects and each Unit is sub-divided into Chapters Chapter organization for each Unit Subject is very constructive and covers the complete syllabus Each Chapter contains an average of 40 questions and there are approximate problems for each subject The questions match to the level of GATE examination Solutions are well-explained, tricky and consume less time.
Solutions are presented in such a way that it enhances your fundamentals and problem solving skills There are a variety of problems on each topic Gate Previous Year Solved Questions has been added for each subject Engineering Mathematics and General Aptitude GA are also included in the book Included Numerical Type Questions. Basic Concept 2. Basic Laws 3. Graph Theory 4. Nodal and Loop Analysis 5.
Circuit Theorems 6. Inductor and Capacitor 7. Second Order Circuit 9. Sinusoidal Steady State Analysis AC Power Analysis Three-phase Circuits Magnetically Coupled Circuits Frequency Response Circuit Analysis Using Laplace Transform Coulomb's Law and electric Field Intensity 2. Electric Potential 4. Dielectrics and Capacitance 5. Steady Magnetic Field 6. Magnetic Forces, Materials and Inductance 7.
Continuous Time Signals 2. Continuous Time Systems 3. Discrete Time Signals 4. Practice Exercise Practice exercises covers variety of problems from each topic that enhance your confidence level. Practice exercises are divided into two levels on the basis of complexity.
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This book is published with the understanding that Jhunjhunuwala and its author are supplying information but are not attempting to render engineering or other professional services. Sampling theorem. Signal transmission through LTI systems. Impulse response, transfer function and frequency response of first- and second order systems. Convolution, correlation and characteristics of linear time invariant systems.
Discrete time system, impulse and frequency response. Pulse transfer function. Amplitude and frequency modulation and demodulation. Sampling theorem, pulse code modulation. Frequency and time division multiplexing. Amplitude shift keying, frequency shift keying and pulse shift keying for digital modulation.
System modelling in terms of differential and difference equations; State variable representation; Fourier series; Fourier transforms and their application to system analysis; Laplace transforms and their application to system analysis; Convolution and superposition integrals and their applications; Z-transforms and their applications to the analysis and characterisation of discrete time systems; Random signals and probability, Correlation functions; Spectral density; Response of linear system to random inputs.
Similarly, z -transforms enables us to analyze The properties of z -transform are similar discrete time signals and systems in the z -domain. Like, the Laplace transform, it is also classified as bilateral z -transform and unilateral z -transform. The bilateral or two-sided z -transform is used to analyze both causal and non-causal LTI discrete systems, while the unilateral z -transform is defined only for causal signals.
This corresponds to a circle in z plane with radius r as shown in figure 6. Therefore we conclude that the range of values of the variable z for which the sum in equation 6. Thus, z -transform of a sequence is completely specified if both the expression [X z ] and ROC are given to us. Let X z be the z -transform of sequence x [n] , expressed as a ratio of two polynomials N z and D z. The poles and zeros of X z are shown in pole-zero plot of figure 6. Chapter 6 The Z-Transform Page 6.
These properties can be proved by taking appropriate examples of different DT signals.
Property 1: The ROC is a concentric ring in the z -plane centered about the origin. Hence the ROC will consists of concentric rings centered at zero. Property 2: The ROC cannot contain any poles.
ROC is defined as the values of z for which z -transform X z converges. We know that X z will be infinite at pole, and, therefore X z does not converge at poles.
Hence the region of convergence does not include any pole. Property 3: Consider a finite duration sequence x [n] shown in figure 6. If N1 is negative and N2 is positive, then X z will have both positive and negative powers of z. The negative powers of z becomes unbounded infinity if z " 0.
Similarly positive powers of z becomes unbounded infinity if z " 3. Property 4: Consider a right-sided sequence x [n] shown in figure 6. Let, it is bounded by some value Mx , then equation 6.
Property 5: Consider a left-sided signal x [n] Here N 2 can be either positive or negative. Let it is bounded by some value Mx , then equation 6. The ROC of a left-sided sequence is illustrated in figure 6. Property 6: For any time N 0 , a two-sided sequence can be divided into sum of left-sided and right-sided sequences as shown in figure 6.
The z -transform of x [n] converges for the values of z for which the transform of both xR [n] and xL [n] converges. From property 4, the ROC of a right-sided sequence is a region which is bounded on the inside by a circle and extending outward to infinity i.
From property 5, the ROC of a left sided sequence is bounded on the outside by a circle and extending inward to zero i. The ROC for the right-sided sequence xR [n], the left- sequence xL [n] and their combination which is a two sided sequence x [n] are shown in figure 6. If the z -transform X z of x [n] is rational, then its ROC is bounded by poles or extends to infinity. Property 8: If the z -transform X z of x [n] is rational and x [n] is a right-sided sequence then the ROC is the region in the z -plane outside the outermost pole i.
ROC is the region outside a circle with a radius greater than the magnitude of largest pole of X z. This property can be be proved by taking property 4 and 7 together. Property 9: If the z -transform X z of x [n] is rational and x [n] is a left-sided sequence then the ROC is the region in the z -plane inside the innermost pole i. ROC is the region inside a circle with a radius equal to the smallest magnitude of poles of X z. This property can be be proved by taking property 5 and 7 together.
TABLE 6. To obtain the sequence x [n] from its z -transform is called the inverse z -transform. There are other commonly used methods to evaluate the inverse z -transform given as follows 1. Partial fraction method 2. Power series expansion 6. The values of coefficients A1 , A2 , A Three cases are given as follows Case I: The coefficients are obtained by equation 6.
Refer property 8, section 6. Property 9, section 6. So this terms will corresponds to a left sided equation. To obtain inverse z -transform i. The proofs of properties are given for bilateral transform only and can be obtained in a similar way for the unilateral transform. Page The Z-Transform Chapter 6 6. Like Laplace transform, the linearity Proof: Since, the z -transform X1 z is finite within the between x1 [n] and x 2 [n], which may specified ROC, R1.
Time reversal property states that time Proof: Rx or z! Rx dz For both unilateral and bilateral z -transforms. This property states that multiplication Proof: This operation does not affect the ROC.
Rx X a z k, with ROC: Multiplication of a time sequence with an Proof: Since the expanded sequence xk [n] is zero everywhere except when n is a multiple of k. As defined in equation 6. For both unilateral and bilateral transform. Time convolution property states that Proof: The z -transform of signal x [n] is given by equation 6. Thus final value theorem 4 applies. As the Laplace transform is used in solving differential equations which describe continuous LTI systems, the z -transform is used to solve difference equation which describe the discrete LTI systems.
Similar to Laplace transform, for CT domain, the z -transform gives transfer function of the LTI discrete systems which is the ratio of the z -transform of the output variable to the z -transform of the input variable. These applications are discussed as follows 6.
The solution or total response y [n] consists of two parts as discussed below. Zero-input Response or Free Response or Natural Response The zero input response yzi [n] is mainly due to initial output in the system.
The zero-input response is obtained The zero input response is also called the from system equation 6. The zero-input response yzi [n] is given by inverse z -transform of Yzi z. Zero-State Response or Forced Response The zero-state response yzs [n] is the response of the system due to input signal and with zero initial conditions.