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Z-TransformThe Z-transform is a mathematical tool which is used to convert the difference equations in discrete time domain into the algebraic equations in z-domain. Mathematically, if $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a discrete time function, then its Z-transform is defined as, $$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}}$$Multiplication by Exponential Sequence Property of Z-TransformStatement - The exponential multiplication property of Z-transform states that the exponential sequence multiplied in time domain corresponds to the scaling in z-domain. The exponential multiplication property is also known as scaling in z-domain property of the Z-transform. Therefore, if$$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}\overset{\mathit{ZT}}{\leftrightarrow}\mathit{X}\mathrm{\left(\mathit{z}\right)};\:\mathrm{ROC}\:\mathrm{=}\:\mathit{R}}$$Then, according to the exponential multiplication property, $$\mathrm{\mathit{a^{\mathit{n}}}\mathit{x}\mathrm{\left(\mathit{n}\right)}\overset{\mathit{ZT}}{\leftrightarrow}\mathit{X}\mathrm{\left(\mathit{\frac{\mathit{z}}{\mathit{a}}}\right)};\:\mathrm{ROC}\:\mathrm{=}\:\left| \mathit{a}\right|\mathit{R}}$$Where, a is a complex number.ProofFrom the definition of ... Read More
Z-TransformThe Z-transform is a mathematical tool which is used to convert the difference equations in discrete time domain into the algebraic equations in z-domain. Mathematically, if $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a discrete time function, then its Z-transform is defined as, $$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}}$$Convolution Method to Find Inverse Z-TransformThe inverse Z-transform can be calculated using the convolution theorem. In the convolution integration method, the given Z-transform X(z) is first split into $\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{z}\right)}$ and $\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{z}\right)}$ such that $\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{z}\right)}\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{z}\right)}$.The signals $\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}$ and $\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}$ are then obtained by taking the inverse Z-transform of $\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{z}\right)}$ and $\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{z}\right)}$ respectively. Finally, the function $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is obtained by performing the convolution of ... Read More
The inverse discrete-time Fourier transform (IDTFT) is defined as the process of finding the discrete-time sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ from its frequency response X(ω).Mathematically, the inverse discrete-time Fourier transform is defined as −$$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}\:\mathrm{=}\: \frac{1}{2\pi}\int_{-\pi}^{\pi}\mathit{X}\mathrm{\left(\mathit{\omega}\right)}\mathit{e}^{\mathit{j\omega n}}\:\mathit{d\omega}\:\:\:\:\:\:...(1)}$$The solution of the equation (1) for $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is useful for the analytical purpose, but it is very difficult to evaluate for typical functional forms of function X(ω). Therefore, an alternate method of determining the values of the discrete-time sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ follows directly from the definition of the Fourier transform, i.e., $$\mathrm{\mathit{X}\mathrm{\left(\mathit{\omega}\right)}\:\mathrm{=}\:\sum_{n=-\infty }^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{e}^{-\mathit{j\omega n}}\:\mathrm{=}\:...\:\mathrm{+}\:\mathit{x}\mathrm{\left(\mathrm{-3}\right)}\mathit{e}^{\mathit{j}\mathrm{3}\omega}\:\mathrm{+}\:\mathit{x}\mathrm{\left(\mathrm{-2}\right)}\mathit{e}^{\mathit{j}\mathrm{2}\omega}\:\mathrm{+}\:\mathit{x}\mathrm{\left(\mathrm{-1}\right)}\mathit{e}^{\mathit{j}\omega}\:\mathrm{+}\:\mathit{x}\mathrm{\left(\mathrm{0}\right)}\:\mathrm{+}\:\mathit{x}\mathrm{\left(\mathrm{1}\right)}\mathit{e}^{\mathit{-j}\omega}\:\mathrm{+}\:\mathit{x}\mathrm{\left(\mathrm{2}\right)}\mathit{e}^{\mathit{-j}2\omega}\:\mathrm{+}\:\mathit{x}\mathrm{\left(\mathrm{3}\right)}\mathit{e}^{\mathit{-j}3\omega}\:\:\:\:\:\:...(2)}$$Hence, from the equation of X(ω) we can say that, if X(ω) can be expressed ... Read More
Z-TransformThe Z-transform is a mathematical tool which is used to convert the difference equations in discrete time domain into the algebraic equations in z-domain. Mathematically, if $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a discrete time function, then its Z-transform is defined as, $$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}}$$Final Value Theorem of Z-TransformThe final value theorem of Z-transform enables us to calculate the steady state value of a sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$, i.e., $\mathit{x}\mathrm{\left(\mathit{\infty}\right)}$ directly from its Z-transform, without the need for finding its inverse Z-transform.Statement - If $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a causal sequence, then the final value theorem of Z-transform states that if, $$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}\overset{\mathit{ZT}}{\leftrightarrow}\mathit{X}\mathrm{\left(\mathit{z}\right)}}$$And if the Z-transform X(z) has no poles outside ... Read More
Z-TransformThe Z-transform is a mathematical tool which is used to convert the difference equations in discrete time domain into the algebraic equations in z-domain. Mathematically, if $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a discrete time function, then its Z-transform is defined as, $$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}}$$Initial Value Theorem of Z-TransformThe initial value theorem enables us to calculate the initial value of a signal $\mathit{x}\mathrm{\left(\mathit{n}\right)}$, i.e., $\mathit{x}\mathrm{\left(\mathrm{0}\right)}$ directly from its Z-transform X(z) without the need for finding the inverse Z-transform of X(z).Statement - The initial value theorem of Z-transform states that if$$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}\overset{\mathit{ZT}}{\leftrightarrow}\mathit{X}\mathrm{\left(\mathit{z}\right)}}$$Where, $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a causal sequence. Then, $$\mathrm{\mathit{x}\mathrm{\left(\mathrm{0}\right)}\:\mathrm{=}\:\displaystyle \lim_{\mathit{n} \to 0}\mathit{x}\mathrm{\left(\mathit{n}\right)}\:\mathrm{=}\:\displaystyle \lim_{\mathit{z} \to \infty }\mathit{X}\mathrm{\left(\mathit{z}\right)}}$$ProofFrom the definition ... Read More
In this problem, we are given two values lValue and hValue. Our task is to find the largest twins in given range.Two numbers are said to be twin numbers if both of them are prime numbers and the difference between them is 2.Let's take an example to understand the problem, Input : lValue = 65, rValue = 100 Output : 71, 73Solution ApproachA simple solution to the problem is by looping from rValue - 2 to lValue and checking each pair of i and (i+2) for twins and print the first occurred twin.Another Approach is by finding all prime numbers ... Read More
In this problem, we are given an arr[] consisting of N unsorted elements. Our task is to find the largest pair sum in an unsorted array.We will find a pair whose sum is the maximum.Let's take an example to understand the problem, Input : arr[] = {7, 3, 9, 12, 1} Output : 21Explanation −Pair with largest sum, (9, 12). Sum = 21 Solution ApproachA simple solution to the problem is by making a pair of maximum and second maximum elements of the array.For this we will initialise the max and secondMax elements of the array with the first and ... Read More
In this problem, we are given two integer values, n and m. Our task is to find the largest number with n set and m unset bits in the binary representation of the number.Let's take an example to understand the problemInput : n = 3, m = 1 Output : 14Explanation −Largest number will have 3 set bits and then 1 unset bit. (1110)2 = 14Solution ApproachA simple solution to the problem is by finding the number consisting of (n+m) set bits. From this number, toggle off the m bits from the end (LSB). To create a number with (n+m) ... Read More
In this problem, we are given two integer values, N denoting the count of digits of a number and sum denoting the sum of digits of the number. Our task is to find the largest number with given number of digits and sum of digits.Let's take an example to understand the problem, Input : N = 3, sum = 15 Output : 960Solution ApproachA simple approach to solve the problem is by traversing all N digit numbers from the largest to smallest. The find the digit sum numbers, if it's equal to sum return the number.ExampleProgram to illustrate the working ... Read More
In this problem, we are given a doubly linked list LL. Our task is to find the largest node in Doubly Linked List.Let's take an example to understand the problem, Input : linked-list = 5 -> 2 -> 9 -> 8 -> 1 -> 3 Output : 9Solution ApproachA simple approach to solve the problem is by traversing the linked-list and if the data value of max is greater than maxVal's data. After traversing the linked-list, we will return the macVal nodes data.ExampleProgram to illustrate the working of our solution#include using namespace std; struct Node{ int data; ... Read More