 Hello, welcome all. Welcome to video lecture on properties of Z-transform. Myself, Mr. S. N. Chamath Goudar from Walchand Institute of Technology, Solapur. At the end of this session, students will be able to explain the properties of Z-transform. The various properties of the Z-transform are as listed below. Linearity, time shifting, scaling in the Z-domain, time reversal, the convolution property and differentiation in the Z-domain. Now, let us discuss time reversal property. Time reversal property states that if x of n is a discrete time signal, then Z-transform of x of n is equal to x of z with ROC is equal to r. Then, x of minus n be a discrete time signal, then Z-transform of x of minus n is equal to x of 1 by z with ROC is equal to 1 by r. Let us prove this property. We know that x of z is equal to summation n is equal to minus infinity to infinity x of n z to the power minus n, where x of n is equal to x of minus n. By substituting x of n is equal to x of minus n in the above expression, x of z will be written as x of z is equal to summation n equal to minus infinity to infinity x of minus n minus n into z to the power minus n. Put minus n is equal to k in the above expression, then n will be equal to minus k and n is equal to minus infinity minus n will be equal to plus infinity and minus n is replaced by k. So, k is equal to plus infinity. Similarly, n is equal to plus infinity minus n will be equal to minus infinity minus n is replaced by k, k is equal to minus infinity. So, we substitute k equal to minus infinity k is equal to plus infinity and minus n is equal to k in the above expression. We get x of z is equal to summation k is equal to infinity to minus infinity x of k z to the power k. So, the limits are interchanged here because the lower limits must be below the summation and upper limit must be above the summation. So, summation is equal to k equal to minus infinity to infinity x of k into z inverse to the power minus k. So, we know that summation k equal to minus infinity to infinity x of k z inverse to the power minus k is equal to x of z inverse. This is also written as x of 1 by z hence proved. Let us discuss the differentiation in z domain property. Let us discuss the differentiation in z domain property. This property states that if x of n is a discrete time signal then z transform of x of n is equal to x of z with R O C is equal to R. Then n into x of n be a discrete time signal then z transform of n into x of n will be equal to minus z d x of z by d z with R O C is equal to R. Let us prove this one. We know that x of z is equal to summation n equal to minus infinity to infinity x of n into z to the power minus n. Now differentiate both LHS and RHS by z here. Therefore, d x of z by d z is equal to differentiation of summation n equal to minus infinity to infinity x of n into z to the power minus n. d x of z differentiation of x of z is written as summation n equal to minus infinity to infinity x of n d by d z of z to the power minus n. Now pause the video for some time and write down the differentiation of z to the power minus n here. I hope all of you have written the differentiation of z to the power minus n. So, this will be written as summation n equal to minus infinity to infinity x of n into minus n into z to the power minus n minus 1. So, this will be written as summation n is equal to minus infinity to infinity x of n into minus n into z to the power minus n into z to the power minus 1. So, z to the power minus 1 is a constant term for this summation. So, we will write down. So, z to the power minus 1 is a constant term for this summation. So, we will write that one outside minus z inverse summation n equal to minus infinity to infinity n into x of n into z to the power minus n. So, summation n equal to minus infinity to infinity n into x of n z to the power minus n will be written as z operating on n into x of n. So, this will be written as applying z transform on hence proved. These are the references. Thank you.