 Hello, welcome all. Welcome to video lecture on properties of Z-transform. Myself Mr. S. N. Chamath Goudar from Walchand Institute of Technology, Sulapur. 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 the convolution property. The convolution property states that if x 1 of n is a discrete time signal, then Z-transform of x 1 of n is equal to x 1 of z with R O C is equal to R 1. Similarly, if x 2 of n is a discrete time signal, then Z-transform of x 2 of n is equal to x 2 of z with R O C is equal to R 2. Then x 1 of n convolved x 2 of n be a discrete time signal, then Z-transform of x 1 of n convolved x 2 of n is equal to x 1 of z into x 2 of z with R O C is equal to R 1 intersecting R 2. So, 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 z to the power minus n, where x of n is equal to x 1 of n convolved x 2 of n. Now, we all know that the definition of convolution between two signals x 1 of n and x 2 of n. Now pause the video for some time and recall the definition of convolution between two signals x 1 of n and x 2 of n. I hope you have recalled the convolution definition here. x 1 of n convolved x 2 of n is defined as summation k is equal to minus infinity to infinity x of k x 1 of k into x 2 of n minus k. So, this is the definition of convolution here. So, we substitute x of n is equal to summation k equal to minus infinity to infinity x 1 of k into x 2 of n minus k in the above expression x of z here. So, this will be written as summation n equal to minus infinity to infinity summation k is equal to minus infinity to infinity x of k x 1 of k x 1 of k into x 2 of n minus k into z to the power minus n. So, let me put this summation inside the bracket here. Now by ordering the summation here, we can write this one as k is equal to minus infinity to infinity x 1 of k into summation n equal to minus infinity to infinity x 2 of n minus k into z to the power minus n. Now, let us put n minus k is equal to l here. So, n is equal to l plus k and minus n is equal to minus n is equal to minus l minus k. Similarly, n is equal to minus infinity minus n is equal to plus infinity n is equal to minus infinity n minus k is equal to minus infinity minus k and n minus k is replaced by L. So, L is equal to minus infinity, subtracting k from minus infinity will not affect the infinite value. So, it will remain same that is minus infinity. Now, n is equal to plus infinity, n minus k is equal to plus infinity minus k. Now, replace n minus k by L. So, L is equal to plus infinity, subtracting k from plus infinity will not affect the plus infinite value. So, L is equal to plus infinity. Now, we substitute these L is equal to plus infinity, L is equal to minus infinity, minus n is equal to minus L minus k and n minus k is equal to L in the above expression. So, this will be written as summation k is equal to minus infinity to infinity, x 1 of k summation L is equal to minus infinity to infinity, x 2 of L z to the power minus L minus k. So, this will be written as summation k is equal to minus infinity to infinity, x 1 of k summation L is equal to minus infinity to infinity, x 2 of L z to the power minus L into z to the power minus k. So, this will be written as summation k is equal to minus infinity to infinity, x 1 of k z to the power minus k into summation L is equal to minus infinity to infinity, x 2 of L z to the power minus L. We know that summation k is equal to minus infinity to infinity, x 1 of k into z to the power minus k is equal to x 1 of z and summation L equal to minus infinity to infinity, x 2 of L into z to the power minus L is equal to x 2 of z, hence proved. Next, we discuss the property scaling in z domain, this property states that if x of n is a discrete time signal and z transform of x of n is equal to x of z with R O C equal to R, then z naught to the power n x of n be a discrete time signal, then z transform of z naught to the power n x of n is equal to x of z divided by z naught with R O C equal to mod z naught into 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 z to the power minus n, where x of n is equal to z naught to the power n into x of n. By substituting x of n is equal to z naught to the power n x of n in the above expression, x of z will be written as summation n equal to minus infinity to infinity z naught to the power n into x of n into z to the power minus n n. So, this will be written as summation n equal to minus infinity to infinity x of n z naught to the power n into z to the power minus n. This will be written as summation n equal to minus infinity to infinity x of n into z naught into z inverse to the power n. This will be written as summation n equal to minus infinity to infinity x of n z naught inverse into z to the power minus n. So, summation n equal to minus infinity to infinity x of n into z naught inverse into z to the power minus n is nothing but x of z naught inverse into z, which is also written as x of z divided by z naught hence proved. These are the references. Thank you.