Problem 2. (Filtering in the spatial domain, 10...

Problem 2. (Filtering in the spatial domain, 10 points) Consider a 2-D linear, shift-invariant system with impulse response \begin{equation*} h = \begin{pmatrix} 0 & -1 & 0\\ -1 & 4 & -1\\ 0 & -1 & 0 \end{pmatrix}, \end{equation*} where \boxed{} represents the location of the origin. Determine the output when the input to the system is $f[m, n] = 0.25 \cos(0.2\pi m + 0.3\pi n) + 0.5 \cos(0.7\pi m - 0.2\pi n)$, $m, n \in \mathbb{Z}$.

Transcript

00:01 In this question, we have been given a 2d impulse signal which is given by delta of xy, it is equals to 1.

00:10 This happens when x equals to y equals to 0 and otherwise it is equals to 0, correct? then we have been given that the impulse response of a 2d system which is given by h of xy is equals to t of delta x, comma, correct? we have been given that input discrete space signal okay so input discrete space signal is given by the function f of xy is equals to x plus y whole raise to three in the first part we need to show that the output signal which is g xy obtained by this function f of xy okay is so we have to show that g of xy it is nothing but 27 times delta of x minus 1 comma y minus 2 so let us see how we are going to obtain this so property of impulse signal we are going to apply correct so the property of impulse signal by which what we know that x t x t times delta of t plus minus t not is nothing but so this is nothing but x of minus plus t not delta of t plus minus t not then x t times delta of t minus t not is equivalent to x of t not delta of t minus t not correct so this was the multiplication property and then we will make use of the convolution property which says that xt multiplied with or convolution of xt with delta of t plus minus t0 is same as x of t plus minus t not.

02:23 Similarly integral from minus infinity to infinity delta of t d t is equals to one or we say summation and going from minus infinity to infinity.

02:36 Delta of n equals to 1 so this is the convolution property now what else we can do so we are going to apply the multiplication property so apply the multiplication property what we are going to get we have been given this function f of x y delta of x minus 1 comma y minus 2 so from this what i can say this will be nothing but x plus y whole cube and here i get delta of x minus 1 minus 2.

03:10 So this can be written as x is 1, y is 2.

03:14 So this is 1 plus 2 cube delta of x minus 1, y minus 2...

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